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Economic Quarterly— Volume 103, Numbers 1-4— First-Fourth Quarter 2017— Pages 1–35  Inequality Across and Within US Cities around the Turn of the Twenty-First Century Felipe Schwartzman  Y  ou have just …nished your PhD, and you have excellent o¤ers. One of them is in a major city such as Washington, DC, and the other is in a smaller place like Charlottesville, Virginia, or Durham, North Carolina. In terms of the quality of each department, all o¤ers look like excellent moves, and your advisor would be thrilled to see you in either place. It comes down to where you would rather live. If you move to the larger city, opportunities might look better down the line. There are multiple great universities in and around Washington, increasing the number of people you can interact with and learn from. International organizations in Washington, such as the IMF and the World Bank, are willing to pay high salaries to people with your quali…cations, bringing up your market wage. At the same time, a large city o¤ers unique amenities like superb restaurants and great art. On the other hand, rent is expensive: you will probably need to settle for a smaller house or a longer commute. Suppose instead that you have never pursued a PhD. In fact, you barely graduated from high school. Your choices may look fairly different. Washington, DC, will have few, if any, good jobs for you, since there is no space for the large industrial facilities that are likely to o¤er good jobs for people without college degrees. You do not have much interest in the elitist art emphasized in posh neighborhoods and cerThe views expressed in this paper are those of the author and should not necessarily be interpreted as those of the Federal Reserve Bank of Richmond or the Federal Reserve System.  DOI: https://doi.org/10.21144/eq1030101  2  Federal Reserve Bank of Richmond Economic Quarterly  tainly no disposable income to go to nice restaurants. Smaller towns may o¤er you better prospects, and you can purchase a better house there with your wage. In the end, irrespective of your education, all of these factors are also in‡uenced by your own personal preferences. Durham might be closer to relatives, or you might have a special appreciation for monuments and memorials on the National Mall.1 Together, all these factors determine a spatial equilibrium, in which people choose where to live and wages and rental prices adjust accordingly. What recent research has shown is that the nature of this spatial equilibrium has changed in the US in the past few decades. Large cities such as Washington, DC, New York, or San Francisco are increasingly places for the skilled elites. Those cities have experienced higher wage growth, and their growth has been unequal; concentrating income among educated professionals. At the same time, much of that wage growth has been o¤set by increased rents. Not surprisingly, the share of college-educated workers in these cities has increased. Those facts can be accounted for by a spatial equilibrium framework as consequences from relative increases in the demand for skilled labor by …rms in large, skilled cities. An important part of the trends may have to do with adoption of computer technology. The mechanism is also most likely related to greater spillovers among those skilled workers in those cities, but the precise nature of those spillovers is still open to more research. In what follows, I describe in greater detail the research documenting those facts and the lessons that one can derive about the underlying mechanisms. In Section 1, I lay out the facts, and in Section 2, I lay out the explanations. Section 2 includes the presentation of a canonical urban equilibrium model with two occupations that can be used to think through di¤erent mechanisms and discuss the evidence surrounding alternative hypotheses.  1.  KEY FACTS  The key facts about inequality across and within US cities can be summarized as follows: if you have a college degree or, more generally, are a more skilled worker, you are more likely to live in a larger city. Wages in those cities are generally higher, although they are also o¤set by higher rental prices. Those relative wage gains are particularly pronounced 1 Perhaps the answer is to go for the happy medium and come work in Richmond, Virginia!  Schwartzman: Inequality Across and Within US Cities  3  among more highly skilled workers, making those large, skilled cities more unequal. The literature typically identi…es skills with individual characteristics correlated with productivity. High-skilled workers can be identi…ed as ones who have high levels of education, work in high-wage occupations such as management or law, or work in high-wage industries such as professional and business services. One alternative to this vertical de…nition of skill is adopted by Bacolod, Blum, and Strange (2009), who examine how salaries vary with di¤erent skill dimensions such as cognitive processing or personal interaction. Alternatively, skills can be estimated from structural models (Baum-Snow and Pavan 2013; Eeckhout, Pinheiro, and Schmidheiny 2014). In spite of this variety of measures, di¤erent papers have recently documented fairly robust facts about spatial inequality in the US around the turn of the twenty-…rst century: 1. Larger cities have a greater concentration of high-skilled workers. This fact is true regardless of how one measures skill. Baum-Snow and Pavan (2013), Eeckhout et al. (2014), and Davis and Dingel (2017) provide evidence for measures of skill quality based on education and occupation. For example, Baum-Snow and Pavan (2013) report that in the late 2000s, about 40 percent of the workforce of cities in the top size decile had a college degree, whereas in cities in the bottom size decile only about 20 percent of the workforce had a college degree. The extent to which these relationships have changed since the 1980s appears to be dependent on the exact de…nition of skills. Berry and Glaeser (2005) and Diamond (2016) …nd that cities with a high share of college graduates have experienced a larger increase in that share, but Baum-Snow, Freedman, and Pavan (forthcoming) argue that such a relationship is not apparent if skilled workers are rede…ned to include those with “some college.”This seems to suggest that much of the change over this period has occurred due to more people …nishing college. The sensitivity of changes over time to de…nitions is probably also a re‡ection of high persistence of the educational composition of cities, a fact emphasized by Beaudry, Doms, and Lewis (2010). Such a sensitivity of time trends to the boundaries between highand low-skilled workers appears to call for a more disaggregated view. Disaggregation reveals that in recent periods both the highest- and lowest-skilled workers tend to concentrate in large cities, with smaller cities exhibiting a more concentrated skill distribution. For example, when measuring skills by education, Eeckhout et al. (2014) …nd that large cities include more college graduates, but also more high school dropouts. Of the latter, many (but not all) appear to be recent  4  Federal Reserve Bank of Richmond Economic Quarterly  international immigrants.2 The di¤erences in dispersion were not always present. Davis and Dingel (2017) show evidence to the e¤ect that this phenomenon is relatively new, with larger cities exhibiting uniformly more skilled workers in 1980. This move toward a more extreme distribution of skills in recent decades is consistent with Autor and Dorn’s (2013) description of a geographic dimension to job market polarization, with cities that have a high share of workers in occupations with intermediate wage levels (“routine intensive” occupations) seeing large shifts in their labor composition toward low-wage occupations (“service” occupations). 2. Nominal wages are overall higher and increasing in larger and in more skill-intensive cities, but real wages are not necessarily. A key distinction when interpreting geographic data is between nominal and real wages (or income, more broadly), where the latter incorporates a local price adjustment. Because not all goods consumed by households can be freely traded between cities, the law of one price does not necessarily hold everywhere. In particular, land is the ultimate nontradable good and corresponds to a large fraction of households’ consumption baskets. One of the most important stylized facts of urban economics is that larger cities exhibit higher nominal wages. Most recently, Baum-Snow et al. (forthcoming) have calculated that from 2005–07 nominal wages increase 0.065 percent for each percentage increase in city size. Relatedly, Glaeser and Maré (2001) pin the wage di¤erences between urban and rural areas to around 33 percent in 1990. Those relationships have strengthened over time. The elasticity of wages to city size reported by Baum-Snow et al. (forthcoming) for 2005–07 is about 50 percent larger than what they report for 1980. Also, in the working paper version of their 2013 paper, Baum-Snow and Pavan point out that the wage gap between the largest cities (1.5 million people or more) and rural areas increased from 24 percent in 1980 to 33 percent in 2000. These gaps re‡ect in part the increasing correlation between city size and skill mix, but after controlling for those they remain sizable at 17 percent and 24 percent, respectively, and the trend remains noticeable. There is, moreover, a strengthening of the relationship between the skill intensity of a city and wages, with more skill-intensive cities exhibiting higher wages for both skilled and unskilled workers (Diamond 2016). 2 Interestingly, they do not …nd any di¤erences in average skills between cities even as they con…rm the …ndings by others that those cities feature a higher concentration of college-educated workers. This is because college-educated workers concentrate in the higher quantiles of the skill distribution.  Schwartzman: Inequality Across and Within US Cities  5  While the relationship between nominal wages and city size is a clear and robust fact of urban economics, this relationship does not necessarily extend itself to real wages. For a recent example, Eeckhout et al. (2014) report that there is no systematic di¤erence in average real wages between cities. Moretti (2013) and Diamond (2016) show, moreover, that cities with a large share of skilled workers are also cities in which rent prices are higher and have increased the most in recent decades. In e¤ect, Moretti (2013) shows that, while inequality of nominal wages across cities has clearly increased, it was met by an increased dispersion in rents, so cross-city inequality in real wages has not increased as strongly. One important caveat to those …ndings is that the measurement of local price levels is itself fraught. In a recent paper, Handbury and Weinstein (2015) show that typical price indices measured to compare standards of living across cities are biased because they do not properly account for di¤erences in the quality and variety of goods. They …nd that after one properly controls for those, there is a negative relationship between the price of tradable goods and city size. Given existing evidence, this would imply real wages that increase with city size. 3. The skill premium is higher and increasing in larger cities or cities with more skilled workers. Larger cities appear to be more unequal. When comparing rural areas and the three largest metropolitan areas, Baum-Snow and Pavan (2013) report that in 2004–07, the variance of log hourly wages was 0.28 in the former and 0.53 in the latter. This, they show, is a relatively new phenomenon, since in 1979 the variances of log hourly wages for rural areas and the three largest metropolitan areas were 0.19 and 0.24, respectively. A major focus of the recent literature has been the evolution of the skill premium across and within cities. One robust …nding is that wage premia increase with city size. Eeckhout et al. (2014) and Davis and Dingel (2017) report those relationships for recent data using a variety of skill measures based on education, occupation, or observed real wages. Baum-Snow et al. (forthcoming) also point out that the relationship between skill premium and city size has become more pronounced over time, doubling in its strength over that period. Finally, recent data also show more skilled cities exhibiting larger skill premia (Moretti 2013), although the fact does not appear to be robust to the exact de…nition of skill. For example, Beaudry et al. (2010) and Hendricks (2011) do not appear to …nd a robust relationship in recent data. There appears to be more consensus around an increasingly positive relationship between skill composition and skill premia over time. In fact, Beaudry et al. (2010) show evidence that  6  Federal Reserve Bank of Richmond Economic Quarterly  skilled cities had lower skill premia in 1980 and before, but this negative correlation disappeared in the early 2000s. Together, these facts point to the aggregate inequality trends as having an important geographic component, with large, skill-intensive cities leading the charge.3  Implications for Interpretation of Wage Inequality The facts above suggest a reinterpretation of observed trends in wage inequality. A high wage in New York will sound appealing until one realizes how much one needs to pay for rent. Since both wages and rents are higher in large, skill-intensive cities (Fact 2), this suggests that adjusting for the local cost of living could imply less inequality in standards of living than is implied by wages alone, a point explored by Moretti (2013). He …nds that the real wage di¤erential between college and noncollege workers has increased 20 to 30 percent less than the nominal wage di¤erentials, as rents have increased more quickly in skill-intensive cities. At the same time, New York may o¤er more than smaller cities in terms of the quality of its restaurants and art scene. Diamond (2016) presents evidence that the increase in rental prices was more than matched by an increase in the amenities provided to residents of more skill-intensive cities. She estimates the e¤ect based on a structural model similar to one we will present in Section 2 below, but in her model, local amenities change endogenously in response to the population composition. She estimates the model using measures of amenities such as quality of public schools, crime rates, and restaurant density, and she …nds that once one accounts for those e¤ects, the inequality of standards of living increases by 30 percent more than what is implied by wage inequality trends alone.  2.  EXPLAINING THE FACTS  The most natural explanation for the set of facts described above, advanced by Berry and Glaeser (2005), is that the demand for skilled workers has increased more in cities that are larger and more skill intensive, while the demand for unskilled workers has not increased much 3 Again, the direction and strength of correlations appears to depend on exactly how skill is measured. Moretti (2013) …nds a very clear positive correlation between college share and college premium in 2000, whereas Beaudry et al. (2010) and Hendricks (2011) do not …nd a statistically signi…cant positive correlation between a “college equivalent” share and returns to education. The main di¤erence appears to be again in how skilled labor is de…ned.  Schwartzman: Inequality Across and Within US Cities  7  anywhere. In this section, we explain why such a view is a natural …t for the data. We will then examine di¤erent theories behind that increase in demand, including the rise of computers and externalities. The most important alternative to labor demand increases is an increase in endogenous sorting for unobserved worker characteristics, so that, for example, among college-educated workers, it is the most productive ones who choose to live in large cities such as New York and San Francisco. Nevertheless, the most recent literature appears to indicate that such sorting is unlikely to be an important driving force behind the observed facts. In order to build the argument, we rely on a class of equilibrium models that have been originally proposed by Rosen (1979) and Roback (1982). In those models, cities exist in …xed locations and are characterized by a production technology for a fully tradable good and by their land availability. At a given wage, …rms in more productive locations seek to attract more workers. However, as workers move into those cities, their demand for housing pushes rents up. Because workers are free to choose where to live, they will only choose to live in cities with high rents if wages are commensurably high. In spatial equilibrium, rents in more productive cities are just high enough to o¤set the productivity advantage of …rms in those cities. The congestion coming from scarce land ensures that all cities are populated in equilibrium, irrespective of the productivity of their workers. We now build a variant of such a model with workers of di¤erent skills. Similar variants have been used in recent work by Moretti (2013) and Diamond (2016), among others.  Model Setup There are N cities, indexed n 2 f1; :::; N g. Each of these cities is equipped with a production technology for a tradable good that depends on the number of high- and low-skilled workers in the city. A representative …rm in city n can produce quantity Yn of the …nal good L by employing LH n high-skilled workers and Ln low-skilled workers according to the constant returns to scale production function: H Yn = Fn (LL n ; Ln ):  Note that we allow the production function to be city-speci…c. Differences in the production function may also lead certain cities to produce the …nal good using more of one or the other type of labor. Those di¤erences can capture “natural advantages”that can make a city more productive than another. The clearest examples of such advantages include proximity to waterways or to fertile terrain, but one could be  8  Federal Reserve Bank of Richmond Economic Quarterly  naturally skeptical as to whether such natural advantages directly explain the productivity di¤erences between modern cities. In subsequent discussions on the causal mechanisms behind the facts surrounding inequality and geography, we allow for variation in capital stock and for externalities. Also note that we assume …rms are native to individual cities and stay there. This goes counter to a long-standing emphasis of the spatial economics literature on location decisions of …rms. The assumption is inoccuous, however, because of our assumptions of constant returns to scale and of city-speci…c production technology. One could similarly postulate a model where individual …rms are free to establish themselves in any city and produce using the local technology. In equilibrium, the zero-pro…t condition would imply the same spatial distribution of production. A less inoccuous alternative, which we do not explore, would be to allow entrepreneurs with di¤erent abilities to choose which city to live in. For an example of a framework with this feature, see Behrens, Duranton, and Robert-Nicoud (2014). Labor markets are competitive, so …rms pay wages equal to the marginal product of labor. For each skill level k 2 fL; Hg, this induces the labor demand equation: wnk =  H @Fn (LL n ; Ln ) ; @Lkn  where wnk is the wage of workers of type k in city n in terms of the …nal tradable good. Workers have preferences over the tradable good, housing, and location. For a given worker i, those idiosyncratic experiences are captured by vector worker-speci…c amenity parameters f"1 (i); "2 (i); :::; "N (i)g, where "1 (i) parameterizes the worker-speci…c preference for living in city n. Those capture the extent to which di¤erent workers have preferences for di¤erent cities. They can incorporate, for example, proximity to family or to the place where the worker grew up. A worker indexed i with skill k 2 fL; Hg living in city n enjoys a utility equal to: Unk (i) = Gn (Xnk ; Hnk ; "n (i)); where Xnk and Hnk are, respectively, the amount a worker in city n consumes of the tradable good and of housing. Note again that the worker’s utility function is allowed to depend on the city n where the household chooses to live. This captures the notion that amenities, such as weather, can make some cities overall more pleasant than others. Those amenity e¤ects operate in addition to the idiosyncratic preference shifts captured by "n (i). Understanding the origin of such  Schwartzman: Inequality Across and Within US Cities  9  amenities and the extent to which they are endogenous has also been an important topic of research (see Albouy [2012] and Diamond [2016] for recent contributions). As we will see, preferences for housing play an important role in the model: they introduce a source of congestion at the city level, generating a reason for population to spread across cities in spite of di¤erences in the marginal product of workers. Similar sources of congestion could arise in the presence of other nontradable goods (such as certain traditional services) or trade costs generating home bias in consumption. Other “nonpecuniary” sources of congestion, including pollution and crime, could be captured by endogenizing the amenity parameters. The workers’ purchases of …nal goods and housing has to satisfy their budget constraints. We assume the only income workers receive is their wage, so that Xnk + rn Hnk  wnk ;  where rn is the rental price of housing, quoted relative to the tradable good. We assume workers can rent houses but not buy them. In a static framework such as the one presented here, this di¤erence is mostly unconsequential. In a dynamic framework, the di¤erence matters since the wealth of workers who purchase housing would become a function of the history of shocks to housing values in the cities where they lived. This has the potential to generate interesting e¤ects over the wealth distribution but would be computationally challenging and has yet to be extensively explored in the literature. Workers can freely choose in which city to live. We can solve their problem in two stages: …rst, we solve for the optimal choice of housing and …nal goods consumption given that the household lives in some city n. This induces a value function Vnk (rn ; wnk ; "n (i)), satisfying: Vn (rn ; wnk ; "n (i)) = max Gn (Xnk ; Hnk ; "(i)) s:t: Xn + rn Hn  wnk :  Given those value functions, the household then selects as its living location the city where it attains the highest value. This induces a labor supply as a function of wages and rental prices. The number of workers of type k in city n is thus given by the fraction of workers who have a draw of the worker-speci…c amenity parameter f"1 (i); "2 (i); :::; "N (i)g, such that Lkn = Pr Vn (rn ; wnk ; "n (i)) > max Vn (rn0 ; wnk 0 ; "n0 (i)) Lk ; 0 n 6=n  where  Lk  is the total number of workers of type k.  10  Federal Reserve Bank of Richmond Economic Quarterly  The model is closed by housing market clearing conditions, implying that housing demand within each city has to be equal to housing supply within each city. H L L LH n Hn + Ln Hn = Hn ;  (1)  where Hn is the supply of housing in city n. Finally, we assume that all rental income is appropriated by absentee landlords who only have preferences for the …nal good and do not supply labor. This assumption ensures that the market for the tradable good clears.  Model Parameterization In what follows we argue that Facts 1, 2, and 3 in Section 1 can be largely explained by cross-city variations in the demand for skilled labor. In particular, we specialize the model by assuming that all di¤erences in production functions across locations stem from a skilled-labor augmenting component:  H Fn LL n ; Ln =  1  LL n  H n Ln  +  1  1  ;  > 1;  (2)  where is the elasticity of substitution between di¤erent skill levels, and n is a labor-augmenting parameter speci…c to the labor of collegeeducated workers. The assumption > 1 is consistent with common estimates of the elasticity of substitution between di¤erent types of labor.4 Under this parameterization, labor demand functions become: L LL n = wn  LH n =(  n)  Yn 1  (3)  wnH  Yn :  (4)  We parameterize the household utility of di¤erent types of workers k living in di¤erent cities n as: Gkn (Xnk ; Hnk ; "n (i)) = An "n (i)  Xnk 1  1  Hnk  :  The parameterization assumes a unit elasticity of substitution between …nal goods and housing. The term An "n (i) captures di¤erences in amenities between cities, with An incorporating amenities that a¤ect 4 See Ciccone and Peri (2005) for a useful summary of estimates of the elasticity of substitution between college- and noncollege-educated workers.  Schwartzman: Inequality Across and Within US Cities  11  all workers (such as climate or quality of public schools) and "n (i) incorporating worker-speci…c amenities (such as proximity to family). Under this parameterization, the value function for a household living in city n can be written as:  Vnk (rn ; wnk ; "n (i)) = "n (i)An  wnk (rn )  :  We also assume that "n (i) has a Fréchet distribution with shape parameter and is drawn independently for each city. This distribution is commonly used in trade models and has the property that it is stable under the max operator, i.e., the max of two random variables that have a Fréchet distribution is also distributed according to a Fréchet. This property makes it particularly convenient for use in aggregate models where agents make discrete choices. Furthermore, under certain conditions, it emerges naturally as the limiting distribution for the max of a sequence of random variables. It can therefore be motivated by the notion that, when considering a given city, individuals are also choosing the best of several living situations that they have available to them within that city. As we show in the Appendix, one can then derive the labor supply function as:  Lkn =  An P  n0  k wn (rn )  An0  k wn 0  Lk :  (5)  (rn0 )  The parameter controls the degree of heterogeneity in tastes. This in turn governs the supply elasticity of the labor supply. A high value of corresponds to low heterogeneity. Thus, small variations in wages received (or rents paid) by workers in some city n imply large changes in the number of workers willing to live in that city. In particular, for the extreme case in which there is no heterogeneity in tastes, ( ! 