The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.

Authorized for public release by the FOMC Secretariat on 1/12/2024 BOARD OF GOVERNORS OF THE FEDERAL RESERVE SYSTEM DIVISION OF MONETARY AFFAIRS FOMC SECRETARIAT Date: June 5, 2018 To: Research Directors From: Matthew M. Luecke Subject: Supporting Documents for DSGE Models Update The attached documents support the update on the projections of the DSGE models. Note that the Chicago Fed’s DSGE documentation has not changed since the last quarter, so their guide from December is being redistributed. Page 1 of 1 Authorized for public release by the FOMC Secretariat on 1/12/2024 The Current Outlook in EDO: June 2018 FOMC Meeting Class II FOMC – Restricted (FR) Diego Vilán∗ June 1, 2018 1 The EDO Forecast from 2018 to 2020 The EDO model’s forecast is conditional on data through the first quarter of 2018 and on a preliminary Tealbook forecast for the second quarter of 2018. Real GDP growth is about 2.6 percent, on average, over the projection horizon, somewhat below its long-run value of 3 percent. Inflation reaches the Committee’s 2 percent objective in the third quarter of 2020. Below-trend real GDP growth is driven by the slow fading of risk premium shocks and the waning effects of the currently accommodative stance of monetary policy. On the nominal side, for a number of years wages have been below the level consistent with the models wage Phillips curve, holding down marginal cost and depressing inflation over that time period. The persistence of these wage shocks accounts for most of the weakness in inflation over the forecast. By contrast, the rise in inflation at the beginning of 2018 is attributed by the model to transient shocks to the price Phillips curve, which largely dissipate by the first half of 2019. The output gap is currently estimated to be −0.7 percent, and the economy slowly reaches its full potential by the end of 2020. The real natural rate of interest is estimated to be around 0.9 percent in the third quarter of 2018 and hovers around 1.7 percent thereafter until the end of 2020, 0.4 percentage point below its steady-state value of 2.1 percent. The trajectories of the natural rate of interest and the output gap are heavily driven by the model’s view that capital stocks are currently well below those that would have prevailed in the absence of nominal rigidities, and that the investment-related shocks responsible for this condition are likely to dissipate slowly. Consistent with the gradual return of inflation and the output gap to their long-run values, the federal funds rate is projected to increase gradually over the forecast horizon, reaching 3 34 percent by the end of 2020. At the end of the projection horizon, the federal funds rate is still below its ∗ The author is affiliated with the Division of Research and Statistics of the Federal Reserve Board. Sections 2 and 3 contain background material on the EDO model, as in previous rounds. These sections were co-written with Hess Chung and Jean-Philippe Laforte. 1 Page 1 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) Figure 1: Recent History and Forecasts long-run value of 4.1 percent, reflecting the inertia in the policy rule and the persistently negative output gap. Compared with the March 2018 projection, the EDO model’s forecast of real GDP growth in this round is stronger until the first half of 2019 because of various transitory factors; thereafter, real GDP growth is only a touch weaker relative to March. Core PCE inflation is, on average, unchanged over the forecast horizon. The output gap has been revised up, on average, 15 basis points over the projection period since March, while the forecast of the real natural rate of interest has been revised up 17 basis points, on average, since March, with a noticeable upward revision in 2017 due to positive revisions to the contribution of TFP growth shocks. The path of the federal funds is higher this round than in March, following the upward revision of the output gap. 2 of 16 Page 2 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) Figure 2: Recent History and Forecasts: Latent Variables 2 An Overview of Key Model Features Figure 3 provides a graphical overview of the model. While similar to most related models, EDO has a more detailed description of production and expenditure than most other models.1 Specifically, the model possesses two final good sectors in order to capture key long-run growth facts and to differentiate between the cyclical properties of different categories of durable expenditure (for example, housing, consumer durables, and nonresidential investment). For example, technological progress has been faster in the production of business capital and consumer durables (such as computers and electronics). The disaggregation of production (aggregate supply) leads naturally to some disaggregation of expenditures (aggregate demand). We move beyond the typical model with just two categories of (private domestic) demand (consumption and investment) and distinguish between four categories of private demand: consumer nondurable goods and nonhousing services, consumer durable goods, residential investment, and nonresidential investment. The boxes surrounding the producers in the figure illustrate how we structure the sources of each demand category. Consumer nondurable goods and services are sold directly to households; consumer durable goods, residential capital goods, and nonresidential capital goods are intermediated through capital-goods intermediaries (owned by the households), who then rent these capital stocks to households. Consumer nondurable goods and services and residential capital goods are purchased (by households and residential capital goods owners, respectively) from the first of economy’s two final goods-producing sectors, while consumer durable goods and nonresidential capital goods are purchased (by consumer durable and residential capital goods owners, respectively) from the second sector. In addition to consuming the nondurable goods and services that they purchase, households supply labor to the intermediate goods-producing firms in both sectors of the economy. The remainder of this section provides an overview of the main properties of the model. In 1 Chung, Kiley, and Laforte (2010) provide much more detail regarding the model specification, estimated parameters, and model properties. 3 of 16 Page 3 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) Figure 3: Model Overview particular, the model has five key features: • A New-Keynesian structure for price and wage dynamics. Unemployment measures the difference between the amount workers are willing to be employed and firms’ employment demand. As a result, unemployment is an indicator of wage and, hence, price pressures as in Gali (2011). • Production of goods and services occurs in two sectors, with differential rates of technological progress across sectors. In particular, productivity growth in the investment and consumer durable goods sector exceeds that in the production of other goods and services, helping the model match facts regarding long-run growth and relative price movements. • A disaggregated specification of household preferences and firm production processes that leads to separate modeling of nondurables and services consumption, durables consumption, residential investment, and business investment. • Risk premiums associated with different investment decisions play a central role in the model. These include, first, an aggregate risk premium, or natural rate of interest, shock driving a wedge between the short-term policy rate and the interest rate faced by private decisionmakers 4 of 16 Page 4 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) (as in Smets and Wouters (2007)) and, second, fluctuations in the discount factor/risk premiums faced by the intermediaries financing household (residential and consumer durable) and business investment. 2.1 Two-sector production structure It is well known (for example, Edge, Kiley, and Laforte (2008)) that real outlays for business investment and consumer durables have substantially outpaced those on other goods and services, while the prices of these goods (relative to others) has fallen. For example, real outlays on consumer durables have far outpaced those on other consumption while prices for consumer durables have been flat and those for other consumption have risen substantially; as a result, the ratio of nominal outlays in the two categories has been much more stable, although consumer durable outlays plummeted in the Great Recession. Many models fail to account for this fact. EDO accounts for this development by assuming that business investment and consumer durables are produced in one sector and other goods and services in another sector. Specifically, production by firm j in each sector s (where s equals kb for the sector producing business investment and consumer durables and cbi for the sector producing other goods and services) is governed by a Cobb-Douglas production function with sector-specific technologies: 1−α Xts (j) = (Ztm Zts Lst (j)) α (Ktu,nr,s (j)) , for s = cbi, kb. (1) In equation (1), Z m represents (labor-augmenting) aggregate technology, while Z s represents (laboraugmenting) sector-specific technology; we assume that sector-specific technological change affects the business investment and consumer durables sector only. Ls is labor input and K u,nr,s is capital input (that is, utilized nonresidential business capital (and hence the nr and u terms in the superscript). Growth in this sector-specific technology accounts for the long-run trends, while highfrequency fluctuations allow for the possibility that investment-specific technological change is a source of business cycle fluctuations, as in Fisher (2006). 2.2 The structure of demand EDO differentiates between several categories of expenditure. Specifically, business investment spending determines nonresidential capital used in production, and households value consumer nondurables goods and services, consumer durable goods, and residential capital (for example, housing). Differentiation across these categories is important, as fluctuations in these categories of expenditure can differ notably, with the cycles in housing and business investment, for example, occurring at different points over the last three decades. Valuations of these goods and services, in terms of household utility, is given by the following utility function: 5 of 16 Page 5 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) E0 ∞ X cnn β t ς cnn ln(Etcnn (i)−hEt−1 (i))+ς cd ln(Ktcd (i)) t=0 +ς r ln(Ktr (i)) −ΛLpref ΘH t t X Z s=cbi,kb 1 ς l,s Lst (i) 1+σN σN 1+ 1+σh 0 di , (2) where E cnn represents expenditures on consumption of nondurable goods and services, K cd and K r represent the stocks of consumer durables and residential capital (housing), ΛLpref represents a t labor supply shock, Θt is an endogenous preference shifter whose role is to reconcile the existence of a long-run balance growth path with a small short-term wealth effect2 , Lcbi and Lkb represent the labor supplied to each productive sector (with hours worked causing disutility), and the remaining terms represent parameters (such as the discount factor, relative value in utility of each service flow, and the elasticity of labor supply). Gali, Smets, and Wouters (2011) state that the introduction of the endogenous preference shifter is key in order to match the joint behavior of the labor force, consumption, and wages over the business cycle. By modeling preferences over these disaggregated categories of expenditure, EDO attempts to account for the disparate forces driving consumption of nondurables and durables, residential investment, and business investment —thereby speaking to issues such as the surge in business investment in the second half of the 1990s or the housing cycle in the early 2000s recession and the most recent downturn. Many other models do not distinguish between developments across these categories of spending. 2.3 Risk premiums, financial shocks, and economic fluctuations The structure of the EDO model implies that households value durable stocks according to their expected returns, including any expected service flows, and according to their risk characteristics, with a premium on assets that have high expected returns in adverse states of the world. However, the behavior of models such as EDO is conventionally characterized under the assumption that this second component is negligible. In the absence of risk adjustment, the model would then imply that households adjust their portfolios until expected returns on all assets are equal. Empirically, however, this risk adjustment may not be negligible and, moreover, there may be a variety of factors, not explicitly modeled in EDO, that limit the ability of households to arbitrage away expected return differentials across different assets. To account for this possibility, EDO features several exogenous shocks to the rates of return required by the household to hold the assets in question. Following such a shock —an increase in the premium on a given asset, for example —households will wish to alter their portfolio composition to favor the affected asset, leading to changes in the prices of all assets and, ultimately, to changes in the expected path of production underlying these claims. 2 The cnn , where Z = endogenous preference shifter is defined as ΘH t t = Zt Λt 1−ν Zt−1 Λcnn t and Λcnn is the shadow price of t nondurable consumption. The importance of the short-term wealth effect is determined by the parameter ν ∈ (0, 1]. 6 of 16 Page 6 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) The “sector specific” risk shocks affect the composition of spending more than the path of GDP itself. This occurs because a shock to these premiums leads to sizable substitution across residential, consumer durable, and business investment; for example, an increase in the risk premiums on residential investment leads households to shift away from residential investment and toward other types of productive investment. Consequently, it is intuitive that a large fraction of the noncyclical, or idiosyncratic, component of investment flows to physical stocks will be accounted for by movements in the associated premiums. Shocks to the required rate of return on the nominal risk-free asset play an especially large role in EDO. Following an increase in the premium, in the absence of nominal rigidities, the households’ desire for higher real holdings of the risk-free asset would be satisfied entirely by a fall in prices, that is, the premium is a shock to the natural rate of interest. Given nominal rigidities, however, the desire for higher risk-free savings must be offset, in part, through a fall in real income, a decline which is distributed across all spending components. Because this response is capable of generating co-movement across spending categories, the model naturally exploits such shocks to explain the business cycle. Reflecting this role, we denote this shock as the “aggregate risk-premium.” Movements in financial markets and economic activity in recent years have made clear the role that frictions in financial markets play in economic fluctuations. This role was apparent much earlier, motivating a large body of research (for example, Bernanke, Gertler, and Gilchrist (1999)). While the range of frameworks used to incorporate such frictions has varied across researchers studying different questions, a common theme is that imperfections in financial markets —for example, related to imperfect information on the outlook for investment projects or earnings of borrowers —drives a wedge between the cost of riskless funds and the cost of funds facing households and firms. Much of the literature on financial frictions has worked to develop frameworks in which risk premiums fluctuate for endogenous reasons (for example, because of movements in the net worth of borrowers). Because the risk-premium shocks induces a wedge between the short-term nominal risk-free rate and the rate of return on the affected risky rates, these shocks may thus also be interpreted as a reflection of financial frictions not explicitly modeled in EDO. The sector-specific risk premiums in EDO enter the model in much the same way as does the exogenous component of risk premiums in models with some endogenous mechanism (such as the financial accelerator framework used Boivin, Kiley, and Mishkin (2010)), and the exogenous component is quantitatively the most significant one in that research.3 2.4 Labor market dynamics in the EDO model This version of the EDO model assumes that labor input consists of both employment and hours per worker. Workers differ in the disutility they associate with employment. Moreover, the labor market is characterized by monopolistic competition. As a result, unemployment arises in equilibrium – some workers are willing to be employed at the prevailing wage rate, but cannot find employment because firms are unwilling to hire additional workers at the prevailing wage. 3 Specifically, the risk premiums enter EDO to a first-order (log)linear approximation in the same way as in the cited research if the parameter on net worth in the equation determining the borrowers cost of funds is set to zero; in practice, this parameter is often fairly small in financial accelerator models. 7 of 16 Page 7 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) As emphasized by Gali (2011), this framework for unemployment is simple and implies that the unemployment rate reflects wage pressures: When the unemployment rate is unusually high, the prevailing wage rate exceeds the marginal rate of substitution between leisure and consumption, implying that workers would prefer to work more. The new preference specification and the incorporation of labor force participation in the information set impose discipline in the overall labor market dynamics of the EDO model. The estimated short-run wealth effect on labor supply is relatively attenuated with respect to previous versions of the EDO model. Therefore, the dynamics of both labor force participation and employment are more aligned with the empirical evidence. In addition, in our environment, nominal wage adjustment is sticky, and this slow adjustment of wages implies that the economy can experience sizable swings in unemployment with only slow wage adjustment. Our specific implementation of the wage adjustment process yields a relatively standard New Keynesian wage Phillips curve. The presence of both price and wage rigidities implies that stabilization of inflation is not, in general, the best possible policy objective (although a primary role for price stability in policy objectives remains). While the specific model on the labor market is suitable for discussion of the links between employment and wage/price inflation, it leaves out many features of labor market dynamics. Most notably, it does not consider separations, hires, and vacancies, and is hence not amenable to analysis of issues related to the Beveridge curve. The decline in employment during the Great Recession primarily reflected, according to the EDO model, the weak demand that arose from elevated risk premiums that depressed spending, as illustrated by the light blue and red bars in figure 1. The role played by these demand factors in explaining the cyclical movements in employment is only determinant during the 1980s and during the Great Recession. As apparent in figure 1, the most relevant drivers of employment in the remaining of the sample are labor supply (preference) and markup shocks as shown by the blue bars. Specifically, favorable supply developments in the labor market are estimated to have placed upward pressure on employment until 2010; these developments have reversed, and some of the currently low level for employment growth is, according to EDO, attributable to adverse labor market supply developments. As discussed previously, these developments are simply exogenous within EDO and are not informed by data on a range of labor market developments (such as gross worker flows and vacancies). 