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Working Paper Series Seasonal Solow Residuals and Christmas: A Case for Labor Hoarding and Increasing Returns R. Anton Braun and Charles L. Evans Working Papers Series Issues in Macroeconomics Research Department Federal Reserve Bank of Chicago October 1991 (W P -91-20) FEDERAL RESERVE BANK OF CHICAGO Seasonal Solov Residuals and Christmas: A Case for Labor Hoarding and Increasing Returns * R. Anton Braun _ , ** Charles L. Evans Revised October 1991 Abstract In aggregate unadjusted data, measured Solov residuals exhibit large seasonal variations. Total Factor Productivity grows rapidly in the fourth quarter at an annual rate of 24% and regresses sharply in the first quarter at an annual rate of -30%. This paper considers two potential explanations for the measured seasonal variation in the Solow residual: labor hoarding and increasing returns to scale. Using a specification that allows for no exogenous seasonal variation in technology and a single seasonal demand shift in the fourth quarter, we ask the following question: How much of the total seasonal variation in the measured Solow residual can be explained by Christmas? The answer to this question is surprising. With increasing returns and time varying labor effort, Christmas is sufficient to explain the seasonal variation in the Solow residual, consumption, average productivity and output in all four quarters. Our analysis of seasonally unadjusted data uncovers important roles for labor hoarding and increasing returns which are difficult to identify in adjusted data. JEL Classification(s): ★ 131, 023 Department of Economics Rouss Hall University of Virginia Charlottesville, VA 22901 (804) 924-7845 Research Department Federal Reserve Bank of Chicago P.0. Box 834 Chicago, IL 60690-0834 (312) 322-5812 Ve thank Marty Eichenbaum, Jim Hamilton, Bob Chirinko and seminar participants at the Federal Reserve Bank of Chicago for helpful comments. All opinions expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Chicago, or the Federal Reserve System. 1. Introduction Prescott [1986] has argued that the variability of Solow's residual is a reasonable estimate of the variability of exogenous technology shocks. When Solow's residual is measured using seasonally unadjusted data for the postwar U.S. economy, it exhibits large seasonal variations, growing rapidly in the fourth quarter at an annual rate of 24% and falling sharply in the first quarter at an annual rate of -30%. This paper starts from the premise that it is implausible to attribute seasonal variation of this magnitude to changes in the state of technology.^ We present a model in which fluctuations arise from a single demand shift, Christmas. all seasonal This demand shift together with misspecification of the traditional production function leads to large seasonal variation in the Solow residual. for misspecification, scale. Even when labor hoarding technological growth and is We consider two candidates external increasing aseasonal, either returns to candidate in isolation can induce spurious seasonality in the Solow residual. Our general equilibrium analysis indicates that: (1) the economy's seasonal patterns in all four quarters may be a response simply to a fourth quarter consumption demand shift, and (2) a combination of labor hoarding and external increasing returns are important for replicating these patterns variables for the postwar U.S. economy. Since in aggregate our analysis quantity identifies important roles for labor hoarding and increasing returns, these results have implications for nonseasonal macroeconomic models. By focusing on seasonal fluctuations we are able to exploit information Beaulieu the first Argentina no reason and Miron [1991] cast doubt on weather explanations by showing that quarter is a period of negative output growth in Australia and as well as the United States. In the Southern Hemisphere, there is to expect a negative technology seasonal in the first quarter. 2 that is typically ignored in empirical macroeconomic research, 2 This is important because the research strategy of selecting a macroeconomic theory based upon its ability to match certain statistical moments adjusted data example, often fails to discriminate Murphy-Shleifer-Vishny [1989] among competing in seasonally theories. survey the many similarities between technology shock-driven and increasing returns equilibrium theories. seasonality, For Ignoring they conclude that both theories can be modified in plausible ways to generate the same implications for aggregate quantity variables. In a similar vein, Cooper and Haltiwanger [1989] conclude that production bunching may arise due to either nonconvexities in production or the arrival of 3 technology shocks in bunches. The literature on procyclical productivity also produces mixed results. labor hoarding and Rotemberg and Summers [1990] show that the combination of labor hoarding and price inflexibility can generate procyclical total factor productivity without appealing to technology shocks. On the other hand, Burnside, Eichenbaum, and Rebelo [1990] find that labor hoarding, price flexibility, and relatively small exogenous technology shocks can generate procyclical total factor productivity. The principal difficulty in selecting one theory, of course, is that each school of thought interprets the postwar period as being dominated by either supply shocks (RBC) or demand shocks (Keynesian labor hoarding, increasing 2 Other researchers have noted this also. Barsky and Miron [1989] argue that the seasonal cycle facts for the U.S. contain information for evaluating alternative macroeconomic theories. Ghysels [1991] disputes the claim that the seasonal cycle is like the business cycle, but agrees that seasonally unadjusted data contain useful information for identifying propagation mechanisms. 3 Evidence favoring increasing returns is uncovered by Hall [1989], Ramey [1991], and Caballero and Lyons [1990]. Their conclusions rely on the assumed exogeneity of their instruments, which is controversial. Further evidence of increasing returns which does not rely on these instruments would strengthen their case. Chirinko [1991] concludes that increasing returns is more important than either labor hoarding or technology shocks. 