1), all workers have to be indi¤erent between all locations, so An wnk =rn is the same for all n. Conversely, a low value of corresponds to high heterogeneity. In that case, most workers choose where to live based entirely o¤ their idiosyncratic preferences, and variations in the wages (or rents paid) have little bearing on the number of workers living in each city. In the Appendix, we also show that the average utility of a household that chooses to live in city n is  12  Federal Reserve Bank of Richmond Economic Quarterly  Vk =  1  1  " X  An  1  wnk  n  wnk (rn )  ! #1  ;  where is a gamma function. Note that the average utility does not depend on the location, so that we can denote average worker welfare V k without the city subscript. The reason is that as real wages increase in a city, workers who have lower amenity values for that city decide to live there, so that high real wage cities will include more workers with low idiosyncratic preferences for that city. Given the Fréchet distribution of tastes, the positive impact of the real wage on city welfare is exactly o¤set by the negative impact of worker selection. This calculation allows us to write the labor supply condition more compactly as:  Lkn =  1  V k (rn )  !  Lk :  (6)  Finally, we assume that housing supply does not vary across cities, Hn = 1. This rules out land endowment as a key determinant of city size in the model and is consistent with the casual observation that some of the largest cities in the US, such as New York or San Francisco, are con…ned on relatively small land masses.5  Explaining the Facts with Variation in Skilled-Labor Augmenting Technology We now show how, at least qualitatively, one can explain the facts in Section 1 entirely as a function of variations in demand. In order to do this, we make the stark assumption that An = 1 for all n, so that cities do not di¤er by intrinsic amenities. The only exogenous di¤erence between cities is thus given in the skilled-labor augmenting productivity parameter n . That serves as a shifter in the demand for skilled labor in di¤erent cities. Algebraically, the way in which variation in n implies Facts 1 and 2 is easiest to see in the case of homogeneous preferences ( ! 1).6 In that case, inspection of equation (6) implies that real wages have to 5 Hsieh and Moretti (2017) explore the e¤ect of land restriction regulations on city size and …nd that because of those restrictions, there is less concentration of population in large cities than there should be, leading to substantial output losses. 6 See the Appendix for a derivation of the results in the general heterogeneous case.  Schwartzman: Inequality Across and Within US Cities  13  be the same in all locations; that is, for all n where workers choose to live, it must be the case that, for k 2 fL; Hg and n 2 f1; :::; N g, wnk  = V k:  rn  (7)  Substituting this expression in the labor demand equations (3) and (4), we have that: L LL n = V rn  LH n =(  n)  Yn ;  1  V H rn  Yn :  Taking the ratio of both labor demand functions, we have that / ( n ) 1 , so that with > 1, cities in which skilled workers are more productive will have a greater proportion of those workers. Substituting the labor demand equations into the CES production function 2, canceling out Yn , and rearranging yields an expression for rents in each city n,  L LH n =Ln  rn =  h  VL  1  +(  n)  1  VH  1  i1  1 1  :  It follows that, along the cross-section of cities, rents increase in . n From the indi¤erence condition (7), nominal wages for both worker types must increase with rents. Thus, cities with a higher fraction of skilled workers are also cities with higher nominal wages for all workers, consistent with Fact 2. At the same time, real wages in those cities are not necessarily higher. Optimal worker demand for housing implies they will spend a fraction of their income on housing. Using the indi¤erence condition (7), we can write housing demand in terms of rents only: Hnk = V k rn  1  :  Thus, from housing market clearing condition 1, H L LH + LL nV nV  rn  1  = 1:  This last expression appears to suggest that population increases with rents, consistent with Fact 1. However, this is not necessarily true, since wages of low-skilled workers are generally smaller than those of high-skilled workers, that is, V L < V H . In the Appendix, we show that, for small variations in n around a cross-city average, the population  14  Federal Reserve Bank of Richmond Economic Quarterly  of both types of workers increases with n if and only if < 1. To see why this is necessary, note that as n increases, …rms will hire fewer low-skilled workers for each unit of output they produce. How much substitution occurs depends on the elasticity parameter . However, higher n also implies higher overall output. Thus, the net e¤ect in the demand for low-skilled workers is ambiguous. Output will increase more with productivity if rents are a small share of wage income (low ). Otherwise, workers will demand large wage increases in order to o¤set small increases in rent, thus limiting output variation. To summarize, LL n increases with n if …rms are not too ready to substitute between the two types of workers (i.e., if is low) and if wages are not too sensitive to rents (i.e., if is low). Fortunately, those two parameters have been amply estimated. Typically, the housing share of consumption is pinned at = 1=3, and the elasticity of substitution between collegeand noncollege-educated workers is smaller than = 2.7 Thus, we can safely assume that < 1, so that demand for both types of employment rises with productivity of skilled workers. It follows that variations in n can thus also account for Fact 1.8 Finally, in order to explain Fact 3, we need to depart from the model with homogeneous preferences. If workers have identical preferences for living in all cities, the wage premium has to be the same in all cities (wnH =wnL = V H =V L ). Fact 3 emerges once one allows for such labor heterogeneity. Manipulating the labor demand and labor supply equations (3), (4), and (6), we thus have that wnH =V H wnL =V L  =  LH n = LL n  wnH wnL  (  n)  1  ;  where the …rst equation we obtain from labor supply and the second from labor demand. Combining the two equations then yields the wage premium: wnH = wnL  VH VL  +  (  n)  1 +  :  It follows that, so long as there is some heterogeneity in preferences ( < 1) and > 1, the wage premium increases with n . 7 See Ciccone and Peri (2005) for estimates of : A value of close to 1/3 has been used by Hsieh and Moretti (2017) among others. 8 One interesting question is what happens if there are multiple tradable industries with di¤erent intensities of use for di¤erent workers. Then substitution at the city level may be higher than at the …rm level or in the aggregate, since it can operate through changes in the industrial composition.  Schwartzman: Inequality Across and Within US Cities  15  Sources of Variation in Demand for Skilled Labor The recent literature has identi…ed two main sources of variation in the demand for skilled labor: di¤erential biased technical progress, expressed by di¤erences in adoption of computing technology, and externalities. The …rst source has been emphasized by Beaudry et al. (2010) and Autor and Dorn (2013), whereas the second has been emphasized by Baum-Snow et al. (forthcoming). We discuss those two in turn.9 A third source of variation in the demand for skilled labor that has been less explored in the literature comes from the industrial composition of cities and, in particular, the importance of the business services sector (Hendricks 2011). We discuss this third source last. Computers One explanation for the increase in the productivity of skilled workers is the adoption of computing technology (Krusell et al. 2000). This is a natural hypothesis, since computers have become cheaper in the same period in which wage inequality has increased. At the same time, in that same period, many of the trends associated with Facts 1, 2, and 3 have been in play. This has motivated Beaudry et al. (2010) and Autor and Dorn (2013) to propose computerization as an explanation for cross-city variation in wage inequality trends. To see the role of computer adoption, extend the model to allow the production function to include three inputs, the third of which is capital:  F  H LL n ; Ln ; Kn  2  = 4  > 1  LL n  1  +  1  Kn  1 1  1  LH n  !  1  3 5  Note that capital is complementary to skilled labor but not to unskilled labor. In the spatial setting, Autor and Dorn (2013) have motivated this complementarity through a task-based approach. Computers are particularly adept at enhancing the ability of workers performing abstract tasks involving creativity, coordination, and problem solving, whereas automation can serve as a substitute for workers performing routine tasks such as bookkeeping, clerical work, and repetitive production. 9 Giannone (2017) is a recent contribution analyzing the implications of biased technical progress and local externalities in a common framework.  1  ;  16  Federal Reserve Bank of Richmond Economic Quarterly  Suppose computers are a perfectly tradable good, which can be purchased in any city n at the same price p. The price should be understood to incorporate the “quality-adjusted” cost of computers, so that its time variation would include all the large gains in computational power in the past few decades.10 Then cost minimization implies H that Kn = 1 wpn LH n , so that substituting into the production function we have: 2  Yn = 4  1  1  LL n  +  1  wnH p  1  3  1  5  LH n  :  The resulting function on the right-hand side is isomorphic to the one in our basic model, with the labor-augmenting technology parameter substituted for an increasing function of the relative cost of skilled labor and computers ( n = 1 1 wnH =p ). As the relative price of computers decreases, …rms use more of those, increasing the relative productivity of skilled workers. With homogeneous preferences for location ( ! 1), we can rewrite the production function in terms of rents: 2  Yn = 4  LL n  1  1  +  1  (rn ) H V p  !  LH n  !  1  3 5  1  :  Cities with higher rents are also cities where wages are higher and where …rms have the most incentives to adopt computers. This, in turn, generates a correlation between rents and the labor mix of cities. In the presence of heterogeneity in preferences for di¤erent cities ( < 1), cities with higher rents will also feature more wage inequality. Of course, once we endogenize n , we need to have another source of exogenous variation in order to explain di¤erences in rents across cities. One possibility is to still allow for some exogenous variation in the skill premium, possibly due to the location of universities or to allow for citywide productivity di¤erentials. In order to explain the observed trends, Beaudry et al. (2010) examine an environment in which they can explain observed trends just from the reduction in the price of computers (p). In their environment, apart from the three factors of production, …rms have the option to 10 Suppose a unit of an input purchased at a price p~ produces z units of output according to a linear technology. Then one can de…ne the quality-adjusted value of the input by p = p~=z. In our example, the output would be computational power, and z would capture the increase in quality of computers.  Schwartzman: Inequality Across and Within US Cities  17  choose between technologies with higher or lower intensity in skilled labor. As computers become cheaper, the incentive to use a technology that is more intensive in skilled labor increases. Beaudry et al. (2010) also show, however, that this incentive is stronger in cities where the supply of skilled labor is higher to begin with.11 This generates a correlation between the adoption of computers and the cross-city supply of skilled labor before computers were adopted, which they verify in the data. Intriguingly, they also show that before computers became an important part of production, the cross-city correlation between skill composition and skill premium was negative, as predicted by our model if there are no important di¤erences in n across cities but there are di¤erences in relative labor supply. As time has progressed and computer adoption has increased, they …nd that correlation ‡ipping and becoming insigni…cantly di¤erent from zero. One limitation of their explanation for the observed trends is that it cannot account for the positive correlation between skill composition and wage premia observed in recent data (or with other de…nitions of skill). In their framework, as the price of computers declines, all cities adopt the skill-intensive technology. Externalities An alternative explanation for the di¤erences in demand for skilled labor in di¤erent cities involves the di¤erential impact of local externalities on di¤erent types of workers. Moretti (2004) separates relevant theories into those involving learning and those involving labor markets.12 Learning-based theories emphasize the role of geographic proximity in facilitating the transfer of knowledge.13 High-skilled workers perform tasks that are more knowledge-intensive, so they would stand to bene…t more from those transfers. Matching theories exploit pecuniary externalities that emerge in imperfect labor markets. A deeper 11  In terms of our model, this would involve extending the utility function to contain a city/skill-speci…c e¤ect, so that k Gkn (Xnk ; Hn ; "n (i)) = Akn "n (i)  Xnk 1  1  k Hn  :  12 See Duranton and Puga (2004) for a detailed overview of theories of agglomeration externalities more generally, including the role of sharing, matching, and learning. 13 In Marshall’s (1890) words “Great are the advantages that people following the same trade get from near neighborhood to one another: the mysteries of the trade become no mysteries; but are, as it were, in the air.” Lucas (1988) follows up by stating that “Most of what we know we learn from other people (. . .) most of it we get for free. We know that this kind of external e¤ect is common to all arts and sciences – the ‘creative professions.’”  18  Federal Reserve Bank of Richmond Economic Quarterly  pool of skilled workers provides an environment where …rms can use them more e¢ ciently.14 There is an ample empirical literature pointing to human capital externalities as a source of di¤erences in wages across cities (see Moretti [2004] for a review). This literature focuses on the e¤ects of the skill composition of the workforce. Most recently, the role of externalities in explaining observed trends has been examined by Baum-Snow et al. (forthcoming). In their paper, they focus on what the urban economics literature has called “agglomeration externalities,”i.e., external e¤ects associated with overall city size. They allow for city size to a¤ect the productivity of skilled and unskilled labor separately (and also of capital, which they allow for explicitly). They …nd a robust signi…cant increase over time in the positive agglomeration e¤ect on skilled labor but do not …nd a robust change in the agglomeration (or congestion) e¤ect L on unskilled labor. In terms of our model, they …nd n = LH n + Ln , 15 where > 0. They state that such a change in agglomeration externalities can account for 80 percent of the more rapid increase in wage inequality in large cities, with capital accumulation, the other leading alternative discussed above, accounting for less than 20 percent. Recent work has also tried to disentangle the two sources of local spillovers. Learning and matching theories have di¤erent predictions at the microeconomic level. Learning theories imply that individual productivity is likely to increase with the time that individuals spend in a city and that productivity gains will be embodied in workers who will carry those gains with them if they immigrate. In contrast, matching externalities are unlikely to change with the time a worker has spent in a city and remain speci…c to the city. The evidence is consistent with learning among the high-skilled workers playing an important role, with Glaeser and Maré (2001), Baum-Snow and Pavan (2013), and De La Roca and Puga (2017) pointing out that high-skilled workers tend to experience faster wage increases when they live in large cities.16 Understanding why local externalities have become stronger for skilled workers is challenging. One interesting mechanism is explored by Michaels, Rauch, and Redding (2017). Investigating changes over long periods of time and using a detailed breakdown of occupations into a variety of tasks, they document a rising concentration of occupationintensive “interactive tasks”in cities. Those tasks are ones “concerned 14 For example, it could reduce the importance of holdout problems, allowing for more investment in skill-complementary capital (Acemoglu 1996). 15 In reality, they are measuring changes in . All the comparative statics presented before also hold for changes in n . 16 See also Davis and Dingel (2016) for a microfounded spatial model with knowledge spillovers that captures most of the facts above.  Schwartzman: Inequality Across and Within US Cities  19  with thought, communication, and inter-social activity” and are typically associated with highly skilled occupations such as nurses, accountants, and statisticians. In their model, those tasks tend to become concentrated as the cost of trading them across cities decreases and large cities are able to exploit their comparative advantage in those tasks. Industrial and Functional Composition One potential source of variation in demand for skilled labor is variation in industry composition of cities. One extreme case, which is nested in our model, has each city specialize in a single industry. Since industries may have di¤erent skill intensities, such industrial specialization would imply cross-city variation in n . For example, Brinkman (2014) points to the increased concentration of certain skill-intensive industries, such as …nance, in large cities as an important factor explaining some of the trends discussed. However, as argued by Hendricks (2011), cross-city variation in industrial composition accounts for only a small fraction of cross-city variation in skill composition. Moreover, he shows that most of the variation in skill composition across cities can be tied to variation in a high-skill-speci…c productivity component that is common to all industries. At the same time, Hendricks (2011) also …nds that cities with a high fraction of skilled workers are also ones with large business services sectors. He proposes a model where, like computers, the output of those services is complementary to high-skilled labor. When external accountants become cheaper, …rms hire those to work with their internal sta¤ rather than as a substitute. In terms of our model, cities where business services are cheaper would thus have higher n . He endogenizes the variation in business services productivity by allowing for increasing returns to scale in that sector. One advantage of this focus on the business services sector over a focus on learning or matching externalities is that it provides a mechanism through which the external e¤ects on the employment of skilled workers spread somewhat uniformly over a fairly heterogeneous set of sectors, in consonance with Hendricks’s (2011) data analysis. A special role for the business services sector is shared by Duranton and Puga (2004). They show that this is associated with an increasing specialization of cities by function, meaning that …rms have increasingly concentrated executives and managers in larger cities and production activities in smaller ones. They therefore develop a model in which there are gains to concentrating management in cities with a high concentration of business services. While they do not explicitly tie their model to facts about the skill composition of cities and their wage  20  Federal Reserve Bank of Richmond Economic Quarterly  di¤erentials, it is natural to map workers active in their management function as relatively high-skilled workers and those active in production as relatively low skilled. Given this mapping, their model could also deliver a concentration of high-skilled workers in large cities, as in the data.  Sorting for Unmeasured Skill Di¤erences in labor demand need not be the only source of the systematic di¤erences in skill composition, wages, and skill premia across cities. An alternative explanation relies on sorting for unobserved worker characteristics. To see how that works, suppose there are multiple types of workers rather than just two, and assume for simplicity that they are all perfect substitutes in production, so that17 Yn =  X  k  Lkn :  k  Note that now the production function is the same for all cities. The parameters k capture the marginal product of workers of type k in any city. Since the di¤erent types of workers are perfect substitutes, for any worker type k in any city n, it will be the case that wnk = k . Observed wage variation across cities can occur if di¤erent types of workers are combined in common bins over the course of empirical analysis. For example, a broad “college-educated” group of workers might include workers with some college, with four-year college degrees, and with postgraduate degrees; within each of these bins, workers may have heterogeneous innate abilities. One can then explain many of the empirical facts described in the literature through such sorting (for a detailed exposition see, for example, Davis and Dingel [2017]). To see how such an e¤ect of skill sorting could come about, reparameterize the utility function of individual workers to: ( An "(i)Xnk if Hnk Gkn (Xnk ; "(i)) = 1 otherwise  1  ;  that is, a worker requires a minimal amount of housing to survive, which we normalize to 1, but beyond that how much housing they purchase is irrelevant to their utility. This is an extreme form of nonhomotheticity 17  There are other ways of generating such sorting. See Eeckhout et al. (2014) for an alternative sorting mechanism based on complementarities between di¤erent types of workers in nonhomothetic production technology.  Schwartzman: Inequality Across and Within US Cities  21  in preferences, with housing considered a “necessity.” Note that, in order to show that in this setup it is possible to account for many of the key facts as stemming from di¤erences in labor supply, we now allow for an exogenous city-speci…c variation in amenities, captured by An . Under this parameterization, the value for a worker of type k of living in city n is Vnk wnk ; rn ; "(i) = "(i)An wnk  rn :  Given the Fréchet distribution of "(i), the fraction of workers of type k living in city n is An wnk rn Lkn = P k Lk rn0 n0 An0 wn0  :  As before, the fraction increases with wages and decreases with rents. The di¤erence is that now it increases more rapidly with wages when wages are smaller. Substituting in the labor demand condition wnk = k yields: An k Lkn P = Lk n0 (An0 (  rn k  rn0 ))  :  As rents in a city increase, the labor composition shifts toward workers with higher productivity. Finally, note that since all households buy one unit of housing, housing market equilibrium implies that: X  Lkn = H:  k  Since labor supply is increasing in the city-speci…c amenity and decreasing in rents, it follows that rents have to increase with the cityspeci…c amenity. Thus, cities with higher amenities also have higher rents and a more skilled workforce. If skill di¤erences are not directly observable, this could translate into higher measured wages for each skill level. If skill di¤erences are particularly hard to observe among the most highly skilled workers, this sorting would translate into a higher wage premium for those cities. The mechanism described here could be criticized on a priori grounds, since it relies on assuming that housing is a necessity. As a matter of fact, while the share of expenditures on shelter does decrease with income, it decreases by relatively little, with higher-income households spending more dollars. Furthermore, one implication of the sorting mechanism is that more-skilled workers are less sensitive to city-speci…c  22  Federal Reserve Bank of Richmond Economic Quarterly  prices. This is contrary to much evidence that …nds high-skilled workers to be more mobile.18 While such a priori criticism could potentially be accommodated with suitable changes to the model without abandoning some of the key insights, more direct empirical assessments have not been favorable to sorting for unobservable characteristics. A …rst approach, adopted by Combes, Duranton, and Gobillon (2008), is to assess the e¤ects of sorting on city wage di¤erences through worker …xed e¤ects. Those exercises rely on changes in the wages of workers who migrate between cities to separate the e¤ects of sorting from other city characteristics.19 While this …rst approach indicates a large role for sorting, it could underestimate or overestimate those e¤ects to the extent that changes in the wages of migrants are not representative of the di¤erences in wages of the overall population of cities. For example, if individuals migrate from small cities to large cities only when they receive particularly good wage o¤ers, this selection e¤ect would lead to an overestimation of the relationship between city size and wages. On the other hand, if workers move to large cities when they are young, take advantage of local learning opportunities to slowly become better workers, and then move back to smaller cities carrying their new abilities with them, such evidence would tend to underestimate the contribution of large cities to worker productivity since wages would change little at the time of moving. Baum-Snow and Pavan (2012) and De La Roca and Puga (2017) speci…cally control for these e¤ects and …nd that allowing for learning within cities is especially important. Once that learning is explicitly allowed for, selection for unobservable skills ceases to be an important source of bias in estimating the e¤ect of city size on productivity.  3.  CONCLUSION  Inequality in the United States has an important spatial component. More-skilled workers tend to reside in larger cities where they earn higher wages. In the meantime, less-skilled workers make lower wages even when they live in those cities. Those relationships appear to have become more pronounced as inequality has increased. The evidence points to externalities among high-skilled workers as a signi…cant contributor to those patterns. This suggests that policy may face an equity e¢ ciency trade-o¤. The presence of positive externalities among 18 See Notowidigdo (2011) for a discussion of how this fact can be rationalized by allowing for transfers that occur to all households irrespective of their income. This is the mirror image of the mechanism discussed here. 19 See Combes and Gobillon (2015) for a thorough discussion of these and other econometric issues involved in estimating the impact of city size on wages.  