2.5 New Keynesian price and wage Phillips curves As in most of the related literature, nominal prices and wages are both “sticky” in EDO. This friction implies that nominal disturbances —that is, changes in monetary policy —have effects on real economic activity. In addition, the presence of both price and wage rigidities implies that stabilization of inflation is not, in general, the best possible policy objective (although a primary role for price stability in policy objectives remains). Given the widespread use of the New Keynesian Phillips curve, it is perhaps easiest to consider the form of the price and wage Phillips curves in EDO at the estimated parameters. The price 8 of 16 Page 8 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) Phillips curve (governing price adjustment in both productive sectors) has the form p,s p,s πtp,s = 0.22πt−1 + 0.76Et πt+1 + .017mcst + θts (3) where mc is marginal cost and θ is a markup shock. As the parameters indicate, inflation is primarily forward looking in EDO. The wage (w) Phillips curve for each sector has the form s s s w 4wts = 0.014wt−1 + 0.95Et 4wt+1 + .012 mrsc,l − w t t + θt + adj. costs. (4) where mrs represents the marginal rate of substitution between consumption and leisure. Wages are primarily forward looking and relatively insensitive to the gap between households’ valuation of time spent working and the wage. The top right panel of figure 1 presents the decomposition of inflation fluctuations into the exogenous disturbances that enter the EDO model. As can be seen, aggregate demand fluctuations, including aggregate risk premiums and monetary policy surprises, contribute little to the fluctuations in inflation according to the model. This is not surprising: In modern DSGE models, transitory demand disturbances do not lead to an unmooring of inflation (so long as monetary policy responds systematically to inflation and remains committed to price stability). In the short run, inflation fluctuations primarily reflect transitory price and wage shocks, or markup shocks in the language of EDO. Technological developments can also exert persistent pressure on costs, most notably during and following the strong productivity performance of the second half of the 1990s, which is estimated to have lowered marginal costs and inflation through the early 2000s. More recently, disappointing labor productivity readings over the course of 2011 have led the model to infer sizable negative technology shocks in both sectors, contributing noticeably to inflationary pressure over that period (as illustrated by the blue bars in figure 1). 2.6 Monetary authority and a long-term interest rate We now turn to the last agent in our model, the monetary authority. It sets monetary policy in accordance with an Taylor-type interest rate feedback rule. Policymakers smoothly adjust the actual interest rate Rt to its target level R̄t ρr Rt = (Rt−1 ) R̄t 1−ρr exp [rt ] , (5) where the parameter ρr reflects the degree of interest rate smoothing, while rt represents a monetary policy shock. The central bank’s target nominal interest rate, R̄t depends on the deviation of output from the level consistent with current technologies and “normal” (steady-state) utilization of capital and labor (X̃ pf , the “production function” output gap). Also, the change in the output gap and consumer price inflation enter the target. The target equation is 9 of 16 Page 9 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) R̄t = X̃t pf r y dX̃t pf rdy Πc rπ t Πc∗ R∗ . (6) pf In equation (6), R∗ denotes the economy’s steady-state nominal interest rate, dX̃t denotes the change in the output gap and ry , rdy and rπ denote the weights in the feedback rule. Consumer price inflation, Πct , is the weighted average of inflation in the nominal prices of the goods produced in each sector, Πp,cbi and Πp,kb : t t Πct = (Πp,cbi )1−wcd (Πp,kb )wcd . t t (7) The parameter wcd is the share of the durable goods in nominal consumption expenditures. The model also includes a long-term interest rate (RLt ), which is governed by the expectations hypothesis subject to an exogenous term premiums shock: RLt = Et ΠN τ =0 Rτ · Υt . (8) where Υ is the exogenous term premium, governed by Ln (Υt ) = 1 − ρΥ Ln (Υ∗ ) + ρΥ Ln (Υt−1 ) + Υ t . (9) In this version of EDO, the long-term interest rate plays no allocative role; nonetheless, the term structure contains information on economic developments useful for forecasting (for example, Edge, Kiley, and Laforte (2010)), and hence RL is included in the model and its estimation. 2.7 Summary of model specification Our brief presentation of the model highlights several points. First, although our model considers production and expenditure decisions in a bit more detail, it shares many similar features with other DSGE models in the literature, such as imperfect competition, nominal price and wage rigidities, and real frictions like adjustment costs and habit-persistence. The rich specification of structural shocks (to aggregate and investment-specific productivity, aggregate and sector-specific risk premiums, and markups) and adjustment costs allows our model to be brought to the data with some chance of finding empirical validation. Within EDO, fluctuations in all economic variables are driven by 13 structural shocks. It is most convenient to summarize these shocks into five broad categories: • Permanent technology shocks: This category consists of shocks to aggregate and investmentspecific (or fast-growing sector) technology. • A labor supply shock: This shock affects the willingness to supply labor. As was apparent in our earlier description of labor market dynamics and in the presentation of the structural drivers below, this shock captures the dynamics of the labor force participation rate in the sample and those of employment. While EDO labels such movements labor supply shocks, an alternative 10 of 16 Page 10 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) interpretation would describe these as movements in the labor force and employment that reflect structural features not otherwise captured by the model. • Financial, or intertemporal, shocks: This category consists of shocks to risk premiums. In EDO, variation in risk premiums —both the premium households receive relative to the federal funds rate on nominal bond holdings and the additional variation in discount rates applied to the investment decisions of capital intermediaries —are purely exogenous. Nonetheless, the specification captures aspects of related models with more explicit financial sectors (for example, Bernanke, Gertler, and Gilchrist (1999)), as we discuss in our presentation of the model’s properties below. • Markup shocks: This category includes the price and wage markup shocks. • Other demand shocks: This category includes the shock to autonomous demand and a monetary policy shock. 3 Estimation: Data and Properties 3.1 Data The empirical implementation of the model takes a log-linear approximation to the first-order conditions and constraints that describe the economy’s equilibrium, casts this resulting system in its state-space representation for the set of (in our case, 13) observable variables, uses the Kalman filter to evaluate the likelihood of the observed variables, and forms the posterior distribution of the parameters of interest by combining the likelihood function with a joint density characterizing some prior beliefs. Since we do not have a closed-form solution of the posterior, we rely on Markov-Chain Monte Carlo (MCMC) methods. The model is estimated using 13 data series over the sample period from 1984:Q4 to 2015:Q3. The series are the following: 1. The growth rate of real gross domestic product (∆GDP ); 2. The growth rate of real consumption expenditure on nondurables and services (∆C); 3. The growth rate of real consumption expenditure on durables (∆CD); 4. The growth rate of real residential investment expenditure (∆Res); 5. The growth rate of real business investment expenditure (∆I); 6. Consumer price inflation, as measured by the growth rate of the Personal Consumption Expenditure (PCE) price index (∆PC,total ); 7. Consumer price inflation, as measured by the growth rate of the PCE price index excluding food and energy prices (∆PC,core ); 8. Inflation for consumer durable goods, as measured by the growth rate of the PCE price index for durable goods (∆Pcd ); 9. Hours, which equals hours of all persons in the nonfarm business sector from the Bureau of Labor Statistics (H); 11 of 16 Page 11 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) 10. Civilian employment-population ratio, defined as civilian employment from the Current Population Survey (household survey) divided by the noninstitutional population, age 16 and over (N ); 11. Labor force participation rate; 12. The growth rate of real wages, as given by compensation per hour in the non-farm business sector from the Bureau of Labor Statistics divided by the GDP price index (∆RW ); and 13. The federal funds rate (R). Our implementation adds measurement error processes to the likelihood implied by the model for all of the observed series used in estimation except the short-term nominal interest rate series. 3.2 Estimates of latent variable paths Figures 4, 5, and 6 report estimates of the model’s persistent exogenous fundamentals (for example, risk premiums and autonomous demand). These series have recognizable patterns for those familiar with U.S. economic fluctuations. For example, the risk premiums jump at the end of 2008, reflecting the financial crisis and the model’s identification of risk premiums, both economy-wide and for housing, as key drivers. Of course, these stories from a glance at the exogenous drivers, yield applications for alternative versions of the EDO model and future model enhancements. For example, the exogenous risk premiums can easily be made to have an endogenous component, following the approach of Bernanke, Gertler, and Gilchrist (1999) (and, indeed, we have considered models of that type). At this point, we view incorporation of such mechanisms in our baseline approach as premature, pending ongoing research on financial frictions, banking, and intermediation in dynamic general equilibrium models. Nonetheless, the EDO model captured the key financial disturbances during the last several years in its current specification, and examining the endogenous factors that explain these developments will be a topic of further study. 12 of 16 Page 12 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) Figure 4: Model Estimates of Risk Premiums Aggregate Risk Capital Risk 2.5 6 2 4 1.5 2 1 0 0.5 -2 0 -4 -0.5 -6 -1 1990 2000 2010 2020 1990 Housing Risk 2000 2010 2020 Durables Risk 6 40 4 20 2 0 0 -20 -2 -40 -4 -60 -6 1990 2000 2010 2020 1990 2000 2010 2020 Black line: modal parameters. Red line: posterior median. Dark blue intervals: 68 percent credible set. Light blue intervals: 95 percent credible set. 13 of 16 Page 13 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) Figure 5: Model Estimates of Key Supply-side Variables Aggregate Tech Capital Tech 3 2 2 1 1 0 0 -1 -1 -2 -2 1990 2000 2010 2020 2010 2020 1990 2000 2010 2020 Labor Pref 60 40 20 0 -20 1990 2000 Black line: modal parameters. Red line: posterior median. Dark blue intervals: 68 percent credible set. Light blue intervals: 95 percent credible set. 14 of 16 Page 14 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) Figure 6: Model Estimates of Selected Other Exogenous Drivers Wage Markup Exog Spending 50 10 5 0 0 -5 -10 -50 1990 2000 2010 2020 1990 2000 2010 2020 Black line: modal parameters. Red line: posterior median. Dark blue intervals: 68 percent credible set. Light blue intervals: 95 percent credible set. 15 of 16 Page 15 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Class II FOMC – Restricted (FR) References [Bernanke, Gertler, and Gilchrist (1999)] Bernanke, B., M. Gertler, and S. Gilchrist. 1999. The financial accelerator in a quantitative business cycle framework, in: John B. Taylor and Michael Woodford, Editor(s), Handbook of Macroeconomics, Elsevier, 1999, volume 1, part 3, pages 13411393. [Boivin, Kiley, and Mishkin (2010)] Boivin, J., M. Kiley, and F.S. Mishkin. 2010. How Has the Monetary Transmission Mechanism Evolved Over Time? In B. Friedman and M. Woodford, eds., The Handbook of Monetary Economics, Elsevier. [Chung, Kiley, and Laforte (2010)] Chung, H., M. Kiley, and J.P. Laforte. 2010. Documentation of the Estimated, Dynamic, Optimization-based (EDO) model of the U.S. economy: 2010 version. Finance and Economics Discussion Series, 2010-29. Board of Governors of the Federal Reserve System (U.S.). [Edge, Kiley, and Laforte (2008)] Edge, R., M. Kiley, and J.P. Laforte. 2008. Natural rate measures in an estimated DSGE model of the U.S. economy. Journal of Economic Dynamics and Control, vol. 32(8), pages 2512-2535. [Edge, Kiley, and Laforte (2010)] Edge, R., M. Kiley, and J.P. Laforte. 2010. A comparison of forecast performance between Federal Reserve staff forecasts, simple reduced-form models, and a DSGE model. Journal of Applied Econometrics vol. 25(4), pages 720-754. [Fisher (2006)] Fisher, Jonas D. M. 2006. The Dynamic Effects of Neutral and Investment-Specific Technology Shocks. Journal of Political Economy, University of Chicago Press, vol. 114(3), pages 413-451. [Gali (2011)] Gali, J. 2011. The Return Of The Wage Phillips Curve. Journal of the European Economic Association, vol. 9(3), pages 436-461. [Gali, Smets, and Wouters (2011)] Gali, J., F. Smets, and R. Wouters. 2011. Unemployment in an Estimated New Keynesian Model. NBER Macroeconomics Annual vol. 26(1), pages 329-360. [Smets and Wouters (2007)] Smets, F., and R. Wouters. 2007. Shocks and Frictions in the US Busines Cycles: A Bayesian DSGE Approach. American Economic Review, American Economic Association, vol. 97(3), pages 586-606. 16 of 16 Page 16 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Introduction This document describes the New York Fed DSGE model, which we use both for internal forecasting and for creating our contributions to the System DSGE memo distributed quarterly to the FOMC. The document is structured as follows. First, we provide a description and interpretation of the forecast for the current forecast horizon. Next, we describe the structure of the DSGE model followed by the impulse response functions to various shocks. Model Forecast The New York Fed model forecasts are obtained using data released through 2018Q1, augmented for 2018Q2 with the New York Fed staff forecasts (as of May 31) for real GDP growth and core PCE inflation, and with values of the federal funds rate, the 10-year Treasury yield and the spread between Baa corporate bonds and 10-year Treasury yields based on 2018Q2 averages up to May 31. Table 1 shows both the conditional and unconditional forecasts of real GDP growth, core PCE inflation, federal funds rate, real natural rate of interest and the output gap. Unconditional forecasts are obtained using data up to the quarter for which we have the most recent GDP release, as well as the federal funds rate, 10-year Treasury yield, and spreads data for the following (“current”) quarter. Conditional forecasts further include the current-quarter New York Fed staff projections for GDP growth and core PCE inflation as additional data points. Figure 1 plots the conditional and unconditional forecasts of real GDP growth, core PCE inflation and the federal funds rate. Figure 2 provides a comparison of current and previous quarterly forecasts while Figure 3 depicts the shock decomposition of the conditional forecasts, where different colored bars indicate the contribution of different shocks to the conditional forecast of GDP growth, inflation and federal funds rate. Finally, Figure 4 plots the historical estimates and forecast of the output gap in the top panel, and the real natural rate in the bottom panel. New York Fed DSGE Team, Research and Statistics Page 17 of 112 1 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 The output gap is defined as the difference between actual output and potential output. Potential output is defined as the level that output would take in a world where capital and labor are fully utilized, i.e., where there are no nominal rigidities or shocks to markups.1 A positive (negative) output gap indicates that output is above (below) its potential. The natural rate of interest is a concept analogous to potential output: it represents the rate of interest that would prevail in the economy absent nominal rigidities and markup shocks. Current Forecast We project real GDP growth of 2.3 percent in 2018 on a Q4/Q4 basis, in line with the March forecast. GDP growth is anticipated to decline to 1.9 percent in 2019 and 2020, slightly below the March forecast of 2.0 percent and 2.1 percent in 2019 and 2020 respectively. Inflation is forecast to be slightly higher in the short term, at 1.9 percent relative to 1.8 percent in the March projection. While inflation in 2018 is close to the FOMC’s longer run goal of 2 percent, the model projects that inflation will decline to 1.5 percent in 2019 and 2020 (unchanged from the March projection). The output gap is projected to be -0.4 percent in 2018, -0.3 percent in 2019 and -0.3 percent in 2020. This represents an improvement relative to the March forecast of the output gap (-0.7 percent, -0.6 percent and -0.4 percent in 2018, 2019 and 2020 respectively), especially in the near term. The forecast for the natural rate of interest is unchanged from March, rising from 1.0 percent in 2018 to 1.4 percent by the end of 2020. The federal funds rate is forecast to be 0.1 percentage points lower than the March projections throughout the forecast horizon: 2.2, 2.6 and 2.8 percent in 2018, 2019 and 2020 respectively. This shallower path translates into approximately two more hikes in 2018, one more in 2019 and another in 2020. The projections for all the variables are surrounded by significant uncertainty. For instance, the 68 percent posterior probability interval for GDP growth includes negative readings for the years between 2019 and 2020. In comparison, the posterior probability intervals for inflation are tighter, with their upper bound never exceeding 3 percent throughout the forecast horizon. As in March, the model explains the above long-run average real GDP growth rate in 2018 with continued improvement in financial conditions, as captured by positive contributions of 1 Markup shocks represent exogenous fluctuations in price and wage inflation arising from various sources, such as variations in the degree of market power, or in the price of commodities. New York Fed DSGE Team, Research and Statistics Page 18 of 112 2 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 both the financial and marginal efficiency of investment shocks. Over the medium term however, these factors are offset by lower TFP growth and the gradual withdrawal of monetary accommodation (see the top panel of Figure 3). Sluggish TFP growth also explains, at least in part, the negative output gap over the forecast horizon. The model projects near-target inflation for 2018 (see middle panel of Figure 3), driven by a temporary increase in price markups. However, as in March, the model projects below-target inflation in 2019 and 2020, driven by a confluence of several factors. These factors include the lingering effects of the financial headwinds that have hampered the recovery, as well as negative shocks to wage and price markups. Finally, the federal funds rate is projected to remain below its long-run level of 4 percent throughout the forecast horizon owing to persistence in the interest rate rule, weak projected inflation and a persistently negative output gap (see lower panel of Figure 3). New York Fed DSGE Team, Research and Statistics Page 19 of 112 