3 returns)-- but not both. If economists could identify demand shocks, however, the predicted countercyclical response of labor productivity in RBC models could be compared with the predicted returns and labor hoarding theories. is difficult procyclical response in increasing Needless to say, such an identification to achieve unambiguously in postwar seasonally adjusted time series data. Seasonal fluctuations offer valuable identifying restrictions. For instance, few economists would dispute that Christmas is an important seasonal event which produces fourth quarter. an increased demand On the other hand, seasonal shifters of technology. seasonal finding impulse that is weakened seasonal patterns in the U.S. and that Christmas for it is difficult in by Southern Beaulieu Hemisphere [1990] in the identify bonafide and Miron's countries [1991] resemble Together our assumptions that technology is aseasonal is an important shift in demand restrictions that have strong discriminatory power. Evans to services The contention that weather is an important considerably patterns consumption 4 provide identifying For instance, Braun and find that seasonal Real Business Cycle models predict totally aseasonal patterns in output under these identifying assumptions. transient and anticipated nature of Christmas, households Due to the simply draw down their savings and increase consumption by equal amounts in the fourth quarter leaving the level of output unchanged. RBC models require implausibly large seasonal variations in technology to explain the seasonal pattern in output. Thus, our identifying assumptions offer strong explanation for the seasonal pattern in output. 4 evidence against one In this paper we demonstrate4 Our use of seasonal identifying restrictions is similar to Bernanke and Parkinson's analysis. Using interwar data, Bernanke and Parkinson [1991] investigate procyclical productivity in industrial markets. Under the plausible identifying assumption that the Great Depression was not caused by a series of large technology shocks, they find evidence in favor of increasing returns and labor hoarding. 4 that labor hoarding in conjuction with increasing returns can generate many of the observed patterns in output, consumption, and the measured Solow residual. Modeling seasonal fluctuations with a single Christmas demand shift requires us to model economic agents' responses to anticipated and transitory impulses. First, the anticipated nature of the Christmas seasonal shift leads us to model variations in labor effort as driven by convex costs of adjusting employment. Burnside, Eichenbaum and Rebelo [1990] model labor hoarding by assuming that employment is fixed at the beginning of the period and only labor effort can respond within a period to shocks. effort responds persistence."* only To to induce unanticipated seasonal shocks, In their framework labor exhibiting labor hoarding, quasi-fixed factors must be modeled explicitly. the costs no of noticeable adjusting Second, the transitory nature of seasonal shifts leads us to consider convex costs of adjusting capital. In an economy with external increasing returns, Baxter and King [1990] found a negligible response of output to a purely transitory increase in consumption demand: consumption rose but investment fell, leaving output unchanged. In the absence of adjustment costs, a similar result is to be expected for the case of a fourth quarter Christmas seasonal. If increasing returns is to have a chance, it must be costly to adjust investment. Third, the nontime-separable preferences emphasized by Kydland and Prescott [1982], Eichenbaum, Hansen, and Singleton [1988], and others for business cycle variability also play an important role in propagating the Christmas demand shock beyond simply the fourth quarter. Thus, modeling seasonal fluctuations leads to a specification that incorporates the same propagation mechanisms that receive wide attention in models of the business cycle. In Burnside-Eichenbaum-Rebelo [1990], the impulse response functions of labor effort to innovations in technology and government purchases appear to be zero after the initial period's response. 5 Many of the model's parameters governing returns to scale, the magnitude of adjustment costs and elasticities for work effort are difficult to pin down on a priori grounds. A Generalized Method of Moments (GMM) estimation strategy is used to produce estimates of these parameters. These estimates are then used to evaluate the seasonal growth rates implied by the model. We find that both labor hoarding and increasing returns mechanisms are important for capturing the U.S. variables in the economy's seasonal fluctuations. model, the hypothesis that For each of the real the predicted fluctuations match the data's seasonals cannot be rejected. seasonal The estimated parameterization proves to be remarkably successful at capturing the seasonal pattern in the measured Solow residual as well as the seasonal pattern in output, consumption, and suggest further that average productivity. labor hoarding and Results reported in section 4 nontime-separabilities biggest role in propagating the Christmas demand shock. play the Increasing returns prove to be important for amplifying the seasonal patterns generated by the other features of the model. Finally, this model also offers an explanation for the similar seasonal patterns across countries which Beaulieu and Miron Christmas-like celebrations induce fourth quarter [1991] have documented. preference shifts consumption demand in many Northern and Southern Hemisphere countries. extent that increasing returns to scale and labor hoarding are in To the important features of these other economies as well, then our model predicts a seasonal pattern similar to that found in the U.S. An outline of the remainder of the paper follows. In section 2 the model is described and the seasonal equilibrium growth path is defined. Section 3 contains a description of the data, the estimation strategy and a summary of the estimation results. Section 4 evaluates the seasonal implications of the estimated parameterization and explores the role of the various components of 6 th e m o d e l. I n s e c t io n 5 w e 2. c o n c lu d e b y s u m m a r iz in g o u r r e s u lt s . The Economic Model In this section we describe the model economy. The presentation of the economy leads naturally competitive equilibrium to an optimization problem whose allocation. to a productive externality. take account of solution is the This solution is not Pareto optimal due As we pose the problem, the planner does not the externality. A benevolent social planner could do better by allowing agents to coordinate. This strategy for calculating competitive allocations in distorted economies is discussed at length in Romer[1988]. Preferences The household's period preferences depend upon consumption services c^, leisure services 1^, and (negatively upon) the intensity of labor effort v . The period utility function is: rt log c* where