Schwartzman: Inequality Across and Within US Cities  23  high-skilled workers would imply gains to policies that incentivize them to become more concentrated. However, this would tend to increase inequality across and within cities. As pointed out by Fajgelbaum and Gaubert (2018), who discuss these trade-o¤s in detail, the inability to separate the place one lives from the place one works may lead a utilitarian planner to choose to redistribute income across cities so as to compensate workers who have high amenity value of living in lowproductivity cities. While the aforementioned paper provides a substantial …rst step in that direction, the literature thoroughly exploring these trade-o¤s from various angles is still in its infancy.  24  Federal Reserve Bank of Richmond Economic Quarterly  APPENDIX Appendix 1: Derivation of Labor Supply and Average Utility with Frechet Distributed Preferences Under a Fréchet distribution, "n has a support between zero and in…nity with c.d.f. "n is e "n ; and the p.d.f. is "n 1 e "n . Note that in order for any value x to be smaller than the max of several variables, it must be the case that it is smaller than each one of them. Given that individual draws of "n are independent, we can calculate this probability by multiplying the individual c.d.f.’s. Speci…cally, for any given ^"n . Pr ^"n wnk =(rn )  max "n0 wnk 0 =(rn0 ) 0  =  n 6=n  Y  n0 6=n  =  Y  h Pr "n0 (wn0 ) =rnk 0 "  =  Y  (rn0 ) =wnk 0  Pr "n0  n0 6=n  (rn ) =wnk  =wk 0 n k (rn ) =wn  (rn0 ) e  (wn ) =rnk ^"n  ^ "n  ^"n  #  !  n0 6=n  =e where  "n n^  P 1 k . To …nd Pr n0 6=n wn0 = (rn0 ) (wnk =(rn ) ) maxn0 6=n "n0 wnk 0 =(rn0 ) integrate over all ^"n , i.e., cal-  n  "n wnk =(rn ) culate Z  1  n"  e  dF ("n ) =  1  Z  1  "n  0  0 P  1  e  "n  d"n ;  (wnk 0 =(rn0 ) ) . The integrand is equal to the pdf of a (wnk =(rn ) ) Fréchet with scale parameter and which must therefore integrate to 1. It follows that  where now  Z  0  1  e  n0  n"  dF ("n ) =  1  =P  n0  wnk = (rn ) wnk 0 = (rn0 )  :  i  Schwartzman: Inequality Across and Within US Cities  25  h We kcan also derive average utility in each i city. This is given by wn k k maxn0 6=n "n0 wn0 =(rn0 ) . Again, we can write E "n j"n wn =(rn ) rn this as Z  0  1  E "n  wnk rn  j"n  wnk 0 =(rn0 ) 0 max " n n0 6=n wnk =(rn )  dF (^"n ) =  Z  1  0  = =  =  wnk  (rn ) wnk (rn ) " X  wnk  "n Z  (rn ) 1  "n  0 1  "n  1  "n  1  1  wnk 0 = (rn0 )  #1  where the …rst equality follows from the de…nition of conditional expectation, and the third equality follows from the fact that the integrand is the expected value of a Fréchet with scale parameter .  Appendix 2: Comparative Statics with Heterogeneous Preferences The model can be summarized by the following system of equations: 1  LL n  L LL n = wn  LH n =(  Lkn  =  Vk =  "  1  H n Ln  +  1  wnH  wnk k  V (rn )  n0  ;  Yn ;  n)  X  1  wnk 0 (rn0 )  Yn ;  !  Lk  ! #1  L H H LL n Hn + Ln Hn = 1;  ;  > 1;  e  1  n0  Yn =  e  ;  "n  d"n  "n  d"n  26  Federal Reserve Bank of Richmond Economic Quarterly  rn Hnk = wnk : Consider an equilibrium where all cities have the same linearize around that equilibrium to get: dyn = dlnL + (1 dlnL = dlnH dlnk  = =  dv k =  dwnH dwnk  X Ln Ln  (dlnL + dhL n ) + (1 drn +  =  )d  dwnL + dyn ;  n0  dhkn  )dlnH + (1  n;  n.  Log-  (8) (9)  + dyn + (  1)d  n;  (10)  dv k  drn ;  (11)  (dwnk 0  drn0 );  (12)  )(dlnH + dhH n ) = 0;  dwnk ;  (13) (14)  where we use dyn to denote the log deviation of Yn from the identical city case and where w L LL = P n k nk ; wn L n  and  1  LL n  1 1 : H) (LL ) + ( L n n n We can show that = . To see this, solving the demand equations for wnk and multiplying both sides by Lkn yields  =  1  L wnL LL n = Ln  wnH LH n =  1  (Yn ) ;  H n Ln  1  1  (Yn ) :  Plugging those equations in the formula for yields the formulas for . Next, use the log-linearized housing demand equation (14) to eliminate dhkn from the housing market clearing equation (13) and rearrange to get (dlnL + dwnL ) + (1  )(dlnH + dwnH ) = drn :  Use the labor supply equation (10) to substitute out dwnk from the other equations in the system and rearrange. De…ne ddyn = dyn dy;  Schwartzman: Inequality Across and Within US Cities  27  where dy denotes the cross-city average log-change in Yn and analogously for other variables. The system becomes: ddyn = ddlnL + (1 ddlnL =  +  ddlnH =  +  0= 1+  ddrn +  Ln  ddlnL + (1  )  1+  ddyn +  +  (ddwnk 0  )dd  n;  (15)  ddyn ;  +  ddrn +  X Ln n0  )ddlnH + (1  (16) (  1) +  dd  n;  (17)  ddrn0 );  (18)  ddlnH = (1  ) ddrn :  (19)  Using equations (15), (16), and (17) to substitute out ddlnk from equation (19) and using = yields an expression for rents as a function of productivity. 1+ (1 ) > 0: 1+ Manipulating (15), (16), and (17) to obtain an expression ddyn as a function of ddrn , we can substitute the expression above to get: ddrn =dd  n  =  ddyn =dd  n  = (1  For the limiting case with ddyn =  +1 > 0: 1+ ! 1, this yields 1  )  (1  For the other limiting case, with ddyn = (1 We can also obtain expressions for ddlnL =dd  n  =  ddlnH =dd  n  =  + +  (1 (1  )  1+ [1 1+ 1+ ) 1+  )dd  n:  =0 )dd ddlnL  n:  and ddlnH :  ]; 1  1 +(  1) 1  (1  )  + 1+  > 0:  28  Federal Reserve Bank of Richmond Economic Quarterly  It follows that ddlnL > 0 if 1 > , i.e., if land share is su¢ ciently small or the two types of labor are not too strongly substitutable. Normal calibrations feature ' 13 and ' 2, so that the condition is satis…ed and population grows for both types of workers with n , although most strongly for the high type. Again, with ! 1 the expressions simplify to ddlnL =dd  n  = (1  ddlnH =dd  n  = (1  and we verify that with  1  )  ; 1  )  1 + (  1) > 0;  = 0 employment does not change:  ddlnL =dd  n  = 0;  ddlnH =dd  n  = 0:  Calculating real wages, we get: ddwnL =dd  n  ddrn =  ddwnH =dd  n  ddrn =  1 (1 + 1 (1 +  )  1+ [1 1+ 1+ ) 1+  ]; 1  1 +(  1) 1  Thus, real wages will increase or decrease in relative terms with relative employment. With ! 1; we have that real wages do not change in relative terms at all. With = 0,  ddwnH  1 ddrn = (1  ddwnH  ddrn =  1  [(1  ) [1 ) [1  ]; ]+(  Finally, we can obtain nominal wages:  1) (1  (1  ) )] :  (1  )  + 1+  :  Schwartzman: Inequality Across and Within US Cities  29  1+ 1+ (1 ) ; + 1+ 1 1+ + 1 ddwnH =dd n = (1 ) 1 +( 1) 1 (1 ) + 1+ 1+ 1+ + (1 ); 1+ so that both nominal wages increase unambiguously. Taking limits, for !1 ddwnH =dd  n  whereas for  =  ddwnL =dd  n  =1  :  ddwnH =dd  n  =1  ;  =0 1  ddwnL =dd  n  =  ddwnH =dd  n  =1  (1  ); :  Comparative Statics with Housing as a Necessity With housing as a necessity good as in subsection 2.5, we can derive the labor supply function following the same steps as in Appendix 1 to get: Lkn = P  wnk  n0  rn  wnk 0  rn0  Lk :  It follows that, using the same “deviation” notation as above ddLkn =  wk  ddwnk  r  ddrn : r r Since all workers buy one unit of housing, the housing market equilibrium equation can be written as: wk  X  wk  Lkn = Hn :  k  Suppose the production function is the same in all cities but housing supply is not. The equations de…ning the equilibrium of the economy in log-linearized form become:  30  Federal Reserve Bank of Richmond Economic Quarterly  ddyn = ddlnL + (1 ddlnk =  (20)  ddwnk + ddyn k 2 fL; Hg; wk  ddlnk = 0=  )ddlnH ;  wk  r  X Ln n0  (  Ln  ddlnL + (1  r  ddwnk  wk  wk wk  r  r  ddwnk  (21)  ddrn ; k 2 fL; Hg; r  wk  r  (22)  ddrn );  (23)  )ddlnH = ddhn ;  (24)  LL n L Ln +LH n  where now = . Use the new labor supply equation (22) to eliminate wages from the labor demand equations and the production function to eliminate output: wk r k r ddln + k ddrn + ddlnL + (1 k w w  ddlnk =  )ddlnH :  Combine those with the production function and substitute out the new housing clearing equilibrium (24) to obtain: +  wH r ddlnH wH  ddlnL =  )ddlnH + 1  (1  +  r ddrn ; wH wL r ddlnL = wL  r ddrn : wL  Solve the system for ddlnL and ddlnH as functions of ddrn . From inspection of the structure of the problem, it follows that:  where  'H  0 with D 'H > 'L if  h  h  +  +  wL r wL  i  ddlnH =  'H ddrn  ddlnL =  'L ddrn ;  r wH  +  r wL  D ih wH r 1 wH  wH wL  1  > 0 and +  > (1  h  'L  wL r wL  i  1  +  wH wL  i  r +(1 wL  )  D  + (1  2 ) 1+  wH r wH  ) .  Note that  :  Substituting back into the housing market clearing equation yields:  r wH  >  Schwartzman: Inequality Across and Within US Cities  'H + (1  31  )'L ddrn = ddhn :  It follows that interest rates fall with housing supply, and employment in both sectors rises. At the same time, the labor mix will change. H So long as wwL is su¢ ciently greater than 1, skilled labor will increase more rapidly with housing supply than unskilled labor. At the same time, because the elasticity of labor supply at the city level is not in…nity, wages for skilled labor will have to grow faster. Consider now an environment with three worker types, L, HL; and HH. The production function is now  Yn =  LL n  1  HH LHL n + Ln  +  1  1  :  For simplicity, we will focus on the case with ! 1 so that highand low-skilled workers are also perfect substitutes. Suppose now there are common city-speci…c amenities, so that their utility of living in city n is now: Gkn (Xnk ; "(i))  ( An "(i)Xnk if Hnk = 1 otherwise  1  ;  where An represents amenities such as good public schools or nice weather. Those factors are common for all types of labor. Labor supply for k 2 fLH; HHg Lkn = P  An wnk  rn  Lk : An0 rn0 Using the fact that di¤erent types of labor are perfect substitutes so that their wages do not vary in the cross-section, the log-linearized system in deviation form becomes: wnk 0  n0  ddyn = (1  LH  HH  )ddlnL + r  ddlnk =  wk  r  LH  ddlnLH +  HH  ddlnHH ;  (25)  ddrn + ddakn ; k 2 fL; HL; HHg; (26)  0=  X Ln n0  (1  HL  HH  )ddlnL +  HL  r  ddrn ;  Ln wk  r  ddlnHL +  HH  ddlnHH = 0;  (27) (28)  32  Federal Reserve Bank of Richmond Economic Quarterly  Substitute the new labor supply equations into the market clearing equation to get:  (1  HL  HH  )  r wL  r  +  HL  r wHL  r  +  HH  r wHH  r  ddrn = ddan :  It follows that rental rates rise with amenities. At the same time, from equation (26) it follows immediately that employment reacts more to interest rates if wages are smaller. Thus, a given increase in amenities will imply larger increases in the population of workers with higher wages. At the same time, even though wages do not change, if one cannot observe HL and HH type workers separately, one will see their average wage rise, as the composition of high-skilled workers shifts toward the ones with a very high skill. Thus, cities with higher amenities will have a more skilled labor force and higher wage premia. One can further generate the correlation with city size by allowing housing supply to be rent elastic.  Schwartzman: Inequality Across and Within US Cities  33  REFERENCES Acemoglu, Daron. 1996. “A Microfoundation for Social Increasing Returns in Human Capital Accumulation.” Quarterly Journal of Economics 111 (August): 779–804. Albouy, David. 2012. “Are Big Cities Bad Places to Live? Estimating Quality of Life across Metropolitan Areas.”Working Paper (May). Autor, David H., and David Dorn. 2013. “The Growth of Low-Skill Service Jobs and the Polarization of the US Labor Market.” American Economic Review 103 (August): 1553–97. Bacolod, Marigee, Bernardo S. Blum, and William C. Strange. 2009. “Skills in the City.” Journal of Urban Economics 65 (March): 136–53. Baum-Snow, Nathaniel, Matthew Freedman, and Ronni Pavan. Forthcoming. “Why Has Urban Inequality Increased?” American Economic Journal: Applied Economics. Baum-Snow, Nathaniel, and Ronni Pavan. 2012. “Understanding the City Size Wage Gap.” Review of Economic Studies 79 (January): 88–127. Baum-Snow, Nathaniel, and Ronni Pavan. 2013. “Inequality and City Size.” Review of Economics and Statistics 95 (December): 1535–48. Beaudry, Paul, Mark Doms, and Ethan Lewis. 2010. “Should the Personal Computer be Considered a Technological Revolution? Evidence From U.S. Metropolitan Areas.” Journal of Political Economy 118 (October): 988–1036. Behrens, Kristian, Gilles Duranton, and Frédéric Robert-Nicoud. 2014. “Productive Cities: Sorting, Selection, and Agglomeration.” Journal of Political Economy 122 (June): 507–53. Berry, Christopher R., and Edward L. Glaeser. 2005. “The Divergence of Human Capital Levels Across Cities.” Papers in Regional Science 84 (August): 407–44. Brinkman, Je¤rey. 2014. “The Supply and Demand of Skilled Workers in Cities and the Role of Industry Composition.” Federal Reserve Bank of Philadelphia Working Paper 14-32 (October).  34  Federal Reserve Bank of Richmond Economic Quarterly  Ciccone, Antonio, and Giovanni Peri. 2005. “Long-Run Substitutability Between More and Less Educated Workers: Evidence from U.S. States, 1950–1990.” Review of Economics and Statistics 87 (November): 652–63. Combes, Pierre-Philippe, Gilles Duranton, and Laurent Gobillon. 2008. “Spatial Wage Disparities: Sorting matters!” Journal of Urban Economics 63 (March): 723–42. Combes, Pierre-Philippe, and Laurent Gobillon. 2015. “The Empirics of Agglomeration Economies.” In Handbook of Urban and Regional Economics Vol. 5, edited by Gilles Duranton, J. Vernon Henderson, and William Strange. Amsterdam: North Holland, 247–348. Davis, Donald R., and Jonathan I. Dingel. 2016. “A Spatial Knowledge Economy.” Working Paper (August). Davis, Donald R., and Jonathan I. Dingel. 2017. “The Comparative Advantage of Cities.” Working Paper (March). De La Roca, Jorge, and Diego Puga. 2017. “Learning by working in big cities.” Review of Economic Studies 84 (January): 106–42. Diamond, Rebecca. 2016. “The Determinants and Welfare Implications of US Workers’Diverging Location Choices by Skill: 1980–2000.” American Economic Review 106 (March): 479–524. Duranton, Gilles, and Diego Puga. 2004. “Micro-foundations of Urban Agglomeration Economies.” In Handbook of Urban and Regional Economics Vol. 4, edited by J. Vernon Henderson and Jacques-François Thisse. Amsterdam: Elsevier, 2063–117. Duranton, Gilles, and Diego Puga. 2005. “From Sectoral to Functional Urban Specialisation.” Journal of Urban Economics 57 (March): 343–70. Eeckhout, Jan, Roberto Pinheiro, and Kurt Schmidheiny. 2014. “Spatial Sorting.” Journal of Political Economy 122 (June): 554–620. Fajgelbaum, Pablo D., and Cecile Gaubert. 2018. “Optimal Spatial Policies, Geography and Sorting.” Working Paper 24632. Cambridge, Mass.: National Bureau of Economic Research. (May). Giannone, Elisa. 2017. “Skilled-Biased Technical Change and Regional Convergence.” University of Chicago Working Paper (January). Glaeser, Edward L., and David C. Maré. 2001. “Cities and Skills.” Journal of Labor Economics 19 (April): 316–42.  Schwartzman: Inequality Across and Within US Cities  35  Handbury, Jessie, and David E. Weinstein. 2015. “Goods Prices and Availability in Cities.” Review of Economic Studies 82 (January), 258–96. Hendricks, Lutz. 2011. “The Skill Composition of U.S. Cities.” International Economic Review 52 (February): 1–32. Hsieh, Chang-Tai, and Enrico Moretti. 2017. “Housing Constraints and Spatial Misallocation.” Working Paper (May). Krusell, Per, et al. 2000. “Capital-Skill Complementarity and Inequality: A Macroeconomic Analysis.” Econometrica 68 (September): 1029–53. Lucas Jr., Robert E. 1988. “On the Mechanics of Economic Development.” Journal of Monetary Economics 22 (July): 3–42. Marshall, Alfred. 1890. Principles of Economics, 1st ed. London: Macmillan. Michaels, Guy, Ferdinand Rauch, and Stephen Redding. 2017. “Task Specialization in U.S. Cities from 1880-2000.” Working Paper 18715. Cambridge, Mass.: National Bureau of Economic Research. (August). Moretti, Enrico. 2004. “Human Capital Externalities in Cities.” In Handbook of Urban and Regional Economics Vol. 4, edited by J. Vernon Henderson and Jacques-François Thisse. Amsterdam: Elsevier, 2243–91. Moretti, Enrico. 2013. “Real Wage Inequality.” American Economic Journal: Applied Economics 5 (January): 65–103. Notowidigdo, Matthew J. 2011. “The Incidence of Local Labor Demand Shocks.” Working Paper 17167. Cambridge, Mass.: National Bureau of Economic Research. (December). Roback, Jennifer. 1982. “Wages, Rents, and the Quality of Life.” Journal of Political Economy 90 (December): 1257–78. Rosen, Sherwin. 1979. “Wage-Based Indexes of Urban Quality of Life.” In Current Issues in Urban Economics Vol. 3, edited by Peter Mieszkowski and Mahlon Straszheim. Baltimore: Johns Hopkins University Press, 324–45.  Economic Quarterly— Volume 103, Numbers 1-4— First-Fourth Quarter 2017— Pages 37–79  The Fed's Discount Window: An Overview of Recent Data Felix P. Ackon and Huberto M. Ennis  T  he Dodd-Frank Act of 2010 requires the Federal Reserve to publish data on discount-window transactions with approximately two years’delay, starting from the passage of the Act in July 2010. The availability of these data provides an opportunity, for policymakers and researchers alike, to get a more detailed perspective on the nature of lending in this important and traditional central bank credit facility. The Fed provides credit through the discount window using three di¤erent programs: primary credit, secondary credit, and seasonal credit. Primary credit and secondary credit are emergency credit programs. These programs constitute a backup source of short-term funding for eligible …nancial institutions. Seasonal credit is aimed at smaller institutions with a predictable (and demonstrable) seasonal pattern in their funding needs. Each loan must be secured by collateral from the borrowing institution. The primary-credit program is a standing facility in which depository institutions in good …nancial conditions (in the form of a high score on their examination rating) can access (mainly) overnight funding with “no questions asked” and at an interest rate higher than the target policy short-term rate. Those institutions not eligible for primary credit can receive secondary credit. Normally, secondary credit is o¤ered at a rate that is …fty basis points above the primary-credit rate. Furthermore, secondary-credit We would like to thank Chris Mast for helpful discussions and Caroline Davis, Doug Diamond, Borys Grochulski, Beth Klee, Steve McCarther, John Walter, and John Weinberg for comments on a previous draft. The views expressed in this article are those of the authors and do not necessarily represent the views of the Federal Reserve Bank of Richmond or the Federal Reserve System. E-mails: Felix.Ackon@rich.frb.org; Huberto.Ennis@rich.frb.org. DOI: https://doi.org/10.21144/eq1030102  38  Federal Reserve Bank of Richmond Economic Quarterly  loans are subject to an extra degree of scrutiny (by the Fed) and cannot be used for certain purposes, such as interest rate arbitrage. To receive seasonal credit, a …nancial institution must have a demonstrated seasonal need for funding and an approved seasonal line of credit with its corresponding Reserve Bank. The interest rate is a ‡oating rate that is calculated based on the average of some selected rates in the money market— it is not intended to be a penalty rate like the others (the interest rates at the primary- and secondary-credit programs). A bank that is able to receive seasonal credit can also tap the primarycredit program if the need arises and the bank is in sound …nancial condition. There is an extensive theoretical literature exploring issues related to the provision of discount-window credit by a central bank (see Ennis [2016] for a recent review). However, the empirical literature is much more scant.1 One obvious reason for this situation is that, over the years, disaggregated data on discount window activity have not been made available to the public or researchers on a regular basis. There are (potentially) good reasons for this lack of transparency: central banks are often concerned about the possibility that some degree of stigma is associated with accessing the discount window (for a discussion of this issue see, for example, Courtois and Ennis [2010]). By keeping loan information private, central banks seek to minimize the incidence of stigma. On the other hand, more information is a constant demand from stakeholders looking to oversee the central bank’s activities. The desire for more transparency was duly recognized in the Dodd-Frank Act, and, as a result, a detailed description of the Fed’s discount-window actions is now easily accessible on the internet with a two-year lag, approximately. We take advantage of the new data availability in this paper and provide an overview of the main patterns identi…able in such data so far. The paper is organized as follows. In the next section, we discuss the data used as well as some preliminary statistics. In Section 2, we present information about primary-credit loans. In Section 3, we study 1 There is some empirical work on emergency lending by government institutions in the US. The early work by Fur…ne (2003), for example, uses data on market transactions and interest rates and compares that information with the interest rate and volume at the Fed’s discount window to infer the attitudes of market participants toward central bank liquidity provision. More recently, several studies have used data on the Term Auction Facility to study di¤erent aspects of how emergency credit provision worked during the recent …nancial crisis (see, for example, Benmelech [2012], Berger et al. [2017], and Armantier et al. [2015]). From a historical perspective, Anbil (2015) and Vossmeyer (2017) study borrowing from the Reconstruction Finance Corporation during the Great Depression. Finally, for a detailed study of the European experience during the recent crisis, see Drechsler et al. (2016).  Ackon & Ennis: The Fed's Discount Window  39  Table 1 Discount Window Lending{Totals  All Primary credit Secondary credit Seasonal credit  Total number of loans 16514 11429 650 4435  Total number of loans $10,000 5277 4655 611 11  Total amount lent ($mm) 36124.977 26286.218 118.976 9719.781  Note: We use all the data in our sample for these calculations. In a few cases, a given loan entry is composed of multiple loans to the same institution with the same term but, possibly, di¤erent amounts, which are consolidated in one entry for the purpose of reporting.  secondary-credit loans, and in Section 4, seasonal credit. Section 5 provides an analysis of the collateral pledged by borrowing banks to their corresponding Reserve Banks. Finally, Section 6 concludes.  1.  THE DATA  Our data are a comprehensive list of discount-window loans made by Federal Reserve Banks between July 22, 2010, and June 30, 2015. The Federal Reserve releases the information on a quarterly basis, with approximately a two-year lag. The data for the second quarter of 2015 are the most recent data available at the time of this writing. There are 16,514 individual loan observations. In some cases, an observation is the result of the consolidation of several loans granted to the same borrower on the same day with the same term to maturity. Furthermore, some loans in the sample are the consequence of a bank rolling over a previous loan. In those cases, loans are counted separately (i.e., a loan taken for one day and rolled over for a second day counts as two independent loans if at origination each of the loans was granted as an overnight loan). For each loan, the data include: the date the loan was granted, the term to maturity, the date the loan was repaid, the name and location of the borrower and the Reserve Bank playing the role of lender, the type of credit (primary, secondary, or seasonal), the interest rate and loan amount, whether the bank had other loans outstanding, and a description of the collateral pledged by the borrowing bank. Because the target policy rate (or range) did not change during the sample period, all primary-credit loans in the sample were granted at the same interest rate of 0.75 percent and all secondary-credit loans but one were granted at the interest rate of 1.25 percent (see Section 3 for a detailed discussion of the exceptional loan).  40  Federal Reserve Bank of Richmond Economic Quarterly  Many of the loans in the sample are likely to be “test”loans. These are loans that depository institutions take in order to test whether the systems involved in processing a discount-window loan are working as expected. Test loans are generally for a very small amount: $1,000 is probably the most common amount, but most of the $10,000 loans in our sample are likely to also be tests, and a few loans of larger amounts may also be tests. Unfortunately, the data available do not include information about whether a given loan is a test or not. There are a lot of $1,000 loans in the data: 27 percent of the primary-credit loans in the data are $1,000 or less. For secondary-credit loans, the percentage of loans with amounts at or below $1,000 is much higher, 82 percent. If we take an even more conservative approach and consider test loans as all loans for amounts equal to or less than $10,000 in order to identify more accurately actual credit events, then we can see from Table 1 that about a third of the loans were in that category. Table 1 also reports the number of loans granted under the di¤erent programs. Two-thirds of the loans were granted under the primarycredit program. Interestingly, there seem to be a lot fewer test loans happening in the seasonal-credit program. Using our threshold for test loans (amounts less than or equal to $10,000), we calculate that the number of nontest loans in the primary-credit and seasonal-credit programs is comparable. In contrast, the number of secondary-credit nontest loans is much less (around forty). The total amount lent, instead, is much higher in the primarycredit program. For calculating the total amount lent, excluding the test loans does not make much of a di¤erence. For example, the total value of all primary-credit loans with a face value equal to or below $10,000 is approximately $18 million, which is much less than 1 percent of the total amount lent ($26.3 billion, as reported in the third column of Table 1). It seems clear from these numbers, then, that the primary-credit program is the more active and signi…cant lending program operated by the Fed’s discount window. Since the proportion of test loans is very di¤erent across di¤erent programs, to get a sense of the size of the typical loans in each program, it is important to take into account the implied composition e¤ect. To this end, Table 2 reports some basic statistics when we exclude from the sample all loans of $10,000 or less. We see in the table that the average size of loans is similar for primary and secondary credit and relatively smaller for seasonal-credit loans. Interestingly, the largest primary-credit loan in the sample is several orders of magnitude larger than the largest loans in the other two programs.  Total amt. lent ($mm) 36105.762 26268.305 117.775 9719.681  Average size of loans ($mm) 3.213 3.818 3.020 2.197  Median size of loans ($mm) 1.000 1.000 2.000 1.000  Max. loan size ($mm) 1017.018 1017.018 17.000 26.000  Note: We de…ne as “tests” all loans of size equal or smaller than $10,000. These loans are excluded from the sample used to compute these statistics. In a few cases, a given loan entry is composed of multiple loans to the same institution with the same term but, possibly, di¤erent amounts, which are consolidated in one entry for the purpose of reporting.  