3 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Table 1: Forecasts 2018 Real GDP Growth (Q4/Q4) Core PCE Inflation (Q4/Q4) Federal Funds Rate (Q4) Real Natural Rate (Q4) Output Gap (Q4) Jun. 1.9 (−0.2,3.8) 1.8 (1.2,2.3) 2.1 (0.7,3.6) 1.0 (−0.6,2.6) −0.6 (−2.5,1.1) Mar. 2.6 (0.1,5.0) 1.4 (0.7,2.1) 2.2 (0.8,3.6) 1.0 (−0.6,2.7) −0.5 (−2.5,1.3) Unconditional Forecast 2019 Jun. Mar. 1.8 2.1 (−1.1,4.4) (−0.7,4.7) 1.4 1.4 (0.5,2.4) (0.4,2.3) 2.6 2.6 (1.0,4.4) (1.0,4.4) 1.3 1.3 (−0.5,3.0) (−0.5,3.1) −0.6 −0.4 (−3.5,1.9) (−3.2,2.3) 2020 Jun. Mar. 1.9 2.1 (−0.9,4.6) (−0.7,4.7) 1.5 1.5 (0.4,2.6) (0.3,2.6) 2.8 2.8 (1.0,4.7) (1.0,4.8) 1.3 1.4 (−0.5,3.2) (−0.5,3.3) −0.6 −0.2 (−4.0,2.5) (−3.7,2.9) Conditional Forecast 2019 2020 Mar. Jun. Mar. Jun. Mar. 2.3 1.9 2.0 1.9 2.1 (0.2,4.2) (−1.0,4.5) (−0.8,4.7) (−0.9,4.7) (−0.8,4.8) 1.8 1.5 1.5 1.5 1.5 (1.3,2.4) (0.5,2.4) (0.5,2.5) (0.4,2.6) (0.4,2.7) 2.3 2.6 2.7 2.8 2.9 (0.9,3.7) (1.0,4.4) (1.0,4.4) (1.1,4.7) (1.1,4.8) 1.0 1.3 1.3 1.4 1.4 (−0.7,2.6) (−0.5,3.1) (−0.6,3.1) (−0.5,3.2) (−0.6,3.3) −0.7 −0.3 −0.6 −0.3 −0.4 (−2.6,1.0) (−3.1,2.1) (−3.3,1.9) (−3.6,2.6) (−3.8,2.6) 2018 Real GDP Growth (Q4/Q4) Core PCE Inflation (Q4/Q4) Federal Funds Rate (Q4) Real Natural Rate (Q4) Output Gap (Q4) Jun. 2.3 (0.8,3.8) 1.9 (1.5,2.3) 2.2 (0.7,3.7) 1.0 (−0.6,2.6) −0.4 (−2.1,1.2) The unconditional forecasts use data up to the quarter for which we have the most recent GDP release, as well as the federal funds rate, 10-year Treasury yield, and spreads data for the following (“current”) quarter. In the conditional forecasts, we further include the current-quarter New York Fed staff projections for GDP growth and core PCE inflation as additional data points. Numbers in parentheses indicate 68 percent probability intervals. New York Fed DSGE Team, Research and Statistics Page 20 of 112 4 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Figure 1: Forecasts Unconditional Conditional Real GDP Growth 7.5 Percent Q/Q Annualized Percent Q/Q Annualized Real GDP Growth 5.0 2.5 0.0 -2.5 2012 2014 2016 2018 7.5 5.0 2.5 0.0 -2.5 2020 2012 2014 3 2 1 0 2012 2014 2016 2018 2020 2 1 0 2012 2014 2016 2018 2020 Nominal FFR 6 6 5 5 Percent Annualized Percent Annualized 2020 3 Nominal FFR 4 3 2 4 3 2 1 1 2012 2018 Core PCE Inflation Percent Q/Q Annualized Percent Q/Q Annualized Core PCE Inflation 2016 2014 2016 2018 2020 2012 2014 2016 2018 2020 Quarterly forecasts, both unconditional (left panels) and conditional (right panels). The black line represents data, the red line indicates the mean forecast, and the shaded areas mark the 50, 60, 70, 80 and 90 percent probability intervals for the forecasts, reflecting both parameter and shock uncertainty. New York Fed DSGE Team, Research and Statistics Page 21 of 112 5 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Figure 2: Change in Forecasts Unconditional Conditional Real GDP Growth Percent Q/Q Annualized Percent Q/Q Annualized Real GDP Growth 7.5 5.0 2.5 0.0 -2.5 2012 2014 2016 2018 7.5 5.0 2.5 0.0 -2.5 2020 2012 2014 3 2 1 0 2012 2014 2016 2018 2020 2 1 0 2012 2014 2016 2018 2020 Nominal FFR 6 6 5 5 Percent Annualized Percent Annualized 2020 3 Nominal FFR 4 3 2 1 2012 2018 Core PCE Inflation Percent Q/Q Annualized Percent Q/Q Annualized Core PCE Inflation 2016 4 3 2 1 2014 2016 2018 2020 2012 2014 2016 2018 2020 Comparison of current and previous quarterly forecasts. Solid (dashed) red and blue lines represent the mean and the 90 percent probability intervals, respectively, of the current (previous) forecast. New York Fed DSGE Team, Research and Statistics Page 22 of 112 6 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Figure 3: Shock Decomposition Real GDP Growth Percent Q/Q Annualized (deviations from mean) 3 2 1 0 -1 -2 -3 -4 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2018 2019 2020 2021 2018 2019 2020 2021 Core PCE Inflation Percent Q/Q Annualized (deviations from mean) 0.5 0.0 -0.5 -1.0 -1.5 2012 2013 2014 2015 2016 2017 Nominal FFR Percent Annualized (deviations from mean) 1 0 -1 -2 -3 -4 -5 2012 Gov't Financial TFP Mark-Up Policy MEI 2013 2014 2015 2016 2017 Shock decomposition of the conditional forecast. The solid lines (black for realized data, red for mean forecast) show each variable in deviation from its steady state. The bars represent the shock contributions; specifically, the bars for each shock represent the counterfactual values for the observables (in deviations from the mean) obtained by setting all other shocks to zero. New York Fed DSGE Team, Research and Statistics Page 23 of 112 7 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Figure 4: Output Gap and Natural Interest Rate Output Gap 2.5 0.0 -2.5 -5.0 2012 2014 2016 2018 2020 Natural Rate & Ex-Ante Real Rate 4 Percent Annualized 2 0 -2 Ex-Ante Real Rate Real Natural Rate -4 2012 2014 2016 2018 2020 Historical estimates and forecasts of the output gap (upper panel) and the real natural rate of interest and the ex-ante real interest rate (lower panel). In the upper panel, the black line represents the mean historical estimate, the red line the mean forecast. In the lower panel, the solid lines represent historical estimates and the dashed lines represent forecasts of the natural rate (red) and ex-ante rate (black). In both panels, the shaded areas mark the 50, 60, 70, 80, and 90 percent probability intervals for the historical estimates and forecasts, reflecting both parameter and shock uncertainty. New York Fed DSGE Team, Research and Statistics Page 24 of 112 8 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 The Model The following section contains a description of the New York Fed DSGE model and plots of impulse response functions. General structure The New York Fed DSGE model is a medium scale, one-sector dynamic stochastic general equilibrium model which is based on the New Keynesian model with financial frictions used in Del Negro et al. (2015). The core of the model is based on the work of Smets and Wouters (2007) (henceforth SW) and Christiano et al. (2005): It builds on the neo-classical growth model by adding nominal wage and price rigidities, variable capital utilization, costs of adjusting investment, habit formation in consumption. The model also includes credit frictions as in the financial accelerator model developed by Bernanke et al. (1999b) where the actual implementation of credit frictions follows closely Christiano et al. (2014), and accounts for forward guidance in monetary policy by including anticipated policy shocks as in Laseen and Svensson (2011). The current version of the model has several features that improve upon the version presented in the New York Fed Staff Report no. 647. It features both a deterministic and a stochastic trend in productivity and allows for exogenous movements in risk premia; the inflation target is time-varying, following Del Negro and Schorfheide (2012); households preferences are non-separable in consumption and leisure; the Dixit-Stiglitz aggregator of intermediate goods has been replaced by the more flexible Kimball aggregator; we include indexation in the price and wage adjustment processes. Here is a brief overview. The model economy is populated by eight classes of agents: 1) a continuum of households, who consume and supply differentiated labor; 2) competitive labor aggregators that combine labor supplied by individual households; 3) competitive final goodproducing firms that aggregate the intermediate goods into a final product; 4) a continuum of monopolistically competitive intermediate good producing firms; 5) competitive capital producers that convert final goods into capital; 6) a continuum of entrepreneurs who purchase capital using both internal and borrowed funds and rent it to intermediate good producing firms; 7) a representative bank collecting deposits from the households and lending funds to the entrepreneurs; and finally 8) a government, composed of a monetary authority that sets short-term interest rates and a fiscal authority that sets public spending and collects taxes. New York Fed DSGE Team, Research and Statistics Page 25 of 112 9 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Growth in the economy is driven by technological progress. We specify a process for technology Zt∗ which includes both a deterministic and a stochastic trend, and a stationary component: 1 Zt∗ = e 1−α z̃t Ztp eγt , (1) where γ is the steady state growth rate of the economy, Ztp is a stochastic trend and z̃t is the stationary component. The production function is Yt (i) = max{ez̃t Kt (i)α Lt (i)eγt Ztp 1−α − ΦZt∗ , 0}, (2) where ΦZt∗ is a fixed cost. Trending variables are divided by Zt∗ to express the model’s equilibrium conditions in terms of the stationary variables. In what follows we present a summary of the log-linearized equilibrium conditions, where all variables are expressed in log deviations from their nonstochastic steady state. Log-linear equilibrium conditions The stationary component of productivity z̃t evolves as: z̃t = ρz z̃t−1 + σz εz,t . (3) p Since Ztp is a non stationary process, we define its growth rate as ztp = log(Ztp /Zt−1 ) and assume that it follows an AR(1) process: p ztp = ρzp zt−1 + σzp zp ,t , zp ,t ∼ N (0, 1). (4) It follows that ∗ zt ≡ log(Zt∗ /Zt−1 )−γ = 1 1 (ρz − 1)z̃t−1 + σz z,t + ztp , 1−α 1−α (5) where γ is the steady-state growth rate of the economy. Steady-state values are denoted by ∗-subscripts, and steady-state formulas are provided in the technical appendix of Del Negro and Schorfheide (2012), which is available online. New York Fed DSGE Team, Research and Statistics 10 Page 26 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 The optimal allocation of consumption satisfies the following consumption Euler equation: ct = − he−γ (1 − he−γ ) (R − I E [π ] + b ) + (ct−1 − zt ) t t t+1 t σc (1 + he−γ ) (1 + he−γ ) 1 (σc − 1) w∗ L∗ + I E [c + z ] + (Lt − IE t [Lt+1 ]) , (6) t t+1 t+1 (1 + he−γ ) σc (1 + he−γ ) c∗ where ct is consumption, Lt is labor supply, Rt is the nominal interest rate, and πt is inflation. The exogenous process bt drives a wedge between the intertemporal marginal utility of consumption and the riskless real return Rt −IE t [πt+1 ], and is meant to capture risk-premium shocks.2 This shock follows an AR(1) process with parameters ρb and σb . The parameters σc and h capture the degree of relative risk aversion and the degree of habit persistence in the utility function, respectively. The optimal investment decision satisfies the following relationship between the level of investment it , measured in terms of consumption goods, and the value of capital in terms of consumption qtk : it = 1 β̄ qtk + (i − z ) + IE t [it+1 + zt+1 ] + µt . t−1 t S 00 e2γ (1 + β̄) 1 + β̄ 1 + β̄ (7) This relationship shows that investment is affected by investment adjustment costs (S 00 is the second derivative of the adjustment cost function) and by an exogenous process µt , which we call “marginal efficiency of investment”, that alters the rate of transformation between consumption and installed capital (see Greenwood et al. (1998)). The shock µt follows an AR(1) process with parameters ρµ and σµ . The parameter β̄ depends on the intertemporal discount rate in the household utility function, β, on the degree of relative risk aversion σc , and on the steady-state growth rate γ: β̄ = βe(1−σc )γ . The capital stock, k̄t , which we refer to as “installed capital”, evolves as k̄t = i∗ 1− k̄∗ i∗ i∗ 00 k̄t−1 − zt + it + S e2γ (1 + β̄)µt , k̄∗ k̄∗ (8) where i∗ /k̄∗ is the steady state investment to capital ratio. Capital is subject to variable capacity utilization ut ; effective capital rented out to firms, 2 In the code, the bt shock is normalized to be in the same units as consumption, i.e., we estimate the −γ ) shock b̃t = − σ(1−he −γ ) bt . c (1+he New York Fed DSGE Team, Research and Statistics Page 27 of 112 11 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 kt , is related to k̄t by: kt = ut − zt + k̄t−1 . (9) The optimality condition determining the rate of capital utilization is given by 1−ψ k r = ut , ψ t (10) where rtk is the rental rate of capital and ψ captures the utilization costs in terms of foregone consumption. Real marginal costs for firms are given by mct = wt + αLt − αkt , (11) where wt is the real wage and α is the income share of capital (after paying mark-ups and fixed costs) in the production function. From the optimality conditions of goods producers it follows that all firms have the same capital-labor ratio: kt = wt − rtk + Lt . (12) We include financial frictions in the model, building on the work of Bernanke et al. (1999a), Christiano et al. (2003), De Graeve (2008), and Christiano et al. (2014). We assume that banks collect deposits from households and lend to entrepreneurs who use these funds as well as their own wealth to acquire physical capital, which is rented to intermediate goods producers. Entrepreneurs are subject to idiosyncratic disturbances that affect their ability to manage capital. Their revenue may thus turn out to be too low to pay back the loans received by the banks. The banks therefore protect themselves against default risk by pooling all loans and charging a spread over the deposit rate. This spread may vary as a function of entrepreneurs’ leverage and riskiness. The realized return on capital is given by: R̃tk − πt = (1 − δ) r∗k k rtk + k qtk − qt−1 , k r∗ + (1 − δ) r∗ + (1 − δ) (13) where R̃tk is the gross nominal return on capital for entrepreneurs, r∗k is the steady state value of the rental rate of capital rtk , and δ is the depreciation rate. The excess return on capital (the spread between the expected return on capital and the New York Fed DSGE Team, Research and Statistics Page 28 of 112 12 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 riskless rate) can be expressed as a function of the entrepreneurs’ leverage (i.e. the ratio of the value of capital to nominal net worth) and exogenous fluctuations in the volatility of entrepreneurs’ idiosyncratic productivity: Et h k R̃t+1 i − Rt = bt + ζsp,b qtk + k̄t − nt + σ̃ω,t , (14) where nt is entrepreneurs’ net worth, ζsp,b is the elasticity of the credit spread to the entrepreneurs’ leverage (qtk + k̄t − nt ), and σ̃ω,t captures mean-preserving changes in the crosssectional dispersion of ability across entrepreneurs (see Christiano et al. (2014)). σ̃ω,t follows an AR(1) process with parameters ρσω and σσω . Entrepreneurs’ net worth nt evolves according to: k + k̄t−1 + ζn,n nt−1 nt = ζn,R̃k R̃tk − πt − ζn,R (Rt−1 − πt + bt−1 ) + ζn,qK qt−1 −γ∗ nv∗∗ zt − ζn,σω ζsp,σω (15) σ̃ω,t−1 , where the ζ’s denote elasticities, that depend among others on the entrepreneurs’ steadystate default probability F (ω̄), where γ∗ is the fraction of entrepreneurs that survive and continue operating for another period, and where v∗ is the entrepreneurs’ real equity divided by Zt∗ , in steady state. The production function is yt = Φp (αkt + (1 − α) Lt ) , where Φp = y∗ +Φ , y∗ (16) and the resource constraint is: yt = g∗ gt + c∗ i∗ rk k∗ c t + it + ∗ u t . y∗ y∗ y∗ (17) ∗ where gt = log( Z ∗Gy∗t g∗ ) and g∗ = 1 − c∗y+i . ∗ t Government spending gt is assumed to follow the exogenous process: gt = ρg gt−1 + σg εg,t + ηgz σz εz,t . New York Fed DSGE Team, Research and Statistics Page 29 of 112 13 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 The price and wage Phillips curves are, respectively: πt = κ mct + ιp β̄ πt−1 + IE t [πt+1 ] + λf,t , 1 + ιp β̄ 1 + ιp β̄ (18) and wt = 1 + ιw β̄ (1 − ζw β̄)(1 − ζw ) 1 wth − wt − πt + (wt−1 − zt + ιw πt−1 ) (1 + β̄)ζw ((λw − 1)w + 1) 1 + β̄ 1 + β̄ β̄ IE t [wt+1 + zt+1 + πt+1 ] + λw,t , (19) + 1 + β̄ p β̄)(1−ζp ) , the parameters ζp , ιp , and p are the Calvo parameter, the where κ = (1+ιp(1−ζ β̄)ζp ((Φp −1)p +1) degree of indexation, and the curvature parameter in the Kimball aggregator for prices, and ζw , ιw , and w are the corresponding parameters for wages. wth measures the household’s marginal rate of substitution between consumption and labor, and is given by: wth = 1 ct − he−γ ct−1 + he−γ zt + νl Lt , −γ 1 − he (20) where νl characterizes the curvature of the disutility of labor (and would equal the inverse of the Frisch elasticity in the absence of wage rigidities). The mark-ups λf,t and λw,t follow exogenous ARMA(1,1) processes: λf,t = ρλf λf,t−1 + σλf ελf ,t − ηλf σλf ελf ,t−1 , and λw,t = ρλw λw,t−1 + σλw ελw ,t − ηλw σλw ελw ,t−1 , respectively. Finally, the monetary authority follows a generalized policy feedback rule: πt∗ ) Rt = ρR Rt−1 + (1 − ρR ) ψ1 (πt − + ψ2 (yt − f +ψ3 (yt − ytf ) − (yt−1 − yt−1 ) + rtm . ytf ) (21) where ytf is the flexible price/wage output, obtained from solving the version of the model without nominal rigidities and markup shocks (that is, Equations (6) through (20) with New York Fed DSGE Team, Research and Statistics Page 30 of 112 14 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 ζp = ζw = 0, and λf,t = λw,t = 0), and the residual rtm follows an AR(1) process with parameters ρrm and σrm . In this version of the model we have replaced a constant inflation target with a timevarying inflation target πt∗ , to capture the rise and fall of inflation and interest rates in the estimation sample. Although time-varying target rates have been frequently used for the specification of monetary policy rules in DSGE model (e.g., Erceg and Levin (2003) and Smets and Wouters (2003), among others), we follow the approach of Aruoba and Schorfheide (2008) and Del Negro and Eusepi (2011) and include data on long-run inflation expectations as an observable for the estimation of the model. At each point in time, long-run inflation expectations essentially determine the level of the target inflation rate. To the extent that long-run inflation expectations at the forecast origin contain information about the central bank’s objective function, e.g. the desire to stabilize inflation at 2%, this information is automatically included in the forecast. The time-varying inflation target evolves according to: ∗ πt∗ = ρπ∗ πt−1 + σπ∗ π∗ ,t , (22) where 0 < ρπ∗ < 1 and π∗ ,t is an iid shock. We model πt∗ as a stationary process, although our prior for ρπ∗ will force this process to be highly persistent. The assumption that the changes in the target inflation rate are exogenous is, to some extent, a short-cut. For instance, the learning models of Sargent (1999) or Primiceri (2006) imply that the rise in the target inflation rate in the 1970’s and the subsequent drop is due to policy makers learning about the output-inflation trade-off and trying to set inflation optimally. We are abstracting from such a mechanism in our specification. Anticipated policy shocks This section describes the introduction of anticipated policy shocks in the model, which follows Laseen and Svensson (2011). We modify the exogenous component of the policy rule (21) as follows: K X m m R rt = ρrm rt−1 + t + R (23) k,t−k , k=1 R where R t is the usual contemporaneous policy shock, and k,t−k is a policy shock that is known to agents at time t − k, but affects the policy rule k periods later, that is, at time t. New York Fed DSGE Team, Research and Statistics Page 31 of 112 15 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 2 We assume that R k,t−k ∼ N (0, σk,r ), i.i.d. In order to solve the model we need to express the anticipated shocks in recursive form. For this purpose, we augment the state vector st (described below) with K additional states R whose law of motion is as follows: νtR ,. . . ,νt−K R ν1,t R ν2,t R = ν2,t−1 + R 1,t R = ν3,t−1 + R 2,t .. . R = R νK,t K,t and rewrite the exogenous component of the policy rule (23) as3 m R rtm = ρrm rt−1 + R t + ν1,t−1 . Parameters The following tables describe the parameters used in the New York Fed DSGE model. Table 2 gives the prior distributions for each parameter. Table 3 gives the posterior mean, 5th percentile, and 95th percentile for each parameter. PK R R It is easy to verify that ν1,t−1 = k=1 R k,t−k , that is, ν1,t−1 is a “bin” that collects all anticipated shocks that affect the policy rule in period t. 3 New