r + a log 1* - ( v t - v )2 is a seasonal preference shifter. , £ > 0, a > 0 [1] The preference shifter captures the household's increased desire to consume during the Christmas season: rt * where \ ' Q lt + 7 Q 2t + 7 Q3t + '4 Q4 f > 7 > 0 121 is a quarterly seasonal dummy variable taking on the value of 1 when period t corresponds to season i and zero otherwise. Consumption and leisure services are defined as follows: •fa ct - 1* where cpt + a cpt l , T - nt + b ( T - cpt allocation. are consumption Ia 1 < 1 nt l ) |b | < 1 , expenditures and T [3] [4 ] represents the total time If a > 0 consumption expenditures have a durable quality and are substitutable across adjacent periods. If a < 0 consumption expenditures are complements across adjacent periods, and consumption preferences exhibit habit 7 p e r s is t e n c e . T he sam e in t e r p r e t a t io n s h o ld fo r b an d le is u r e The interpretation of labor effort in the period utility function depends upon the value of v. If v>0, v is a bliss point for labor effort. case, deviations from v provide disutility for the household. interpreted as a normal level of labor effort: workers In this Here v can be view periods of inactivity on the job with the same dissatisfaction as comparable periods of overactivity. If v<0, labor effort provides increasing disutility. In this case, less work effort is strictly preferred to more. Production The representative household has access to a technology which produces goods (y) using capital (k), labor hours (n), and labor intensity (v): 0 < $ < 1 zt - z t l [5] exp( A + ct ) [6] 4t “ *1 < V Zt / 2 ’ *1* *2 > 0 > Aside from the choice of factor inputs k, n, and v, the level of production is influenced by three additional factors: exogenous variation in the state of technology z^, a productive externality 0 , and convex adjustment costs J capital and labor (with the specification described below). on Each of these factors will now be discussed separately. The technology variable z^ is a random walk process in logarithms with constant drift A. random variable. noteworthy. The impulse is an independent, serially uncorrelated Three observations on the role of z^ in our analysis are First, the constant drift term A is nonseasonal-- this is our identifying restriction that the true technology is aseasonal. growth in this economy originates with z^ since our Second, all specification of the productive externality exhibits local increasing returns In the balanced growth equilibrium that we analyze, 8 (discussed below). therefore, all trending p r e variables share the same trend as zf . c role in our analysis of perfect Third, the variability of c foresight seasonal growth paths, plays no but its presence satisfies a necessary condition for our econometric relationships to be well-posed in Section 3. Increasing returns externality variable capita output. Murphy, production where y are captured by Marshallian represents the economy-wide level of per Shleifer, and Vishny [1989], Caballero and Lyons influence the Marshallian productive externalities have been considered by and King [1990], to in [1990], and Baxter In our framework, the representative household is too small the economy-wide household's control. output, so ^ is taken to be beyond the Since the externality is expressed relative to the level of technology z^, this specification embodies local increasing returns-- as aggregate economic activity rises relative to trend, the economy becomes more productive.^ The variable is a factor which relates to the cost of adjusting capital and labor hours in terms of lost output: kt+i - k texp(A+et) \2 exp { where ^ - and ^ )2 - M 4 - ( ^ [8 ] n are positive, and A is the average growth rate of capital as well as the technology z^. The first term states that it is costly to increase the capital stock at a rate other than its growth rate. ^Alternatively, if z The firm has in place were deleted from the specification of in [7], the externality would grow with economy-wide output, and this would be global increasing returns. Given our econometric methodology in Section 3, these two specifications are observationally equivalent. Specifically, for local IR all growth is exogenous; whereas for global IR the exogenous growth is magnified by the process so that some growth is endogenous. In the global case, there is a lower value of X which interacts with the same value of ^ as the local case to produce the same equilibrium as we report in Section 4. Applying our estimation procedure to the global case would produce this lower value of A. 9 a technology for assimilating new capital into the production process. This technology costlessly accepts the normal level of new investment, but other levels create congestion in the production process. Likewise, the second term states that it is costly to increase labor hours at a rate other than its unconditional growth rate, adjustment cost factor which is zero. For this specification, the is in the interval (0,1] and in a nonseasonal steady state J^-l.^ Period Budget Constraint The household's period budget constraint is given by: k t ^ t V V 1 ** J t “ cp t + kt + i * a ' 6) kt [9] where S is the rate of capital depreciation per quarter. be introduced into the model and constraint [9] (as Fiscal policy could in Braun and Evans [1991]), but our focus in this paper is the single demand seasonal Christmas since that is a relative constant across countries. Planner's Problem The competitive equilibrium allocations in a decentralized version of this economy are identical to the solution of the following optimization g problem. At time 0, choose a sequence of contingencies { cpt , n^, v , k t^O } to solve the following: Due to the inclusion of the growth term in J^, the nonseasonal steady state of this economy will be the same as an economy which omits adjustment costs. Besides being plausible, the growth term allows greater comparability with previous studies. g This is a solution strategy previously employed by Romer [1986]. 