All Primary c. Secondary c. Seasonal c.  Number of loans 11237 6774 39 4424  Table 2 Loan Size, Excluding Tests{Basic Statistics  Ackon & Ennis: The Fed's Discount Window 41  2801 3021 3491 3996  w/  All 1941 2049 2390 2554  w/o  All 6656.703 7859.556 6176.920 6849.539  2026 2115 2326 2679  Secondary c. 50.251 7.868 3.743 4.855  1316 1290 1357 1333  Seasonal c. 784.659 1764.286 2378.050 3425.271  169 150 129 98  21 4 1 5  Secondary credit w/ w/o  Panel B – Number of loans  Primary c. 5821.794 6087.402 3795.127 3419.413  Primary credit w/ w/o  All w/o 6653.530 7856.076 6172.846 6844.392  606 756 1036 1219  604 755 1032 1216  Seasonal c. w/ w/o  Note: We use “w/” to denote the columns where the number of loans includes “test” loans and “w/o” to denote where “test” loans are excluded. We aggregate at the annual level to compare across time the recent evolution of discount-window lending. For these calculations, we do not use the available data for 2010 and 2015 because we do not have full-year information for those years and, as we will see later, the data display seasonal yearly patterns that can make the comparison of full-year and half-year aggregates misleading. We de…ne as “tests” all loans equal to or smaller than $10,000.  2011 2012 2013 2014  Year  Year 2011 2012 2013 2014  Panel A – Total amount lent ($mm)  Table 3 Discount-Window Lending Over Time  42 Federal Reserve Bank of Richmond Economic Quarterly  Ackon & Ennis: The Fed's Discount Window  43  Table 3 gives a sense of the evolution over (recent) time of the amount of credit provided through the discount window. We choose to aggregate loans on an annual basis because, as we will discuss later, credit activity exhibits some yearly seasonal patterns. The total amount lent is relatively stable between the years 2011 and 2014 (see Panel A of Table 3).2 This relative stability is the combination of two trends: on one side, the amount of lending in the primary-credit facility decreased signi…cantly (a 40 percent drop) after 2012. On the other side, the total amount lent at the seasonal-credit facility has been increasing over time. The total amount lent through the secondary-credit facility was relatively large in 2011 but became much smaller in the years after that. It is possible that the repercussions of the …nancial crisis were still driving banks to borrow secondary credit, as this is the facility available for banks in weaker …nancial condition. In support of that possibility, note that, for example, the number of “problem banks” in the US (as reported by the Federal Deposit Insurance Corporation [FDIC]) peaked in 2011 at (around) 900 and since then has been experiencing a steady decline, reaching 200 in 2015. Panel B of Table 3 shows the evolution over time of the total number of loans granted through the di¤erent programs. The total number of loans in all three programs is growing over time. This phenomenon is basically explained by two main factors: …rst, the number of test loans in the primary-credit program has grown over time and more than proportionally (to the total number of loans) in 2014— note that the number of loans in the primary-credit program, net of our measure of the number of test loans, is basically stable over the years at around 1,300 loans. The second factor is that the number of seasonal loans has been increasing consistently over the years under consideration. Figure 1 plots the monthly total amount lent at the discount window during our sample period. Since the data are volatile, we include in the …gure a six-month moving average to highlight any trends present in the data. It is evident in the …gure that lending generally increases in the second half of the year. As we will see later, this pattern is present both at the primary-credit and seasonal-credit programs. Aside from those ‡uctuations, lending seems relatively stable over the …ve-year period. 2 Relatively speaking, as a source of funding for banks, the discount window is miniscule during normal times (such as the years covered in our sample). For example, total deposits at FDIC-insured institutions in the last quarter of 2014 were close to $12 trillion.  44  Federal Reserve Bank of Richmond Economic Quarterly  Figure 1 Total Discount-Window Lending  Note: For each month in our sample, we aggregate all loans in that month to generate the monthly series plotted as the solid line. We also compute a symmetric, six-month moving average, which is the smoother dashed line.  Given that now the expectation in …nancial markets is that information about activity at the discount window will become public after two years, one natural question to ask is whether banks have used the discount window less as a result of this added transparency. To get a sense of this, it would be good to compare the levels of lending during our sample period with lending in previous periods of similar characteristics— mainly, noncrisis periods. While loan-level data are not publicly available for the period before 2007, the Federal Reserve has been publishing for a much longer time the average daily amount of outstanding loans at the discount window in its H.4.1 weekly data release. Table 4 uses the data from the H.4.1 release to compare the annual average of daily outstanding loans at the three main discount-window programs for the period 2003–07 and for the period 2011–15. We see in the table that the level of discount-window activity is relatively lower  Ackon & Ennis: The Fed's Discount Window  45  Table 4 Total Discount-Window Lending Before and After the Crisis BEFORE THE CRISIS Average daily loan Year amount outstanding ($mm)  AFTER THE CRISIS Average daily loan Year amount outstanding ($mm)  2003 2004 2005 2006 2007  2011 2012 2013 2014 2015  98.00 153.038 199.346 224.654 587.788  62.058 72.442 78.654 117.585 124.673  Note: The annual number is the average of the daily numbers reported for each week of the year in the Federal Reserve H.4.1 Data Release.  in the period after the Dodd-Frank Act. This is consistent with the view that disclosure tends to discourage borrowing (Kleymenova 2016). However, other changes in the …nancial landscape after the crisis may have also contributed to the reduction in discount-window activity. For example, the high levels of interest-paying excess reserves (Ennis and Wolman 2015) tend to reduce the risk for depository institutions to …nd themselves short of funds due to unexpected payment shocks. Similarly, more recently, heightened regulatory focus on balance sheet liquidity might be inducing banks to hold liquidity bu¤ers that can also protect them from unexpected late-in-the-day payment shocks. Presumably, then, these liquidity enhancements make banks less likely to need to tap the discount window as a backup source of funding. A look at more disaggregated data shows (see Table A1 in the Appendix) that both primary and seasonal credit have decreased after the crisis— secondary credit is very small (and volatile) in both periods. Interestingly, after a signi…cant decrease during the crisis, seasonal credit seems to be converging back to numbers common in the precrisis period. Primary credit, instead, has been consistently smaller and decreasing in time after the crisis (see also Table 3). Note, …nally, in Table 4 that discount-window lending increased signi…cantly in 2007. In fact, the corresponding amounts for 2008 and 2009 (see Table A1) are an order of magnitude larger than the numbers in Table 4 ($32 billion and $40 billion, respectively), and the second half of 2007 was already showing the signs of those forthcoming increases. These “crisis” numbers put in perspective the levels of lending that occur in “normal” times (as in our sample period), which is not zero but relatively moderate.  46  Federal Reserve Bank of Richmond Economic Quarterly  Going back to the transactions data, it is also possible to get a sense of the variation in activity across Federal Reserve districts. In Panel A of Table 5, we show the total amount lent by each Reserve Bank over our sample period. The largest lenders in terms of total amount lent are the Atlanta, Chicago, and San Francisco Feds. The San Francisco Fed lent mostly through the primary-credit program, and the Atlanta Fed lent more than half of its total through the seasonal-credit program. Indeed, the Atlanta Fed is the largest lender of seasonal credit among all the Reserve Banks, followed by Minneapolis. Based on the information on the number of loans granted, as provided in Panel B, it seems that the Chicago and Minneapolis Reserve Banks grant a lot more loans at the seasonal-credit facility than the other Banks, and in particular, Atlanta. This, in turn, implies that the average size of the seasonal-credit loans is much smaller in the case of the Chicago than in the case of Atlanta: the …rst grants a lot of small seasonal loans, and the second grants fewer loans but larger ones. Philadelphia, San Francisco, and Chicago are the top providers of secondary credit in terms of total amount lent. While the Atlanta Fed provided a large number of secondary-credit loans (244), all of them were $1,000 and, hence, most likely test loans. If we restrict the sample to those loans that are greater than $10,000, we see (in the column labeled “w/o”) that San Francisco and New York rank at the top in terms of the number of secondary-credit loans granted during the sample period. It should be said, though, that the small sample size makes it particularly hard in this case to draw strong inferences about general patterns. The largest lender through the primary-credit program (in terms of total amount lent) is the San Francisco Fed. The Chicago Fed made the largest number of loans through this program, but almost half of them were loans that are very likely to have been test loans (for amounts equal or below $10,000 dollars). For most of the Reserve Banks, between a third and a half of the primary-credit loans can be safely categorized as test loans. Perhaps surprisingly, when we restrict the sample to loans greater than $10,000 and look at the average size of primary-credit loans across districts (not reported in the table), we see that the Dallas Fed is providing the largest loans on average ($6.4 million), followed by Philadelphia ($5.9 million), and St. Louis ($5.6 million). The New York Fed is fourth (with an average loan amount equal to $5.5 million). In principle, since the highest concentration of large banks is in the New York district, one might have expected that the largest discount-window loans were being originated there. That is not the case in our sample period.  989 932 456 399 865 1963 3616 1279 2080 1624 645 1666  Boston New York Philadelphia Cleveland Richmond Atlanta Chicago St. Louis Minneapolis Kansas City Dallas San Francisco  All 542 615 284 230 333 1194 2333 737 1931 1286 639 1113  w/o  Primary c. 970.536 3354.844 1678.804 482.530 1546.627 2176.948 3892.651 2105.557 1464.924 1053.308 3044.297 4515.191  701 909 429 390 825 1133 2194 871 826 1079 480 1592  262 604 281 230 331 608 1111 373 715 745 474 1040  Primary c. w/ w/o  10 23 27 9 40 244 202 42 38 1 0 14  2 11 3 0 2 0 6 0 0 1 0 14  w/o  Seasonal c. 684.730 0 0 0 0 3571.444 968.219 857.024 2027.972 707.360 720.850 182.182  Secondary c. w/  Secondary c. 1.608 20.496 34.432 0.009 3.856 0.244 28.196 0.420 0.118 0.500 0 29.100  Panel B – Number of loans  All 1656.874 3375.340 1713.236 482.539 1550.483 5748.636 4889.066 2963.001 3493.014 1761.168 3765.147 4726.473  278 0 0 0 0 586 1220 366 1216 544 165 60  278 0 0 0 0 586 1216 364 1216 540 165 59  Seasonal c. w/ w/o  Note: We use “w/” to denote the columns where the number of loans includes “test” loans and “w/o” to denote where “test” loans are excluded. There are twelve Federal Reserve districts, each with its corresponding Reserve Bank (which takes the name of the city where it is located). Reserve Banks make loans to institutions headquartered in their district.  w/  District  District Boston New York Philadelphia Cleveland Richmond Atlanta Chicago St. Louis Minneapolis Kansas City Dallas San Francisco  Panel A – Total amount lent ($mm)  Table 5 Discount-Window Lending Across Federal Reserve Districts  Ackon & Ennis: The Fed's Discount Window 47  48  Federal Reserve Bank of Richmond Economic Quarterly  Finally, note (Panel B of Table 5) that there are very few loans of $10,000 or less granted by the Dallas Fed (in all three programs). Upon closer inspection of the transactions data, one can see that 264 loans (out of the 645 loans) were for exactly $100,000. This suggests that di¤erent Reserve Banks may have di¤erent practices when it comes to the size of the loans used for testing, with the Dallas Fed perhaps being a Bank that recommends larger test loans than other Feds.  2.  PRIMARY CREDIT  In this section, we explore further the main characteristic of loans granted through the primary-credit program. We look at the distribution of loan sizes and maturity terms. We also investigate the extent to which lending displays seasonal patterns and, lastly, whether some banks are more prone than others to tap the primary-credit facility (that is, the intensity of use across banks). Loan sizes: as was discussed in Section 1, the median primarycredit loan is much smaller than the average. This is because the largest loans in our sample were granted through the primary-credit facility, which also made a signi…cant proportion of relatively small loans. Even when we abstract from the majority of the small test loans, as we did in Table 2, we see that the average size of primary-credit loans is four times larger than the median. Figure 2 plots the distribution of loan sizes for all loans smaller than or equal to $10 million. It seems clear from the …gure that most loans are in the size range that is below $5 million. In fact, during our sample period, 40 percent of all primary-credit loans were $10,000 or less, and more than half were $100,000 or less. On the other end of the distribution, 10 percent of loans were greater than $5 million. The distribution of loan sizes in Figure 2 appears to be approximately log-normal (with additional concentration of mass at certain amounts). To shed further light on these patterns, we plot in the Appendix (see Figure A1) the distribution of the log size of loan amounts, which should have the shape of a normal distribution if the distribution in Figure 2 was log-normal. While the resulting distribution is relatively symmetric, the …gure makes more evident the high concentration of mass around certain speci…c amounts, such us $1,000, $10,000, $100,000, and $1 million. Some of these spikes in the distribution are likely to be explained by the preponderance of small-amount test loans, but there may also be a tendency from banks to borrow round-number amounts. To gain some perspective on the …nancial stakes at play when a bank decides to borrow from the primary-credit program, we calculate  Ackon & Ennis: The Fed's Discount Window  49  Figure 2 Distribution of Loan Sizes. Primary Credit  Note: We plot the distribution of loan sizes for loans of $10 million or less. We omit loans greater than $10 million from the plot to reduce the length of the right tail. We then scaled the distribution by the proportion of loans less than $10 million. There are 436 loans that are greater than $10 million in the sample. The vertical dashed line represents the mean of all loans ($2.3 million).  the interest expense on a $10 million overnight loan at the interest rate relevant for all the primary-credit loans in our sample, 75 basis points. The amount charged is approximately $200 in interest payments per day. Given that almost 90 percent of primary-credit loans are overnight loans (see Table 6), we can conclude that a signi…cant portion of the activity at the primary-credit facility constitutes relatively small loans with no signi…cant interest cost for the borrower. However, there are also some very large primary-credit loans being made in the period under consideration. For example, the largest primary-credit loan in the sample was for $1.02 billion and was granted by the St. Louis Fed on Wednesday, November 24, 2010, to First Tennessee Bank, N.A. The term of the loan was two days, which is relatively unusual, as most primary-credit loans are overnight. First Tennessee  50  Federal Reserve Bank of Richmond Economic Quarterly  Figure 3 Distribution of Loan Sizes{Evolution Over Time  Note: We consider the …rst and the last twelve-month period in our data. For the period July 2010 to June 2011, we exclude from our density calculations one clear outlier: a loan for (approximately) $1 billion (which is ten times larger than the second-largest loan in this subsample). As in Figure 2, we plot only the portion of the distribution that corresponds to loans of $10 million or less.  Bank had $24.5 billion in assets and $3.2 billion total equity as of December 2010, so the discount-window loan was a signi…cant …nancial transaction for the bank. The second-largest primary-credit loan in the sample was for $900 million and was granted by the New York Fed to the New York branch of the Belgian bank KBC on Friday, December 9, 2011. The term of the loan was three days and was repaid at maturity on the following Monday. Evolution of the distribution over time: As indicated in Table 3, total lending at the primary-credit facility was signi…cantly lower toward the end of our sample period relative to what it was at the beginning. It is also interesting to see that the number of loans increased moderately and, consequently, the mean size of primary-credit loans decreased substantially.  Ackon & Ennis: The Fed's Discount Window  51  Figure 3 plots the distribution of loan sizes for the …rst twelve months in our sample period (from July 2010 to June 2011) and compares it with the distribution of loan sizes for the last twelve months of that period (from July 2014 to June 2015). This con…rms the pattern suggested by Table 3: in the later part of our sample period, the proportion of loans that are small is much higher. Figure A2 in the Appendix provides a more complete picture of the evolution of the distribution of loan size over time. During 2010 and 2011, the US banking system was still recovering from the …nancial crisis. Furthermore, economic turmoil in Europe at the time impacted the …nancial condition of some banks in the US (recall that foreign-bank subsidiaries have access to the Fed’s discount window). This may partly explain the elevated levels of primary-credit activity during that period and the relatively large size of the average loan. In contrast, by the second half of 2014, the banking system in the US was in much better …nancial shape, a fact that is likely to have contributed to reducing banks’demand for emergency credit. Term to maturity: Primary-credit loans are mainly overnight loans. Loans made on a Friday, however, are held for a minimum of three days and are charged interest accordingly (that is, for at least three days). Furthermore, in certain speci…c situations, loans are granted for longer periods of time. While primary credit is provided relatively automatically (“no questions asked”), when banks request loans of a longer maturity than overnight, some extra administrative oversight may occur. Table 6 provides a description of the maturity pro…le of primarycredit loans in our sample. There are a lot of loans in the two- to four-day maturity term. Many of those are associated with holidays and weekends. Of the 962 loans with a three-day maturity term, 955 were granted on a Friday. Similarly, many (but not all) of the four-day loans involved a weekend, preceded or followed by a holiday.  12.380  2.011  3.707  3 962 269 3566.472 5.628  1.200  9.000  Term of loan (days) 4 5 6 122 5 3 21 0 0 686.669 6.000 27.000  5.857  7 15 0 87.850  4.000  8 2 0 8.000  10.000  12 1 0 10.000  4.000  14 1 0 4.000  Note: The term of the loans is expressed in days, and the amounts are all in millions of dollars. We call “test” loans all loans for amounts less than or equal to $10,000.  Number of loans Number of “test” loans Total amt. lent ($mm) Avg. size of “nontest” loans ($mm)  2 110 26 1361.845  1 10208 4339 20528.381  Table 6 Primary-Credit Loans. Maturity Term  52 Federal Reserve Bank of Richmond Economic Quarterly  Ackon & Ennis: The Fed's Discount Window  53  We see in Table 6 that there are a few loans lower than or equal to $10,000 and with maturity terms higher than one day. This may be indicative of the fact that not all loans in that size range constitute test loans, in the sense that it seems unlikely that a bank would choose a Friday to conduct a test given that the interest cost is higher than when the test is done with an overnight loan. The longest maturity term for a primary-credit loan in our sample was a fourteen-day loan granted by the Dallas Fed to First National Bank of Rotan, Texas, in January 2015. The loan was for $4 million, which is close to the average amount of all overnight loans greater than $10,000 and not much di¤erent from many of the other longer-term loans in our sample. Interestingly, the loan was actually paid back early, after nine days. Of the more than 10,000 primary-credit loans in our sample, only ten of them were paid before maturity. We provide a list of these loans in the Appendix (Table A2). The loans that are paid before maturity tend to be relatively large: the average size is $4.8 million, and all but one of them are larger than $100,000. Their maturity is not concentrated in any particular term, and it does not appear to be the case that only the longest maturity loans get repaid earlier. Seasonality: The average amount lent in each month of the year during our sample period seems to increase as the end of the year approaches. Figure 4 shows that after fairly high levels of total borrowing in January, borrowing slows down in February and only gradually increases as the year progresses. The high level of lending observed in September is partly driven by an event in September 2012 when a bank in Texas took out several loans of a signi…cant amount during that month. This episode is discussed in more detail below.3 Even when we remove the large loans to the Texas bank, the month of September exhibits more lending than August and October. In e¤ect, June, September, and December (endof-quarter months) seem to be months where lending tends to increase. We con…rm this below in Table 7. 3  It is important to realize here that the way these numbers are calculated tends to overweight the importance of short-term loans that get rolled over. As an example, if a bank borrows $100,000 for two days and another bank takes a loan of $100,000 for one day and then another loan of the same amount the following day, the second lending strategy would add up to $200,000 of lending, while the …rst will only count as $100,000 toward the total amount lent in that month.  54  Federal Reserve Bank of Richmond Economic Quarterly  Figure 4 Average Total Amount Lent of Primary Credit in Each Month  Note: Each month appears …ve times in our sample period. For each month, we sum the total amount lent during that month and average that amount among the …ve corresponding totals. That is, for example, for the month of January we sum all the lending done in January in each of the years in our sample and then take the average over the …ve Januaries in our sample.  Quarterly and monthly frequency: The …rst row of Table 7 shows how much primary-credit lending takes place during the beginning, middle, and end days of the month. The pattern suggests that lending is higher at the beginning and end of the month and slows down during the days in the middle. A similar pattern arises when we compute the average amount lent in the …rst, second, and third month of the quarter (second row of Table 7): lending picks up noticeably toward the end of the quarter. Daily frequency: Table 8 shows the average total number of loans granted in each day of the week during our sample period and the average number of test loans. We see that banks tend to avoid “testing” on Fridays and other special days such as the end of the month, quarter,  Ackon & Ennis: The Fed's Discount Window  55  Table 7 Total Amount Lent at Di erent Times of the Month and the Quarter ($mm)  Month Quarter  Beginning 136.316 387.220  Middle 111.110 378.519  End 194.801 548.571  Note: For the monthly calculations, we divide the month into three periods of (approximately) ten days each and call the …rst ten days the beginning of the month, the next ten days the middle of the month, and the last ten days the end of the month. For the quarterly calculations, the beginning of the quarter is the …rst month of the quarter, the middle is the second month, and the end is the third month. We calculate the total amount lent in each subperiod, and we report the average across all the corresponding subperiods in the sample.  and year. This makes sense to the extent that conducting a test on a Friday would be more costly. We also report in Table 8 the total amount lent on average in each of those days (third row) and the average size of loans granted each day that were greater than $10,000 (fourth row). The last three columns report similar statistics for the day that corresponds to the end of a month, a quarter, or a year in our sample period. In general, there is less lending happening on regular Fridays, but the size of the loans are larger on average. Also, the table shows that there is a lot more lending happening on days that are the end of a month, quarter, or year, and the average size of those loans is larger than for other regular weekdays. Intensity of use: A lot of banks accessed the primary-credit facility during our sample period. Overall, a total of 2,758 di¤erent banks (identi…ed by the corresponding ABA number) borrowed at least once from the facility. Of these, 1,756 of them borrowed at least twice, and some banks borrowed over sixty times during our …ve-year sample. Many of these loans are likely to be tests, of course. For this reason, we also look at intensity of use after restricting our sample to loans greater than $10,000. The total number of di¤erent banks that borrowed from the facility (at least once) reduces to 1,450 after we restrict our sample to nontest loans. Figure 5 presents the histogram of the number of times that individual banks borrowed loans for amounts greater than $10,000 from the primary-credit facility during our sample period.  