York Fed DSGE Team, Research and Statistics Page 32 of 112 16 Authorized for public release by the FOMC Secretariat on 1/12/2024 Table 2: Priors Dist Policy Parameters ψ1 ψ2 ψ3 ρR Normal Normal Normal Beta Mean Std Dev Dist Mean Std Dev 1.50 0.12 0.12 0.75 0.25 0.05 0.05 0.10 ρr m σr m σant1 Beta InvG InvG 0.50 0.10 0.20 0.20 2.00 4.00 0.10 0.15 ζw ιw w Beta Beta - 0.50 0.50 10.00 0.10 0.15 S 00 ψ π∗ Normal Beta Normal Normal Normal - 4.00 0.50 0.50 1.00 0.00 -45.00 1.50 0.18 1.50 0.15 Nominal Rigidities Parameters ζp ιp p Beta Beta - 0.50 0.50 10.00 fixed fixed Other Endogenous Propagation and Steady State Parameters 100γ α 100(β −1 − 1) σc h νl δ Φp Normal Normal Gamma Normal Beta Normal Normal 0.40 0.30 0.25 1.50 0.70 2.00 0.03 1.25 0.10 0.05 0.10 0.37 0.10 0.75 fixed 0.12 γgdpdef δgdpdef L̄ λw g∗ fixed 2.00 2.00 5.00 fixed fixed Financial Frictions Parameters F (ω̄) SP∗ Gamma 0.03 2.00 fixed 0.10 ζsp,b γ∗ Beta - 0.05 0.99 0.00 σg σb σµ σz σσω σπ∗ σz p σλf σ λw ηgz InvG InvG InvG InvG InvG InvG InvG InvG InvG Beta 0.10 0.10 0.10 0.10 0.05 0.03 0.10 0.10 0.10 0.50 2.00 2.00 2.00 2.00 4.00 6.00 2.00 2.00 2.00 0.20 fixed Exogenous Process Parameters ρg ρb ρµ ρz ρσ ω ρπ ∗ ρz p ρλ f ρλ w ηλf η λw Beta Beta Beta Beta Beta Beta Beta Beta Beta Beta 0.50 0.50 0.50 0.50 0.75 0.99 0.50 0.50 0.50 0.50 0.50 0.20 0.20 0.20 0.20 0.15 fixed 0.20 0.20 0.20 0.20 0.20 Measurement Error Parameters Note: For Inverse Gamma prior mean and SD, τ and ν reported. σant1 through σant12 all have the same distribution. Page 33 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft Cme ρgdp ρgdi ρ10y ρtf p ρgdpdef ρpce Dist Normal Normal Beta Beta Beta Beta Mean 1.00 0.00 0.00 0.50 0.50 0.50 0.50 May 31, 2018 Std Dev fixed 0.20 0.20 0.20 0.20 0.20 0.20 %gdp σgdp σgdi σ10y σtf p σgdpdef σpce Dist Normal InvG InvG InvG InvG InvG InvG Mean 0.00 0.10 0.10 0.75 0.10 0.10 0.10 Std Dev 0.40 2.00 2.00 2.00 2.00 2.00 2.00 Note: For Inverse Gamma prior mean and SD, τ and ν reported. New York Fed DSGE Team, Research and Statistics Page 34 of 112 18 Authorized for public release by the FOMC Secretariat on 1/12/2024 Table 3: Posteriors Mean Policy Parameters ψ1 ψ2 ψ3 ρR ρr m σr m 1.45 0.05 0.25 0.66 0.31 0.23 (p5, p95) (1.21, (0.03, (0.20, (0.59, (0.20, (0.21, Mean 1.68) 0.08) 0.31) 0.73) 0.42) 0.25) (p5, p95) σant1 σant2 σant3 σant4 σant5 σant6 0.09 0.09 0.09 0.08 0.09 0.11 (0.07, (0.07, (0.07, (0.06, (0.07, (0.08, 0.11) 0.10) 0.10) 0.10) 0.11) 0.13) ζw ιw w 0.92 0.55 10.00 (0.90, 0.94) (0.34, 0.76) (2.11, 4.12) (0.42, 0.69) γgdpdef δgdpdef L̄ λw g∗ 3.15 0.56 0.50 1.03 0.00 -48.03 1.50 0.18 ζsp,b γ∗ 0.05 0.99 (0.05, 0.06) σg σb σµ σz σσω σπ∗ σz p σλf σ λw ηgz 2.26 0.03 0.44 0.57 0.04 0.03 0.15 0.08 0.40 0.38 (2.07, (0.02, (0.38, (0.52, (0.03, (0.02, (0.10, (0.06, (0.35, (0.10, %gdp -0.15 Nominal Rigidities Parameters ζp ιp p 0.93 0.22 10.00 (0.91, 0.95) (0.10, 0.35) fixed fixed Other Endogenous Propagation and Steady State Parameters 100γ α 100(β −1 − 1) σc h νl δ Φp 0.35 0.18 0.15 1.00 0.48 2.30 0.03 1.10 (0.28, (0.15, (0.07, (0.77, (0.38, (1.50, 0.43) 0.20) 0.23) 1.25) 0.58) 3.08) fixed (1.04, 1.16) S 00 ψ π∗ fixed (0.95, 1.10) (-0.05, 0.05) (-50.06, -46.03) fixed fixed Financial Frictions Parameters F (ω̄) SP∗ 0.03 1.81 fixed (1.67, 1.94) fixed Exogenous Process Parameters ρg ρb ρµ ρz ρσω ρπ ∗ ρz p ρλ f ρλ w ηλf η λw 0.99 0.95 0.81 0.96 0.98 0.99 0.89 0.80 0.36 0.65 0.40 (0.98, (0.94, (0.75, (0.94, (0.97, 1.00) 0.96) 0.87) 0.98) 1.00) fixed (0.84, (0.69, (0.09, (0.46, (0.18, 0.95) 0.91) 0.61) 0.85) 0.62) 2.45) 0.03) 0.50) 0.63) 0.05) 0.04) 0.21) 0.10) 0.44) 0.64) Measurement Error Parameters Cme 1.00 fixed Page 35 of 112 (-0.79, 0.43) Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft ρgdp ρgdi ρ10y ρtf p ρgdpdef ρpce Mean 0.04 0.93 0.96 0.21 0.51 0.30 (p5, p95) (-0.17, 0.25) (0.89, 0.97) (0.94, 0.99) (0.10, 0.32) (0.38, 0.64) (0.08, 0.51) May 31, 2018 σgdp σgdi σ10y σtf p σgdpdef σpce Mean 0.24 0.31 0.12 0.77 0.16 0.10 (p5, p95) (0.20, 0.28) (0.28, 0.35) (0.11, 0.13) (0.70, 0.84) (0.15, 0.18) (0.08, 0.12) Impulse Responses The following figures depict impulse response functions to various shocks. Figure 5 depicts the response of the economy to a discount factor shock, Figure 6 to a spread shock, Figure 7 to a shock to the marginal efficiency of investment (MEI), Figure 8 to a TFP shock, Figure 9 to a price markup shock, and Figure 10 to a monetary policy shock. New York Fed DSGE Team, Research and Statistics Page 36 of 112 20 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Figure 5: Responses to a Discount Factor Shock bt Hours Per Capita Real GDP Growth 0.0 -0.1 -0.2 -0.3 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 10 20 30 40 10 Percent Change in Wages 0.00 -0.01 -0.02 -0.03 10 20 20 30 40 30 40 30 40 GDP Deflator 30 40 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 -0.035 10 Core PCE Inflation 20 Nominal FFR -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 -0.025 -0.050 -0.075 -0.100 10 20 30 40 10 Consumption growth per capita 20 Real Investment per capita 0.0 -0.1 -0.2 -0.3 0.0 -0.1 -0.2 -0.3 10 20 30 40 10 20 30 40 Long term inflation expectations BAA - 10yr Treasury Spread 0.08 0.06 0.04 0.02 -0.005 -0.010 -0.015 -0.020 10 20 30 40 10 Long term interest rate expectations 20 30 40 Total Factor Productivity 0.00 -0.01 -0.02 -0.03 -0.01 -0.02 -0.03 -0.04 -0.05 10 20 30 40 30 40 10 20 30 40 Real GDI Growth 0.0 -0.1 -0.2 -0.3 10 20 New York Fed DSGE Team, Research and Statistics Page 37 of 112 21 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Figure 6: Responses to a Spread Shock σ̃ω,t Real GDP Growth Hours Per Capita 0.01 0.00 -0.01 -0.02 -0.03 -0.04 0.00 -0.05 -0.10 -0.15 10 20 30 40 10 Percent Change in Wages 30 40 30 40 30 40 GDP Deflator 0.002 0.000 -0.002 -0.004 -0.006 0.001 0.000 -0.001 -0.002 10 20 30 40 10 Core PCE Inflation 20 Nominal FFR 0.005 0.000 -0.005 -0.010 -0.015 0.001 0.000 -0.001 -0.002 10 20 30 40 10 0.04 0.03 0.02 0.01 0.00 -0.01 10 20 20 Real Investment per capita Consumption growth per capita 30 40 0.0 -0.1 -0.2 -0.3 -0.4 BAA - 10yr Treasury Spread 0.05 0.04 0.03 0.02 0.01 0.00 20 10 20 30 40 Long term inflation expectations 0.0010 0.0005 0.0000 -0.0005 10 20 30 40 10 Long term interest rate expectations 20 30 40 Total Factor Productivity 0.002 0.000 -0.002 -0.004 0.002 0.000 -0.002 -0.004 10 20 30 40 30 40 10 20 30 40 Real GDI Growth 0.01 0.00 -0.01 -0.02 -0.03 -0.04 10 20 New York Fed DSGE Team, Research and Statistics Page 38 of 112 22 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Figure 7: Responses to an MEI Shock µt Real GDP Growth 0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25 Hours Per Capita 0.2 0.0 -0.2 -0.4 10 20 30 40 10 20 30 40 30 40 30 40 GDP Deflator Percent Change in Wages 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.005 0.000 -0.005 -0.010 -0.015 10 20 30 40 10 Core PCE Inflation 20 Nominal FFR 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.02 0.00 -0.02 -0.04 -0.06 10 20 30 40 10 Real Investment per capita Consumption growth per capita 0.025 0.000 -0.025 -0.050 10 20 20 30 40 0.0 -0.5 -1.0 -1.5 BAA - 10yr Treasury Spread 10 20 30 40 Long term inflation expectations 0.00 -0.01 -0.02 -0.03 -0.04 -0.05 0.004 0.003 0.002 0.001 10 20 30 40 10 Long term interest rate expectations 20 30 40 Total Factor Productivity 0.012 0.009 0.006 0.003 0.000 10 20 30 40 30 40 0.01 0.00 -0.01 -0.02 -0.03 10 20 30 40 Real GDI Growth 0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25 10 20 New York Fed DSGE Team, Research and Statistics Page 39 of 112 23 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Figure 8: Responses to a TFP Shock z̃t Real GDP Growth Hours Per Capita 0.4 0.3 0.2 0.1 0.0 0.1 0.0 -0.1 -0.2 10 20 30 40 10 20 30 40 30 40 30 40 GDP Deflator Percent Change in Wages -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 -0.035 0.03 0.02 0.01 0.00 10 20 30 40 10 20 Nominal FFR Core PCE Inflation -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 -0.035 10 20 30 40 10 Consumption growth per capita 20 Real Investment per capita 0.3 0.2 0.1 0.0 0.20 0.15 0.10 0.05 0.00 10 20 30 40 10 BAA - 10yr Treasury Spread 20 30 40 Long term inflation expectations 0.010 0.008 0.006 0.004 0.002 0.000 -0.002 0.000 -0.005 -0.010 -0.015 10 20 30 40 10 Long term interest rate expectations 20 30 40 Total Factor Productivity 0.000 -0.005 -0.010 -0.015 -0.020 0.6 0.4 0.2 0.0 10 20 30 40 30 40 10 20 30 40 Real GDI Growth 0.4 0.3 0.2 0.1 0.0 10 20 New York Fed DSGE Team, Research and Statistics Page 40 of 112 24 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Figure 9: Responses to a Price Markup Shock λf,t Real GDP Growth Hours Per Capita 0.00 -0.03 -0.06 -0.09 -0.12 0.0 -0.1 -0.2 -0.3 10 20 30 40 10 Percent Change in Wages 20 30 40 0.15 0.10 0.05 0.00 10 Core PCE Inflation 0.15 0.10 0.05 0.00 30 40 30 40 30 40 GDP Deflator 0.00 -0.05 -0.10 -0.15 10 20 20 Nominal FFR 0.06 0.05 0.04 0.03 0.02 0.01 0.00 10 20 30 40 10 Consumption growth per capita 20 Real Investment per capita 0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25 0.00 -0.02 -0.04 -0.06 -0.08 -0.10 -0.12 10 20 30 40 10 BAA - 10yr Treasury Spread 20 30 40 Long term inflation expectations 0.010 0.008 0.006 0.004 0.002 0.000 -0.002 0.0000 -0.0025 -0.0050 -0.0075 -0.0100 10 20 30 40 10 Long term interest rate expectations 20 30 40 Total Factor Productivity 0.015 0.010 0.005 0.000 0.00 -0.01 -0.02 -0.03 10 20 30 40 30 40 10 20 30 40 Real GDI Growth 0.00 -0.03 -0.06 -0.09 -0.12 10 20 New York Fed DSGE Team, Research and Statistics Page 41 of 112 25 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Figure 10: Responses to a Monetary Policy Shock rtm Real GDP Growth Hours Per Capita 0.5 0.4 0.3 0.2 0.1 0.0 0.3 0.2 0.1 0.0 10 20 30 40 10 20 30 40 GDP Deflator Percent Change in Wages 0.005 0.004 0.003 0.002 0.001 0.000 -0.001 0.015 0.010 0.005 0.000 10 20 30 40 10 Core PCE Inflation 20 30 40 30 40 Nominal FFR 0.005 0.004 0.003 0.002 0.001 0.000 -0.001 -0.025 -0.050 -0.075 -0.100 -0.125 10 20 30 40 10 Consumption growth per capita 20 Real Investment per capita 0.4 0.3 0.2 0.1 0.0 0.6 0.4 0.2 0.0 10 20 30 40 10 BAA - 10yr Treasury Spread 20 30 40 Long term inflation expectations 0.005 0.000 -0.005 -0.010 10 20 30 40 0.0010 0.0005 0.0000 -0.0005 -0.0010 Long term interest rate expectations 10 20 30 40 Total Factor Productivity 0.000 -0.005 -0.010 -0.015 0.04 0.03 0.02 0.01 0.00 10 20 30 40 30 40 10 20 30 40 Real GDI Growth 0.3 0.2 0.1 0.0 10 20 New York Fed DSGE Team, Research and Statistics Page 42 of 112 26 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 References Aruoba, S. B. and F. Schorfheide (2008): “Insights from an Estimated Search-Based Monetary Model with Nominal Rigidities,” Working Paper. Bernanke, B., M. Gertler, and S. Gilchrist (1999a): “The Financial Accelerator in a Quantitative Business Cycle Framework,” in Handbook of Macroeconomics, ed. by J. B. Taylor and M. Woodford, North Holland, Amsterdam, vol. 1C. Bernanke, B. S., M. Gertler, and S. Gilchrist (1999b): “The Financial Accelerator in a Quantitative Business Cycle Framework,” in Handbook of Macroeconomics, ed. by J. B. Taylor and M. Woodford, Amsterdam: North-Holland, vol. 1C, chap. 21, 1341–93. Christiano, L., R. Motto, and M. Rostagno (2003): “The Great Depression and the Friedman-Schwartz Hypothesis,” Journal of Money, Credit and Banking, 35, 1119–1197. Christiano, L. J., M. Eichenbaum, and C. L. Evans (2005): “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy, 113, 1–45. Christiano, L. J., R. Motto, and M. Rostagno (2014): “Risk Shocks,” American Economic Review, 104, 27–65. De Graeve, F. (2008): “The External Finance Premium and the Macroeconomy: US Post-WWII Evidence,” Journal of Economic Dynamics and Control, 32, 3415 – 3440. Del Negro, M. and S. Eusepi (2011): “Fitting Observed Ination Expectations,” Journal of Economic Dynamics and Control, 35, 2105–2131. Del Negro, M., M. P. Giannoni, and F. Schorfheide (2015): “Inflation in the Great Recession and New Keynesian Models,” American Economic Journal: Macroeconomics, 7, 168–196. Del Negro, M. and F. Schorfheide (2012): “DSGE Model-Based Forecasting,” Federal Reserve Bank of New York Working Paper. Erceg, C. J. and A. T. Levin (2003): “Imperfect Credibility and Inflation Persistence,” Journal of Monetary Economics, 50, 915–944. New York Fed DSGE Team, Research and Statistics Page 43 of 112 27 Authorized for public release by the FOMC Secretariat on 1/12/2024 New York Fed DSGE Model: Research Directors Draft May 31, 2018 Greenwood, J., Z. Hercovitz, and P. Krusell (1998): “Long-Run Implications of Investment-Specific Technological Change,” American Economic Review, 87, 342–36. Laseen, S. and L. E. Svensson (2011): “Anticipated Alternative Policy-Rate Paths in Policy Simulations,” International Journal of Central Banking, 7, 1–35. Primiceri, G. (2006): “Why Inflation Rose and Fell: Policymakers Beliefs and US Postwar Stabilization Policy,” Quarterly Journal of Economics, 121, 867–901. Sargent, T. J. (1999): The Conquest of American Inflation, Princeton University Press, Princeton. Smets, F. and R. Wouters (2003): “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area,” Journal of the European Economic Association, 1, 1123 – 1175. ——— (2007): “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach,” American Economic Review, 97, 586 – 606. New York Fed DSGE Team, Research and Statistics Page 44 of 112 28 Authorized for public release by the FOMC Secretariat on 1/12/2024 Detailed Philadelphia (PRISM) Forecast Overview June 2018 Keith Sill Forecast Summary The FRB Philadelphia DSGE model denoted PRISM, projects that real GDP growth will be slightly above its trend pace over the next three years with real output growth running at about a 3 percent pace. Core PCE inflation is only marginally below the FOMC target of 2 percent over most of the next three years. The funds rate rises to 2.7 percent in 2018Q4 and increases steadily to reach 3.7 percent at the end of 2020. The current gap between the level of output and its trend level remains significant in the estimated model and, absent any shocks, the model continues to predict a recovery to the trend level. The relatively slow pace of growth and low inflation that have tended to characterize U.S. economic performance over the past few years require the presence of shocks to offset the strength of the model’s internal propagation channels. The PRISM model does not take into account the recent tax reform except to the extent that it is represented in current data observations. The Current Forecast and Shock Identification The PRISM model is an estimated New Keynesian DSGE model with sticky wages, sticky prices, investment adjustment costs, and habit persistence. The model is similar to the Smets & Wouters 2007 model and is described more fully in Schorfheide, Sill, and Kryshko 2010. Unlike in that paper though, we estimate PRISM directly on core PCE inflation rather than projecting core inflation as a non-modeled variable. Details on the model and its estimation are available in a Technical Appendix that was distributed for the June 2011 FOMC meeting or is available on request. The current forecasts for real GDP growth, core PCE inflation, and the federal funds rate are shown in Figures 1a-1c along with 68 percent probability coverage intervals. The forecast uses data through 2018Q1 supplemented by a 2018Q2 nowcast. The model takes the 2018Q2 nowcast for output growth of 3.2 percent as given and the projection begins with 2018Q3. PRISM anticipates that output growth edges down to 3 percent in 2018Q3, with growth then running at about that pace through the end of 2019. Growth tapers off slightly in 2020 to run at 2.8 percent by the end of the year. Overall, the growth forecast for this round is very similar to that from the March projection. While output growth is fairly robust going forward, core PCE inflation stays contained and runs at a pace slightly below the 2 percent target over the forecast horizon. Based on the 68 percent coverage interval, the model sees a minimal chance of deflation or recession (measured as negative quarters of real GDP growth) over the next 3 years. The federal funds rate is determined by an estimated policy rule and the funds rate rises from 1.7 Page 1 of 22 Page 45 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 percent in 2018Q2, to 2.7 percent in 2018Q4, 3.4 percent in 2019Q4, and 3.7 percent in 2020Q4. This path for the funds rate is similar to that in the March projection. Figures 1d and 1e plot the model’s estimates of the output gap and the natural real rate of interest. The output gap is defined as the log deviation of the level of output from the level that obtains under flexible wages and prices. The real natural rate of interest is the short term real interest rate that obtains in the model when prices and wages are flexible. The model currently estimates the output gap at -1.6 percent, a more or less continued improvement from the -3 percent reached during the recession. The output gap is expected to improve slowly, reaching about -0.9 percent at the end of 2020. The estimated real natural rate of interest is low at 0.13 percent in 2018Q2. The natural rate then rises over the next three years to reach 1.4 percent at the end of 2020. Note though that the natural rate estimate is extremely volatile and the 68 percent confidence bands are quite large. The key factors driving the projection are shown in the forecast shock decompositions (Figures 2a-2e) and the smoothed estimates of the model’s primary shocks (shown in Figure 3, where they are normalized by standard deviation). GDP growth is at an above trend pace in 2018Q2, boosted by shocks to labor supply, investment, and government spending (which includes net exports). Thereafter, growth is at a close to a trend pace over the forecast horizon. TFP, financial, and monetary policy shocks act as a drag on growth, but are largely offset by rebounds in government spending, investment, and labor supply. Over the course of the recession and recovery PRISM estimated a series of large positive shocks to leisure (negative shocks to labor supply) that have a persistent effect on hours worked and so pushed hours well below steady state. As these shocks unwind hours worked continue to rebound over the forecast horizon and so support higher output growth. Similarly, the unwinding of investment shocks contribute to output growth over the forecast horizon. Consumption is expected to grow at a slightly below-trend pace over the forecast horizon (Figure 2d). Consumption growth is pulled down by shocks to technology, investment, monetary policy, and government spending. Consumption gets a boost from the unwinding of financial and labor shocks, but not enough to offset the downward pull from the other shocks in the model. Financial shocks exert a considerable drag on investment growth in early 2018 – these same shocks make a positive contribution to consumption growth (Figure 2d-e). However, strong and offsetting investment shocks are enough to keep investment growth near its trend over the forecast horizon. All told, the model now forecasts trend growth in investment (gross private domestic + durable goods consumption) in 2018 as the gradual unwinding of MEI shocks (see Figures 2e and 3) are offset by the effects of financial shocks. The forecast for core PCE inflation continues to be a story of upward pressure from the unwinding of negative labor supply shocks and MEI shocks being offset by downward pressure from the waning of discount factor shocks. Negative discount factor shocks have a strong and persistent negative effect on marginal cost and inflation in the estimated model. But labor supply shocks that push down aggregate hours also serve to put upward pressure on the real wage and hence marginal cost. The effect is persistent -- as the labor supply shocks unwind over the Page 2 of 22 Page 46 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 forecast horizon they exert a waning upward push to inflation. On balance the effect of these opposing forces keep inflation slightly below target over the next 3 years. The federal funds rate is projected to rise fairly quickly over the forecast horizon. The model attributes the current level of the funds rate primarily to a combination of monetary policy, discount factor and labor supply dynamics. Looking ahead, the positive contribution from labor supply shocks is more than offset by discount factor shock dynamics over the medium term, but as these shocks wane the funds rate gradually rises to 3.7 percent by the end of 2020. Page 3 of 22 Page 47 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 References Schorfheide, Frank, Keith Sill, and Maxym Kryshko. 