10 max E0£ ^t|rtlog(cpt+acpt l ) + alog(T-nt+b(T-nt l )) - |-(vt-v)2 - t-0 + where are • cpc • V i ] } + i9 10i is a Lagrange multiplier, and the initial values k^, cp given. treating Notice that as given: the planner ignores the allocations externality, while this is suboptimal, it is the analagous problem to the one faced by small households and firms. optimal productive and n ^ which solve this problem It is well-known that the are characterized by the first-order conditions for cpt , nt> v , kt+^, and a transversality condition related to capital (for an example, see Braun and Evans [1991]). Furthermore, assets can be pricedusing intertemporal marginal rates of substitution in the usual way. 4 Perfect Foresight Seasonal Equilibrium Growth Path This economy grows over time at the progress which is given by A per period. calendar year, however, foresight seasonal rate of exogenous technological Since preferences shift over the these growth rates may vary seasonally. equilibrium growth path for this A perfect economy is a 9 generalization of the standard definition of a balanced growth path. The relevant new feature is that the seasonal growth path is indexed by season. Thus, consumption in year t and quarter i is linked to consumption in year t+1 quarter i by the following relationship: ct+i i “ 4A e ct i* Along the seasonal growth path, consumption will always grow x^% in the winter, Xj% in 9 For example, see King, Plosser, and Rebelo [1988] for a standard definition of a balanced growth path. 11 th e s p r in g , x^% in th e su m m er, an d x^% in th e f a l l . ^ Seasonality 1b Measured golow Res,iduals Suppose that a researcher attempts to measure Solow residuals for this economy as Prescott [1986] does. Armed with the precise knowledge of 0, the measured Solow residual will b e : ^ St ■ [ A lo g yt - 0 A log kt - (1-0) A log nt ] / (1-0) - [ (l-0 -^2) Assuming that A l o 8 zt + ^ 2 Alog yt * the deterministic Alog v t + Alog Jt ^ / C1**) component of technological aseasonal, then seasonality in S^ can arise from: output is seasonal, seasonal, [11] growth (z^) is (1 ) increasing returns if (2 ) labor hoarding if variations in labor effort are and (3) seasonal adjustments in capital and labor hours. If the fourth quarter increased desire to consume is strong enough to generate a seasonal increase in fourth quarter output, then measured Solow residuals will be proseasonal due to the productive externality. achieved by a seasonal increase in work effort If the higher output is (without a correspondingly large increase in adjustment costs), then the demand effect is reinforced. Whether or not a single Christmas seasonal in preferences can 12 explain seasonality in Solow residuals, in all four quarters, depends upon the model's ability to generate seasonality in output and labor effort across the entire calendar year.* 2 1 ^ F o r an explicit characterization of this type of seasonal equilibrium path, see Braun and Evans or Chatterjee and Ravikumar [1989]. X1Ve assume that S^ is an attempt to measure Alog z^ rather than (l-0)Alog z^. 12 Evans [1991] documents that Prescott's measure of the Solow residual is not exogenous when seasonally adjusted data is used. The finding that money, interest rates, and government spending Granger-cause Prescott's residual could be due to increasing returns or unobserved variations in labor effort of the form modeled here. 12 3. Econometric Estimation of the Model's Structural Parameters The vector of structural parameters $ contains 16 elements: * - ( 6, 0, T, r, r4 , a, a, b, £, v, 6, A, ). In assigning parameter values, there are three categories of parameters: parameters which can be normalized a priori because influence upon the analysis; (2 ) parameters which their values are parameters Moments. Since which First, utility are econometrically the parameters is ordinal, we estimated (a, T, a«l and estimate whose level and Method a (3) of the consumption The time allocation is set to T-1369 hours per quarter (as in Christiano and Eichenbaum [1991]). index variable Generalized set v) are inherently unidentified. normalize preference parameters r and r^. by have no customarily priori because their values are not well-identified in the data; (1) depends upon v: we guarantees that average labor effort will be 1. 13 Labor effort v set v at a level is an which The parameter < simply f >^ defines the units of measure for commodities (thousands of dollars, billions of yen, etc.): analysis. its value can be selected arbitrarily without affecting the Second, the discount factor 0 is not well-identified in aggregate time series data. We set 0 equal to 1.03 - 25 as in Christiano and Eichenbaum. Third, the lack of seasonally unadjusted quarterly data on the capital stock leads us to depreciation construct rate S is capital 2.5% from per investment quarter. The flows assuming remaining that the parameters are econometrically estimated by Generalized Method of Moments. 3.1 GMM estimation The parameters a * b, ( 0, r, r^, 13 ^ ^ are estimated by1 3 This normalization ensures that the average labor input corresponds to the average level of labor hours in the data. 13 in the model imposing jointly two sets of moment conditions: based upon the household's consumption and leisure; intratemporal (1 ) orthogonality conditions Euler equation for choosing and (2 ) explicitly equating a set of first moments in the data with the model's predictions for these moments. For the first set of moments, the Euler equation can be written (in terms of observables) as: where - ^/(cp^ + a cp^ ^). a valid instrument instrument set Xt l^Xt-2 ^ for includes Any variable in the time t information set is estimating the time the t parameters and t- in this growth 1 equation. rates (x^/x^ ^ The and ^ak°r hours, capital, consumption, and output, as well as four seasonal dummy variables. To describe the second set of moment restrictions, let H(x^) refer to the following transformations of the data: cpt)qt (Alog yt/nt)qt (Alog kt)qt / (Alog (l+rt))qt where kt/yt is a 4x1 vector of seasonal dummies, r ] is a real interest rate, and the symbol ' denotes transposition, so H(xt) is a 22x1 vector. the first rates of 2 0 elements of the expected value of H(xt> are the seasonal growth output, consumption, labor productivity, seasonal change in the real interest rate. to the average capital-output ratio H(xt) - h(¥) + ut 14 and capital, and the The last two elements correspond and average definition, the model predicts that Accordingly, labor hours. Given this where MU') corresponds to the model's predicted first moments of H(xt) and is a vector mean zero, serially correlated random variable. Based upon these moment restrictions, our estimator of V attempts to set the sample mean of u^ to zero, as well as the sample moments based upon equation [1 2 ]. Our choice of moment restrictions is motivated by two concerns. First, since labor effort is unobserved, the parameter £ cannot be estimated by Euler equation methods. Second, estimating ^ from production function residuals seems hopeless due to the presence of unobserved variations in labor effort: no exogenous instruments are available. 