56  Federal Reserve Bank of Richmond Economic Quarterly  Figure 5 Histogram of the Number of Primary-Credit Loans by Individual Banks  We compute the number of individual nontest transactions (loans greater than $10,000) that each individual bank entered at the primary-credit facility and plot the histogram of these numbers; that is, each bar in the histogram denotes the number of banks that borrow n amounts of times from the primary-credit facility, with n = 1,2,3. . . up to sixty-six, which is the maximum number of nontest primary-credit loans taken by an individual bank in our sample during the …veyear period under consideration. We identify a bank by its ABA number because it is a more consistent identi…er than the bank’s name.  We see in the …gure that almost 600 banks took one “nontest” loan during the period, but there were also twenty-eight banks that took thirty or more “large” (nontest) loans from the discount-window primary-credit program in the span of …ve years. In other words, some banks appear to be relatively frequent users of the facility.4  4 Even if we restrict attention to loans greater than $100,000 (given that it is possible that some Reserve Banks, such as Dallas, ask for test loans of an amount equal to $100,000), the number of banks that borrowed at least thirty loans greater than $100,000 in the space of …ve years is equal to twenty-six.  11 4 23.773 3.806  Number of loans made Number of “test” loans Daily amount lent ($mm) Size of nontest loans ($mm)  11 4 18.502 2.851  Tues. 10 4 26.968 4.603  Wed. 10 4 20.444 3.554  Thurs. 4 1 16.499 5.392  Fri.  8 1 42.892 6.581  End of month  9 1 57.493 7.141  End of quarter  12 0 98.217 8.184  End of year  Note: We identify the number of days of each day category (Monday, Tuesday, etc.) when the discount window was open during our sample period. We then count the number of loans and the amount lent in each of those days and report the average for each day category (taking into account that, for example, most of the holidays— for which the discount window is closed— fall on a Monday). We also compute the average size of the nontest loan made in each day category, where by “nontest” we mean loans greater than $10,000.  Mon.  Averages  Table 8 Total Amount Lent in a Day (Average Over Sample)  Ackon & Ennis: The Fed's Discount Window 57  58  Federal Reserve Bank of Richmond Economic Quarterly  In terms of intensity of use of primary credit, one noticeable case is the situation of Texas Capital Bank, N.A., which in September 2012 borrowed signi…cant amounts overnight for several consecutive days from the Dallas Fed, with balances as high as $250 million (on September 5). Between October 2011 and January 2013, Texas Capital Bank borrowed nineteen times, and in ten of those instances the loans were for an amount larger $100 million (See Table A3 in the Appendix for details). While the calculations used to produce Figure 5 can be informative, they are not able to reveal some particularly relevant details of the lending patterns observed in the data. For example, some institutions may be rolling over a loan for several days with each loan counted as a new loan while, in a sense, the credit event could be considered to be just one event. Other institutions, instead, may be “repeat users” in that they periodically take an overnight loan and repay it at maturity (and only take another loan after some time). Additionally, some institutions may be repeat users, yet of loans of a very small amount. There are, of course, many ways to present information that speaks to these issues. We pursue several ones here. For example, we compute the number of primary-credit loans that were the result of a rollover of a previous loan. There were 1,731 loans that were followed by a loan by the same institution on the day that the …rst loan matured. This amounts to approximately 15 percent of all primary-credit loans in our sample. Of the loans that were rolled over, only 302 were rolled over for the same dollar amount. Figure 6 plots the amount of the original loan and the amount of the subsequent loan (the rolled-over amount) to get a sense of whether or not loans get rolled over to smaller amounts (as a way to pay down the loan gradually). To make the …gure readable, we only plot those loans that were $20 million or less.5 The …gure shows that there is no clear pattern: some loans get rolled over into larger amounts, some into smaller amounts, and some into the same amount (these last are represented by the dots that fall exactly over the forty-…ve-degree line plotted in the …gure).6 Of the 11,429 primary-credit loans, 9,000 of them were loans that were not followed by another loan from the same institution on the maturity day (that is, they were not rolled over). Of these loans, 4,650 were loans of an amount less than or equal to $10,000. Given that, in 5 Note also that loans that were initially larger than $20 million but were rolled over into amounts smaller than $20 million do not appear in the …gure. 6 We looked at larger loans than $20 million, and there is no indication that larger loans produce a more de…nitive pattern.  Ackon & Ennis: The Fed's Discount Window  59  Figure 6 Rollovers in Primary Credit  Note: The solid diagonal line is the forty–…ve-degree line. We only consider consecutive loans that were originally for an amount less than or equal to $20 million.  total, there were 4,655 primary-credit loans of an amount lower than or equal to $10,000, that tells us most of those smaller loans, predominantly “test” loans, were not followed by a consecutive loan. If we call a credit event a sequence of consecutive loans taken by a bank where one loan is followed by another loan (of potentially a di¤erent amount) at the time of the maturity of the previous loan, then there were 350 events involving one rollover (see Appendix), 146 events involving two rollovers, and there are events of all numbers of rollovers up to sixteen, the maximum observed in our sample (only one event involved sixteen consecutive loan rollovers). The average size of the loans that get rolled over at least once is $4.7 million— somewhat larger than the average for all primary-credit loans with amounts greater than $10,000, which is equal to $3.8 million, as reported in Table 2. This indicates that banks that end up rolling over primary-credit loans tend to borrow larger amounts.  60  Federal Reserve Bank of Richmond Economic Quarterly  We also look at situations where a bank takes a loan that we can consider an isolated situation, in the sense that the same bank does not take a loan in the two months around the loan in question (one month prior and one month after). The number of isolated loans identi…ed this way is 7,078, but presumably many of them were taken as test loans. Indeed, only 2,871 of these isolated loans were for an amount greater than $10,000. This means that within the subsample of loans for amounts greater than $10,000, 42 percent of them were isolated. When we look at this subsample of loans (i.e., loans that are greater than $10,000 and isolated), the average size was $3.7 million, which is similar to the average size of all loans greater than $10,000 (Table 2). In other words, isolated loans that are unlikely to be test loans do not tend to be much di¤erent in size than other nontest loans.  3.  SECONDARY CREDIT  Secondary credit is used much less often than primary credit. There are 650 loans in our …ve-year sample, of which only thirty-nine are greater than $10,000. Virtually all secondary-credit loans were overnight loans. The volume in the secondary-credit program has been trending down over the years under consideration, with the total amount lent and the number of loans both decreasing over time (see Table 3). The number of test loans has probably been decreasing as a result of banks moving out of the “poorly capitalized” category, either by becoming eligible for primary credit— as banks (and the economy) exit the crisis state— or by exiting the industry (through mergers or liquidations). Similarly, the intensity of use of the secondary-credit program, beyond just testing, is also likely to be strongly in‡uenced by prevailing …nancial conditions: as those improved over the years in the sample period, usage declined. Loan Sizes: In general, secondary-credit loans tend to be smaller than primary-credit loans. The largest loan at the secondary-credit facility during our sample period is much smaller than the largest primary-credit loan ($17 million and $1 billion, respectively). Except for a loan granted by the Philadelphia Fed to Nova Bank in December 2010 for $17 million, all other secondary-credit loans in our sample were under $10 million in value. This pattern is stronger toward the later part of our sample period. Indeed, since the end of 2012, there has been very little lending of signi…cant amounts at the secondary-credit program. If we consider only larger loans, using a $1 million threshold, then there was only one loan at or above that amount in 2013, three in 2014, and again only one in the …rst half of 2015. For the whole sample, only twenty-nine  Ackon & Ennis: The Fed's Discount Window  61  Table 9 Secondary-Credit Loans. Maturity Term  Number of loans Number of “test” loans Total amount lent ($mm) Average size of “nontest” loans ($mm)  Term of loan 1 2 587 4 558 3 73.249 17.003 2.489 17.000  (days) 3 48 43 6.010 1.180  4 11 7 22.716 5.675  Note: The term of the loans is expressed in days, and the quantities are all in millions of dollars.  loans were for amounts greater than or equal to $1 million. Clearly, secondary credit is used in a meaningful way outside crisis periods only in rare occasions. Term to maturity: Most secondary-credit loans have a one- or three-day maturity term (and all the three-day loans were taken on Fridays). Contrary to primary credit, there are no secondary-credit loans of longer maturity than four days (see Table 9) and all four-day loans were made on Fridays followed by a holiday. Potentially, the extra scrutiny by loan o¢ cers at Reserve Banks, and the fact that the …nancial institution seeking the loan is not in obviously sound …nancial condition, drives this pattern of low maximum maturity length. When reading Table 9, it is worth noting that there are only four secondary-credit loans of two-day maturity (middle column in the table), of which three of them are for $1,000, and one of them is for $17 million. This fact drives the high average value of loans of the two-day maturity term but it seems reasonable to attribute the anomaly to the small sample size. The $17 million two-day loan is unusual in that it is the only secondary-credit loan in our sample that was charged an interest rate higher than the standard 125 basis points (that is, …fty basis points higher than the primary-credit rate). The loan was granted by the Philadelphia Fed to Nova Bank, of Berwyn, Pennsylvania, on December 28, 2010, and the interest rate charged was 6.25 percent. Interestingly, this loan was a rollover loan from a loan of the same amount granted for four days on December 24, 2010, at the standard interest rate of 125 basis points. There are no other loans to Nova Bank around that time, which suggests that the bank paid the loan in full on December 30, 2010.7 7 On Friday, October 26, 2012, the Pennsylvania Department of Banking and Securities closed NOVA Bank, and the FDIC was named receiver.  62  Federal Reserve Bank of Richmond Economic Quarterly  Figure 7 Average Total Amount Lent of Secondary Credit in Each Month  Note: This …gure is constructed in the same manner as Figure 4. Each month appears …ve times in our sample period. For each month, we sum the total amount lent during that month and average that amount among the …ve corresponding totals. That is, for example, for the month of January, we sum all the lending done in January in each of the years in our sample and then take the average over the …ve Januaries in our sample. For the month of December, the grey bar represents the average after we exclude the two large loans to Nova Bank in December 2010. The white bar is the average including those two large loans.  The initial loan for $17 million to Nova Bank on December 24, 2010, is also responsible for the high average size of the four-day loan category (see last column of Table 9). Excluding that loan, the average four-day loan size is much lower ($1.9 million). Here, again, the small sample size is a signi…cant limiting factor, but it is worth pointing out that this amount is smaller than the average size for all overnight secondary-credit loans ($2.489 million). Seasonality: Figure 7 shows that secondary credit is concentrated around the middle of the year, with activity being very moderate in the initial months of the year. The December average is heavily in‡uenced  Ackon & Ennis: The Fed's Discount Window  63  Table 10 Total Amount Lent at Di erent Times of the Month and the Quarter  Month Quarter  Beginning 0.464 1.866  Middle 0.384 1.603  End 1.149 2.479  Note: We use the same methodology as used in the construction of Table 7.  by the large loans to Nova Bank in December 2010. After excluding these two loans, the average total amount lent in December at the secondary-credit program is similar to the average level for November and relatively small compared with the middle months of the year. The sensitivity of the December average to the exclusion/inclusion of two loans, however, underscores the fact that the sample for secondary credit is relatively small and, as a result, idiosyncratic events may be biasing some of the statistics computed here. Just for completeness, Table 10 shows the average level of lending at the secondary-credit program at the beginning, middle, and end of the month and of the quarter (to compare it with Table 7). The pattern shows that lending tends to increase at the end of the month and the quarter. We do not compute a table comparable to Table 8 in the primarycredit section because the subsamples needed to deal with daily patterns (given the presence of test loans) are too small to draw any meaningful inferences. For computing total amounts lent, the test loans play very little role: for example, the 611 test loans (equal to or lower than $10,000) at the secondary-credit program account for $1.2 million of the total $119 million lent over the sample period (see Table 1). E¤ectively, then, the total amounts lent re‡ect the thirty-nine nontest loans in the sample. As we split the sample into subsamples, the number of (nontest) loans in each category becomes too small to make any reliable inference.  4.  SEASONAL CREDIT  Even though seasonal credit presents a strong annual seasonal pattern, it is clearly discernable from Figure 8 that there has been an upward trend in the total amount lent over the years of our sample. This is also evident in Table 3. Of course, the strong seasonal pattern is not surprising given the objective of this program: to provide ample access  64  Federal Reserve Bank of Richmond Economic Quarterly  Figure 8 Total Monthly Seasonal Credit ($ million)  Note: For each month in our sample, we aggregate all loans in that month to generate the monthly time series plotted as the lighter dashed line. We also compute a symmetric, twelve-month moving average, which is the darker dashed line. We overlay the most frequent (discount window) interest rate in the period as the solid line.  to credit to smaller institutions with a predictable (and demonstrable) seasonal pattern in their funding needs. We included in the …gure the interest rate charged for seasonal credit. The rate is calculated as an average of market rates, and it has ‡uctuated over the years while exhibiting a moderate downward trend. It may be the case that the gradual decrease in the cost of borrowing over the sample period is partly responsible for the (also gradual) increase in lending shown in the …gure (although, of course, the cost of other sources of funding probably move together with market rates as well, making those other sources of funding also more attractive). Loan Sizes: The distribution of loan sizes is plotted in Figure 9. As we know from Table 2, the median size of seasonal-credit nontest loans is similar to the median for primary-credit nontest loans, but the  Ackon & Ennis: The Fed's Discount Window  65  Figure 9 Distribution of Loan Sizes. Seasonal Credit  Note: We plot the distribution of loan sizes for all seasonal-credit loans in the sample. The dotted vertical line is the mean.  maximum loan amount is much smaller. As a result, the distribution shows more density in intermediate values, such as loans between $1 million and $10 million. Since seasonal credit is not provided at a penalty interest rate, the nature of its usage is likely to be very di¤erent than for primary credit, with midsize loans being more common (as re‡ected in the distribution of loan sizes). Testing also seems less common in the case of seasonal credit. In fact, there are no seasonal-credit loans in our sample for an amount equal to or less than $1,000 and eleven loans for an amount between $1,000 and $10,000. This is of course in sharp contrast with the patterns observed in primary and secondary credit where a large proportion of the loans fell within that range of very small amounts. Term to maturity: Aside from the di¤erence in the distribution of loan sizes between primary, secondary, and seasonal credit, the maturity of seasonal-credit loans is also much less concentrated around  66  Federal Reserve Bank of Richmond Economic Quarterly  Figure 10 Seasonal Credit Loans. Maturity Term  Note: We compute the number of loans of each maturity term from one day (overnight) to thirty-four days, the maximum maturity observed in the sample, and we plot the histogram of the resulting data.  short periods (overnight and three days) than the maturity term of primary- and secondary-credit loans. Figure 10 presents a histogram of the maturity term of seasonalcredit loans in our sample. Interestingly, loans that are approximately a month long tend to be relatively common among this class of loans. But this relative concentration does not mean that other maturities are not used. In fact, there are many loans for each of the maturity terms in the range between one and thirty-four days. It is still the case, though, that overnight loans are the most common in this subsample, representing 15 percent of the total. Seasonality: As a complement to Figure 8, in Figure 11 we plot the average amount of seasonal credit provided during each month of the year. The seasonal pattern at the annual frequency is again evident in this …gure, with the amount of seasonal credit increasing during the second half of the year and being minimal during the winter months  Ackon & Ennis: The Fed's Discount Window  67  Figure 11 Average Total Amount Lent of Seasonal Credit in Each Month  Note: See note for Figure 4.  of January and February. While one might suspect that some of this credit follows the agricultural cycle, it is interesting to note that some of the Reserve Banks in agricultural areas, such as St. Louis and Kansas City, are not large providers of seasonal credit (see Table 5).  5.  COLLATERAL  Depository institutions can choose to enter credit agreements with their respective Reserve Banks, allowing them to borrow from the discount window if they need to do so at some point. Not all institutions enter credit agreements with their Reserve Bank. When they do, as part of such agreements, depository institutions pledge collateral with Reserve Banks, a process that requires the submission of all the relevant information that would allow loan o¢ cers at Reserve Banks to assess the value of the corresponding collateral. Many institutions do not take any loans from the discount window in a given period of time even  68  Federal Reserve Bank of Richmond Economic Quarterly  when they have an outstanding credit agreement and potentially large amounts of pledged collateral. Our data only include collateral information as of the time when a bank actually takes a loan from the discount window. In that sense, the information about pledged collateral in our dataset is not comprehensive. To be more speci…c, for each loan in the sample, the dataset includes information on the value and composition of the collateral available to the borrower with the corresponding Reserve Bank at the time of the loan. The reported value of collateral is adjusted using the appropriate margins (haircuts) so that, in principle, the borrower could receive a loan for the total amount of the reported collateral. To get a sense of how collateral is being used in these credit relationships, we look at the average composition of the collateral and the level of utilization (how much lending actually occurs relative to the total amount of collateral available). For this latter calculation, we use the total amount of loans outstanding, including the new loan, which is also reported in our data. The collateral pledged at the discount window can be used in any of the discount-window programs so, for example, a bank can take a loan at the primary-credit program and another loan, simultaneously, at the seasonal-credit program. The reported amount of collateral in our dataset is usable in both types of loans. None of the banks that took a secondary-credit loan during our sample period had any other outstanding loans from the discount window. Eligibility criteria for borrowing at the seasonal credit porgram generally rule out secondary-credit institutions. For those banks taking loans at the primary-credit program, instead, having loans outstanding is more common but still not very prevalent. The situation is much di¤erent for those banks taking seasonalcredit loans, as more than 40 percent of those banks have loans outstanding at the time of taking the loan reported in our dataset. While most of the outstanding loans are previous seasonal-credit loans, we also veri…ed that some banks were borrowing simultaneously from the primary and seasonal programs during our sample period. Collateral composition. The composition of collateral varies systematically across credit programs, as re‡ected in Table 11. For primary-credit loans, almost 50 percent of the collateral is accounted for by commercial and consumer loans, with high-quality securities such as US Treasuries and agency debt and mortgage-backed securities taking roughly another 20 percent of the total. Loans granted under the secondary-credit program, instead, have almost 60 percent of the collateral in the form of commercial real estate (CRE) loans. Finally,  Ackon & Ennis: The Fed's Discount Window  69  Table 11 Type of Collateral (as a Proportion of Total)  Type of collateral  Primary credit  Secondary credit  Seasonal credit  Commercial loans Consumer loans CRE loans US Treasury/Agency securities Agency MBS ABS Other  0.277 0.199 0.069 0.067 0.123 0.083 0.182  0.135 0.046 0.585 0.070 0.057 0.039 0.068  0.422 0.002 0.281 0.091 0.092 0.000 0.112  Note: For each individual loan taken by a bank, the data include the total amount of collateral available to the borrowing bank and its composition. We aggregate across all loans in our dataset and compute the proportion of each collateral component. There are other categories of collateral not reported in the table and for which the proportions were relatively small. These categories are all aggregated under the label “other.” Note that if a bank takes several loans, its collateral is counted multiple times in the aggregation, one time for each time that the bank took a loan.  the collateral of banks taking loans in the seasonal-credit program is mainly composed of commercial loans and CRE loans. Collateral utilization. We measure collateral utilization as the ratio of total outstanding loans for a given bank and its total collateral pledged with the corresponding Reserve Bank. To understand collateral utilization, it is important to keep in mind that the large proportion of test loans in the primary- and secondary-credit programs, by their nature, tend to use a very low proportion of the collateral available to the borrowing bank. For primary credit, we concentrate attention on loans that are greater than $1 million, since the predominance of test loans can be expected to be very low at those levels of borrowing. Figure 12 shows the level of utilization in the vertical axis and the amount of outstanding loans in the horizontal axis (we plot outstanding amounts up to $100 million to make the …gure more readable, but the pattern is similar for all loans over $1 million).8 We can see from the …gure that (contrary to what might be expected) there are a lot of large primarycredit loans that use a signi…cant proportion of the collateral available 8 The amount of outstanding loans includes the amount of the loans for which the data are being reported. For this reason, the amount of outstanding loans is always greater than $1 million in the …gure. Since, for this restricted subsample, only thirtythree out of 2,829 loans were made when previous loans to the borrowing bank were outstanding, the …gure basically re‡ects the utilization ratio calculated using the amount of the current loan.  70  Federal Reserve Bank of Richmond Economic Quarterly  Figure 12 Collateral Utilization Ratio. Primary Credit  Note: We consider loans at the primary-credit program that, added to the outstanding loans, sum to an amount between $1 million and $100 million. For each loan, we compute the ratio of the total loan amount outstanding and the total collateral available to the borrower. For each loan, then, we plot the level of utilization on the vertical axis and the outstanding loan amount on the horizontal axis.  to the borrowing bank. Smaller loans also can have relatively high utilization ratios. Furthermore, the …gure suggests that there is no strong correlation between the size of the loan and the level of collateral utilization. For secondary credit, we restrict attention to loans greater than $10,000, which in our sample amount to a total of thirty-nine loans. Of those thirty-nine loans, nineteen of them used more than half of the collateral available to the borrower. Collateral utilization for the thirty-nine loans is not concentrated in any particular value and instead is (roughly) evenly spread in the unit interval. Just as with primary-credit loans, for secondary-credit loans there is no indication of a tight correlation between size of the loans and  Ackon & Ennis: The Fed's Discount Window  71  Figure 13 Distribution of Collateral Utilization Ratio. Seasonal Credit  Note: For each loan, we compute the ratio of the total loan amount outstanding and the total collateral available to the borrower.  