2010. “DSGE model-based forecasting of non-modelled variables.” International Journal of Forecasting, 26(2): 348-373. Smets, Frank, and Rafael Wouters. 2007. “Shocks and Frictions in U.S. Business Cycles: A Bayesian DSGE Approach.” American Economic Review, 97(3): 586-606. Page 4 of 22 Page 48 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 1a Real GDP Growth 10 8 6 4 2 0 -2 -4 -6 -8 -10 2014 2015 2016 2017 2018 Page 5 of 22 Page 49 of 112 2019 2020 2021 Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 1b Core PCE Inflation 6 5 4 3 2 1 0 -1 2014 2015 2016 2017 2018 Page 6 of 22 Page 50 of 112 2019 2020 2021 Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 1c Fed Funds Rate 8 6 4 2 0 -2 -4 2014 2015 2016 2017 2018 Page 7 of 22 Page 51 of 112 2019 2020 2021 Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 1d Output Gap 4 3 2 1 0 -1 -2 -3 -4 -5 1980 1985 1990 1995 2000 2005 Page 8 of 22 Page 52 of 112 2010 2015 2020 2025 Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 1e Natural Rate of Interest 12 10 8 6 4 2 0 -2 -4 -6 -8 1980 1985 1990 1995 2000 2005 Page 9 of 22 Page 53 of 112 2010 2015 2020 2025 Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 2a Shock Decompositions percent Real GDP Growth 8 8 6 6 4 4 2 2 0 0 -2 -2 -4 -4 -6 -6 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 Date tech gov mei mrkp shocks: TFP: Gov: MEI: MrkUp: Labor: Fin: Mpol: Total factor productivity growth shock Government spending shock Marginal efficiency of investment shock Price markup shock Labor supply shock Discount factor shock Monetary policy shock Page 10 of 22 Page 54 of 112 labor fin mpol Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 2b Shock Decompositions percent Core PCE Inflation 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 -3 -3 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 Date tech gov mei mrkp shocks: TFP: Gov: MEI: MrkUp: Labor: Fin: Mpol: Total factor productivity growth shock Government spending shock Marginal efficiency of investment shock Price markup shock Labor supply shock Discount factor shock Monetary policy shock Page 11 of 22 Page 55 of 112 labor fin mpol Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 2c Shock Decompositions percent Fed Funds Rate 6 6 4 4 2 2 0 0 -2 -2 -4 -4 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 Date tech gov mei mrkp shocks: TFP: Gov: MEI: MrkUp: Labor: Fin: Mpol: Total factor productivity growth shock Government spending shock Marginal efficiency of investment shock Price markup shock Labor supply shock Discount factor shock Monetary policy shock Page 12 of 22 Page 56 of 112 labor fin mpol Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 2d Shock Decompositions percent Real Consumption Growth 6 6 4 4 2 2 0 0 -2 -2 -4 -4 -6 -6 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 Date tech gov mei mrkp shocks: TFP: Gov: MEI: MrkUp: Labor: Fin: Mpol: Total factor productivity growth shock Government spending shock Marginal efficiency of investment shock Price markup shock Labor supply shock Discount factor shock Monetary policy shock Page 13 of 22 Page 57 of 112 labor fin mpol Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 2e Shock Decompositions Real Investment Growth 30 20 20 10 10 percent 30 0 0 -10 -10 -20 -20 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 Date tech gov mei mrkp shocks: TFP: Gov: MEI: MrkUp: Labor: Fin: Mpol: Total factor productivity growth shock Government spending shock Marginal efficiency of investment shock Price markup shock Labor supply shock Discount factor shock Monetary policy shock Page 14 of 22 Page 58 of 112 labor fin mpol Authorized for public release by the FOMC Secretariat on 1/12/2024 Figure 3 Smoothed Shock Estimates for Conditional Forecast Model (normalized by standard deviation) labor shock discount factor shock 4 5 2 0 0 -2 -4 2010 -5 2012 2014 2016 2018 2010 2012 TFP shock 2014 2016 2018 2016 2018 mei shock 2 2 0 0 -2 -2 -4 2010 2012 2014 2016 2018 2010 Page 15 of 22 Page 59 of 112 2012 2014 Authorized for public release by the FOMC Secretariat on 1/12/2024 Impulse Responses to TFP shock output growth consumption growth 1 1 0.5 0.5 0 0 5 10 15 0 0 investment growth 0.5 0 0 0 5 10 15 -0.5 0 inflation 0.05 0 0 0 5 15 5 10 15 nominal rate 0.05 -0.05 10 aggregate hours 2 -2 5 10 15 -0.05 0 Page 16 of 22 Page 60 of 112 5 10 15 Authorized for public release by the FOMC Secretariat on 1/12/2024 Impulse Response to Leisure Shock output growth consumption growth 2 2 0 0 -2 0 5 10 15 -2 0 investment growth 5 0 0 -1 -5 0 5 10 15 -2 0 inflation 0.4 0.2 0.2 0 5 10 15 5 10 15 nominal rate 0.4 0 5 aggregate hours 10 15 0 0 Page 17 of 22 Page 61 of 112 5 10 15 Authorized for public release by the FOMC Secretariat on 1/12/2024 Impulse Responses to MEI Shock output growth consumption growth 2 0.2 0 0 -2 0 5 10 15 -0.2 0 investment growth 1 0 0.5 0 5 10 15 0 0 inflation 0.4 0 0.2 0 5 15 5 10 15 nominal rate 0.1 -0.1 10 aggregate hours 10 -10 5 10 15 0 0 Page 18 of 22 Page 62 of 112 5 10 15 Authorized for public release by the FOMC Secretariat on 1/12/2024 Impulse Responses to Financial Shock output growth consumption growth 1 2 0 0 -1 0 5 10 15 -2 0 investment growth 0.5 0 0 0 5 10 15 -0.5 0 inflation 1 0.2 0.5 0 5 15 5 10 15 nominal rate 0.4 0 10 aggregate hours 5 -5 5 10 15 0 0 Page 19 of 22 Page 63 of 112 5 10 15 Authorized for public release by the FOMC Secretariat on 1/12/2024 Impulse Responses to Price Markup Shock output growth consumption growth 0.5 0.5 0 0 -0.5 0 5 10 15 -0.5 0 investment growth 0 0 -0.1 0 5 10 15 -0.2 0 inflation 0.5 0 0 0 5 15 5 10 15 nominal rate 1 -1 10 aggregate hours 1 -1 5 10 15 -0.5 0 Page 20 of 22 Page 64 of 112 5 10 15 Authorized for public release by the FOMC Secretariat on 1/12/2024 Impulse Responses to Unanticipated Monetary Policy Shock output growth consumption growth 0.5 0.5 0 0 -0.5 0 5 10 15 -0.5 0 investment growth 0.2 0 0 0 5 10 15 -0.2 0 inflation 1 0 0 0 5 15 5 10 15 nominal rate 0.1 -0.1 10 aggregate hours 1 -1 5 10 15 -1 0 Page 21 of 22 Page 65 of 112 5 10 15 Authorized for public release by the FOMC Secretariat on 1/12/2024 Impulse Responses to Govt Spending Shock output growth consumption growth 2 0.5 0 0 -2 0 5 10 15 -0.5 0 investment growth 0.4 0 0.2 0 5 10 15 0 0 inflation 0.04 0.01 0.02 0 5 15 5 10 15 nominal rate 0.02 0 10 aggregate hours 0.2 -0.2 5 10 15 0 0 Page 22 of 22 Page 66 of 112 5 10 15 Authorized for public release by the FOMC Secretariat on 1/12/2024 Research Directors’ Guide to the Chicago Fed DSGE Model∗ Jeffrey R. Campbell Filippo Ferroni Jonas D. M. Fisher Leonardo Melosi November 30, 2017 This guide describes the construction and estimation of the Chicago Fed’s DSGE model, which we use both for internal forecasting and for creating our contributions to the System DSGE memo distributed quarterly to the FOMC. The model has been in use and under ongoing development since 2010. Originally, it was largely based on Justiniano, Primiceri, and Tambalotti (2010). We published results based on simulations from the estimated model in Campbell, Evans, Fisher, and Justiniano (2012) and in Campbell, Fisher, Justiniano, and Melosi (2016). The model contains many features familiar from other DSGE analyses of monetary policy and bussiness cycles. External habit in preferences, i-dot costs of adjusting investment, price and wage stickiness based on Calvo’s (1983) adjustment probabilities, and partial indexation of unadjusted prices and wages using recently observed price and wage inflation. The features which distinguish our analysis from many otherwise similar undertakings are • Forward Guidance Shocks: An interest-rate rule which depends on recent (and expected future) inflation and output and is subject to stochastic disturbances governs our model economy’s monetary policy rate. Standard analysis prior to the great recession restricted the stochastic disturbances to be unforecastable. Our model deviates from this historical standard by including forward guidance shocks, as in Laséen and Svensson (2011). A j-quarter ahead forward guidance shock revealed to the public at time t influences the interestrate rule’s stochastic intercept only at time t + j. Each period, the model’s monetary authority reveals a vector of these shocks with one element for each quarter from the present until the end of the forward guidance horizon. The ∗ The views expressed herein are the authors’. They do not necessarily represent those of the Federal Reserve Bank of Chicago, the Federal Reserve System, or its Board of Governors. 1 Page 67 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 vector’s elements may be correlated with each other, so the monetary authority could routinely reveal persistent shifts in the interest-rate rule’s stochastic intercept. However, the forward guidance shocks are serially uncorrelated over time, as is required for them to match the definition of “news.” • Investment-Specific Technological Change: As in the Real Business Cycle models from which modern DSGE models decend (King, Plosser, and Rebelo, 1988a), stochastic trend productivity growth both short-run and longrun fluctuations. Our model features two such stochastic trends, one to Hicks-neutral productivity (King, Plosser, and Rebelo, 1988b) and one to the technology for converting consumption goods into investment goods (as in Fisher (2006)). This investment-specific technological change allows our model to reproduce the dynamics of the relative price of investment goods to consumption goods, which is a necessary input into the formula we use to create Fisher-ideal chain-weighted index of real GDP. • A Mixed Calibration-Bayesian Estimation Empirical Strategy: Bayesian estimation of structural business cycle models attempts to match all features of the data’s probability distribution using the model’s parameters. Since no structural model embodies Platonic “truth,” this exercise inevitably requires trading off between the model’s ability to replicate first moments with its fidelty to the business cycles in second moments. Since the criteria for this tradeoff are not always clear, we adopt an alternative “first-moments-first” strategy. This selects the values of model parameters which govern the model’s steady-state growth path, such as the growth rates of Hicks-neutral and investment-specific technology, to match estimates of selected first moments. These parameter choices are then fixed for Bayesian estimation, which chooses values for model parameters which only influence second moments, such as technology innovation variances. (Since we employ a log linear solution of our model and all shocks to its primitives have Gaussian distributions, our analysis has no non-trivial implications for third and higher moments of the data.) The guide proceeds as follows. The next section presents the model economy’s primitives, while Section 2 presents the agents’ first-order conditions. Section 3 gives the formulas used to remove nominal and technological trends from model variables and thereby induce model stationarity, and Sections 4 and 5 discuss 2 Page 68 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 the stationary economy’s steady state and the log linearization of its equilibrium necessary conditions around it. Section 6 discusses measurement issues which arise when comparing model-generated data with data measured by the BEA and BLS. Section 7 describes our mixed Calibration-Bayesian Estimation empirical strategy and presents the resulting parameter values we use for model simulations and forecasting. 1 The Model’s Primitives Eight kinds of agents populate the model economy: • Households, • Investment producers, • Competitive final goods producers, • Monopolistically-competitive differentiated goods producers, • Labor Packers, • Monopolistically-competitive guilds, • a Fiscal Authority and • a Monetary Authority. These agents interact with each other in markets for • final goods used for consumption • investment goods used to augment the stock of productive capital • differentiated intermediate goods • capital services • raw labor • differentiated labor • composite labor 3 Page 69 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 • government bonds • privately-issued bonds, and • state-contingent claims. The households have preferences over streams of an aggregate consumption good, leisure, and the real value of the fiscal authority’s bonds in their portfolios. Our specification for preferences displays balanced growth. They also feature external habit in consumption; which creates a channel for the endogenous propagation of shocks. Our bonds-in-the-utility-function preferences follow those of Fisher (2015), and they allow us to incorporate a persistent spread between the monetary policy rate and the return on productive capital. The aggregate consumption good has a single alternative use, as the only input into the linear production function operated by investment producers. These firms sell their output to the households. In turn, households produce capital services from their capital stocks, which they then sell to differentiated goods producers. Producers of final goods operate a constantreturns-to-scale technology with a constant elasticity of substitution between its inputs, which are differentiated goods produced by the monopolistically-competitive firms. These firms operate technologies with affine cost curves (a constant fixed cost and linear marginal cost), which employs capital services and composite labor as inputs. The labor packers produce composite labor using a constant-returns-toscale technology with a constant elasticity of substitution between its inputs, the differentiated labor sold by guilds. Each of these produces differentiated labor from the raw labor provided by the households with a linear technology, and they sell their outputs to the labor packers. There is a nominal unit of account, called the “dollar.” The fiscal authority issues one-period nominally risk-free bonds, provides public goods through government spending, and assesses lump-sum taxes on households. The monetary authority sets the interest rate on the fiscal authority’s one-period bond according to an interest-rate rule. All non-financial trade is denominated in dollars, and all private agents take prices as given with two exceptions: the monopolistically-competitive differentiatedgoods producers and guilds. These choose output prices to maximize the current value of expected future profits taking as given their demand curves and all relevant input prices. Financial markets are complete, but all securities excepting equities in differentiated-goods producers are in zero net supply. These producers’ profits 4 Page 70 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 and losses are rebated to the households (who own the firms’ equities) lump-sum period-by-period, as are the profits and losses of the guilds. Given both a process for government spending and taxes and a rule for the monetary authority’s interest rate choice, a competitive equilibrium consists of allocations and prices that are consistent with households’ utility maximization, firms’ profit maximization, guilds’ profit maximization, and market clearing. The economy is subject to stochastic disturbances in technology, preferences, and government policy. Without nominal rigidities, the economy’s real allocations in competitive equilibrium can be separated from inflation and other dollardenominated variables. Specifically, monetary policy only influences inflation. To connect real and nominal variables in the model and thereby consider the impact of monetary policy on the business cycle, we introduce Calvo-style wage and price setting. That is, nature endows both differentiated goods producers and guilds with stochastic opportunities to incorporate all available information into their nominal price choices. Those producers and guilds without such a opportunity must set their prices according to simple indexing formulas. These two pricing frictions create two forward-looking Phillips curves, one for prices and another for wages, which form the core of the new Keynesian approach to monetary policy analysis. The model economy is stochastic and features complete markets in statecontingent claims. To place these features on a sound footing, we base all shocks on a general Markovian stochastic process st . Denote the history of this vector from an initial period 0 through τ with sτ ≡ (s0 , s1 , . . . , sτ ). The support of sτ is Στ , and the probability density of sτ given st for some t < τ is ( סsτ st ). (The Hebrew letter ס, pronounced “samekh,” corresponds to the Greek letter σ.) All model shocks are implicit functions of st , and all endogenous variables are implicit functions of st . We refer to all such implicit functions as “state-contingent sequences.” We use braces to denote such a sequence. For example, {Xt } represents the state-contingent sequence for a generic variable Xt . 1.1 Households Our model’s households are the ultimate owners of all assets in positive net supply (the capital stock, differentiated goods producers, and guilds). They provide labor and divide their current after-tax income (from wages and assets) between current 5 Page 71 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 consumption, investment in productive capital, and purchases of financial assets, both those issued by the government and those issued by other households. The individual household divides its current resources between consumption and the available vehicles for intertemporal substitution (capital and financial assets) to maximize a discounted sum of current and expected future felicity. ∞ Et [ ∑ β τ εbt+τ (Ut+τ + εst+τ L ( τ =0 Bt+τ ))] Pt+τ Rt+τ with Ut = 1 (1−γc ) ((Ct − %C̄t−1 )(1 − Ht1+γh )) 1 − γc (1) The function L(⋅) is strictly increasing, concave, and differentiable everywhere on [0, ∞). In particular, L′ (0) exists and is finite. Without loss of generality, we set L′ (0) to one. The argument of L(⋅) equals the real value of government bonds in the household’s portfolio: their period t + 1 redemption value Bt divided by their nominal yield Rt expressed in units of the consumption good with the nominal price index Pt . The time-varying coefficient multiplying this felicity from bond holdings, εst , is the liquidity preference shock introduced by Fisher (2015). A separate shock influences the household’s discounting of future utility to the present, εbt . Specifically, the household discounts a certain utility in t + τ back to t with β τ Et [εbt+τ /εbt ]. In logarithms, these two preference shocks follow independent autoregressive processes. ln εbt = (1 − ρb ) ln εb∗ + ρb ln εbt−1 + ηtb , ηtb ∼ N(0, σb ) (2) ln εst = (1 − ρs ) ln εs∗ + ρs ln εst−1 + ηts , ηts ∼ N(0, σs ). (3) A household’s wealth at the beginning of period t consists of its nominal government bond holdings, Bt , its net holdings of privately-issued financial assets, and its capital stock Kt−1 . The household chooses a rate of capital utilization ut , and the capital services resulting from this choice equal ut Kt−1 . The cost of increasing utilization is higher depreciation. An increasing, convex and differentiable function δ(U ) gives the capital depreciation rate. We specify this as δ(u) = δ0 + δ1 (u − u⋆ ) + δ2 2 (u − u⋆ ) . 