14 However, these parameters can be estimated by forcing the model to confront the seasonal growth rates in the data by choosing Finally, ^ Sims an<* £, as well as the other parameters. [1990] and Hansen and Sargent [1991] have argued that econometricians who use seasonally unadjusted data and misspecify the seasonal mechanisms may do much worse than econometricians who discard the potential information content at seasonal frequencies and simply use seasonally adjusted data. On the other hand, Ghysels [1991] has pointed out that great efficiency gains may be possible if seasonally unadjusted data is used. Thus, there is a potential trade-off involved in using seasonally unadjusted data, efficiency gains versus misspecification bias. Ue try to address the bias issue by comparing our parameter estimates with other econometric studies which used seasonally adjusted or annual data.^ 1 4 14 Hall [1988] has noted that his set of instruments would fail to be exogenous in this setting. ^Another interesting statistical issue is whether the unadjusted time series data are better characterized by purely indeterministic seasonality, purely deterministic seasonality, or a mixture of both. Ve identify Christmas effects with a fourth quarter mean of r which is larger than the other three quarters. This single nonzero seasonal mean induces nonzero seasonal means in other economic aggregates. Even if r is stochastic, our economic theory predicts that economic time series will possess some deterministically seasonal components. This argues against purely indeterministic models of 15 3 .2 D a ta The original data set employed in this study is the Barsky-Miron [1989] data for the sample period 1964-1985: adjusted for theoretical seasonality. constructs of U.S. quarterly data which has not been For the empirical our model, analysis however, we to conform redefine to of some the the variables as follows (and convert to per capita values). Output (y) is Gross National Product per capita. is nondurables plus Private consumption (cp) services consumption expenditures per capita. Fixed Investment plus Durable consumption Investment (i) is the sum of expenditures, per capita. The capital stock is computed using the flow investment expenditures, a quarterly depreciation rate of 2.5%, and an initial capital stock value for 1950. Labor hours are computed as the product of total nonagricultural employment times average hours per week of nonagricultural production workers times 13 weeks per quarter (per capita). The real interest rate is the ex post return on three-month Treasury Bills, not seasonally adjusted as reported in Citibase. The data is converted to per capita values by using the civilian population, 16 years and older. 3.3 Estimation Results Table 1 presents moment restrictions our estimation results. in estimating 1 0 The estimation structural parameters; imposes 34 in principle, there are 24 overidentifying restrictions which are tested by Hansen's [1982] J-statistic. The statistic is 20.86 with a probability uncovering no evidence against these restrictions. restrictions involve matching the model's seasonality. 16 value of 0.65, Recall that 20 of the 34 seasonal predictions against the data's seasonal growth rates. Informally, this test suggests that the model successfully captures the data's seasonal properties. This claim is examined in more detail in Section 4 where we consider a variety of other tests that focus explicitly Turning to on the the model's individual seasonal predictions. parameters, our estimates using seasonally unadjusted data are similar to other estimates in the literature which have employed seasonally adjusted data. Our estimate of 0 is .279 which is close to Prescott's [1986] value of .25 (when output is identified with GNP and does not include the services of durable consumption goods). value of r and is .2363. The inverse 1/r The weighted average corresponds to preference parameters estimated by Christiano and Eichenbaum; 4.23 falls within the range 3.92 and 5.15 consumption effect is estimated to be r^/r " they 1 .0 2 2 report. ; the leisure our value of The Christmas this is the percentage increase in the marginal utility of consumption services, holding consumption services fixed. This value does not seem to be implausibly large. A A The unusual precision of A 0, r, and is due to the two moment restrictions in H(x^) which are related to the capital-output ratio and the level of labor hours. re-estimated, the If these two moment conditions are dropped and ¥ is parameter estimates are essentially unchanged, but the A A standard errors respectively. for 0, A r, and rise to .0352, .0105, and .0107, Therefore, the unusual precision of these parameter estimates is due to the inclusion of strong identifying restrictions from the model's equilibrium predictions. The nontime-separability parameters a and b are similar to other researchers' estimates. The value of a-.445 indicates that consumption goods have a durable quality: in seasonally adjusted data, this has been found by 17 The Eicheribaum, Hansen, and Singleton [1988] and Gallant and Tauchen [1990].^ value of b--.517 indicates that leisure preferences exhibit habit-persistence: in seasonally adjusted data, this has been found by Eichenbaum, Hansen, Singleton as well as Braun [1990]. relatively smooth; habit-persistence in and This feature makes leisure and labor hours addition to in leisure will adjustment smooth costs labor hours for and labor lead to hours, greater variations in labor effort in response to exogenous shocks. The adjustment cost parameters are significantly different from zero.^ The capital and labor estimates are 28.45 and 0.258, but these numbers are a poor indication of their relative effects. On a quarterly basis, the standard deviations for the growth rates of capital and labor are 0.35% and 2.21%. percentage reduction in output due to adjusting capital (only) and (only) by one standard deviation above average is -0.018% and -0.006%. capital penalty. adjustment Also, penalty is only about 3 times larger than the The labor So the labor these numbers indicate that the direct effect of adjustment costs on measured Solow residuals is negligible. That is, the effects of Alog in equation [11] are small. indirect effect may be large: As was noted in response in Section 2, however, to a consumption demand the shock, reducing investment may now be costly enough to induce a large response in output. Our estimate of the output elasticity with respect to external increasing returns is ^ “ .2389. The elasticity is significantly different from zero. This value is within the range of estimates reported by Caballero and Lyons On the other hand, Constantinides and Ferson [1990] find evidence of habit-persistence in consumption goods preferences (a<0). In simulations of an equilibrium business cycle model with seasonality, Braun and Evans [1990] found that durability in consumption (a>0 ) helped the model match key business cycle moments better than habit-persistence. ^Ghysels [1988] observes that there is a lot of spectral power at seasonal frequencies for identifying adjustment cost parameters. 