their collateral utilization. The two largest loans in this subsample of thirty-nine loans— those made to Nova Bank in 2010 for $17 million each— have a collateral utilization ratios of 55 percent. On the upper end of the utilization margin (for loans greater than $10,000), there are three loans with utilization ratio greater than 90 percent. For seasonal credit, collateral utilization is spread (roughly) evenly over values between zero and one. Figure 13 shows the density of the collateral utilization ratio for seasonal-credit loans. Since test loans seem to be much less common in the seasonal-credit program, we include all loans in the …gure. For seasonal credit, it is critical to measure utilization using outstanding loan amounts, given that in most cases these loan amounts can be much higher than the amount of the new individual loan that originates the reporting.  72 6.  Federal Reserve Bank of Richmond Economic Quarterly  CONCLUSION  In the …ve years of now-available discount-window transactions data, from mid-2010 to mid-2015, the Federal Reserve made over 16,000 loans for a total amount of more than $36 billion. Most of the lending was done through the primary-credit program. The seasonal-credit program was also a signi…cant source of funding for banks during our sample period. In the span of those …ve years, the amount of primary credit shows a slow secular decline and the amount of seasonal credit has been gradually trending up but with large seasonal swings. A signi…cant proportion of the transactions reported in the data are likely to be “test”loans. These test loans tend to be for very small amounts and do not have a signi…cant impact on the total amounts lent when aggregated across all loans. Still, for understanding the typical size of loans and other relevant aspects of the data, it is helpful to minimize the in‡uence of test loans in the results. We choose a threshold of $10,000 and exclude the loans at or below that threshold when we want to focus on the characteristics of nontest loans. In the case of secondary credit, this procedure leaves us with a very small number of nontest loans, which reduces our ability to draw robust inferences. In contrast, for the case of the primary-credit program, the sample of nontest loans is large. The size of primary-credit loans is relatively widespread with some very large loans present in the sample. Most loans are overnight, but there are loans with up to a fourteen-day maturity term. A lot of banks accessed the primary-credit program during our sample period, and many of those banks used the program on several occasions. Primary-credit loans are more common at the end of the month, the quarter, and the year, but in general there is credit extended at all times. Most of the primary-credit loans constitute onetime events, in the sense that they do not get rolled over. Those loans that do get rolled over tend to be somewhat larger on average and do not always get rolled over into equal or smaller amounts (sometimes they are rolled over into larger amounts). Very large secondary-credit loans are rare. Abstracting from test loans, there are actually not many loans being granted through this program. Comparing secondary-credit loans with primary-credit loans, we see that the former tend to be smaller and of shorter term. This is consistent with the fact that those are loans o¤ered at a higher interest rate, and the corresponding borrowers are subject to more supervisory scrutiny. Consistent with its name, the volume of seasonal credit presents a strong annual seasonal pattern with lending generally picking up signi…cantly in the second half of the year. Test loans are a lot less common in the seasonal-credit program, and loan size is more uniformly  Ackon & Ennis: The Fed's Discount Window  73  spread in the range of $1 million to $15 million. The maturity term is also widespread between one and thirty-plus days. Collateral utilization is not concentrated around a certain level and does not tend to re‡ect a correlation with the size of the loans. Because collateral is measured adjusted for the appropriate haircuts and the utilization ratio is below unity for most of the loans, the level of credit risk involved in these transactions seems likely to be fairly low. It is, of course, di¢ cult to determine the appropriate amount of discount-window credit needed by US banks. Those needs are likely to also depend on the state of aggregate …nancial conditions. The period we study includes some episodes of heightened …nancial turmoil (such as at the peak of the European crisis in 2011) but not a fullblown …nancial crisis (in the US). Also, during our sample period, banks were holding signi…cant amounts of excess reserves (and liquidity more generally), which likely reduced banks’ needs for emergency funding. Still, the data we have analyzed here show that many banks do access the discount window regularly, suggesting that the routine provision of backup funding by the central bank is a valuable option for many participants in the US banking sector.  74  Federal Reserve Bank of Richmond Economic Quarterly  APPENDIX  Table A1 Annual Average of Daily Loan Amount Outstanding ($mm) Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017  Primary credit  Secondary credit  Seasonal credit  Total  34.038 42.115 52.673 58.962 479.750 31,817.943 40,221.212 4,429.154 25.942 22.981 13.173 12.830 17.250 17.808 13.212  1.264 0 4.212 0.154 3.192 34.642 192.731 309.923 0.173 0.019 0.019 0 0.019 0 0.019  62.698 110.923 142.462 165.538 104.846 39.698 45.115 39.212 35.942 49.442 65.462 104.755 107.404 83.615 81.846  98.000 153.038 199.346 224.654 587.788 31,892.283 40,459.058 4,778.288 62.058 72.442 78.654 117.585 124.673 101.423 95.077  Ackon & Ennis: The Fed's Discount Window  75  Figure A1 Distribution of Primary-Credit Loan Sizes. Log Amounts  Note: We compute the distribution of primary-credit loan sizes after taking log of each corresponding amount.  Amount ($mm) 0.300 0.950 0.050 0.800 25.000 4.000 1.350 5.000 10.000 0.700  Borrower  Community First Bank Community First Bank Community First Bank Community First Bank Douglas National Bank First National Bank First National Bank First State Bank Lakeside Bank University & Com. FCU 8/5/2010 8/26/2010 9/2/2010 9/7/2010 12/29/2014 1/13/2015 1/5/2015 6/17/2013 1/31/2011 6/28/2011  Date of loan 6 5 5 3 7 14 7 8 4 3  Term 8/6/2010 8/30/2010 9/3/2010 9/9/2010 1/2/2015 1/22/2015 1/12/2015 6/19/2013 2/3/2011 6/30/2011  Date of repayment  Table A2 Primary-Credit Loans Paid Before Maturity  Fairview Heights, Fairview Heights, Fairview Heights, Fairview Heights, Lawrence, KS Rotan, TX Chadron, NE Norton, KS Chicago, IL Stillwater, OK  Location IL IL IL IL  St. Louis St. Louis St. Louis St. Louis Kansas City Dallas Kansas City Kansas City Chicago Kansas City  Lender  76 Federal Reserve Bank of Richmond Economic Quarterly  Ackon & Ennis: The Fed's Discount Window  77  Table A3 Primary-Credit Loans from Texas Capital Bank Date  Term  10/31/2011 11/2/2011 11/15/2011 11/30/2011 12/1/2011 12/30/2011 1/31/2012 4/30/2012 5/1/2012 5/2/2012 5/3/2012 8/31/2012 9/4/2012 9/5/2012 9/6/2012 9/7/2012 9/10/2012 9/12/2012 1/30/2013  1 1 1 1 1 4 1 1 1 1 1 4 1 1 1 3 1 1 1  Loan amount ($mm) 15.000 60.000 35.000 82.000 7.000 132.000 7.000 115.000 115.000 296.000 7.000 115.000 200.000 250.000 215.000 150.000 160.000 50.000 20.000  Table A4 Loan Rollovers Number of rollovers (x)  Number of credit events involving x rollovers  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  9000 350 146 69 37 27 16 18 7 6 7 1 5 4 2 2 1  Average size of loans 1.649 3.277 3.631 6.377 2.594 10.366 6.427 5.209 3.101 8.751 4.405 9.575 3.58 7.075 0.85 3.097 13.929  78  Federal Reserve Bank of Richmond Economic Quarterly  Figure A2 Distribution of Primary Loan Sizes. Evolution over Time  Note: We compute the distribution of primary-credit loan sizes for each twelvemonth period starting in July 2010. The distribution is calculated after excluding from the data all loans greater than $100 million (…fteen loans in total for all …ve years). This helps to make the estimated distributions smoother. We then plot only the portion of the distribution that corresponds to loans of $10 million or less.  Ackon & Ennis: The Fed's Discount Window  79  REFERENCES Anbil, Sriya. Forthcoming. “Managing Stigma During a Financial Crisis.” Journal of Financial Economics. Armantier, Olivier, Eric Ghysels, Asani Sarkar, and Je¤rey Shrader. 2015. “Discount Window Stigma During the 2007–2008 Financial Crisis.” Journal of Financial Economics 118 (November): 317–35. Benmelech, Efraim. 2012. “An Empirical Analysis of the Fed’s Term Auction Facility.” Cato Papers on Public Policy 2: 57–91. Berger, Allen N., Lamont K. Black, Christa HS Bouwman, and Jennifer Dlugosz. 2017. “Bank Loan Supply Responses to Federal Reserve Emergency Liquidity Facilities.” Journal of Financial Intermediation 32 (October): 1–15. Courtois, Renee, and Huberto M. Ennis. 2010. “Is There Stigma Associated with Discount Window Borrowing?” Federal Reserve Bank of Richmond Economic Brief 10-05 (May). Drechsler, Itamar, Thomas Drechsel, David Marques-Ibanez, and Philipp Schnabl. 2016. “Who Borrows from the Lender of Last Resort?” Journal of Finance 71 (October): 1933–74. Ennis, Huberto M. 2016. “Models of Discount Window Lending: A Review.” Federal Reserve Bank of Richmond Economic Quarterly 102 (First Quarter): 1–50. Ennis, Huberto M., and Alexander L. Wolman. 2015. “Large Excess Reserves in the United States: A View from the Cross-Section of Banks.” International Journal of Central Banking 11 (January): 251–89. Fur…ne, Craig. 2003. “Standing Facilities and Interbank Borrowing: Evidence from the Federal Reserve’s New Discount Window.” International Finance 6 (November): 329–47. Kleymenova, Anya. 2016. “Consequences of Mandated Bank Liquidity Disclosures.” Chicago Booth Research Paper 16-04 (November). Vossmeyer, Angela. 2017. “Analysis of Stigma and Bank Behavior.” Manuscript.  Economic Quarterly— Volume 103, Numbers 1-4— First-Fourth Quarter 2017— Pages 81–137  Using the Richmond Fed Manufacturing Survey to Gauge National and Regional Economic Conditions Nika Lazaryan and Santiago M. Pinto  S  everal Regional Banks in the Federal Reserve System conduct regional surveys of business conditions in an e¤ort to obtain realtime information about changes in local economic conditions. To the extent that the performance of the national economy is related to the performance of its regions, the regional surveys may provide useful information about national economic conditions as well. The results of the monthly regional surveys receive attention among analysts and other organizations that assess and forecast economic conditions because they are typically available one or two weeks prior to the release of the national and regional data. Considering how the survey data are used, it is extremely important, …rst, to understand what the surveys actually measure and, second, to determine how well they measure changes in economic conditions. This paper intends to o¤er some insights on these issues by carefully analyzing the underlying survey data and investigating their ability to precisely gauge economic conditions observed at both national and regional levels. The present work focuses on the information content of the Regional Surveys of Business Activity conducted by the Federal Reserve Bank of Richmond (FRBR). We speci…cally examine the survey that Federal Reserve Bank of Richmond: nika.lazaryan@rich.frb.org; Federal Reserve Bank of Richmond: santiago.pinto@rich.frb.org. The views expressed in this paper are those of the authors and should not necessarily be interpreted as those of the Federal Reserve Bank of Richmond or the Federal Reserve System. DOI: https://doi.org/10.21144/eq1030103  82  Federal Reserve Bank of Richmond Economic Quarterly  tracks the manufacturing sector, the Fifth District Survey of Manufacturing Activity. In general, the surveys collect qualitative data from businesses in the Fifth District on several items that are supposed to convey information about recent changes in economic activity. For instance, survey participants are asked if they have observed an increase, a decrease, or no change in their levels of employment, shipments, orders, and wages, as well as other indicators. The responses are summarized into a statistic, called a di¤usion index, that essentially captures the breadth of the changes taking place along the relevant economic dimensions in the time period under consideration. A few individual di¤usion indices are combined into a composite di¤usion index. In this paper, we evaluate the performance of these individual and composite di¤usion indices by examining how closely they track overall national and regional economic conditions. As a measure of national economic activity, this paper uses the manufacturing di¤usion index produced by the Institute of Supply Management (ISM). The individual di¤usion indices calculated by the ISM are based on questions that are very similar to those included in the FRBR survey. Previous work, such as Harris et al. (2004), shows that the composite ISM index is a good gauge of national economic activity based on its ability to track the national gross domestic product (GDP) and personal income. Their work also shows that there is indeed a strong correlation between the ISM and the indices produced from regional surveys by the Richmond and Philadelphia Federal Reserve Banks.1 Our indicator of regional economic activity in the Fifth District is a weighted average of state payroll manufacturing employment growth (MEG) rates. Only a few papers have attempted to evaluate the accuracy of the di¤usion indices produced by regional Reserve Banks in describing economic changes at the regional or local level. The limited work in this area includes Harris et al. (2004) and Pinto et al. (2015b). Even though the analysis in Harris et al. (2004) focuses on the national economy, it also brie‡y assesses the extent to which the FRBR composite di¤usion index helps explain changes in personal consumption expenditures (PCE) in the Fifth District. We use MEG instead of PCE because, among other reasons, the former series are available monthly, whereas the latter are available only at quarterly intervals. Our work is also related to Pinto et al. (2015a). This paper assesses the ability of certain speci…c individual di¤usion indices (employment and wage di¤usion indices) to explain employment and wage growth rates. It argues that, in general, growth rates include information both about 1 It would be useful, in future analysis, to benchmark the FRBR indices against other measures of economic activity, such as the Industrial Production Index.  Lazaryan & Pinto: Richmond Fed Survey and the Economy  83  the “intensive margin,” or the size or magnitude of a change, and the “extensive margin,” or the spread or breadth of a change. As a result, di¤usion indices would describe fairly well how a variable changes over time, if those changes are predominantly explained by a higher proportion of participants reporting a change (up, down, or remain the same). Among other things, this paper shows that while the FRBR employment di¤usion index tends to track quite well regional employment growth, it does not do a good job at tracking wage growth. The analysis performed in this paper departs from the previous work in at least two fundamental ways. First, we carefully examine the behavior of all di¤usion indices currently reported by the FRBR, with the intention of gaining a much broader understanding of the information conveyed by the FRBR survey. The conclusions of this study may be used to verify whether the FRBR indices capture what they intend to capture and to determine which indices are more informative depending on the speci…c objective. Second, this exercise o¤ers useful insights that could guide the design and construction of alternative indicators of economic activity and provide feedback on what kind of data to gather (for instance, which survey questions are more relevant, or how representative the survey sample is). Our analysis proceeds as follows. We …rst evaluate how well the FRBR composite di¤usion index tracks national and regional economic activity, as measured by the ISM di¤usion index and the Fifth District MEG, respectively. We next explore the di¤erential contribution of the individual components of the composite di¤usion index. Finally, we explore the bene…ts of including information that is currently available from the FRBR surveys but is not considered in the calculation of the composite index. Our …ndings show, among other things, that while the reported composite index contains useful information about economic activity at both the national and regional levels, models that incorporate additional survey information may improve the predictive power of the FRBR indices. The rest of this paper is organized as follows. Section 1 describes the survey and the data used in the analysis. Section 2 evaluates the ability of FRBR di¤usion indices to describe the behavior of the national economy. Section 3 examines the relationship between the FRBR di¤usion indices and economic conditions in the Fifth District. Section 4 summarizes and discusses the results.  1.  PRELIMINARY ANALYSIS OF THE DATA  The work in this paper focuses on the Fifth Federal Reserve District, which includes Virginia, most of West Virginia, Maryland, North  84  Federal Reserve Bank of Richmond Economic Quarterly  Carolina, South Carolina, and the District of Columbia. Moreover, the analysis is centered on the manufacturing sector. Three pieces of data are used: the di¤usion indices produced by the FRBR, constructed from the information collected by the Fifth District Survey of Manufacturing Activity; the manufacturing di¤usion indices reported by the ISM, used as a proxy of national economic activity in the manufacturing sector; and Fifth District payroll MEG, used as a measure of regional manufacturing activity. The data are monthly, and the sample covers the period from May 2002 to June 2017. The FRBR conducts monthly surveys within the Fifth Federal Reserve District states to assess business conditions in two sectors: manufacturing and service. The manufacturing survey is designed to approximate the distribution of manufacturing …rms by state, industry type, and …rm size.2 It inquires about various aspects related to the economic conditions faced by …rms, including questions on employment, shipments, new orders, backlogs, inventories, prices, etc.3 The survey is qualitative in nature, in the sense that …rms are asked whether they experienced an increase, decrease, or no change in each variable of interest from the preceding month. The responses to each question are then combined into what is referred to as a di¤usion index.4 The di¤usion index calculated by the FRBR is similar to the manufacturing di¤usion index reported by the ISM. The latter, however, is based on information collected through a survey of more than 300 purchasing managers of manufacturing companies across the US. This survey is nationally representative and captures the various manufacturing categories by their relative contribution to the GDP. In general, di¤usion indices are summary statistics of the form Dt = 100  wu  Ns Nd Ntu + ws t + wd t Nt Nt Nt  ;  (1)  where Ntu ; Nts ; and Ntd denote, respectively, the number of survey participants who report that the relevant economic variable has increased, stayed unchanged, or declined from period (t 1) to period t. The 2  On average, and during the sample period under consideration, the number of respondents in the manufacturing survey has oscillated around 100. 3 The survey also includes questions on vendors, average workweek hours, wages, business expenditures, inventories of raw materials and …nished goods, as well as capacity utilization. 4 The FRBR currently reports di¤usion indices at the Fifth District level. Sample sizes are at the moment too small to report informative di¤usion indices at the state level. In this paper, the focus is precisely on the regional di¤usion index because, as explained earlier, one of the goals is to determine whether information about the economic performance of the region conveys useful information about the national economy. See Pinto et al. (2015b) for a thorough discussion about the information content of statelevel di¤usion indices.  Lazaryan & Pinto: Richmond Fed Survey and the Economy  85  FRBR, for example, reports di¤usion indices with wu = 1; ws = 0; and wd = 1, so the range of the index is [ 100; 100]. Note that, in this case, the di¤usion index is simply the di¤erence between the fraction of respondents who reported an increase and the number of respondents who reported a decrease in a particular measure of economic activity. Therefore, a positive (negative) reading indicates that the proportion of participants who report an increase is higher (lower) than the proportion of those who report a decline in the variable. A larger value of the index (in absolute terms) indicates that the change in the economic variable is widely spread out and broadly observed among respondents.5 The di¤usion indices reported by the ISM use wu = 1; ws = 0:5; and wd = 0. The range of this index is [0; 100], so in this instance, the series are centered at 50: a reading of the index above (below) 50 indicates that the percentage of responses reporting an increase is higher (lower) than the percentage of responses indicating a decline. Both the FRBR and ISM also report composite indices that consist of a weighted average of several individual di¤usion indices, each one tracking di¤erent indicators of economic activity. Speci…cally, the composite index reported by the FRBR is given by RICt = 0:27  RICtE + 0:33  RICtS + 0:40  RICtO ;  (2)  and the ISM composite index by ISMt = 0:20  ISMtE + ISMtS + ISMtO + ISMtP + ISMtI ;  (3)  where RICti and ISMti are the FRBR and ISM individual di¤usion indices for category i, and the superscript i stands for E: employment, S: shipments, O: orders, P : production, and I: inventories.6 When comparing the FRBR di¤usion index to the ISM, we normalize the FRBR di¤usion index in order to compare both indices on the same scale. Finally, to measure regional economic activity in the manufacturing sector, we use the series of payroll MEG obtained from the Bureau of Labor Statistics (BLS). We calculate the Fifth District MEG as a weighted average of the states’ MEG.7 For the purposes of our analysis, the closest counterpart to the manufacturing FRBR di¤usion indices is regional MEG, not only because MEG closely tracks changes 5 See Pinto et al. (2015a) or Pinto et al. (2015b) for a thorough explanation of di¤usion indices. 6 The weights currently used in the FRBR composite index RICt were obtained from Harris et al. (2004). Note, however, that the weights in that paper were chosen with the explicit goal of comparing the ISM and FRBR di¤usion index series. In other words, the FRBR composite index, with those speci…c weights, was intended to track changes at the national level. 7 We apply our own seasonal adjustment process to the ISM, MEG, and FRBR di¤usion index series to preserve uniformity.  86  Federal Reserve Bank of Richmond Economic Quarterly  Table 1 Summary Statistics  ISMt M EGt RICt RICtS RICtO RICtE RICtB RICtC RICtV RICtH RICtW RICtIF RICtIR  Variable Composite ISM Manufacturing employment growth FRBR manu. composite di¤usion index Shipments Orders Employment Backlog Capacity Vendors Hours Wages Inventory …nished goods Inventory raw materials  Mean 52.765 -0.172 50.366 50.820 50.363 49.817 45.410 49.304 52.032 49.531 54.826 59.627 58.006  Std. Dev. 4.810 0.386 5.790 6.462 6.904 4.980 5.667 5.847 3.354 4.978 2.828 4.036 3.422  N 182 181 182 182 182 182 182 182 182 182 182 182 182  in the manufacturing sector, but also because the data are available at monthly intervals.8  Descriptive analysis of the series We start our examination with a simple descriptive analysis of the series. Consider …rst the RICt and ISMt series. The summary statistics reported in Table 1 indicate that, for the period under consideration, the average value of ISMt is higher than the average of RICt , but the volatility of RICt is more pronounced.9 Moreover, from Figures 1a and 1b, it appears that the series follow each other very closely. Figure 2a describes the behavior of M EGt (measured on the left axis) and RICt (measured on the right axis), and Figure 2b shows a scatter plot of the two series.10 It appears from the two …gures that the FRBR composite di¤usion index tracks fairly well the Fifth District MEG. In Pinto et al. (2015b), we …nd that the FRBR employment di¤usion index also follows very closely the behavior of regional employment growth. In Section 3, we will evaluate in more detail the di¤erential contribution of each one of the FRBR di¤usion index series in explaining regional economic changes. 8 When comparing the regional MEG and the FRBR di¤usion indices, we use the currently reported version of the FRBR di¤usion index with wu = 1; ws = 0, and wd = 1. 9 The higher volatility of RICt may be partly attributed to a smaller sample size. 10 In Figures 2a and 2b, we use the FRBR di¤usion index series centered at zero.  Lazaryan & Pinto: Richmond Fed Survey and the Economy  87  Figure 1 ISMt and RICt  Figure 3a shows the evolution of the individual components of the composite index RICt : RICtE , RICtS , and RICtO . While the RICtO series shows the largest volatility, the RICtE series shows the least. Additional di¤usion indices obtained from other questions in the FRBR survey are shown in Figure 3b. Among all the series, RICtW has the lowest standard deviation.  88  Federal Reserve Bank of Richmond Economic Quarterly  Figure 2 M EGt and RICt  A key feature of the series considered in the analysis is that they exhibit high levels of persistence.11 This behavior is evident from the autocorrelation (ACF) and partial autocorrelation functions (PACF) of the series. Figure 4 shows the ACF and PACF of the ISMt , RICt , 11 We calculate and report in Table 11 in the Appendix the results of several unit root tests for all variables. In all cases, the tests reject the presence of a unit root.  Lazaryan & Pinto: Richmond Fed Survey and the Economy  89  Figure 3 FRBR Di usion Indices  M EGt , and the individual di¤usion indices used to calculate RICt . In every case, the ACF and PACF indicate a strong autocorrelation at the …rst three of four lags. The latter is relevant because it suggests that when considering models that explain and predict the evolution of ISMt and M EGt , it would become critical to incorporate their dy-  90  Federal Reserve Bank of Richmond Economic Quarterly  Figure 4 ACF and PACF  namic behavior in addition to the dynamic behavior of RICt and RICti . We evaluate several univariate dynamic models that explain the behavior of the series in Sections 2.1 and 3.1.  