2 6 Page 72 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 A household can augment its capital stock with investment, It . Investment requires paying adjustment costs of the “i-dot” form introduced by Christiano, Eichenbaum, and Evans (2005). Also, an investment demand shock alters the efficiency of investment in augmenting the capital stock. Altogether, if the household’s investment in the previous period was It−1 , and it purchases It units of the investment good today, then the stock of capital available in the next period is Kt = (1 − δ(ut )) Kt−1 + εit (1 − S ( AK t−1 It )) It . K At It−1 (4) In (4), AK t equals the productivity level of capital goods production, described in more detail below, and εit is the investment demand shock. In logarithms, this follows a first-order autoregression with a normally-distributed innovation. ln εit = (1 − ρi ) ln εi∗ + ρi ln εit−1 + ηti , ηti ∼ N(0, σi ) 1.2 (5) Production The producers of investment goods use a linear technology to transform the final good into investment goods. The technological rate of exchange from the final good to the investment good in period t is AIt . We denote ∆ ln AIt with ωt , which we call the investment-specific technology shock and which follows first-order autogregression with normally distributed innovations. ωt = (1 − ρω )ω⋆ + ρω ωt−1 + ηtω , ηtω ∼ N(0, σω2 ) (6) Investment goods producers are perfectly competitive. Final good producers also operate a constant-returns-to-scale technology; which takes as inputs the products of the differentiated goods producers. To specify this, let Yit denote the quantity of good i purchased by the representative final good producer in period t, for i ∈ [0, 1]. The representative final good producer’s output then equals Yt ≡ (∫ 1 0 1 p 1+λ t Yit 1+λpt di) . 7 Page 73 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 With this technology, the elasticity of substitution between any two differentiated products equals 1 + 1/λpt in period t. Although this is constant across products within a time period, it varies stochastically over time according to an ARMA(1, 1) in logarithms. p ln λpt = (1 − ρp ) ln λp⋆ + ρp ln λpt−1 − θp ηt−1 + ηtp , ηtp ∼ N(0, σp ) (7) Given nominal prices for the intermediate goods Pit , it is a standard exercise to show that the final goods producers’ marginal cost equals Pt = (∫ 1 0 − Pit 1 p λ t −λpt di) (8) Just like investment goods firms, the final goods’ producers are perfectly competitive. Therefore, profit maximization and positive final goods output together require the competitive output price to equal Pt . Therefore, we can define inflation of the nominal final good price as πt ≡ ln(Pt /Pt−1 ). The intermediate goods producers each use the technology 1−α α Yit = (Kite ) (AYt Hitd ) − At Φ (9) Here, Kite and Hitd are the capital services and labor services used by firm i, and AYt is the level of neutral technology. Its growth rate, νt ≡ ln(AYt /AYt−1 ), follows a first-order autogregression. νt = (1 − ρν ) ν∗ + ρv νt−1 + ηtν , ηtν ∼ N(0, σν ), (10) The final term in (9) represents the fixed costs of production. These grow with α At ≡ AYt (AIt ) 1−α . (11) We demonstrate below that At is the stochastic trend in equilibrium output and consumption, measured in units of the final good. We denote its growth rate with zt = νt + α ωt 1−α (12) 8 Page 74 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Similarly, define I AK t ≡ At A t (13) In the specification of the capital accumulation technology, we labelled AK t the “productivity level of capital goods production.” We demonstrate below that this is indeed the case with the definition in (13). Each intermediate goods producer chooses prices subject to a Calvo (1983) pricing scheme. With probability ζp ∈ [0, 1], producer i has the opportunity to set Pit without constraints. With the complementary probability, Pit is set with the indexing rule p Pit = Pit−1 πt−1 π⋆ ι 1−ιp . (14) In (14), π⋆ is the gross rate of price growth along the steady-state growth path, and ιp ∈ [0, 1].1 1.3 Labor Markets Households’ hours worked pass through two intermediaries, guilds and labor packers, in their transformation into labor services used by the intermediate goods producers. The guilds take the households’ homogeneous hours as their only input and produce differentiated labor services. These are then sold to the labor packers, who assemble the guilds’ services into composite labor services. The labor packers operate a constant-returns-to-scale technology with a constant elasticity of substitution between the guilds’ differentiated labor services. For its specification, let Hit denote the hours of differenziated labor purchased from guild i at time t by the representative labor packer. Then that packer’s production of composite labor services, Hts are given by Hts 1 = (∫ (Hit ) 1 1+λw t 1+λw t di) . 0 As with the final good producer’s technology, an ARMA(1, 1) in logarithms governs To model firms’ price-setting opportunities as functions of st , define a random variable upt which is independent over time and uniformly distributed on [0, 1]. Then, firm i gets a pricesetting opportunity if either upt ≥ ζp and i ∈ [upt − ζp , upt ] or if upt < ζp and i ∈ [0, upt ] ∪ [1 + upt − ζp , 1]. 1 9 Page 75 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 the constant elasticity of substitution between any two guilds’ labor services. w w w w w 2 ln λw t = (1 − ρw ) ln λ⋆ + ρw ln λt−1 − θw ηt−1 + ηt , ηt ∼ N(0, σw ) (15) Just as with the final goods producers, we can easily show that the labor packers’ marginal cost equals 1 − λ1w Wt = (∫ (Wit ) t −λw t di) . (16) 0 Here, Wit is the nominal price charged by guild i per hour of differentiated labor. Since labor packers are perfectly competitive, their profit maximization and positive output together require that the price of composite labor services equals their marginal cost. Each guild produces it’s differentiated labor service using a linear technology with the household’s hours worked as its only input. A Calvo (1983) pricing scheme similar to that of the differentiated goods producers constrains their nominal prices. Guild i has an unconstrained opportunity to choose its nominal price with probability ζw ∈ [0, 1]. With the complementary probability, Wit is set with an indexing rule based on πt−1 and last period’s trend growth rate, zt−1 . Wit = Wit−1 (πt−1 ezt−1 ) w (π⋆ ez⋆ ) ι 1−ιw . (17) α ω⋆ is the unconditional mean of zt and ιw ∈ [0, 1]. In (17), z⋆ ≡ ν⋆ + 1−α 1.4 Fiscal and Monetary Policy The model economy hosts two policy authorities, each of which follows exogenouslyspecified rules that receive stochastic disturbances. The fiscal authority issues bonds, Bt , collects lump-sum taxes Tt , and buys “wasteful” public goods Gt . Its periodby-period budget constraint is Gt + Bt−1 = Tt + Bt . Rt (18) The left-hand side gives the government’s uses of funds, public goods spending and the retirement of existing debt. The left-hand side gives the sources of funds, taxes 10 Page 76 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 and the proceeds of new debt issuance at the interest rate Rt . We assume that the fiscal authority keeps its budget balanced period-by-period, so Bt = 0. Furthermore, the fiscal authority sets public goods expenditure equal to a stochastic share of output, expressed in consumption units. Gt = (1 − 1/gt )Yt , (19) ln gt = (1 − ρg ) ln sg⋆ + ρg ln gt−1 + ηtg , ηtg ∼ N(0, σg2 ). (20) with The monetary authority sets the nominal interest rate on government bonds, Rt . For this, it employs a Taylor rule with interest-rate smoothing and forward guidance shocks. M j ln Rt = ρR ln Rt−1 + (1 − ρR ) ln Rtn + ∑ ξt−j . (21) j=0 1 M The monetary policy disturbances in (21) are ξt0 , ξt−1 , . . . , ξt−M . The public learns j the value of ξt−j in period t − j. The conventional unforecastable shock to current monetary policy is ξt0 , while for j ≥ 1, these disturbances are forward guidance shocks. We gather all monetary shocks revealed at time t into the vector ε1t . This is normally distributed and i.i.d. across time. However, its elements may be correlated with each other. That is, ε1t ≡ (ξt0 , ξt1 , . . . , ξtM ) ∼ N(0, Σ1 ). (22) The off-diagonal elements of Σ1 are not necessarily zero, so forward-guidance shocks need not randomly impact expected future monetary policy at two adjacent dates independently. Current economic circumstances influence Rt through the notional interest rate, Rtn . ln Rtn = ln r⋆ + ln πt⋆ + 1 1 φ1 φ2 Et ∑ (ln πt+j − ln πt⋆ ) + Et ∑ (ln Yt+j − ln y ⋆ − ln At+j ) . 4 j=−2 4 j=−2 (23) 11 Page 77 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 The constant r⋆ equals the real interest rate along a steady-state growth path, and πt⋆ is the central bank’s intermediate target for inflation. We call this the inflation-drift shock. it follows a first-order autoregression with a normally-distributed innovation. Its unconditional mean equals π⋆ , the inflation rate on a steady-state growth path. ⋆ ln πt⋆ = (1 − ρπ )π⋆ + ρπ ln πt−1 + ηtπ , ηtπ ∼ N(0, σπ2 ) (24) Allowing πt⋆ to change over time enables our model to capture the persistent decline in inflation from the early 1990s through the early 2000s engineered by the Greenspan FOMC. 1.5 Other Financial Markets and Equilibrium Definition All households participate in the market for nominal risk-free government debt. Additionally, they can buy and sell two classes of privately issued assets without restriction. The first is one-period nominal risk-free private debt. We denote the P value of household’s net holdings of such debt at the beginning of period t with Bt−1 P and the interest rate on such debt issued in period t maturing in t + 1 with Rt+1 . The second asset class consists of a complete set of real state-contingent claims. As of the end of period t, the household’s ownership of securities that pay off one unit of the aggregate consumption good in period τ if history sτ occurs is Qt (sτ ), and the nominal price of such a security in the same period is Jt (sτ ). We define an equilibrium for our economy in the usual way: Households maximize their utility given all prices, taxes, and dividends from both producers and guilds; final goods producers and labor packers maximize profits taking their input and output prices as given; differentiated goods producers and guilds maximize the market value of their dividend streams taking as given all input and financial-market prices; differentiated goods producers and guilds produce to satisfy demand at their posted prices; and otherwise all product, labor, and financial markets clear. 2 First Order Conditions In this section we present the first-order conditions associated with the optimization problems that the agents in our model solve. 12 Page 78 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 2.1 Households Given initial financial asset holdings holdings, a stock of productive capital, investment in the previous period (which influences investment adjustment costs), and the external habit stock; households’ choices of consumption, capital investment, capital utilization, hours worked, and financial investments maximize utility subject to the constraints of the capital accumulation and utilization technology and a sequence of one-period budget constraints. To specify these budget constraints, denote the nominal wage-per-hour paid by labor guilds to households with Wth , the nominal rental rate for capital services with Rtk , the nominal price of investment goods with PtI , and the dividends paid by labor guilds added to those paid by differentiated good producers with Dt . With this notation, writing the period t budget constraint with uses of funds on the left and sources of funds on the right yields Ct + P Bt BP Tt Bt−1 Bt−1 W h Ht Rtk ut Kt−1 Dt PtI It + + Pt + ≤ + + t + + Pt Rt Pt Rt Pt Pt Pt Pt Pt Pt Pt (25) Denote the Lagrange multiplier on (25) with β t Λ1t , and that on the capital accumulation constraint in (4) with β t Λ2t . With these definitions, the first-order conditions for a household’s utility maximization problem are 13 Page 79 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Λ1t = εbt ((Ct − %C̄t−1 )(1 − εht Ht1+γh )) −γc (1 − εht Ht1+γh ) Wth −γc = (1 + γh )εbt ((Ct − %C̄t−1 )(1 − εht Ht1+γh )) (Ct − %C̄t−1 )εht Htγh Pt 1 Λ1 Λt εs Bt − εbt+q L′ ( ) t = βEt [ t+1 ] Rt Pt Rt Pt Rt Pt Pt+1 Λ1t Λ1t+1 Λ1t = βE [ ] t Pt+1 RtP Pt Λ2t = βE [Λ1t+1 k Rt+1 ut+1 + Λ2t+1 (1 − δ(ut+1 ))] Pt+1 Λ1t Rtk = Λ2t δ ′ (ut ) Pt Λ1t = εit Λ2t ((1 − St (⋅)) − St′ (⋅) it it−1 ) ′ +βEt [εit+1 e(1−γC )zt+1 λ2t+1 St+1 (⋅) i2t+1 ] i2t In equilibrium, C̄t = Ct always. 2.2 2.2.1 Goods Sector Final Goods Producers The nominal marginal cost of final goods producers equals the right-hand side of (8). A producer of final goods maximizes profit by shutting down if Pt is less than this marginal cost and can make an arbitrarily large profit if Pt exceeds it. When (8) holds, an individual final goods producer’s output is indeterminate. Final goods producers’ demand for intermediate goods takes the familiar constant-elasticity form. If we use Yt to denote total final goods output, then the amount of differentiated good i demanded by final goods producers is Pit − Yit = Yt ( ) Pt p 1+λ t p λ t . Given the choice of a reset price, we wish to calculate the overall price level. All intermediate goods producers with a price-setting opportunity choose P̃t . The 14 Page 80 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 remaining producers use the price-indexing rule in (14). The aggregate price level is given by 1 λp,t −1 Pt = [(1 − ζp )P̃t + ζp ((πt−1 ) (π∗ ) 1−ιp ιp Pt−1 ) 1 λp,t −1 λp,t −1 ] where P̃t is the optimal reset price 2.2.2 Intermediate Goods Producers Intermediate goods producers’ cost minimization reads as follows: d e max Wt Ht,i + Rtk Ki,t e Ht,i ,Ki,t 1−α e α d ) s.t. Yt,i = εat (Kt,i ) (Ayt Ht,i − At Φ We get the following optimal capital-labor ratio. α Wt (Kite )s = d 1 − α Rtk Ht,i Notice how for each firm, the idiosyncratic capital to labor ratio is not a function of any firm-specific component. Hence, each firm has the same capital to labor ratio. In equilibrium, Kte = ut Kt−1 To find the marginal cost, we differentiate the variable part of production with respect to output, and substitute in the capital-labor ratio. −(1−α) M Ct,i = (εat )−1 (Ayt ) Wt1−α Rtkα α−α (1 − α)−(1−α) Again, notice that each firm as the same marginal cost. Given cost minimization, a differentiated goods producer with an opportunity to adjust its nominal price does so to maximize the present-discounted value of profits 15 Page 81 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 earned until the next opportunity to adjust prices arrives. Formally, ∞ max Et ∑ ζps P̃t,i s=0 β s Λ1t+s Pt y [P̃t,i Xt,s − M Ct+s ] Yt+s,i 1 Λt Pt+s λp,t 1−λp,t y P̃t,i ) s.t. Yt (i) = (Xt,s Yt Pt ⎧ ⎪ ∶s=0 ⎪ 1 y where Xt,s = ⎨ s ιp 1−ιp ⎪ ∶ s = 1, . . . , ∞ ⎪ ⎩ ∏l=1 πt+l−1 π∗ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ This problem leads to the following price-setting equation for firms that are allowed to reoptimize their price: 0= ⎡ s 1 ⎢ s β Λt+s Pt Yit+s ⎢⎢λp,t+s M Ct+s Et ∑ ζp 1 Λt Pt+s ⎢ s=0 ∞ ⎣ ⎤ ⎥ − Xt,s P̃it ⎥⎥ ⎥ ⎦ It can be shown that the producers that are allowed to reoptimize choose the same price. So henceforth, P̃it = P̃t . 2.2.3 Investment Goods Producers Characterizing the profit-maximizing choices of investment goods and final goods producers is straightforward. If PtI > Pt /AIt , then each investment goods producer can make infinite profit by choosing an arbitrarily large output. On the other hand, if PtI < Pt /AIt , then investment goods producers maximize profits with zero production. Finally, their profit-maximizing production is indeterminate when PtI = Pt /AIt . (26) −1 The relative price of investment to consumption is equal to (AIt ) . Making this substitution into the household f.o.c and noting that Pt YtI is an intermediate input that will not be reflected in the aggregate resource constraint, it suffices to −1 substitute the relative price (AIt ) in the constraint for the household. 16 Page 82 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 2.3 Labor Sector 2.3.1 Labor Packers The labor packers choose the the labor inputs supplied by guilds, pack them into a composite labor service to be sold to the intermediate goods producers. Formally, labor packers’ problem reads as follows: max Wt Hts − ∫ s Ht ,Hit s.t. [∫ 1 0 1 1 1+λw,t Hit Wit Hit di 0 1+λw,t = Hts di] We obtain the following labor demand equation for guild i: Wit − ) Hit = ( Wt 1+λw,t λw,t Ht (27) As for the goods sector, we can show that aggregate wage is given by the following equation: −λ1 Wt = [(1 − ζw )W̃t w,t + ζw ((ezt−1 πt−1 ) w (π∗ ez∗ ) ι 1−ιw −λ1 Wt−1 ) −λw,t w,t ] where W̃ is the optimal reset wage for guilds. 2.3.2 Guilds Each guild with an opportunity to set its nominal price does so to maximize the current value of the stream of dividends returned to the household. Formally, their problem reads ∞ max Et ∑ ζws ( W̃it s=0 s.t. Hit+s = β s Λ1t+s Pt l h ] Hit+s ) [Xt+s W̃it − Wt+s Λ1t Pt+s l W̃it ⎞ ⎛ Xt,s − 1+λw,t+s λw,t+s Ht+s ⎝ Wt+s ⎠ ⎧ ⎪ ∶s=0 ⎪ 1 l where Xt,s = ⎨ s 1−ι w At+j−1 ⎪ (πeγ )ιw ∶ s = 1, . . . , ∞ ⎪ ⎩ ∏j=1 (πt+j−1 At+j−2 ) 17 Page 83 of 112 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ Authorized for public release by the FOMC Secretariat on 1/12/2024 W̃t is the optimal reset wage. This optimal wage is chosen by the guilds who are allowed, with probability ζw , to change their prices in a given period. Also, we index the nominal wage inflation rate with ιw . This maximization problem gives a wage-setting equation that reads as follows: ∞ 0 = Et ∑ ζws s=0 1 β s Λ1t+s Pt h l ((1 + λw,t+s )Wt+s Hit+s − Xt,s W̃it ) 1 Λt Pt+s λw,t+s It can be shown that the guilds that are allowed to reoptimize choose the same wage. So henceforth, W̃it = W̃t . 3 Detrending To remove nominal and real trends, we deflate nominal variables by their matching price deflators, and we detrend any resulting real variables influenced permanently by technological change. All scaled versions of variables are the lower-case counterparts. ct = kt = wt = p̃t = yt = rtk = Ct At Kt At AIt Wt A t Pt P̃t Pt Yt At Rtk AIt Pt it = kte = w̃t = πt = mct = wth = λ1t = Λ1t Aγt C It At AIt Kte At AIt W̃t At P t Pt Pt−1 M Ct Pt Wth At P t λ2t = Λ2t Aγt C AIt εst = Aγt C εst 3.1 Detrended Equations The detrended equations describing our model are listed in the following sections. 18 Page 84 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Households’ FOC −γc ct−1 h 1+γh h (1 )] (1 − εht h1+γ ) ) − ε h t t t ezt −γc ct−1 ct−1 (1+σ ) λ1t wth = (1 + γh )εbt [(ct − % zt ) (1 − εht ht h )] (ct − % zt ) εht hγt h e e 1 1 −γ z t+1 C λ e λt ] = βEt [ t+1 P πt+1 Rt λ1 λ1t εb εs − L′ (0) t t = βEt t+1 e−zt+1 γC Rt Rt πt+1 i2 it ′ ) + βEt [εit+1 e(1−γC )zt+1 λ2t+1 St+1 (⋅) t+1 ] λ1t = εit λ2t ((1 − St (⋅)) − St′ (⋅) it−1 i2t λ1t = εbt [(ct − % k λ2t = βEt [e−γC zt+1 −ωt+1 (λ1t+1 rt+1 ut+1 + λ2t+1 (1 − δ(ut+1 )))] λ1t rtk = λ2t δ ′ (ut ) kt = (1 − δ(ut )) kt−1 e−zt −ωt + εit (1 − S(⋅)) it kte = ut kt−1 e−zt −ωt Final Goods Price Index 1 1−λp,t 1 = [(1 − ζp )p̃t 1−λp,t 1 ιp ∗(1−ιp ) −1 1−λp,t + ζp (πt−1 π πt ) ] Intermediate Goods Firms: Capital-Labor Ratio kte α wt = d ht 1 − α rtk Intermediate Goods Firms: Real Marginal Costs α mct = wt1−α (rtk ) εat αα (1 − α)1−α 19 Page 85 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Intermediate Goods Firms: Price-Setting Equation ∞ 0 =Et ∑ ζps β s λ1t+s s=0 ỹt,t+s At+s 1−γC p [λp,t+s mct+s − X̃t,s p̃t ] ( ) λp,t+s − 1 At where p X̃t,s ⎧ ⎪ ⎪ ⎪ 1 1−ιp ι = ⎨ ∏sj=1 πt+j−1 π∗p ⎪ ⎪ s ⎪ ⎩ ∏j=1 πt+j ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ∶ s = 1, . . . , ∞ ⎪ ⎪ ⎭ ∶s=0 ỹt,t+s denotes the time t + j output sold by the producers that have optimized at time t the last time they have reoptimized. Since it can be shown that optimizing producers all choose the same price, then we do not have to carry the i-subscript. Labor Packers: Aggregate Wage Index −λ1 wt = [(1 − ζw )w̃t w,t ιw zt−1 −zt (1−ιw )z∗ + ζw (e e −λw,t −λ1 −1 1−ιw ι w,t πt−1 πt π∗ wt−1 ) ] Guilds: Wage-Setting Equation ∞ 0 =Et ∑ ζws βλ1t+s ( s=0 At+s 1−γC h̃t,t+s l h ((1 + λw,t+s )wt+s − X̃t,s w̃t ) ) At λw,t+s where l X̃t,s ⎧ ⎪ ⎪ ⎪ 1 = ⎨ ∏sj=1 (πt+j−1 ezt+j−1 )1−ιw (πγ)ιw ⎪ z ⎪ ⎪ ∏sj=1 πt+j e t+j ⎩ ⎫ ⎪ ⎪ ⎪ ⎬ ∶ s = 1, . . . , ∞ ⎪ ⎪ ⎪ ⎭ ∶s=0 h̃t,t+s denotes the time t + j labor supplied by the guild that have optimized at time t the last time they have reoptimized. Since it can be shown that optimizing guilds all choose the same wage, then we do not have to carry the i-subscript. 