18 [1989] and Baxter and King [1990], although Baxter and King use a larger value of ,33 in their model evaluation. our estimate provides some The size and statistical significance of evidence that external important for explaining seasonal fluctuations; offered differ in Section 4. from those estimate of ^ Nevertheless, of provides are our identifying restrictions and [1990] evidence which returns a quantitative assessment is since Caballero-Lyons increasing Baxter-King [1990], our theirs and is both independent of complementary. Our estimate of £ is 0.0241, so the disutility of labor effort deviations may in fact be small enough to induce sizable variations. is .0348, so the estimate is reasonably imprecise. The standard error It is important to note that the hypothesis that variations in labor effort are small would imply that ? is large: the point estimate and standard error do not support this. As we will see in Section 4, estimates of f in the range reported are capable of yielding substantial variations in work habit-persistence in leisure preferences, relatively disutility effort. small associated costly with adjustment varying provide evidence for the labor hoarding hypothesis. implied by the estimates and Thus, labor Finally, normalization is negative. the of estimated labor, effort and jointly the value of v Thus, utility is strictly decreasing in work effort. Overall, the estimated parameterization seems reasonable. The similarity of many estimates with previous studies suggests that if we had chosen to "calibrate" our parameterization model would using not these have been other very studies, different. the resulting Finally, the overidentifying restrictions cannot be rejected. 4 . In t h is E v a lu a tin g s e c t io n w e th e M o d e l's e v a lu a te 19 th e S e a so n a l s e a s o n a l I m p lic a tio n s p r o p e r tie s o f t parameterization and offer evidence on the relative importance of hoarding and increasing returns in explaining the seasonal patterns data. Two criteria First, a series labor in the are used to evaluate the model's seasonal predictions. of hypothesis tests are reported. These tests have the benefit of taking into consideration sampling error in the summary statistics for the data and sampling error in the estimated parameterization. Second, the seasonal growth rates of the data and the model are simply plotted together. This latter approach provides a visual summary of the seasonal properties of the model relative to the data. Table 2 contains results from a series of hypothesis tests. The results in table questions: predict are 2 aimed providing information on the Is there evidence of seasonality in the data? significant patterns at seasonality? Does the model predict following Does the the model same found in the data? Column one reports Wald statistics three seasonal that offer evidence on the first question for each variable individually. The maintained null underlying the column one results is that the four seasonal dummies for a particular time series are equal (equation (1), Table 2). These statistics are constructed from GMM estimates of the average seasonal growth rates in the data and statistic use a Newey-West indicate that covariance the null estimator. hypothesis The of p-values no for each seasonality is overwhelmingly rejected for each time series. These results are representative of findings reported by Barsky and Miron [1989] 18 . Column two reports Wald statistics that offer evidence on the second question. The maintained null hypothesis is that the model's predicted seasonal growth rates are equal. 19 18 Barsky and Miron also find that there is statistically significant seasonality in the real interest rate although the magnitude of the estimated seasonals (in levels) is small. 19 The model's predicted seasonal growth rates are a highly nonlinear function 20 The null hypothesis of no seasonality is also sharply rejected for each of the time-series that the model offers predictions for. On the basis of the results from these two tests, we conclude that both the model and the data offer strong refutable predictions at seasonal frequencies. Certainly the most important question is the third one: predict the seasonal patterns in the data? Does the model Column three of Table 2 provides one metric for evaluating the model 1 s "fit" at seasonal frequencies. maintained null hypothesis model's in column three is that the The predicted seasonal growth rates for the jth time-series equal the corresponding values in the data in each of the four seasons. This LaGrange multiplier (or LM) test is formally a test of particular moment restrictions that were imposed in the course of estimation. statistics were For hours, the Solow residual, and calculated using the fact time that these investment, series the can be expressed as (log) linear combinations of other time series that were included in the estimation. Eichenbaum, Hansen and Singleton [1984] and Newey and West [1987b] describe the details of implementing LM tests in the context of GMM estimation. Column three contains surprisingly null of a common seasonal pattern in all little instances. evidence As against a check we the also calculated a GMM analog to the likelihood ratio statistic and found that the two statistics were virtually identical. This collection of statistics provides two important conclusions. First, the tests reported in columns one and two demonstrate that the statistics have sufficient power to reject the null hypothesis of no seasonality for the model and the data. Second, the column three results find no evidence against the hypothesis that the model correctly predicts the pattern of seasonality found of the estimated structural parameter vector 4. The asymptotic covariance of the predicted seasonals is computed using the covariance estimator of ¥ and the gradient of the nonlinear function. The Wald statistics are constructed from these objects in the usual way. 21 in th e d a ta . Turning to the specific predictions of the model, we report plots of the seasonal growth rates for the data and model numbers are presented in Table 3). in figures These diagrams 1-3 complement (the actual the previous hypothesis tests in that they offer summary information on the ability of the estimated parameterization to capture