Cross-correlations As a …rst approximation to the analysis of the dynamic relationship between the FRBR di¤usion indices and national and regional economic indicators, we examine the cross-correlations between the variables Xt = ISMt ; M EGt , de…ned as Corr(Xt ; RICt+h ), for di¤erent values of h = 20; :::; 1; 0; 1; :::20. The cross-correlograms between ISMt and RICt and M EGt and RICt are shown in Figure 5. Figure 5a indicates a very strong contemporaneous correlation between ISMt and RICt , with a correlation equal to 0.80 (at h = 0). The highest correlation between the M EGt and RICt series, shown in Figure 5b, occurs at h = 1 and is equal to 0.68. In other words, this last cross-  Lazaryan & Pinto: Richmond Fed Survey and the Economy  91  Figure 5 Cross-correlograms  correlogram seems to indicate that RICt tends to lead M EGt , so RICt may contain information about the future behavior of the M EGt series. We also examine the cross-correlations between ISMt and M EGt and the individual FRBR di¤usion indices, and among the individual FRBR di¤usion indices themselves. The results, which are reported in Table 12 in the Appendix, can be summarized as follows. First, the cross-correlograms between ISMt and the FRBR individual di¤usion indices RICti generally re‡ect a strong contemporaneous relationship  92  Federal Reserve Bank of Richmond Economic Quarterly  (at h = 0), with the exception of RICtE and RICtW (in both cases, the highest correlation occurs at h = 1; it is 0.71 for RICtE and 0.58 for RICtW ). Second, the highest correlation is between ISMt and RICtV , with a value of 0.78 at h = 0, followed by RICtO , with a value of 0.77 also at h = 0. Third, for some di¤usion indices, speci…cally RICtIF and RICtIR , the correlation is negative. Fourth, the results are somewhat di¤erent when comparing the cross-correlations between M EGt and RICti in Table 13. For instance, the highest Corr(M EGt ; Yt+h ) E is observed when Yt+h = RICt+h and h = 0, with a value of 0.76. Other di¤usion indices, however, tend to lead M EGt series (when O , the highest value is observed at h = Yt+h = RICt+h 3, and when S Yt+h = RICt+h the highest value is h = 1). Finally, the RICti series also tend to move together. The cross-correlations between RICtE , RICtO , and RICtS , and the other individual di¤usion indices are reported in the Appendix in Tables 14, 15, and 16. Note that the correlations between RICtE and the other di¤usion indices are relatively low. The correlations are higher for the series RICtO (for instance, the contemporaneous correlations between RICtO and RICtS and between RICtO and RICtC are, respectively, 0.92 and 0.90). The main takeaway from this preliminary analysis is that the information contained in the FRBR survey seems to be highly correlated with changes in both national and regional economic conditions. However, based on the correlations observed between the series, various composite indices based on di¤erent series and weights may be constructed to more accurately explain national and regional economic changes.  2.  PREDICTING THE NATIONAL ECONOMY  We now proceed to a formal analysis of how well the FRBR composite index RICt tracks the national economy. As mentioned earlier, we use the ISMt series as a gauge of national economic activity. We begin by …rst evaluating the predictive ability of a number of univariate dynamic models of ISMt for benchmarking purposes. We next estimate several linear and vector autoregressive models (VARs) that incorporate the di¤usion indices obtained from the FRBR manufacturing survey, and we examine how this additional information improves the models’predictive ability. We speci…cally compare the predictive power of models that use the composite di¤usion index RICt to other less-constrained models in which the components of RICt are considered individually, as well as models that incorporate other di¤usion indices, not part of RICt , calculated from currently available FRBR survey data.  Lazaryan & Pinto: Richmond Fed Survey and the Economy  93  Univariate models of ISM The descriptive analysis of Section 1.1 suggests that the ISMt series is highly persistent. In order to formally examine its behavior, we …rst estimate several univariate dynamic models of ISMt and determine how well these models predict ISMt .12 These univariate dynamic models are used as the benchmark against which we evaluate the performance of models that incorporate the information from the FRBR survey. We consider models that assume a general ARMA(p,q) representation of the form p q X X ISMt = a + (4) j "t j ; j ISMt j + "t + j=1  j=1  where "t is assumed to be an i.i.d. white noise process, and = [ 1 ; : : : ; p ] and = [ 1 ; : : : ; q ] are the autoregressive and moving average coe¢ cients, respectively. Table 17 in the Appendix presents estimates of several univariate models …tted to the ISMt series, together with the goodness-of-…t statistics AIC and BIC. Based on the estimation results, AR(4) has the best AIC statistics, whereas AR(1) produces the best BIC statistics.13 These models produce residuals with RMSE of 1.841 for AR(1) and 1.789 for AR(4). Figure 6 shows one-step-ahead predictions of ISMt for the AR(1) and AR(4) models. Overall, the results suggest that past values of ISMt explain fairly well the behavior of the series. The contribution of the information contained in the FRBR survey should, therefore, be assessed by comparing di¤erent models that include the FRBR di¤usion indices to the performance of these very simple univariate dynamic models.  Linear models We now present estimates of several linear models that explain the behavior of ISMt using the information collected from the FRBR survey series. First, we consider very simple models that assume a contemporaneous relationship between the variables of the form ISMt =  + Xt + "t ;  (5)  where Xt is a vector of regional predictors and "t is an error term assumed to be an i.i.d. white noise process. Table 2 presents the estimates of four alternative model speci…cations depending on the variables 12 Throughout the paper, we compare models based on their predictive accuracy measured by the root mean squared error (RMSE). 13 Several alternative ARMA(p,q) were estimated; those reported in Table 17 have the smallest AIC and BIC statistics.  94  Federal Reserve Bank of Richmond Economic Quarterly  Figure 6 One-Step-Ahead Predictions of ISMt  included in Xt . Model (1) includes only the composite index of the FRBR series constructed as a weighted average of RICtE , RICtS , and RICtO , given by (2); in other words, Xt = RICt in this case. In model (2), each one of the components of the composite index are included in an unconstrained form, meaning that Xt = [RICtE ; RICtS ; RICtO ]. Model (3) adds additional components not used in the composite index but available in the FRBR survey, and model (4) is basically a  Lazaryan & Pinto: Richmond Fed Survey and the Economy  95  Table 2 Linear Models of ISM: Contemporaneous Regressors  RICt  (1) 0.665*** (0.054)  RICtE  (2)  (3)  (4)  0.269** (0.089) 0.184* (0.089) 0.246* (0.101)  0.149* (0.064)  0.521*** (0.082)  182 0.770 2.307  Constant  19.282*** (2.762)  17.618*** (3.898)  0.143* (0.068) 0.106 (0.078) 0.096 (0.096) 0.037 (0.070) -0.061 (0.089) 0.479*** (0.087) -0.007 (0.071) 0.083 (0.095) -0.184* (0.072) -0.239** (0.078) 32.422*** (8.178)  N Adj-R2 RMSE  182 0.638 2.893  182 0.639 2.889  182 0.768 2.318  RICtS RICtO RICtB RICtC RICtV RICtH RICtW RICtIV RICtIR  0.168*** (0.048)  -0.193** (0.067) -0.220** (0.074) 34.024*** (7.659)  Note: Newey-West standard errors in parentheses; *p < 0:05, ** p < 0:01, ***p < 0:001:  re…nement of model (3) obtained through a stepwise procedure of regressor selection.14 14  Throughout the paper, we follow a standard stepwise procedure to select the variables of the model. We typically proceed from general to particular: we start with a general model that includes the largest possible set of predictors (in the dynamic versions of the models, we also include lagged values of the variables),then we remove predictors with the highest p-values and re…t the model. The procedure also takes into account, when comparing models, their respective AIC and BIC values. Standard errors are produced by a Newey-West regression procedure that corrects potential serial correlation in the error terms. While under serial correlation, OLS still produces unbiased parameter estimates, the standard errors in this case are not e¢ cient. We reestimate the model in (5) using the Newey-West regression procedure that produces serial correlation robust standard errors. The adjusted-R2 measure is from the OLS regression.  96  Federal Reserve Bank of Richmond Economic Quarterly  The results show that the behavior of the FRBR series explains considerable variation of ISMt . Speci…cally, model (1) shows that the FRBR composite index RICt explains, by itself, about 64 percent of variation of ISMt . Model (1) also tells us that when RICt = 50, which represents the point at which the percentage of respondents reporting an increase is the same as the percentage reporting a decrease in the FRBR survey, the ISM composite index is 52.53. According to the linear estimates, RICt and ISMt are equal when RICt is approximately 57. Moreover, when RICt is higher (lower) than 57, then RICt > (< )ISMt . We additionally perform the following exercise. Suppose the goal is to construct a composite index RIC t that includes fRICtE ; RICtS ; RICtO g and tracks as closely as possible the ISMt series. Speci…cally, suppose that RIC t takes the functional form RIC t = + E RICtE + E RICtS + E RICtO , and f ; E ; S ; O g are chosen so as to minimize PT 2 RIC t , subject to the constraints E + S + O = 1; t=1 ISMt 0. The values obtained in this case are: = 2:545, O = 0:53, i S E = 0:33, = 0:14. Two remarks are worth making. First, since the ISMt series seems to be displaced upward, as explained before, RIC t includes a positive constant term (note that the current composite FRBR di¤usion index RICt does not have a constant term). Second, RICtO should receive the highest weight and RICtE the lowest weight in the composite index, if the objective is to construct a composite index that tracks as closely as possible the ISMt series. When each individual component is included as separate regressors in an unconstrained way (model [2]), the …t and predictive power of the model improve, but note that such improvement is relatively small. Also, by comparing the estimates of model (2), RICtE seems to be the most important variable at explaining the behavior of ISMt , but in the construction of the composite index RICt , this variable receives the lowest weight of the three individual di¤usion indices. Sizable improvements in …t and predictive ability are observed, however, when we incorporate additional survey information, as evidenced by models (3) and (4). In general, the results of Table 2 con…rm that when the objective is to describe or predict the evolution of the national economy, including other information readily available through the FRBR survey would tend to improve the outcome. Including only contemporaneous values of the FRBR di¤usion indices is somewhat restrictive. It is clear from Section 1.1 that the series show high levels of persistence, which suggests that further improvements could be obtained by using models that include a dynamic structure. Thus, we estimate next a set of models that account for this  Lazaryan & Pinto: Richmond Fed Survey and the Economy  97  more general dynamic behavior of the form ISMt =  +  3 X  j Xt j  j=0  +  3 X  j ISMt j  + "t ;  (6)  j=1  where "t is again assumed to be an i.i.d. distributed white noise process. Table 3 presents the estimates of four models of the type represented by (6): model (1) includes contemporaneous and lagged value of the FRBR composite index RICt ; model (2) adds lagged values of ISMt ; model (3) includes contemporaneous and lagged values of the components of the RICt ; and model (4) includes lagged values of ISMt .15 By simply considering lagged values of RICt , such as in model (1), the RMSE decreases substantially (the static model [1] of Table 2 has a RMSE equal to 2.893, and this one has a RMSE of 2.539). However, once the model incorporates lagged values of ISMt , such as in model (2), the explanatory power of RICt declines. Also, model (2) has a much better predictive accuracy. Model (3) is an improvement relative to model (1) but not relative to model (2). Considering both a less constrained and richer dynamic behavior undoubtedly increases the …t of the model and improves its predictive power. Model (4), which includes lagged values of the individual di¤usion indices RICti , i = E; O; S, and lagged values of ISMt , has the lowest RMSE among all models, with a value of 1.676. Notice that the FRBR di¤usion index that captures changes in orders, RICtO , is always relevant at explaining the behavior of ISMt , even after accounting for past values of ISMt . Finally, Table 4 shows the estimates of a model obtained by a stepwise procedure of regressor selection among the FRBR survey series RICti , and their respective lags, and lagged values of ISMt . In this case, the model explains almost 90 percent of the variation in ISMt . The variables that seem to be most relevant at explaining changes in ISMt include, in addition to ISMt 1 , the regional indicators RICtE 2 , RICtO 2 , and RICtV . The linear predictions of this model are shown in Figure 7. The RMSE is 1.554, and this value is the lowest among all the models considered up to this point.  Vector autoregressive models In this section, we use VAR models to further explore and understand the relationship between the ISMt series and the indices elaborated by the FRBR. Let Zt = [RICt1 ; RICt2 ; ; RICtm ; ISMt ] be a multivariate 15  A.3.2.  Linear predictions of models (1) through (4) are shown in Figure 16 in Appendix  98  Federal Reserve Bank of Richmond Economic Quarterly  Table 3 Linear Models of ISM: Contemporaneous Lagged Regressors  Note: Newey-West standard errors in parentheses; *p < 0:05, ** p < 0:01, ***p < 0:001:  time series, where RICti represents each one of the di¤usion indices at time t. The long-run structural relationship between the FRBR series  Lazaryan & Pinto: Richmond Fed Survey and the Economy  99  Table 4 Linear Model of ISM: Stepwise Selection  Note: Newey-West standard errors in parentheses; *p < 0:05, ** p < 0:01, ***p < 0:001:  and ISMt is modeled by the pth -order VAR process B Zt = a +  p X j=1  Aj Z t  j  + "t ;  (7)  100  Federal Reserve Bank of Richmond Economic Quarterly  Figure 7 Stepwise Selection Model of ISM: Observed Values and Predictions  where B and Aj are (m+1) (m+1) matrices, and "t = ["1t ; "2t ; : : : ; "m+1 ]0 t is a multivariate white noise process with mean zero and variance I(m+1) (m+1) . Multiplying both sides of (7) by the inverse of B, we obtain Zt =  +  p X  j Zt j  + et ;  (8)  j=1  where = B 1 a; j = B 1 Aj and et = [e1t ; e2t ; : : : ; em+1 ]0 is a multit variate mean zero white noise process with variance-covariance matrix 1 1 0 e = B (B ) . The equation in (8) represents a VAR model of order p, which can be estimated by maximum likelihood. With the estimates of the e matrix, we perform a Cholesky decomposition and obtain a lower triangular matrix P such that e = P P 0 . Premultiplying (8) by P 1 yields P  1  Zt = P  1  +P  1  p X j=1  j Zt j  + ut ;  (9)  Lazaryan & Pinto: Richmond Fed Survey and the Economy  101  where ut = P 1 et is a multivariate white noise process with variancecovariance matrix I(m+1) (m+1) . The expression in (9) gives us (m + 1) equations in the FRBR series and ISMt , in addition to their past values. Because P is lower triangular, so is P 1 , thus the (m + 1)-st equation of (9) contains all current and past values of the multivariate time series. Also note that the error term um+1 is the linear combit nation of error terms (e1t ; e2t ; : : : ; em+1 ) weighted by the coe¢ cients of t the matrix P 1 . The expression obtained in the (m + 1)-st equation is what is known as the structural equation of ISMt . This equation represents ISMt as a linear function of its past values (up to p-th lag), as well as contemporaneous and lagged values of the FRBR series. Using this equation, we then construct “predictions” of the value of the ISM di¤usion index under the following premise: at time t, when regional survey results have become known, we can use this information to obtain a reasonable prediction of the value of the ISM di¤usion index for the current time period.  Bivariate VAR model: ISM and FRBR composite di¤ usion indices We …rst estimate a VAR(1) model for bivariate series consisting of the composite di¤usion indices ISMt and RICt . The selection of lags is based on AIC and BIC statistics. The parameter estimates of the VAR(1) model are shown in Table 18 in the Appendix, together with the variance-covariance matrix of the error terms and its Cholesky decomposition.16 Using the inverse of the lower triangular matrix obtained from the Cholesky decomposition, we construct the structural form for ISMt , which is plotted in Figure 17 in the Appendix. The RMSE of this speci…cation is 1.72. We additionally perform a forecast error variance decomposition (FEVD) to interpret the results of the VAR model. The FEVD quanti…es the relative contribution of the variables in the system, in this case ISM and RIC, to the variance of the forecast error of each variable. We focus here on the forecast error variance of ISMt . The top panel in Figure 8 shows the percentage of the forecast error variance of ISM explained by RIC, and the bottom panel shows the percentage explained by itself. The …gure indicates that variations in ISM are mostly explained by shocks to the series itself; the variation explained by RIC is virtually zero. 16 The table and the …gure showing the observed and predicted values are shown in Appendix A.3.3.  102  Federal Reserve Bank of Richmond Economic Quarterly  Figure 8 VAR Model: ISMt and RICt . Forecast Error Variance Decomposition  Multivariate VAR: ISM and FRBR individual di¤ usion indices We now estimate a VAR model that includes ISMt and the individual di¤usion indices used in the FRBR composite index, RICt . The AIC and BIC statistics suggest that a VAR(2) model …ts the data best. The parameter estimates of the VAR(2) model are shown in Table 19 in Appendix A.3.4, along with the variance-covariance matrix and its Cholesky decomposition. Using the inverse of the lower triangular matrix obtained from the Cholesky decomposition, we construct the structural form of ISMt .17 The prediction errors have a standard deviation of 1.64, which is slightly smaller than the RMSE of the bivariate VAR model considered in the previous section. The FEVD in Figure 9 describes the e¤ect of a shock on the variables RIC E ; RIC S ; RIC O , and ISM on the forecast error variance of ISM. Once again, the …gure indicates that shocks on the variable ISM essentially explain most of 17 Figure 18 in Appendix A.3.4 compares the predicted values of the model to the observed values.  Lazaryan & Pinto: Richmond Fed Survey and the Economy  103  Figure 9 ISM and FRBR Individual Di usion Indices. Forecast Error Variance Decomposition  the variation of ISM. However, the variable RIC O now becomes relevant, explaining approximately 8 percent of the variation in ISM after eight periods. We …nally estimate a VAR model that incorporates other di¤usion indices available from the FRBR survey. The variables included in the analysis were selected through a stepwise regression procedure similar to the one followed in Section 2.2. Based on AIC and BIC statistics, the VAR(1) model …ts the data best. The results are reported in Table 20 in Appendix A.3.4. This speci…cation o¤ers a high level of predictive accuracy with the lowest RMSE, which is equal to 0.85. The FEVD in Figure 10 con…rms the importance of the ISM series in explaining its own variation. The FEVD also shows that, among all FRBR di¤usion indices, RIC O is the most important one for explaining variations in ISM.  104  Federal Reserve Bank of Richmond Economic Quarterly  Figure 10 ISM and FRBR Individual Di usion Indices. Forecast Error Variance Decomposition  Summary of  ndings  To summarize our …ndings, in Table 5 we compare the RMSE of the models discussed thus far. We include, for comparison, the RMSE of the model that includes only the composite index RICt currently reported by the FRBR. In light of the predictive accuracy of the models, it is clear that a multivariate VAR model dominates all other alternatives, producing a RMSE equal to 0.85. However, a linear dynamic model that considers information readily available through the FRBR survey but not currently included in the calculation of the composite index RICt , o¤ers more accurate predictions, with a RMSE of 1.55. So even the consideration of this last relatively simpler model would entail an important increase in predictive accuracy compared with a  Lazaryan & Pinto: Richmond Fed Survey and the Economy  105  Table 5 RMSE for Selected Models of ISM Model  RMSE  Composite index  RICt  2.89  Univariate  AR(1) AR(4)  1.84 1.79  Linear  Contemporaneous Dynamic  2.31 1.55  VAR  Bivariate Multivariate  1.72 0.85  model that relies exclusively on the composite index RICt , which has the highest RMSE, equal to 2.89.  3.  PREDICTING THE REGIONAL ECONOMY  While in Section 2 we examined the extent to which the information collected by the FRBR manufacturing survey helps explain changes in the national economy, we now focus on how well the survey tracks the regional economy. As explained earlier, we use payroll MEG as a measure of regional manufacturing activity for two reasons. First, MEG data are available monthly and for all states. Second, MEG is a good indicator of the economic performance of the manufacturing sector, which is the focus of the present analysis, so MEG serves as a reasonable benchmark against which to assess the predictive ability of the information contained in the FRBR manufacturing survey. Our approach is similar to that in the previous section: we begin by estimating several univariate dynamic models of MEG; next, we compare the performance of these models to the performance of linear and VAR models of MEG that incorporate the di¤usion index series from the FRBR manufacturing survey.  Univariate models of MEG We …rst examine the predictive power of simple univariate dynamic models that only include the MEG series. The inspection of the MEG autocorrelation and partial autocorrelation functions in Figure 4 reveal that the series show high levels of persistence. To capture such dynamic behavior more formally, we estimate, as we did earlier with the ISM  106  Federal Reserve Bank of Richmond Economic Quarterly  Table 6 Univariate Models of MEG M EGt  1 2  (1) AR(2)  (2) AR(3)  (3) ARMA (1,1)  (4) ARMA (1,2)  (5) ARMA (2,1)  (6) ARMA (2,2)  0.569*** (0.049) 0.271*** (0.051)  0.526*** (0.054) 0.177** (0.065) 0.162* (0.066)  0.915*** (0.033)  0.910*** (0.042)  0.862*** (0.200) 0.0459 (0.157)  0.300 (1.522) 0.565 (1.381)  -0.374*** (0.069)  -0.331 (0.198) -0.172 (0.128)  0.251 (1.525) -0.253 (0.534) -0.172 (0.134)  181 -10.59 5.40 0.228  181 -8.67 10.52 0.228  3  Constant  -0.173 (0.110)  -0.172 (0.132)  -0.172 (0.129)  -0.386*** (0.071) 0.0328 (0.065) -0.172 (0.127)  N AIC BIC RMSE  181 -9.59 3.21 0.230  181 -12.27 3.72 0.227  181 -12.51 0.29 0.228  181 -10.65 5.34 0.228  1 2  Note: Standard errors in parentheses; *p < 0:05, ** p < 0:01, ***p < 0:001:  series, several ARMA(p,q) models of the form  M EGt = a +  p X j=1  j M EGt j  + "t +  q X  j "t j ;  (10)  j=1  where "t is assumed to be an i.i.d. white noise process, and and are the vectors of autoregressive and moving average coe¢ cients, respectively. Table 6 presents the estimates along with goodness-of…t statistics AIC and BIC.18 Based on the AIC criterion, the best model speci…cation is an ARMA(1,1), but the BIC criterion chooses the ARMA(2,2). The predictions obtained from these models, shown in Figure 11, are very close to each other. In terms of their predictive accuracy, all models are practically identical, with a RMSE approximately equal to 0.23. 18 Several alternative ARMA(p,q) models were estimated; Table 6 reports those with the smallest AIC and BIC statistics  Lazaryan & Pinto: Richmond Fed Survey and the Economy  107  Figure 11 ARMA(1,1) and ARMA(2,2) Models. Observed and Predicted Values  Linear models of MEG We now incorporate the di¤usion indices calculated from the FRBR surveys to assess how well they explain economic changes in the Fifth District. We proceed by estimating several linear models of MEG using contemporaneous and lagged values of the FRBR survey series. These models are generally described by the expression M EGt =  +  3 X j=0  j Xt  j  +  3 X  j M EGt j  + "t ;  (11)  j=1  where Xt is a vector of di¤usion indices produced by the FRBR, and "t is an error term assumed to be an i.i.d. white noise process. Table 7 presents the parameter estimates of …ve alternative model speci…cations that only include contemporaneous values of the FRBR series as explanatory variables.19 In other words, those models assume 19 In this section, we use the normalization wu = wd = 1 and ws = 0. This means that when the percentage of participants reporting an increase is equal to the percentage  108  Federal Reserve Bank of Richmond Economic Quarterly  Table 7 Linear Models of MEG: Contemporaneous Regressors M EGt  (1)  RICt  0.022*** (0.002)  RICtE  (2)  (3)  (4)  (5)  0.030*** (0.002)  0.027*** (0.003) -0.003 (0.004) 0.007 (0.004)  0.018*** (0.003) -0.002 (0.005) 0.007 (0.004) 0.001 (0.004) -0.010* (0.004) 0.006 (0.004) 0.006 (0.004) 0.011* (0.005) -0.012*** (0.003) 0.005 (0.004) -0.167 (0.090) 181 0.640 0.232  0.018*** (0.003)  RICtO RICtS RICtB RICtC RICtV RICtH RICtW RICtIF RICtIR Constant N Adj-R2 RMSE  -0.186*** (0.022) 181 0.429 0.292  -0.160*** (0.019) 181 0.578 0.251  -0.170*** (0.019) 181 0.587 0.248  0.006 (0.003)  -0.010** (0.004) 0.006 (0.004) 0.005 (0.004) 0.011* (0.004) -0.012*** (0.003) 0.005 (0.004) -0.179* (0.075) 181 0.644 0.231  Note: Newey-West standard errors in parentheses; *p < 0:05, ** p < 0:01, ***p < 0:001:  = j = 0, for j = 1; 2; 3. Model (1) only includes the FRBR composite di¤usion index, i.e., Xt = RICt ; in model (2), Xt = RICtE , which is the FRBR di¤usion index that tracks changes in employment; model (3) includes the components of the FRBR composite index, i.e., Xt = [RICtE ; RICtS ; RICtO ]; model (4) incorporates additional di¤usion j  of participants reporting a decrease, the di¤usion index is equal to zero. We use this normalization to avoid working with small numbers with many digits. Ideally, we would want to rescale the ISM di¤usion index. However, this requires using the ISM raw data, which are not publicly available.  