20 Page 86 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Monetary Authority ψ1 ψ2 ⎤1−ρR ⎡ ⎢ 1 M 1 4 4 ⎥ y π t+j t+j ⎥ ρR ⎢ ⎢r∗ πt∗ ( ∏ ∗ ) ( ∏ ∗ ) ⎥ Rt = Rt−1 ξt−j,j ∏ ⎥ ⎢ j=−2 y j=0 j=−2 πt ⎥ ⎢ ⎦ ⎣ The Aggregate Resource Constraint yt =ct + it gt Production Function yt =εat (kte ) (hdt )1−α − Φ α Labor Market Clearing Condition ht = hdt 4 Steady State We normalize most shocks and the utilization rate: u⋆ =1 εi =1 εa =1 εb =1 Next, we set the following restriction on adjustment costs: S(⋅∗ ) ≡ 0 S ′ (⋅∗ ) ≡ 0 21 Page 87 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 4.1 Prices and Interest Rates Given β, z∗ , γC , and π∗ , we can solve for the steady-state nominal interest rate on private bonds R∗P by using the FOC on private bonds: R∗P = π∗ (βe−γC z∗ ) (28) From the definition of δ(u), we have δ(1) =δ0 δ ′ (1) =δ1 . Next, given ω∗ , δ0 , and the above, we can solve for the real return on capital r∗k using the FOC on capital: r∗k = 4.2 eγC z∗ +ω∗ − (1 − δ0 ) β (29) Ratios Moving to the production side, we can use the aggregate price equation to solve for p̃∗ : p̃∗ = 1 Using this result and given λp,∗ , we can use the price Phillips curve to solve for mc∗ : mc∗ = 1 1 + λp,∗ (30) Given values for α and εa∗ , we can use the marginal cost equation to solve for w∗ : 1 w∗ = (mc∗ αα (1 − α)1−α (r∗k )−α ) 1−α (31) 22 Page 88 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 The definition of effective capital gives us a value for k∗e in terms of k∗ : k∗e = k∗ e−z∗ −ω∗ Calculating y∗ using the labor share of output 1 − α: y∗ = w∗ h∗ 1−α Using capital shares based off our value of α, we can calculate the output to capital ratio as follows: y∗ r∗k = k∗e α y∗ −z∗ −ω∗ r∗k =e k∗ α Using the capital accumulation equation, we can get a value for i∗ k∗ i∗ = 1 − (1 − δ0 )e−z∗ −ω∗ k∗ Using the resource constraint, we can get c∗ k∗ : y∗ i∗ c∗ = g − k∗ k∗ s⋆ k∗ These ratios will give us the remaining steady-state levels and ratios: k∗ =y∗ ( c∗ = 4.3 y∗ −1 ) k∗ i∗ = c∗ k∗ k∗ i∗ k∗ k∗ g∗ =gy y∗ Liquidity Premium Using the aggregate wage equation, we can get the following for w̃∗ : w̃∗ = w∗ 23 Page 89 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Combining this result with the wage Phillips curve, we get the following: w∗h = w∗ 1 + λw,∗ We can use the FOC for consumption and the labor supply to pin down εh and λ1∗ % −γc (1+γ ) ε [c∗ (1 − z )] (1 − εh h∗ h ) − λ1∗ = 0 e b (1−γ ) −(1 + γh )εb c∗ c (1 − −γc % (1−γc ) h (1+γh ) εh hγ∗h + λ1∗ w∗h = 0 ) (1 − ε h ) ∗ z ε Finally, the government bond rate is calculated from λ1∗ − εb∗ εs∗ = βR∗ π∗ π∗ − εb∗ εs∗ −γ z 1 = R∗ −γ z C βe βe C λ∗ ´¹¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¶ λ1∗ −γC z e π∗ R∗P Noting that R∗P = π∗ βe−γC z we can write R∗P − R∗ εb∗ εs∗ = 1 . R∗P λ∗ This is the liquidity premium in steady state. 5 Log Linearization Hatted variables refer to log deviations from steady-state (x̂ = ln ( xx∗t )): ln εjt = ρj ln εjt−1 + ηtj In the cases of zt , ωt , and νt , we have that x̂ = xt − x∗ as these variables are already in logs. 24 Page 90 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Households’ First Order Conditions % 1 ez (ĉt−1 − ẑt ) % ĉt + γc 1 − ez 1 − e%z % 1 − γc ez 1 h b h ĉt + (1 − γc ) (ĉt−1 − ẑt ) λ̂t + ŵt − ε̂t − ε̂t − 1 − e%z 1 − e%z ε̂bt − λ̂1t − γc h εh h1+γ ∗ ) ĥt = 0 h 2 ) (1 − εh h1+γ ∗ R P − R∗ R∗ λ̂1t = ∗ P (ε̂st + ε̂bt ) + P (R̂t + Et [(λ̂1t+1 − π̂t+1 − γC ẑt+1 ]) R∗ R∗ (32) (33) − (γh + γc (1 + γh ) (34) λ̂1t = Et [λ̂1t+1 − γC ẑt+1 + R̂t − π̂t+1 ] (35) λ̂1t = (ln εit + λ̂2t ) − S ′′ (ı̂t − ı̂t−1 ) + βe(1−γC )γ S ′′ Et (ı̂t+1 − ı̂t ) (36) k + ût+1 )] + λ2∗ λ̂2t = βe−γC z∗ −ω∗ [λ1∗ u∗ r∗k Et (−γC ẑt+1 − ω̂t+1 + λ̂1t+1 + r̂t+1 (37) + βe−γC z∗ −ω∗ [(1 − δ0 )λ2∗ Et (−γC ẑt+1 − ω̂t+1 + λ̂2t+1 ) − λ2∗ δ1 u∗ Et ût+1 ] δ2 λ̂1t = λ̂2t + u∗ ût − r̂tk δ1 εi∗ i∗ εi i ∗ k̂t = (1 − ) (k̂t−1 − ẑt − ω̂t ) + ∗ (ε̂it + ı̂t ) − δ1 u∗ e−z∗ −ω∗ ût k∗ k∗ k̂te = ût + k̂t−1 − ẑt − ω̂t (38) (39) (40) Capital-Labor Ratio k̂te = ŵt − r̂tk + ĥdt (41) Real Marginal Costs ̂ t = (1 − α) ŵt + αr̂tk − ε̂at mc (42) The New Keynesian Phillips Curve for Inflation (1 − βζp e(1−γC )z∗ )(1 − ζp ) λp,∗ ̂ t] + [ λ̂p,t + mc (1−γ )z ∗ C 1 + λp,∗ (1 + βιp e )ζp ιp βe(1−γC )z∗ + π̂ + Et π̂t+1 t−1 1 + βιp e(1−γC )z∗ 1 + βιp e(1−γC )z∗ π̂t = 25 Page 91 of 112 (43) Authorized for public release by the FOMC Secretariat on 1/12/2024 Wage Mark-Up h µ̂w t = ŵt − ŵt (44) The New Keynesian Phillips Curve for Wages ŵt = 1 1 + βe(1−γC ŵ + )z∗ t−1 βe(1−γC )z∗ βe(1−γC )z∗ ŵ + (Et π̂t+1 + Et ẑt+1 )+ t+1 1 + βe(1−γC )z∗ 1 + βe(1−γC )z∗ (45) ιw 1 + ιw βe(1−γC )z∗ (π̂ + ẑ ) − (π̂t + ẑt )+ t−1 t−1 1 + βe(1−γC )z∗ 1 + βe(1−γC )z∗ λw,∗ 1 − βζw e(1−γC )z∗ 1 − ζw λ̂w,t − µ̂w [ t ] (1−γ )z ζw 1 + λw,∗ 1 + βe C ∗ The Aggregate Resource Constraint y∗ c∗ i∗ (ŷt − ĝt ) = ĉt + ı̂t g∗ c ∗ + i∗ c∗ + i ∗ (46) The Production Function ŷt = 1 (ln εat + αk̂te + (1 − α) ĥdt ) mc∗ (47) Labor Market Clearing Condition ĥt = ĥdt (48) Monetary Authority’s Reaction Function R̂t = ρR R̂t−1 + (1 − ρR ) [(1 − ψ1 ) π̂t∗ + M ψ2 1 ψ1 1 ( ∑ π̂t+j ) + ( ∑ ŷt+j )] + ∑ ξˆt−j,j 4 j=−2 4 j=−2 j=0 (49) 26 Page 92 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 6 Measurement 6.1 National Income Accounts The model economy’s basic structure, with the representative household consuming a single good and accumulating capital using a different good, differs in some important ways from the accounting conventions of the Bureau of Economic Analysis (BEA) underlying the National Income and Product Accounts (NIPA). In particular 1. The BEA treats household purchases of long-lived goods inconsistently. If classifies purchases of residential structures as investment and treats the service flow from their stock as part of Personal Consumption Expenditures (PCE) on services. The BEA classifies households purchases of all other durable goods as consumption expenditures. No service flow from the stock of household durables enters measures of current consumption. In the model, all long-lived investments add to the productive capital stock. 2. The BEA treats all government purchases as government consumption. However, government at all levels makes purchases of investment goods on behalf of the populace. In the model, these should be treated as additions to the single stock of productive capital. 3. The BEA sums PCE and private expenditures on productive capital (Business Fixed Investment and Residential Investment), with government spending, inventory investment, and net exports to create Gross Domestic Product. The model features only the first three of these. To bridge these differences, we create four model consistent NIPA measures from the BEA NIPA data. 1. Model-consistent GDP. Since the model’s capital stock includes both the stock of household durable goods and the stock of government-purchased capital, a model-consistent GDP series should include the value of both stocks’ service flows. To construct these, we followed a five-step procedure. (a) We begin by estimating a constant (by assumption) service-flow rate by dividing the nominal value of housing services from NIPA Table 2.4.5 by the beginning-of-year value of the residential housing stock from the 27 Page 93 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 BEA’s Fixed Asset Table 1.1. We use annual data and average from 1947 through 2014. The resulting estimate is 0.096. That is, the annual value of housing services equals approximately 10 percent of the housing stock’s value each year. (b) In the second step, we estimate estimate constant (by assumption) depreciation rates for residential structures, durable goods, and government capital. We constructed these by first dividing observations of value lost to depreciation over a calendar year by the end-of-year stocks. Both variables were taken from the BEA’s Fixed Asset Tables. (Table 1.1 for the stocks and Table 1.3 for the deprecation values.) We then averaged these ratios from 1947 through 2014. The resulting estimates are 0.021, 0.194, and 0.044 for the three durable stocks. (c) In the third step, we calculated the average rates of real price depreciation for the three stocks. For this, we began with the nominal values and implicit deflators for PCE Nondurable Goods and PCE Services from NIPA Table 1.2. We used these series and the Fisher-ideal formula to produce a chain-weighted implicit deflator for PCE Nondurable Goods and Services. Then, we calculated the price for each of the three durable good’s stocks in consumption units as the ratio of the implicit deflator taken from Fixed Asset Table 1.2 to this deflator. Finally, we calculated average growth rates for these series from 1947 through 2014. The resulting estimates equal 0.0029, −0.0223, and 0.0146 for residential housing, household durable goods, and government-purchased capital. (d) The fourth combines the previous steps’ calculations to estimate constant (by assumption) service-flow rates for household durable goods and government-purchased capital. To implement this, we assumed that all three stocks yield the same financial return along a steady-state growth path. These returns sum the per-unit service flow with the appropriately depreciated value of the initial investment. This delivers two equations in two unknowns, the two unknown service-flow rates. The resulting estimates are 0.29 and 0.12 for household durable goods and governmentpurchased capital. (e) The fifth and final step uses the annual service-flow rates to calculate real 28 Page 94 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 and nominal service flows from the real and nominal stocks of durable goods and government-purchased capital reported in Fixed Asset Table 1.1. This delivers an annual series. Since the stocks are measured as of the end of the calendar year, we interpret these as the service flow values in the next year’s first quarter. We create quarterly data by linearly interpollating between these values. With these real and nominal service flow series in hand, we create nominal model-consistent GDP by summing the BEA’s definition of nominal GDP with the nominal values of the two service flows. We create the analogous series for model-consistent real GDP by applying the Fisher ideal formula to the nominal values and price indices for these three components. 2. Model-consistent Investment. The nominal version of this series sums nominal Business Fixed Investment, Residential Investment, PCE Durable Goods, and government investment expenditures. The first three of these come from NIPA Table 1.1.5, while government investment expenditures sums Federal Defense, Federal Nondefense, and State and Local expenditures from NIPA Table 1.5.5. We construct the analogous series for real Model-consistent Investment by combining these series with their real chain-weighted counterparts found in NIPA Tables 1.1.3 and 1.5.3 using the Fisher ideal formula. By construction, this produces an implicit deflator for Model-consistent investment as well. 3. Model-consistent Consumption. The nominal version of this series sums nominal PCE Nondurable Goods, PCE Services, and the series for nominal services from the durable goods stock. The first two of thse come from NIPA Table 1.1.5. We construct the analogous series for real Modelconsistent consumption by combining these series with their real chainweighted counterparts using the Fisher ideal formula. The two real PCE series come from NIPA Table 1.1.3. Again, this produces an implicit deflator for Model-consistent consumption as a by-product. 4. Model-consistent Government Purchases. Conceptually, the model’s measure of Government Purchases includes all expenditures not otherwise classified as Investment or Consumption: Inventory Investment, Net Exports, and actual Government Purchases. We construct the nominal version of this series simply 29 Page 95 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 by subtracting nominal Model-consistent Investment and Consumption from nominal Model-consistent GDP. We calculate the analogous real series using “chain subtraction.” This applies the Fisher ideal formula to Model-consistent GDP and the negatives of Model-consistent Consumption and Investment. Our empirical analysis requires us to compare model-consistent series measured from the NIPA data with their counterparts from the model’s solution. To do this, we begin by solving the log-linearized system above, and then we feed the model specific paths for all exogenous shocks starting from a particular initial condition. for a given such simulation, the growth rates of Model-consistent Consumption and Investment equal ∆ ln Ct = z∗ + ∆ĉt + zt and ∆ ln It = z∗ + ω∗ + ∆ît + zt + ωt The measurement of GDP growth in the model is substantially more complicated, because the variables Yt and yt denote model output in consumption units. In contrast, we mimic the BEA by using a chain-weighted Fisher ideal index to measure model-consistent GDP. Therefore, we construct an analogus chain-weighted GDP index from model data. Since such an ideal index is invariant to the units with which nominal prices are measured, we can normalize the price of consumption to equal one and employ the prices of investment goods and government purchases relative to current consumption. Our model identifies the first of these relative prices as with investment-specific technology. However, the model characterizes only government purchases in consumption units, because private agents do not care about their division into “real” purchases and their relative price. For this reason, we use a simple autoregression to characterize the evolution of the price of government services in consumption units. Denote this price in quarter t with Ptg . We construct this for the US economy by dividing the Fisher-ideal price index for model-consistent government purchases by that for model-consistent consumption. Then, our model for its evolution is g g g g t ln(Ptg /Pt−1 ) = µg + θgg1 ln(Pt−1 /Pt−2 ) + θgg2 ln(Pt−2 /Pt−3 ) + εgg t . (50) g 2 Here, εgg t ∼ N(0, σgg ). Given an arbitrary normalization of Pt to one for some time period, simulations from (??) can be used to construct simulated values of Ptg for 30 Page 96 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 all other time periods. With these and a simulation from the model of all other variables in hand, we can calculate the simulation’s values for Fisher ideal GDP growth using Qt ≡ Qt−1 √ Q̇Pt Q̇Lt , (51) where the Paasche and Laspeyres indices of quantity growth are Ct + PtI It + PtG (Gt /PtG ) and g ) Ct−1 + PtI It−1 + PtG (Gt−1 /Pt−1 I G Ct + Pt−1 It + Pt−1 (Gt /PtG ) ≡ . I G G Ct−1 + Pt−1 It−1 + Pt−1 (Gt−1 /Pt−1 ) Q̇Pt ≡ (52) Q̇Lt (53) In both (52) and (53), PtI is the relative price of investment to consumption. In equilibrium, this always equals AIt . The above gives a complete recipe for simulating the growth of model-consistent real GDP growth. However, we also embody its insights into our estimation with a log-linear approximation. For this, we start by removing stochastic trends from all variables in (52) and (53), and we proceed by taking a log-linear approximation of the resulting expression. Details are available from the authors upon request. 6.2 Hours Worked Measurement Empirical work using DSGE models like our own typically measure labor input with hours worked per capita, constructed directly from BLS measures of hours worked and the civilian non-institutional population over age 16. However, this measure corresponds poorly with business cycle models because it contains underlying low frequency variation. This fact led us to construct a new measure of hours for the model using labor market trends produced for the FRB/US model and for the Chicago Fed’s in-house labor market analysis. We begin with a multiplicative decomposition of hours worked per capita into hours per worker, the employment rate of those in the labor force, and the laborforce participation rate. The BLS provides CPS-based measures of the last two rates for the US as a whole. However, its measure of hours per worker comes from the Establishment Survey and covers only the private business sector. If we use hours per worker in the business sector to approximate hours per worker in the economy 31 Page 97 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 as a whole, then we can measure hours per capita as Ht HtE EtC LC t = . C Pt EtE LC P t t Here, Ht and Pt equal total hours worked and the total population, HtE /EtE equals hours per worker measured with the Establishment survey, EtC /LC t equals one C C minus the CPS based unemployment rate, and Lt /Pt equals the CPS based laborforce participation rate. Our measure of model-relevant hours worked deflates each component on the right-hand side by an exogenously measured trend. The trend for the unemployment rate comes from the Chicago Fed’s Microeconomics team, while those for hours per worker and labor-force participation come from the FRB/US model files. 6.3 Inflation Our empirical analysis compares model predictions of price inflation, wage inflation, inflation in the price of investment goods relative to consumption goods, and inflation expectations with their observed values from the U.S. economy. We describe our implementations of these comparisons sequentially below. 6.3.1 Price Inflation Our model directly characterizes the inflation rate for Model-consistent Consumption. In principle, this is close to the FOMC’s preferred inflation rate, that for the implicit deflator of PCE. However, in practice the match between the two inflation rates is poor. In the data, short-run movements in food and energy prices substantially influences the short-run evolution of PCE inflation. Our model lacks such a volatile sector, so if we ask it to match observed short-run inflation dynamics, it will attribute those to transitory shocks to intermediate goods’ producers’ desired markups driven by λpt . To avoid this outcome, we adopt a different strategy for matching model and data inflation rates, which follows that of Justiniano, Primiceri, and Tambalotti (2013). This relates three observable inflation rates – core CPI inflation, core PCE inflation, and market-based PCE inflation – to Model-consistent consumption inflation using 32 Page 98 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 auxiliary observation equations. For core PCE inflation, this equation is πtp1 = π∗ + π∗p1 + β1p1 π̂t + β2p1 πtD + εp1 t (54) In (54) as elsewhere, π∗ equals the long-run inflation rate. The constant π∗1 is an adjustment to this long-run inflation rate which accounts for possible long-run differences between realized inflation and the FOMC’s goal of π⋆ . The right-hand side’s inflation rates, π̂t and πtD equal Model-consistent consumption inflation and PCE Durables inflation. We refer to the coefficients multiplying them, β1p1 and β2p1 , as the inflation loadings. We include PCE Durables inflation on the righthand side of (54) because the principle adjustment required to transform Modelconsistent inflation into core PCE inflation is the replacement of the price index for durable goods services with that for durable goods purchases. The disturbance term εp1 t follows a first-order autoregression with autocorrelation ϕp1 and normally distributed innovations with mean zero and standard deviation σp1 . The other two observed inflation measures, market-based PCE inflation and core CPI inflation, have identically specified observation equations. We use p2 and p3 in superscripts to denote these equations parameters and error terms, and we use the same expressions as subscripts to denote the parameters governing the evolution of p2 p3 their error terms. We assume that the error terms εp1 t , εt , and εt are independent of each other at all leads and lags. To produce forecasts of inflation with these these three observation equations, we must forecast their right-hand side variables. The model itself gives forecasts of π̂t . The forecasts of durable goods inflation come from a second-order autoregression. D πtD = θ0D + θ1D πt−1 + εD t (55) Its innovation is normally distributed and serially uncorrelated with standard deviation σD . 