particular aspects of the seasonal pattern in the data. We will focus on two aspects of the seasonal pattern: the magnitude of the model's predicted seasonal in a particular quarter relative to the data and the ability of the model to mimic the sequential relationship of seasons found in the data. The Solow residuals labeled "data" are calculated using 0«.28. As was noted in section 3 this number is qualitatively close to the value of .25 used by Prescott [1986]. One of the principal aims of this paper is to investigate the possibility that increasing returns and/or time-varying labor effort can explain the large seasonal variation reported in table in the 2: Solow Residual. the model is quite Figure 1 confirms successful in this predicted Solow residual has the same sequential pattern magnitudes found in the data. These results offer strong the results respect. The and captures the support for our contention that the observed seasonal pattern in the Solow residual is driven largely by demand shocks. In addition to capturing the seasonal pattern in the Solow residual, the model also mimics important features of seasonality in output, consumption and average productivity. In all of these instances the model correctly predicts the sequential seasonal pattern of the data. For consumption we do see a tendency for the model to understate the third quarter deceleration found in the data and for output the model understates the second quarter rise. However, the hypothesis tests indicate that both of these disparities can be attributed to sampling error. These successes across the entire calendar year 22 are particularly striking given that the only seasonal shifter is a fourth quarter shift in preferences. Figure 3 displays the seasonal patterns in labor effort. an unobservable, the data's seasonals cannot be output rises on the strength of higher combination with increasing returns, than reported. normal work in the first quarter. Fourth quarter labor effort. In fourth quarter effort is only 3% above normal in generating an annualized output growth of 18%. at Since this is These variations 20 in Opposing forces are effort do not seem implausible. If we ignore sampling error, the figures suggest that the model fails to account for some aspects of the seasonal pattern in other variables. For hours and investment the magnitudes are off in all four quarters, for the capital stock they are off in three quarters and for interest rates they are off in two quarters. However, even for these time series the model captures many features of the sequential pattern in the data. The fact that the hypothesis tests in table 2 fail to reject a common seasonal pattern in individual time series suggests that there may be considerable sampling error. The most likely A sources for this sampling error are A in £ and parameters which govern respectively the roles of time-varying labor effort and increasing returns. Both parameters are estimated with sizable standard errors. The case analyses below demonstrate that variations in these two paramters lead to a deterioration in the model's seasonal predictions relative to the data. It is also interesting to examine the role of the various features of the model in explaining these seasonal patterns. increasing returns can be shut down by setting ^ in work effort can be ruled out by setting 20 Notice "0 • and first, 23 the Second, time variation F°r t* 1 6 8 6 values of The fourth quarter growth rate of labor effort is only 3.8%. that £ and ^ adjust it is never labor desirable to vary effort, while in production. By examining these two it is costless to special cases of the estimated paramterization we can get some feel for the contributions of labor hoarding and increasing returns to scale. The results from these exercises are displayed in Figures 4, 5, and 6 along with the data for purposes of comparison. Consider first output and the Solow residual. With only increasing returns the Solow residual and output are essentially flat across all four seasons. Labor hoarding, on the other hand, does in isolation produce the correct sequential pattern for these two series while dramatically understating the magnitudes. On the basis of these diagrams it would appear that labor hoarding plays a crucial role in generating the correct seasonal pattern in output and the Solow residual. The primary role of increasing returns then is to magnify these patterns. In our estimated parameterization increasing returns and labor hoarding are clearly interacting to deliver the seasonal patterns in output and the Solow Residual. Consumption and investment, on the other hand, continue to display some seasonality with properties only increasing of the model. The returns. This fourth quarter can be seasonal attributed demand to two shift helps explain the fluctuations in the fourth and first quarters. In the second and third quarters investment and consumption are adjusting endogenously via the nontime-separabilities in consumption which imply that consumption in adjacent periods are substitutes.5 5. This paper demonstrates Conclusion that the seasonal cycle contains valuable information for uncovering the roles of labor hoarding and increasing returns. In contrast to business cycles which are arguably induced by both demand and technology shocks of varying persistence, 24 seasonal fluctuations are anticipated, Christmas. transient, and easily identified with calendar events like Our findings indicate that increasing returns to scale alone does not directly explain the seasonality in measured Solow residuals. However, it plays an important role in magnifying small variations in work effort. Hall [1988] has argued that labor hoarding requires implausibly large variations in work effort to explain cyclical fluctuations in Solow residuals. fluctuations this labor effort variation of no more increasing fluctuations is not returns these in total observed in the data. the case. Our than variations estimated parameterization 5.5% are factor productivity on a labor hoarding-- quarterly basis. magnified, that are of thereby the same With producing magnitude this model offers an productive externalities, explanation for the similar cross-country seasonal patterns documented by Beaulieu and Miron [1991]. implies Finally, since our explanation rests on phenomena which are not country-specific-- Christmas celebrations, and For seasonal 25 R e fe r e n c e s Barsky, R. and J. Miron, 1989, The Seasonal Cycle and the Business Cycle, Journal of Political Economy. Baxter, M . , and R. King, 1990, Productive Externalities and Cyclical Volatility, Rochester Center for Economic Research Working Paper #245. Beaulieu, J . 