Lazaryan & Pinto: Richmond Fed Survey and the Economy  109  Figure 12 Observed and Predicted Values of MEG: Model (5) (Stepwise Selection)  indices from the FRBR surveys; and model (5) is a re…nement of model (4) obtained after a stepwise procedure of regressor selection.20 A few remarks are worth making. First, by inspecting model (1), it follows that when the composite di¤usion index RICt is equal to zero, M EGt = 0:186. In other words, zero employment growth in the district would be consistent with a value of RICt = 8:5. It is important to note that, in theory, a zero di¤usion index does not imply a zero growth rate. A di¤usion index captures the breadth of a change measured by the number of respondents experiencing no change, an increase, or a decrease in a speci…c variable. In other words, a di¤usion index tracks changes in the extensive margin. A growth rate, however, in addition 20 Standard errors are produced by a Newey-West regression procedure that corrects potential serial correlation in the error terms. While under serial correlation, OLS still produces unbiased parameter estimates; the standard errors in this case are not ef…cient. We reestimate the model in (5) using the Newey-West regression procedure that produces serial correlation robust standard errors. The adjusted-R2 measure is from the OLS regression.  110  Federal Reserve Bank of Richmond Economic Quarterly  to changes in the extensive margin, also captures the intensity of the change, or the intensive margin. Pinto et al. (2015a) and Pinto et al. (2015b) show a decomposition of a growth rate into the extensive margin, or a term that includes a di¤usion index, and the intensive margin. So the explanatory power of the di¤usion indices depends both on the information content of the FRBR surveys, summarized by the di¤usion indices, and, more generally, on the extent to which changes in the extensive margin drive changes in the growth rate. Second, the employment di¤usion index RICtE by itself (model [2]) explains about 60 percent of the variation in MEG. In this case, when RICtE = 5:33, M EGt = 0. Since information about RICtE is available prior to the release of the M EGt monthly data (usually, the value of RICtE is known a few weeks earlier), it becomes important to understand such a relationship in order to anticipate the values of M EGt . Third, adding more information from the FRBR surveys improves the …t of the model, as shown by models (4) and (5). In those cases, the employment di¤usion index RICtE is still the most important variable explaining the behavior of M EGt . Other variables, such as RICtC , RICtW , and RICtIF , also contribute to explaining M EGt . Note that all the models considered in Table 7 have a relatively low adjusted-R2 ; model (5) has the highest one, which is equal to 0.644. Figure 12 plots the observed and predicted values of this model. It shows that the two lines only infrequently overlap, con…rming the model’s low goodness of …t. This suggests, in light of the previous discussion on growth rates and extensive and intensive margins, that in the case of Fifth District employment, di¤usion indices (or the extensive margin) only partially explain its growth rate. In other words, a low adjusted-R2 should not be necessarily used to draw conclusions about the quality of the information content of the FRBR surveys. In fact, Pinto et al. (2015b) show that RICtE tracks fairly well the extensive margin component of the actual employment growth rate in the Fifth District. Next, we perform a similar exercise as in Section 2.2 but for the MEG series. In this case, we obtain the following results: = 0:1, E = 0:83, O = 0:00, S = 0:17. The main conclusion from this exercise is that a composite index that assigns weights to the individual di¤usion indices fRICtS ; RICtS ; RICtO g with the objective of tracking M EGt as closely as possible should give the highest weight to RICtE , a lower but positive weight to RICtO , and zero weight to RICtO . This composite index is de…nitely di¤erent from the one that is supposed to track the national economy and is also di¤erent from the one currently reported by the FRBR. Up to this point, the models assume a contemporaneous relationship between the variables. The models examined next, summarized in  Lazaryan & Pinto: Richmond Fed Survey and the Economy  111  Table 8 Linear Models of MEG: Contemporaneous and Lagged Regressors  Note: Newey-West standard errors in parentheses; *p < 0:05, ** p < 0:01, ***p < 0:001:  Table 8, include both contemporaneous and lagged regressors, as speci…ed in expression (11). The results show that by considering a dynamic relationship between the variables, it is possible to improve the …t of the models. Lagged values of RICtE are relevant for explaining the behavior of M EGt when RICtE is the only explanatory variable (model  112  Federal Reserve Bank of Richmond Economic Quarterly  [3]), when RICtE is combined with M EGt (model [4]), and when RICtE is included in the regression model along with RICtS and RICtO . Note, however, that RICtE becomes statistically insigni…cant when all individual di¤usion indices and their lags, in addition to lagged values of M EGt , are included in the model speci…cation (model [6]). The latter result is consistent with the persistent behavior of the M EGt series described earlier (shown in Figure 4), and the fact that the series RICtE , RICtS , and RICtO are highly correlated. In sum, all the models that include lagged values of M EGt (speci…cally, models [2], [4], and [6]) have relatively low RMSEs. However, the lowest RMSE (and also the highest adjusted-R2 ) is associated with model (2), with a RMSE equal to 0.208. Finally, Table 9 presents the best dynamic speci…cation that includes all the di¤usion indices calculated by the FRBR. We report the estimates for the model that results from a stepwise variable selection process. Of all the models considered up to this point, this last speci…cation has the highest predictive accuracy with a RMSE of 0.189. In addition to the lagged values of M EGt , a number of di¤usion indices not currently included in the reported composite index, speci…cally RICtIR and RICtIF , appear to be signi…cantly di¤erent from zero. The model has, however, an adjusted-R2 equal to 0.76, so it imperfectly …ts the data. Figure 13 shows observed and predicted values from this speci…cation.  VAR Models Bivariate VAR model: MEG and FRBR composite di¤ usion index As in the ISM case, we estimate several VAR models, assess their predictive accuracy, and perform a FEVD. We begin by estimating two bivariate VAR models: one includes the FRBR composite index RICt and the other the FRBR employment index RICtE . The results of the estimation are shown in Tables 21 and 22 in Appendix A.4.1 in addition to the observed and predicted values obtained from each model (Figures 20 and 21, respectively). Comparing the accuracy of the predictions, the speci…cation that uses the individual di¤usion index RICtE has a RMSE equal to 0.211, slightly below the model that includes the composite di¤usion index RICt , with a RMSE equal to 0.217, and lower AIC and BIC statistics. Note, however, that some of the models considered in the previous section outperform, in terms of predictive accuracy, these two VAR models. Finally, the FEVDs for each model, shown in Figures 14a and 14b, are practically identical. They indicate  Lazaryan & Pinto: Richmond Fed Survey and the Economy  113  Table 9 Linear Model of MEG: Stepwise Selection  Note: Newey-West standard errors in parentheses; *p < 0:05, **p < 0:01, ***p < 0:001:  114  Federal Reserve Bank of Richmond Economic Quarterly  Figure 13 Dynamic Linear Model of MEG: Stepwise Selection  that, even though MEG explains most of the variation in the series, the FRBR di¤usion indices are still relevant: RICt and RICtE explain, in each case, about 20 percent of the variation of MEG after eight periods.  Multivariate VAR model: MEG and FRBR individual di¤ usion indices Finally, we estimate a VAR model that includes additional di¤usion indices computed from the FRBR survey. We follow a stepwise regression procedure to select the components considered in the analysis. The estimated values are shown in Table 23, and the predicted values from the structural equation are presented in Figure 22 in Appendix A.4.2. This model has the highest predictive accuracy of all the models considered thus far, with a RMSE of 0.131. Of all the FRBR series included in the model, RICtE is still the one that explains a larger proportion of the variation in M EGt (around 15 percent of the variance), as shown by the FEVD in Figure 15.  Lazaryan & Pinto: Richmond Fed Survey and the Economy  115  Figure 14 Forecast Error Variance Decomposition  Summary of results From all the models considered in the previous sections, we present those with the highest predictive accuracy in Table 10, in addition to the model that includes the composite index RICt currently reported by the FRBR, for comparison. The VAR model that includes all the FRBR individual di¤usion indices has the lowest RMSE. This model,  116  Federal Reserve Bank of Richmond Economic Quarterly  Figure 15 M EGt and FRBR Individual Di usion Indices. Forecast Error Variance Decomposition  Table 10 Comparison of RMSE for Selected Models of MEG Model  RMSE  Composite index  RICt  0.29  Univariate  ARM A(1; 1) ARM A(2; 2)  0.23 0.23  Linear  Contemporaneous Dynamic  0.23 0.19  VAR  Bivariate RICt Bivariate RtE Multivariate  0.22 0.21 0.13  with an RMSE of 0.13, is clearly an improvement compared with the model that relies only on RICt . All the other models, however, have approximately the same RMSEs. As in the ISM case, a linear dynamic model that includes readily available information from the FRBR survey performs reasonably well.  Lazaryan & Pinto: Richmond Fed Survey and the Economy 4.  117  CONCLUSION  In this paper, we evaluate the information content of the FRBR manufacturing survey to determine the extent to which the di¤usion indices based on the survey responses contribute to explaining national and regional economic conditions. We do so by examining the predictive accuracy of a variety of models, some of which include the composite di¤usion index reported by the FRBR, and some of which incorporate additional information available from the FRBR survey but not currently employed in the calculation of the composite index. The …ndings of the exercise can be summarized as follows. First, the di¤usion indices currently reported by the FRBR manufacturing survey perform reasonably well at explaining the national economy, described by the evolution of the ISM di¤usion index, and the regional economy, described by the evolution of the MEG. Second, in order to more accurately predict the behavior of the national and regional economy, it becomes essential to consider models that account for a richer dynamic structure given the high persistence of the series under study. And third, there are grounds for improving the predictive power of the FRBR composite index, both at national and regional levels, by adjusting the weights currently used in the calculation and by including other readily available di¤usion indices. However, it should be kept in mind that the composite indices that track the national and regional economy would not necessarily be the same. This paper provides a few insights on what those di¤usion indices would look like. Future analysis should study more carefully the design of composite indices based on currently available information, including perhaps the possibility of constructing those indices based on a principal component analysis.  118  Federal Reserve Bank of Richmond Economic Quarterly  APPENDIX A.1:  UNIT ROOT TESTS  Table 11 Unit Root Tests Variable ISMt M EGt RICt RtE RtO RtS RtB RtC RtH RtW RtIF RtIR RtV  t-stat  Drift p-value  Drift and Trend t-stat p-value  ADF Test t-stat p-value  -2.591 -4.632 -5.132 -4.557 -5.708 -6.624 -6.082 -6.088 -5.717 -6.346 -4.558 -4.720 -5.276  0.095 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000  -2.801 -5.199 -5.257 -5.130 -5.757 -6.647 -6.176 -6.134 -6.034 -6.660 -4.775 -5.163 -5.294  -3.437 -3.122 -3.902 -3.461 -4.358 -4.069 -4.587 -4.326 -5.348 -3.130 -3.415 -3.697 -3.749  0.058 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000  Note: ADF: Augmented Dickey-Fuller. The number of determined by the autoregressive order.  0.047 0.021 0.012 0.044 0.003 0.007 0.001 0.003 0.000 0.099 0.049 0.023 0.019  terms in the ADF is  RICtE  -0.010 0.040 0.090 0.139 0.192 0.261 0.345 0.443 0.512 0.621  0.676  0.713 0.690 0.633 0.583 0.507 0.437 0.362 0.305 0.269 0.237  Lag  -10 -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1 2 3 4 5 6 7 8 9 10  0.710 0.616 0.481 0.380 0.274 0.189 0.128 0.087 0.080 0.035  0.713 0.646 0.536 0.417 0.313 0.206 0.138 0.099 0.092 0.049  0.749  0.163 0.213 .0215 0.254 0.318 0.398 0.499 0.576 0.649 0.709  RICtS  0.682 0.590 0.460 0.363 0.268 0.175 0.118 0.105 0.093 0.062  0.730  0.107 0.152 0.156 0.179 0.267 0.360 0.470 0.579 0.666 0.705  RICtB  0.698 0.616 0.497 0.399 0.303 0.222 0.148 0.113 0.104 0.050  0.722  0.187 0.225 0.226 0.259 0.313 0.398 0.479 0.546 0.630 0.679  RICtC  0.663 0.606 0.490 0.392 0.287 0.193 0.111 0.092 0.059 0.017  0.675  0.130 0.171 0.167 0.190 0.229 0.326 0.415 0.492 0.578 0.641  RICtH  0.579 0.550 0.527 0.455 0.418 0.367 0.342 0.324 0.318 0.301  0.559  -0.073 -0.025 -0.018 0.020 0.107 0.157 0.226 0.336 0.417 0.492  RICtW  -0.651 -0.616 -0.567 -0.488 -0.383 -0.299 -0.239 -0.152 -0.104 -0.050  -0.655  -0.207 -0.272 -0.337 -0.361 -0.417 -0.467 -0.509 -0.546 -0.615 -0.638  RICtIF  -0.626 -0.578 -0.512 -0.430 -0.335 -0.247 -0.163 -0.106 -0.046 0.007  -0.629  -0.136 -0.176 -0.226 -0.263 -0.315 -0.376 -0.439 -0.518 -0.566 -0.622  RICtIR  0.757 -0.711 0.670 0.585 0.492 0.419 0.367 0.353 0.314 0.257  0.777  0.059 0.105 0.121 0.174 0.245 0.324 0.414 0.530 0.641 0.720  RICtV  APPENDIX A.2:  0.770  0.146 0.200 0.205 0.255 0.331 0.410 0.525 0.613 0.700 0.741  RICtO  Table 12 Cross-correlogram between ISMt and RICtI  Lazaryan & Pinto: Richmond Fed Survey and the Economy  CROSS-CORRELOGRAMS  119  RICtE 0.351 0.371 0.410 0.481 0.481 0.549 0.623 0.679 0.704 0.758 0.762 0.741 0.681 0.637 0.576 0.476 0.436 0.352 0.306 0.228 0.215  Lag  -10 -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1 2 3 4 5 6 7 8 9 10  0.479 0.374 0.259 0.180 0.126 0.048 0.027 -0.009 -0.057 -0.081  0.569  0.245 0.253 0.287 0.344 0.355 0.479 0.524 0.608 0.597 0.597  RICtO  0.482 0.391 0.289 0.205 0.165 0.054 0.055 -0.007 -0.046 -0.086  0.566  0.238 0.221 0.274 0.288 0.334 0.438 0.479 0.570 0.572 0.590  RICtS  0.483 0.390 0.275 0.205 0.134 0.077 0.067 0.017 -0.009 0.055  0.551  0.235 0.217 0.266 0.341 0.338 0.467 0.506 0.582 0.573 0.579  RICtB  0.452 0.358 0.272 0.211 0.146 0.059 0.037 -0.019 -0.067 -0.077  0.539  0.259 0.252 0.309 0.327 0.335 0.431 0.473 0.550 0.562 0.551  RICtC  0.572 0.474 0.388 0.318 0.236 0.168 0.129 0.083 0.028 0.024  0.640  0.286 0.285 0.284 0.351 0.367 0.461 0.485 0.611 0.635 0.634  RICtH  Table 13 Cross-correlogram between M EGt and RICtO  0.611 0.578 0.520 0.466 0.460 0.388 0.395 0.359 0.333 0.307  0.618  0.116 0.156 0.172 0.239 0.272 0.321 0.436 0.502 0.506 0.593  RICtW  -0.527 -0.507 -0.440 -0.352 -0.300 -0.190 -0.143 -0.078 -0.029 0.020  -0.601  -0.409 -0.436 -0.478 -0.517 -0.534 -0.552 -0.563 -0.595 -0.637 -0.587  RICtIF  -0.246 -0.188 -0.096 0.024 0.085 0.160 0.209 0.249 0.300 0.312  -0.278  -0.064 -0.058 -0.115 -0.151 -0.166 -0.224 -0.247 -0.245 -0.301 -0.326  RICtIR  0.520 0.434 0.357 0.269 0.205 0.139 0.122 0.108 0.056 -0.036  0.529  0.110 0.171 0.138 0.216 0.273 0.340 0.381 0.456 0.476 0.533  RICtV  120 Federal Reserve Bank of Richmond Economic Quarterly  RICtO 0.215 0.208 0.239 0.282 0.367 0.423 0.517 0.580 0.635 0.668 0.681 0.556 0.389 0.303 0.175 0.109 0.056 0.036 0.029 0.011 0.001  Lag -10 -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1 2 3 4 5 6 7 8 9 10  0.532 0.399 0.331 0.189 0.121 0.058 0.026 0.031 0.033 0.017  0.635  RICtS 0.209 0.185 0.231 0.260 0.329 0.398 0.485 0.520 0.576 0.613  0.538 0.378 0.289 0.187 0.122 0.053 0.094 0.074 0.077 0.044  0.660  RICtB 0.243 0.210 0.231 0.268 0.341 0.425 0.502 0.553 0.640 0.655  0.559 0.418 0.319 0.191 0.105 0.030 0.030 0.018 0.004 0.020  0.683  RICtC 0.252 0.249 0.271 0.267 0.331 0.377 0.474 0.504 0.580 0.653  0.655 0.481 0.391 0.298 0.195 0.114 0.106 0.117 0.134 0.110  0.774  RICtH 0.259 0.266 0.261 0.264 0.314 0.379 0.502 0.517 0.627 0.709  0.608 0.552 0.496 0.458 0.383 0.384 0.364 0.365 0.351 0.352  0.624  RICtW 0.080 0.053 0.123 0.198 0.278 0.337 0.406 0.483 0.546 0.554  Table 14 Cross-correlogram of RICtE with FRBR Di usion Indices  -0.609 -0.538 -0.485 -0.366 -0.293 -0.211 -0.154 -0.115 -0.097 -0.090  -0.625  RICtIF -0.409 -0.446 -0.501 -0.483 -0.526 -0.532 -0.565 -0.606 -0.640 -0.670  -0.257 -0.197 -0.117 -0.009 0.078 0.138 0.211 0.223 0.242 0.266  -0.332  RICtIR -0.098 -0.123 -0.127 -0.174 -0.172 -0.210 -0.243 -0.315 -0.340 -0.350  0.498 0.458 0.347 0.291 0.237 0.162 0.160 0.151 0.117 0.060  0.564  RICtV 0.101 0.121 0.155 0.199 0.278 0.331 0.389 0.460 0.553 0.567  Lazaryan & Pinto: Richmond Fed Survey and the Economy 121  RICtE 0.001 0.011 0.029 0.036 0.056 0.109 0.175 0.303 0.389 0.556 0.681 0.668 0.635 0.580 0.517 0.423 0.367 0.282 0.239 0.208 0.215  Lag -10 -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1 2 3 4 5 6 7 8 9 10  0.647 0.575 0.457 0.291 0.236 0.153 0.101 0.060 0.114 0.057  0.921  RICtS 0.074 0.088 0.093 0.103 0.124 0.202 0.237 0.374 0.476 0.629  0.670 0.505 0.376 0.241 0.173 0.131 0.106 0.101 0.135 0.108  0.897  RICtB 0.093 0.084 0.031 0.034 0.080 0.175 0.290 0.402 0.553 0.646  0.689 0.555 0.450 0.294 0.227 0.141 0.097 0.046 0.098 0.082  0.903  RICtC 0.101 0.132 0.116 0.095 0.105 0.172 0.244 0.351 0.477 0.646  0.669 0.555 0.420 0.350 0.241 0.139 0.089 0.071 0.067 0.062  0.814  RICtH 0.085 0.083 0.054 0.068 0.058 0.121 0.181 0.298 0.463 0.632  0.541 0.464 0.451 0.379 0.365 0.284 0.285 0.278 0.297 0.315  0.546  RICtW -0.068 -0.071 -0.056 -0.018 0.015 0.067 0.169 0.217 0.359 0.416  Table 15 Cross-correlogram of RICtO with FRBR Di usion Indices  -0.576 -0.536 -0.500 -0.421 -0.339 -0.240 -0.195 -0.143 -0.119 -0.073  -0.577  RICtIF -0.087 -0.161 -0.227 -0.203 -0.270 -0.249 -0.302 -0.399 -0.475 -0.505  -0.517 -0.459 -0.404 -0.318 -0.243 -0.173 -0.048 -0.037 -0.003 0.040  -0.548  RICtIR 0.020 -0.041 -0.054 0.082 -0.121 -0.127 -0.200 -0.315 -0.354 -0.467  0.639 0.600 0.499 0.455 0.373 0.306 0.308 0.315 0.279 0.256  0.676  RICtV 0.019 0.022 -0.025 0.040 0.108 0.135 0.204 0.274 0.437 0.548  122 Federal Reserve Bank of Richmond Economic Quarterly  RICtE 0.017 0.033 0.031 0.026 0.058 0.121 0.189 0.331 0.399 0.532 0.635 0.613 0.576 0.520 0.485 0.398 0.329 0.260 0.231 0.185 0.209  Lag  -10 -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1 2 3 4 5 6 7 8 9 10  0.629 0.476 0.374 0.237 0.202 0.124 0.103 0.093 0.088 0.074  0.921  0.057 0.114 0.060 0.101 0.153 0.236 0.291 0.457 0.575 0.647  RICtO  0.608 0.442 0.347 0.179 0.157 0.113 0.100 0.122 0.142 0.095  0.809  0.085 0.072 0.022 0.045 0.093 0.214 0.321 0.434 0.556 0.596  RICtB  0.634 0.466 0.410 0.267 0.212 0.150 0.114 0.084 0.089 0.105  0.897  0.116 0.152 0.115 0.112 0.131 0.189 0.277 0.388 0.505 0.615  RICtC  0.624 0.483 0.378 0.307 0.235 0.123 0.077 0.074 0.066 0.047  0.780  0.127 0.075 0.041 0.077 0.093 0.142 0.225 0.331 0.473 0.625  RICtH  0.488 0.424 0.413 0.347 0.349 0.274 0.255 0.307 0.299 0.291  0.521  -0.056 -0.071 -0.066 -0.025 0.017 0.059 0.169 0.225 0.385 0.422  RICtW  Table 16 Cross-correlogram of RICtS with FRBR Di usion Indices  -0.589 -0.516 -0.480 -0.424 -0.344 -0.263 -0.230 -0.161 -0.117 -0.112  -0.609  -0.117 -0.183 -0.204 -0.209 -0.271 -0.295 -0.316 -0.406 -0.480 -0.526  RICtIF  -0.526 -0.451 -0.410 -0.317 -0.266 -0.179 -0.061 -0.050 -0.012 -0.002  -0.542  0.015 -0.049 -0.031 -0.073 -0.156 -0.167 -0.235 -0.338 -0.355 -0.460  RICtIR  0.607 0.555 0.489 0.427 0.351 0.288 0.293 0.323 0.263 0.238  0.633  0.039 0.040 -0.035 0.032 0.145 0.164 0.256 0.332 0.463 0.536  RICtV  Lazaryan & Pinto: Richmond Fed Survey and the Economy 123  124  Federal Reserve Bank of Richmond Economic Quarterly  APPENDIX A.3:  NATIONAL ECONOMY: VAR MODELS  A.3.1 UNIVARIATE MODELS OF ISM  Table 17 Univariate Models of ISMt  ISMt  1  ISMt  2  ISMt  3  ISMt  4  (1) AR(1) 0.921*** (0.023)  (2) AR(4) 0.972*** (0.065) 0.059 (0.101) 0.016 (0.120) -0.160* (0.078)  (3) ARMA(1,1) 0.909*** (0.028)  (4) ARMA(1,2) 0.892*** (0.037)  0.084 (0.073)  Constant  53.045*** (1.818)  52.953*** (1.285)  53.018*** (1.696)  0.070 (0.077) 0.137 (0.084) 53.018*** (1.596)  N AIC BIC RMSE  182 746.537 756.150 1.841  182 742.370 761.594 1.789  182 747.163 759.979 1.834  182 746.366 762.386 1.820  t 1 t 2  (5) ARMA(4,1) 0.984 (0.556) 0.047 (0.543) 0.015 (0.128) -0.158 (0.115) -0.012 (0.568) 52.953*** (1.283) 182 744.369 766.797 1.789  Note: Standard errors in parentheses; * p < 0:05, **p < 0:01, ***p < 0:001:  Lazaryan & Pinto: Richmond Fed Survey and the Economy  125  A.3.2 LINEAR MODELS  Figure 16 Linear models of ISM with Contemporaneous and Lagged Regressors. Observed Values and Predictions  126  Federal Reserve Bank of Richmond Economic Quarterly  A.3.3 BIVARIATE VAR: ISM AND FRBR COMPOSITE DIFFUSION INDICES  Table 18 VAR(1): ISMt and RICt . Estimates, Variance-Covariance Matrix, and Cholesky Decomposition  p  1  N AIC BIC RMSE  RICt ISMt  0.35*** 0.05  RICt ISMt  1.62 3.97***  RICt ISMt  12.27 2.31  3.38  RICt ISMt  0.29 -0.11  0 0.58  181 9.33 9.43 1.72  0.59*** 0.88***  Lazaryan & Pinto: Richmond Fed Survey and the Economy  127  Figure 17 VAR(1): ISMt and RICt . Observed and Predicted Values  1  N AIC BIC RMSE  p  2  1  3.06 3.93 5.61 6.99*** 6.86 4.48 3.28 0.67 0.38 -0.16 0.03 0.0001  RICtE RICtO RICtS ISMt RICtE RICtO RICtS ISMt RICtE RICtO RICtS 180 19.17 19.43 1.64  0.29*** -0.35*** -0.28* -0.16***  RICtE RICtO RICtS ISMt  ISMt  0.38*** 0.32*** 0.19 0.07  RICtE RICtO RICtS ISMt  0 0.25 -0.35 -0.10  18.68 15.34 2.65  0.03 0.39*** 0.28** 0.13**  0.12*** 0.31** 0.22* 0.05  Table 19 VAR(2): ISMt , RICtE , RICtO , and RICtS . Estimates, Variance-Covariance Matrix, and Cholesky Decomposition  0 0 0.42 0.02  18.40 2.00  -0.08 -0.36*** -0.21* -0.09  -0.06 -0.07 -0.07 -0.02  0 0 0 0.61  3.05  -0.18 -0.34* -0.07 -0.06  0.43*** 0.98*** 0.80*** 0.94***  128 Federal Reserve Bank of Richmond Economic Quarterly  A.3.4 MULTIVARIATE VAR: ISM AND FRBR INDIVIDUAL DIFFUSION INDICES  Lazaryan & Pinto: Richmond Fed Survey and the Economy  129  Figure 18 VAR(2): ISM, RICtE , RICtO , and RICtS . Observed and Predicted Values  1  N AIC BIC RMSE  p  26.60*** 4.91 29.16*** 12.92** 20.79*** 11.84 15.24*** 3.84 -2.20 1.35 0.27 -0.45 -0.04 -0.33 1.96 -1.12 0.70 0.14 -0.23 -0.02 -0.17 181 31.49 32.48 0.85  RICtIR RICtO RICtIF RICtW RICtV RICtE ISMt RICtIR RICtO RICtIF RICtW RICtV RICtE ISMt RICtIR RICtO RICtIF RICtW RICtV RICtE ISMt  0.54*** -0.01 0.09 0.13** -0.11* 0.25*** -0.04  RICtIR RICtO RICtIF RICtW RICtV RICtE ISMt  0 4.44 -0.28 0.49 0.63 0.92 0.72  21.02 -2.03 2.01 3.01 4.11 3.38  -0.60* 0.35*** -0.30 0.04 0.04 0.08 0.06  0 0 2.17 0.18 0.08 0.05 -0.13  5.31 0.36 -0.16 -0.17 -0.61  0.03 0.01 0.57*** 0.00 0.03 -0.30*** -0.06  0 0 0 1.88 0.50 0.26 0.10  3.82 1.23 0.95 0.50  0.17** -0.10 -0.05 0.33*** 0.10 0.03 -0.06  Table 20 VAR(1): ISM and FRBR Di usion Indices. Estimates, Variance-Covariance Matrix, and Cholesky Decomposition  0 0 0 0 1.78 -0.09 0.36  3.86 0.55 1.18  -0.03 -0.14 0.19* 0.02 0.31*** 0.02 0.01  0 0 0 0 0 2.43 -0.14  6.85 0.31  0.13*** 0.08 -0.10 0.11** -0.05 0.43*** -0.03  0 0 0 0 0 0 1.59  3.26  -0.26*** 0.69*** -0.19** 0.18** 0.29*** 0.29*** 0.84***  130 Federal Reserve Bank of Richmond Economic Quarterly  Lazaryan & Pinto: Richmond Fed Survey and the Economy  Figure 19 VAR(1): ISM and FRBR Di usion Indices. Observed and Predicted Values  131  132  Federal Reserve Bank of Richmond Economic Quarterly  APPENDIX A.4  REGIONAL ECONOMY: VAR MODELS  A.4.1 BIVARIATE VAR: MEG AND FRBR COMPOSITE INDEX  Table 21 Bivariate VAR(1): M EGt and RICt . Estimates, Variance-Covariance Matrix, and Cholesky Decomposition  p  1  N AIC BIC RMSE  RICt M EGt  0.635*** 0.010***  4.851** 0.596***  RICt M EGt  1.081 -0.074***  RICt M EGt  58.520 0.448  0.050  RICt M EGt  0.131 -0.035  0 4.619  180 6.751 6.858 0.217  Figure 20 Bivariate VAR(1): M EGt and RICt . Observed and Predicted Values  Lazaryan & Pinto: Richmond Fed Survey and the Economy  133  Table 22 Bivariate VAR(1): M EGt and RICtE . Estimates, Variance-Covariance Matrix, and Cholesky Decomposition  p  1  N AIC BIC RMSE  RICtE M EGt  0.542*** 0.015***  RICtE M EGt  1.349** -0.079***  RICtE M EGt  32.348 0.349  0.048  RICtE  0.176 -0.051  0 4.761  M EGt  8.534*** 0.490***  180 6.098 6.205 0.211  Figure 21 Bivariate VAR(1): M EGt and RICtE . Observed and Predicted Values  134  Federal Reserve Bank of Richmond Economic Quarterly  A.4.2 MULTIVARIATE VAR MODEL: MEG AND FRBR INDIVIDUAL DIFFUSION INDICES  Figure 22 VAR(1) model: MEG and FRBR Individual Di usion Indices. Observed and Predicted Values  1  N AIC BIC RMSE  p  18.564 -12.655 6.973 -1.30 -4.096 -0.906 -0.029 0.232 0.065 -0.076 0.003 0.037 -0.015 -0.020  RICtIR RICtS RICtIF RICtW RICtV RICtE M EGt RICtIR RICtS RICtIF RICtW RICtV RICtE 180 36.876 37.870 0.131  3.810*** 5.765* 5.151*** 5.777** 4.811*** 2.254 -0.028  RICtIR RICtS RICtIF RICtW RICtV RICtE M EGt  M EGt  0.598*** -0.279* 0.113** 0.002 -0.204*** 0.050 0.001  RICtIR RICtS RICtIF RICtW RICtV RICtE M EGt  0 0.096 0.010 -0.027 -0.023 -0.033 -0.014  117.450 -9.740 13.24 15.485 21.629 0.501  -0.744** 0.311*** -0.045 0.022 0.073** 0.082 0.001  0 0 0.218 -0.015 0.002 -0.004 0.045  23.974 0.32 -1.958 -0.698 -0.225  0.096* -0.201 0.621*** -0.008 0.003 -0.247*** 0.005  0 0 0 0.233 -0.069 -0.028 -0.017  20.07 6.481 5.271 0.119  0.162** 0.308* 0.001 0.454*** 0.115 0.178* 0.003  Table 23 VAR(1): MEG and FRBR Individual Di usion Indices. Estimates, Variance-Covariance Matrix, and Cholesky Decomposition  0 0 0 0 0.267 -0.004 0.003  17.618 3.672 0.074  -0.177** 0.163 0.066 0.058 0.445*** 0.111 0.001  0 0 0 0 0 0.190 -0.020  32.152 0.208  0.11** 0.101 -0.114** 0.049 -0.076 0.398*** 0.009***  0 0 0 0 0 0 4.830  0.048  -0.68 -0.294 0.391 2.855** 3.085*** 4.434*** 0.484***  Lazaryan & Pinto: Richmond Fed Survey and the Economy 135  136  Federal Reserve Bank of Richmond Economic Quarterly  REFERENCES Harris, Matthew, Raymond E. Owens, and Pierre-Daniel G. Sarte. 2004. “Using Manufacturing Surveys to Assess Economic Conditions.” Federal Reserve Bank of Richmond Economic Quarterly 90 (Fall): 65–93. Pinto, Santiago, Pierre-Daniel G. Sarte, and Robert Sharp. 2015a. “Learning About Consumer Uncertainty from Qualitative Surveys: As Uncertain As Ever.” Federal Reserve Bank of Richmond Working Paper 15-09 (August). Pinto, Santiago, Sonya Ravindranath Waddell, and Pierre-Daniel G. Sarte. 2015b. “Monitoring Economic Activity in Real Time Using Di¤usion Indices: Evidence from the Fifth District.” Federal Reserve Bank of Richmond Economic Quarterly 101 (Fourth Quarter): 275–301.
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