6.3.2 Wage Inflation Although observed wage inflation does not feature the same short-run variability as does price inflation, it does include the influences of persistent demographic labor-market trends which we removed ex ante from our measure of hours worked. 33 Page 99 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Therefore, we follow the same general strategy of relating observed measures of wage inflation to the model’s predicted wage inflation with a error-augmented observation equation. For this, we employ two measures of compensation per hour, Earnings per Hour and Total Compensation per Hour. In parallel with our notation for inflation measures, we use w1 and w2 to denote these two wage measures of wage inflation. The observation equation for Earnings per Hour is πtw1 = z∗ + π∗w1 + β w1 π̂tw + εw1 t (56) Just as with the price inflation measurement errors, εw1 follows a first-order t autoregression with autocorrelation ϕw1 and innovation standard deviation σw1 . The observation equation for Total Compensation per Hour is analogous to (56). 6.3.3 Relative Price Inflation To empirically ground investment-specific technological change in the model, we use an error-augmented observation equation to relate the relative price of investment to consumption, both model-consistent measures constructed from NIPA and Fixed Asset tables as described above, with the model’s growth rate of the rate of technological transformation between these two goods, ωt . C/I πt = ωt + ε t C/I Here, we use the superscript C/I to indicate that the variables characterize the price C/I of Consumption relative to Investment. The measurement error εt follows a firstorder autoregression with autocorrelation ϕC/I and normally-distributed innovations with standard deviation σC/I . 6.3.4 Inflation Expectations We also discipline our model’s inferences about the state of the economy by comparing expectations of one-yea and 10-year inflation from the Survey of Professional Forecasters with the analogous expectations from our model. Just as with all of the other inflation measures, we allow these two sets of expectations to differ from each other by including serially correlated measurement errors. The 34 Page 100 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 observation equations are πte4 = π∗ + π∗e4 1 4 + ∑ Et [π̂t+i ] + εe4 t 4 i=1 πte40 = π∗ + π∗e40 + 1 40 ∑ Et [π̂t+i ] + εe40 t 40 i=1 The two measurement errors follow mutually-independent first-order autoregressions with autocorrelations ϕe4 and ϕe40 and innovation standard deviations σe4 and σe40 . 6.4 Interest Rates and Monetary Policy Shocks Since our model features forward guidance shocks, it has non-trivial implications for the current policy rate as well as for expected future policy rates. We use two distinct but complementary approaches to disciplining the parameters governing their realizations, the elements of Σ1 , using data. The first method compares the model’s monetary policy shocks to high-frequency interest-rate innovations informed by event studies, such as that of Gürkaynak, Sack, and Swanson (2005). Those authors applied a factor structure to innovations in implied expected interest rates from futures prices around FOMC policy announcement dates. Specifically, they show that the vector of M implied interest rate changes following an FOMC policy announcement, ∆r, can be written as ∆r = Λf + η Where f is a 2 × 1 vector of factors, Λ is a M × 2 matrix of factor loadings, and η is an M × 1 vector of mutually independent shocks. Denoting the 2 × 2 diagonal variance covariance matrix of f with Σf and the M ×M diagonal variance-covariance matrix of η with Ψ, we can express the observed variance-covariance matrix of ∆r as ΛΣf Λ′ + Ψ. Our model has implications for this same variance covariance matrix. For this, use the model’s solution to express the changes in current and future expected interest rates following monetary policy shocks as ∆r = Γ1 ε1 . Here, ε1 is the vector which collects the current monetary policy shock with M − 1 forward guidance shocks, and Γ1 is an M × M matrix. In general, Γ1 does not simply equal the identity matrix, because current and future inflation and output gaps respond to the 35 Page 101 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 monetary policy shocks and thereby influence future monetary policy “indirectly” through the interest rate rule. Given this solution for ∆r, we can calculate its variance-covariance matrix as Γ1 Σ1 Γ′1 . Equating these two expressions and solving for Σ1 yields ′ ′−1 Σ1 = Γ−1 1 (ΛΣf Λ + Ψ) Γ1 . The second approach to disciplining Σ1 is more traditional: directly compare quarterly observations of the current policy rate and expected future interest rates – from market prices, surveys of market participants, or both – with their implied values from the model given a particular realization of the vector of monetary policy shocks. We use both methods in the estimation procedure described below. 7 Calibration and Bayesian Estimation As we noted in the introduction, we follow a two-stage approach to the estimation of our model’s parameters. In a calibration stage, we set the values of selected parameters so that the model has empirically-sensible implications for long-run averages from the U.S. economy. In this stage, we also enforce several normalizations and a judgemental restriction on one of the measurement error variances. In the second stage, we estimate the model’s remaining parameters using standard Bayesian methods. We employ standard prior distributions, but those governing monetary policy shocks deserve further elaboration. Our estimation requires the variance-covariance matrix of monetary policy shocks to be consistent with the factor-structure of interest rate innovations used by Gürkaynak, Sack, and Swanson (2005), as described above. Therefore, we parameterize Σ1 in terms of Λ, Σf , Ψ, and the model parameters which influence Γ1 . We then center our priors for Λ, Σf , and Ψ at their estimates from event-studies. However, we do not require our estimates to equal their prior values. Our Bayesian estimation procedure employs quarterly data on expected future interest rates, the posterior likelihood function includes Λ, Σf , and Ψ as free parameters. It is well known that Λ and Σf are not separately identified, so we impose two scale normalizations and one rotation normalization on Λ. The rotation normalization requires that the first factor, which we label “Factor A”, is 36 Page 102 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 the only factor influence the current policy rate. That is, the second factor, “Factor B” influences only future policy rates. Gürkaynak, Sack, and Swanson (2005) call Factors A and B the “target” and “path” factors. Our estimation’s sample period begins in the first quarter of 1993 and ends in the fourth quarter of 2016. Of course, the FOMC substantially changed its operating procedures in the aftermath of its persistent stay at the Zero Lower Bound, so it would be unwise to imagine the data from this entire period being generated from our model with time-invariant parameters. For this reason, we estimate the model twice. For the first sample, which runs from 1993Q1 through 2008Q3, we estimate all model parameters while allowing for four quarters of forward guidance. For the second sample, we estimate the parameters governing monetary policy shocks allowing for ten quarters of forward guidance, adjust the average rate of Hicksneutral productivity growth to bring potential GDP growth rate from its first-sample value of 3 percent down to 2 percent, and hold all other model parameters fixed at their first-sample posterior-mode values. We report the results of our two-stage two-sample estimation in a series of tables. Table 1 reports our most notable calibration targets. The long-run policy rate equals 1.1 percent on a quarterly basis. We target a two percent growth rate of per capita GDP. Given an average population growth rate of one percent per year, this implies that our potential GDP growth rate equals three percent. The other empirical moments we target are a nominal investment to output ratio of 26 percent and nominal government purchases to output ratio of 15 percent. Finally, we target a capital to output ratio of approximately 10 on a quarterly basis. Table 2 lists the parameters which we calibrate along with their given values. The table includes many more parameters than there are targets in Table 1. This is because Table 1 omitted calibration targets which map one-to-one with particular parameter values. For example, we calibrate the steady-state capital depreciation rate (δ0 ) using standard methods applied to data from the Fixed Asset tables. It is also because Table 2 lists several parameters which are normalized prior to estimation. Most notable among these are the three factor loadings listed at the table’s bottom. Tables 3 and 4 report prior distributions and posterior modes for the model’s remaining paramters, for the first and second samples respectively. 37 Page 103 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Table 1: Calibration Targets Description Expression Fixed Interest Rate (quarterly, gross) R Per-Capita Steady-State Output Growth Rate (quarterly) Yt+1 /Yt Investment to Output Ratio It /Yt Capital to Output Ratio Kt /Yt Fraction of final good output spent on public goods Gt /Yt Value 1.011 1.005 0.260 10.763 0.153 Table 2: First Sample Calibrated Parameters Parameter Discount Factor Steady-State Measured TFP Growth (quarterly) Investment-Specific Technology Growth Rate Elasticity of Output w.r.t Capital Services Steady-State Wage Markup Steady-State Price Markup Steady-State Scale of the Economy Steady-State Inflation Rate (quarterly) Steady-State Depreciation Rate Steady-State Marginal Depreciation Cost Nominal Output over Nominal Private Purchases Std. Dev Long-Run Inflation Expectations Measurement Error Long-Run Inflation Expectations (Constant CPI Adjustment) Average Earnings Constant Average Total Compensation Constant Loading Compensation Loading Core PCE Constant for Relative Price Inflation Loading 0 Factor A Loading 0 Factor B Loading 4 Factor B 38 Page 104 of 112 Symbol β z∗ ω∗ α λw ∗ λp∗ H∗ π∗ δ0 δ1 g∗ σe40 π∗e40 π∗w1 π∗w2 β1w2 β1p1 π∗G λ0,1 λ0,2 λ4,2 Value 0.986 0.489 0.371 0.401 1.500 1.500 1.000 0.500 0.016 0.039 0.847 0.010 0.122 -0.237 -0.202 1.000 1.000 0.252 0.981 0.000 0.951 Authorized for public release by the FOMC Secretariat on 1/12/2024 Table 3: First Sample Estimated Parameters Parameter Depreciation Curve Active Price Indexation Rate Active Wage Indexation Rate External Habit Weight Labor Supply Elasticity Price Stickiness Probability Wage Stickiness Probability Adjustment Cost of Investment Elasticity of Intertemporal Substitution Interest Rate Response to Inflation Interest Rate Response to Output Interest Rate Smoothing Coefficient Autoregressive Coefficients of Shocks Discount Factor Inflation Drift Exogenous Spending Investment Liquidity Preference Price Markup Wage Markup Neutral Technology Investment Specific Technology Moving Average Coefficients of Shocks Price Markup Wage Markup Standard Deviations of Innovations Discount Factor Inflation Drift Exogenous Spending Symbol Density Prior Mean Std.Dev Posterior Mode δ2 δ1 ιp ιw λ γH ζp ζw ϕ γc ψ1 ψ2 ρR G B B B N B B G N G G B 1.0000 0.5000 0.5000 0.7500 0.6000 0.8000 0.7500 3.0000 1.5000 1.7000 0.2500 0.8000 0.150 0.150 0.150 0.025 0.050 0.050 0.050 0.750 0.375 0.150 0.100 0.100 0.499 0.280 0.082 0.790 0.591 0.833 0.904 4.326 1.915 1.833 0.488 0.791 ρb ρπ ρg ρi ρs ρλp ρλw ρν ρω B B B B B B B B B 0.5000 0.9900 0.6000 0.5000 0.6000 0.6000 0.5000 0.3000 0.3500 0.250 0.010 0.100 0.100 0.200 0.200 0.150 0.150 0.100 0.850 0.998 0.920 0.759 0.841 0.687 0.668 0.496 0.407 θλp θλw B B 0.4000 0.4000 0.200 0.200 0.608 0.306 σb σπ σg U I U 0.5000 0.0150 1.0000 2.000 0.0075 2.000 1.187 0.094 2.500 Notes: Distributions (N) Normal, (G) Gamma, (B) Beta, (I) Inverse-gamma-1, (U) Uniform 39 Page 105 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 First Sample Estimated Parameters (Continued) Parameter Investment Liquidity Preference Price Markup Wage Markup Neutral Technology Investment Specific Technology Relative Price of Cons to Inv Monetary Policy Unanticipated 1Q Ahead 2Q Ahead 3Q Ahead 4Q Ahead Total Earnings Loading 1 Standard Deviation AR(1) Coefficient Total Compensation Standard Deviation AR(1) Coefficient Core PCE Constant Loading 2 Standard Deviation AR(1) Coefficient Market-Based Core PCE Constant Loading 1 Loading 2 Standard Deviation Symbol Density Prior Mean Std.Dev Posterior Mode σi σs σλ p σλ w σν σω σ ci I U I I U I I 0.2000 0.5000 0.1000 0.1000 0.5000 0.2000 0.0500 0.200 2.000 1.000 1.000 0.250 0.100 2.000 0.618 0.390 0.069 0.031 0.504 0.183 0.215 σµ0 σµ1 σµ2 σµ3 σµ4 N N N N N 0.0050 0.0050 0.0050 0.0050 0.0050 0.0025 0.0025 0.0025 0.0025 0.0025 0.012 0.012 0.007 0.009 0.010 β1w1 σw1 ϕw1 N I B 0.8000 0.0500 0.4000 0.100 0.100 0.100 0.824 0.147 0.624 σw2 ϕw2 I B 0.0500 0.4000 0.100 0.100 0.169 0.343 π∗p1 β2p1 σp1 ϕp1 N N I B -0.1000 0.0000 0.0500 0.2000 0.100 1.000 0.100 0.100 -0.087 0.014 0.048 0.091 π∗p2 β1p2 β2p2 σp2 N N N I -0.1000 1.0000 0.0000 0.0500 0.100 0.100 1.000 0.100 -0.123 1.102 0.028 0.039 Notes: Distributions (N) Normal, (G) Gamma, (B) Beta, (I) Inverse-gamma-1, (U) Uniform 40 Page 106 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 First Sample Estimated Parameters (Continued) Parameter AR(1) Coefficient Core CPI Constant Loading 1 Loading 2 Standard Deviation AR(1) Coefficient PCE Durable Goods Inflation Constant 1st Lag Coefficient 2nd Lag Coefficient Standard Deviation Relative Price Inflation 1st Lag Coefficient 2nd Lag Coefficient Standard Deviation Factor A Loading 1 Loading 2 Loading 3 Loading 4 Standard Deviation Factor B Loading 1 Loading 2 Loading 3 Standard Deviation Symbol Density Prior Mean Std.Dev Posterior Mode ϕp2 B 0.2000 0.100 0.128 π∗p3 β1p3 β2p3 σp3 ϕp3 N N N I B 0.0500 1.0000 0.0000 0.1000 0.4000 0.100 0.100 1.000 0.100 0.200 0.047 0.804 0.119 0.076 0.597 θ∗D θD1 θD2 σD N N N I -0.3500 0.4500 0.4000 0.2000 0.100 0.200 0.200 2.000 -0.356 0.430 0.362 0.287 θG1 θG2 σG N N I 0.0000 -0.1000 0.5000 0.500 0.500 2.000 0.279 0.006 0.811 λ1,1 λ2,1 λ3,1 λ4,1 σF 1 N N N N N 0.6839 0.5224 0.4314 0.3243 0.1000 0.200 0.200 0.200 0.200 0.0750 1.256 0.857 0.361 0.032 0.041 λ1,2 λ2,2 λ3,2 σF 2 N N N N 0.3310 0.6525 0.8059 0.1000 0.200 0.200 0.200 0.0750 0.698 1.162 1.199 0.072 Notes: Distributions (N) Normal, (G) Gamma, (B) Beta, (I) Inverse-gamma-1, (U) Uniform 41 Page 107 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 Table 4: Second Sample Estimated Parameters Parameter Total Earnings Constant Loading 1 Standard Deviation AR(1) Coefficient Total Compensation Constant Standard Deviation AR(1) Coefficient Core PCE Loading 2 Standard Deviation AR(1) Coefficient Market PCE Constant Loading 1 Loading 2 Standard Deviation AR(1) Coefficient CPI Constant Loading 1 Loading 2 Standard Deviation AR(1) Coefficient Durable Goods Inflation Constant Standard Deviation Relative Price Inflation 42 Page 108 of 112 Symbol Prior Mean Std.Dev Posterior Mode β∗w1 β1w1 σw1 ϕw1 -0.2370 0.8242 0.1468 0.6239 0.200 0.200 0.100 0.200 -0.096 0.252 0.183 0.529 β∗w2 σw2 ϕw2 -0.2023 0.1687 0.3430 0.200 0.100 0.200 -0.142 0.233 0.351 β2p1 σp1 ϕp1 0.0281 0.0481 0.0913 0.100 0.100 0.150 0.232 0.143 0.256 π∗p2 β1p2 β2p2 σp2 ϕp2 -0.1230 1.1022 0.0139 0.0755 0.5972 0.100 0.150 0.100 0.100 0.150 -0.114 0.358 0.219 0.121 0.527 π∗p3 β1p3 β2p3 σp3 ϕp3 0.0475 0.8039 0.1192 0.0388 0.1278 0.100 0.150 0.100 0.100 0.150 -0.022 0.305 0.206 0.085 0.220 π∗D σD -0.4500 0.5000 0.200 0.150 -0.463 0.291 Authorized for public release by the FOMC Secretariat on 1/12/2024 Second Sample Estimated Parameters (Continued) Parameter Constant Standard Deviation Factor A Loading 0 Loading 1 Loading 2 Loading 3 Loading 4 Loading 6 Loading 7 Loading 8 Loading 9 Loading 10 Standard Deviation Factor B Loading 0 Loading 1 Loading 2 Loading 3 Loading 4 Loading 5 Loading 6 Loading 7 Loading 9 Standard Deviation Standard Deviations of Monetary Policy Innovations Unanticipated 1Q Ahead 2Q Ahead 3Q Ahead 43 Page 109 of 112 Symbol Prior Mean Std.Dev Posterior Mode π∗G σG 0.8900 0.8143 0.400 0.080 -0.146 1.023 λ0,1 λ1,1 λ2,1 λ3,1 λ4,1 λ6,1 λ7,1 λ8,1 λ9,1 λ10,1 σF1 0.0180 0.0574 0.1941 0.3996 0.6520 1.2266 1.5237 1.8139 2.0914 2.3523 0.0442 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.100 0.158 0.356 0.496 0.623 0.820 0.990 1.122 1.102 1.087 2.515 0.072 λ0,2 λ1,2 λ2,2 λ3,2 λ4,2 λ5,2 λ6,2 λ7,2 λ9,2 σF2 -0.0181 0.2211 0.3679 0.4424 0.4612 0.4370 0.3817 0.3032 0.1074 0.0334 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.100 0.029 0.039 0.070 0.095 0.123 0.138 0.167 0.184 0.229 0.429 σµ0 σµ1 σµ2 σµ3 0.0061 0.0021 0.0004 0.0019 0.005 0.005 0.005 0.005 0.011 0.010 0.010 0.009 Authorized for public release by the FOMC Secretariat on 1/12/2024 Second Sample Estimated Parameters (Continued) Parameter 4Q Ahead 5Q Ahead 6Q Ahead 7Q Ahead 8Q Ahead 9Q Ahead 10Q Ahead 44 Page 110 of 112 Symbol Prior Mean Std.Dev Posterior Mode σµ4 σµ5 σµ6 σµ7 σµ8 σµ9 σµ10 0.0001 0.0025 0.0019 0.0011 0.0001 0.0014 0.0028 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.010 0.010 0.010 0.010 0.009 0.010 0.010 Authorized for public release by the FOMC Secretariat on 1/12/2024 References Calvo, G. A. (1983). Staggered Prices in a Utility-Maximizing Framework. Journal of Monetary Economics 12, 383–398. 1, 9, 10 Campbell, J. R., C. L. Evans, J. D. Fisher, and A. Justiniano (2012). Macroeconomic effects of Federal Reserve forward guidance. Brookings Papers on Economic Activity Spring, 1–54. 1 Campbell, J. R., J. D. M. Fisher, A. Justiniano, and L. Melosi (2016). Forward guidance and macroeconomic outcomes since the financial crisis. In M. Eichenbaum and J. Parker (Eds.), NBER Macroeconomics Annual 2016, Volume 31. University of Chicago Press. 1 Christiano, L., M. Eichenbaum, and C. Evans (2005). Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of Political Economy 113 (1), 1–45. 7 Fisher, J. (2015, March-April). On the structural interpretation of the smets-wouters “risk premium” shock. Journal of Money, Credit and Banking 47 (2-3), 511–516. 4, 6 Fisher, J. D. M. (2006). The dynamic effects of neutral and investment-specific technology shocks. Journal of Political Economy 114 (3), 413–451. 2 Gürkaynak, R., B. Sack, and E. Swanson (2005). Do actions speak louder than words? International Journal of Central Banking 1 (1), 55–93. 35, 36, 37 Justiniano, A., G. E. Primiceri, and A. Tambalotti (2010). Investment shocks and business cycles. Journal of Monetary Economics 57 (2), 132 – 145. 1 Justiniano, A., G. E. Primiceri, and A. Tambalotti (2013, April). Is there a tradeoff between inflation and output stabilization? American Economic Journal: Macroeconomics 5 (2), 1–31. 32 King, R. G., C. I. Plosser, and S. T. Rebelo (1988a). Production, growth and business cycles I. the basic neoclassical model. Journal of Monetary Economics 21, 195–232. 2 45 Page 111 of 112 Authorized for public release by the FOMC Secretariat on 1/12/2024 King, R. G., C. I. Plosser, and S. T. Rebelo (1988b). Production, growth and business cycles II. new directions. Journal of Monetary Economics 21, 309–341. 2 Laséen, S. and L. E. Svensson (2011). Anticipated alternative policy-rate paths in policy simulations. International Journal of Central Banking. 1 46 Page 112 of 112