9 and J. Miron, 1991, A Cross Country Comparison of Seasonal Cycles and Business Cycles, manuscript, Boston University. Braun, R . , 1990, The Dynamic Interaction of Distortionary Taxes and Aggregate Variables in Postwar U.S. Data, unpublished Ph.D. thesis, Carnegie Mellon University. Braun, R . , and C. Evans, 1991, Seasonality and Equilibrium Business Cycle Theories, Institute for Empirical Macroeconomics, Discussion Paper #45. Burnside, C., M. Eichenbaum, and S. Rebelo, 1990, Labor Hoarding and the Business Cycle, NBER working paper #3556. Caballero, R . , and R. Lyons, 1989, The Role of External Economies in U.S. Manufacturing, manuscript, Columbia University. Chatterjee, S. and B. Ravikumar, 1989, A Neoclassical Model of Seasonal Fluctuations, manuscript, University of Iowa. Christiano, L. and M. Eichenbaum, 1990, Current Real Business Cycle Theories and Aggregate Labor Market Fluctuations, unpublished manuscript, Institute for Empirical Macroeconomics, Discussion Paper #24. Chirinko, R . , 1991, Non-Convexities, Labor Hoarding, Technology Shocks, Procyclical Productivity: A Structural Econometric Approach, manuscript, University of Chicago. Constantinides, G. and W. Ferson, 1990, Habit Persistence and Durability in Aggregate Consumption: Empirical Tests, unpublished manuscript, University of Chicago. Cooper, R . , and J. Haltiwanger, 1989, Macroeconomic Implications of Production Bunching: Factor and Final Demand Linkages, NBER Working Paper #2976. Eichenbaum, M. and L. Hansen, 1990, Estimating Models with Intertemporal Substitution Using Aggregate Time Series Data, Journal of Business and Economic Statistics. Eichenbaum, M . , L. Hansen, and K. Singleton, 1989, A Time Series Analysis of Representative Agent Models of Consumption and Leisure Choice Under Uncertainty, Quarterly Journal of Economics. 26 Eichenbaum, M . , L. Hansen, and K. Singleton, 1984, Appendix to: A Time Series Analysis of Representative Agent Models of Consumption and Leisure Choice Under Uncertainty, manuscript, Carnegie-Mellon University. Evans, C., 1991, Productivity Shocks and Real Business Cycles, manuscript, Federal Reserve Bank of Chicago. Gallant, R . , L. Hansen, and G. Tauchen, 1990, Using Conditional Moments of Asset Payoffs to infer the Volatility of Intertemporal Marginal Rates of Substitution, Journal of Econometrics. Gallant, R. and G. Tauchen, 1989, Seminonparametric Estimation of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications, Econometrics 57, 1091-1120. Ghysels, E., 1988, A Study Toward a Dynamic Theory of Seasonality for Economic Time Series, Journal of American Statistical Association, 83, 168-172. 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Singleton, K . , 1988, Econometric Issues in the Analysis of Equilibrium Business Cycle Models, Journal of Monetary Economics. 28 T a b le Parameter 1 : Estimate GMM P a r a m e t e r E s tim a te s Standard Error 6 .2790 0.00091 r .2350 0.00045 .2402 0.00047 . 0 0 2 1 0.00005 r4 X 28.448 3.64360 .2576 0.05645 a .4453 0.01123 b -.5173 0.00854 .2389 0.09240 .0241 0.03481 *1 * 2 CM J-statistic Degrees of Freedom P-value 2 0 . 8 6 24 .6469 21 A Newey-West procedure [1987a] with four lags was used to compute the optimal GMM weighting matrix. 29 T a b le (1) 4 x X < f(») - d, 1 >✓ * CM (3) Variable X f(*)'q L + w L T e s t R e s u lts f 22 Hn : The model does NOT exhibit deterministic seasonality . The elements of f(40 are equal. h 1: „d Ho H y p o th e s is H q : The data do NOT exhibit deterministic seasonality The elements of d are equal. — d'q t + w t , t 2 : The model's predicted seasonality equals the data's seasonality. 23 Hj Ho 4 Solow 303.64 (.000) 437.74 (.000) 0.125 (.998) Output 413.90 (.000) 348.08 (.000) 0.201 (.995) Consumption 715.28 (.000) 197.32 (.000) 0.058 (.999) Investment 1097.45 (.000) 27.35 (.000) 0.137 (.997) Capital 280.26 (.000) 31.21 (.000) 0.301 (.989) Labor Hours 831.49 (.000) 112.31 (.000) 24 0.216 (.994) 615.61 (.000) Labor Prod 227.14 (.000) 391.25 (.000) 0.167 (.996) Real Rate 31.06 (.000) 54.03 (.000) 0.034 (.999) 22 Equation (2) is predicted by our theoretical model, but our test is not regression-based. See the text for a description. 23 For both columns 1 and 2, the Wald test statistics are asymptotically distributed x with four degrees of freedom. The numbers in parentheses are probability values of the test statistic. 24 2 The Lagrange Multiplier test statistic is asymptotically distributed x with four degrees of freedom. 30 T a b le 3 : S e a s o n a l 25 G r o w th Variable Season Model Solow Residual Winter Spring Summer Fall -7.451 3.512 - .703 5.466 -7.323 3.889 -2.019 5.889 Output Winter Spring Summer Fall -6.501 2.924 - .345 4.746 -7.356 4.623 - .674 4.529 Consumption Winter Spring Summer Fall -6.744 3.500 -1.322 5.390 -6.876 3.015 .461 4.767 Investment Winter Spring Summer Fall -5.637 .854 3.141 2.465 -14.065 12.365 -1.578 4.896 Capital Winter Spring Summer Fall .297 .136 .155 .235 .640 .219 .572 .506 Labor Hours Winter Spring Summer Fall -1.681 0.491 .165 1.025 -3.126 2.438 .863 .197 Labor Prod. Winter Spring Summer Fall -4.821 2.433 - .510 3.721 -4.229 2.186 -1.537 4.332 Real Rate Winter Spring Summer Fall 2.901 0.812 0.409 -4.122 R a te s .701 1.247 - .397 -1.162 25 Data Quarterly rates of growth in percentages, except for the real interest rate which is the quarterly change in annualized yields (i.e., Alog (l+rt), with r at annual rates). 31 SOLOW RESIDUAL OUTPUT QUARTERLY RATES O F GROWTH QUARTERLY RATES O F GROWTH 50 . 25 . PERCENT PERCENT 0.0 -. 25 -. 50 -. 75 2 3 CONSUMPTION 4 QUARTERLY SEASONS INVESTMENT QUARTERLY RATES O F GROWTH QUARTERLY RATES O F GROWTH 15 10 PERCENT PERCENT 5 0 5 -10 -5 1 1 2 3 QUARTERLY SEASONS 4 1 4 QUARTERLY SEASONS Figure 1 QUARTERLY SEASONS CAPITAL STOCK QUARTERLY RATES O F GROWTH PERCENT LABOR HOURS Q UARTERLY RATES O F GROWTH QUARTERLY SEASONS REAL INTEREST RATE QUARTERLY RATES O F GROWTH CHANGE IN ANNUALIZED YIELDS PERCENT LABOR PRODUCTIVITY 1 2 3 QUARTERLY SEASONS 4 3 QUARTERLY SEASONS F ig u r e OUTPUT QUARTERLY RATES O F GROWTH QUARTERLY RATES O F GROWTH PERCENT PERCENT SOLOW RESIDUAL QUARTERLY SEASONS INVESTMENT QUARTERLY RATES O F GROWTH QUARTERLY RATES O F GROWTH PERCENT PERCENT CONSUMPTION 1 Figure 4 QUARTERLY SEASONS 2 3 QUARTERLY SEASONS 4 1 2 3 QUARTERLY SEASONS 4 CAPITAL STOCK QUARTERLY RATES O F GROWTH PERCENT PERCENT LABOR HOURS QUARTERLY RATES O F GROWTH REAL INTEREST RATE QUARTERLY RATES O F GROWTH CHANGE IN ANNUALIZED YIELDS PERCENT PERCENTAGE POINTS QUARTERLY SEASONS LABOR PRODUCTIVITY Figure 5 QUARTERLY SEASONS 1 4 QUARTERLY SEASONS 1 2 3 QUARTERLY SEASONS 4 6 QUARTERLY SEASONS F ig u r e 130333 dO B V 73 0 X30NI