The Buffer-Stock Model and the Marginal Propensity to Consume. A Panel-Data Study of the U.S. States.

Transcription

1 The Buffer-Stock Model and the Marginal Propensity to Consume. A Panel-Data Study of the U.S. States. María José Luengo-Prado Northeastern University Bent E. Sørensen University of Houston and CEPR March 16, 2005 Abstract We simulate a buffer-stock model of consumption, explicitly aggregate over consumers, and estimate (aggregate) state-level marginal propensities to consume out of current and lagged income using simulated data generated by the model. We calculate the predicted marginal effects of changing state-specific persistence of income shocks, uncertainty of statespecific output, and individual-level risk. Next, we estimate marginal propensities for U.S. states using panel-data methods. We find effects of persistence that clearly correspond to the predictions of the model and while the effect of state-level uncertainty cannot be determined precisely, indicators of individual-level uncertainty have strong effects consistent with the model. Overall, the buffer-stock model clearly helps explain differences in consumer behavior across states. KEYWORDS: Buffer-stock, Consumption, Precautionary saving. JEL classifications: E21 - Consumption; Saving.

2 1 Introduction The buffer-stock model of consumption, pioneered by Deaton (1991) and Carroll (1992), is a promising candidate for replacing the Friedman (1957)-Hall (1978) Permanent Income Hypothesis (PIH) as the benchmark model of consumer behavior. In this paper, we examine if the buffer-stock model predicts the cross-state variation in the marginal propensities to consume (MPC) out of current and past income observed in U.S. state-level aggregate data. Rather than testing if a particular implementation of the model is literally true, we examine if the model predicts the directions in which the MPCs vary across states. Estimating the model structurally for 50 states, using simulation based estimation methods, would require a number of calculations beyond what is feasible at current computational speeds. First, we analyze the buffer-stock model for state-level consumption by simulating individuallevel consumption and explicitly aggregating across consumers of a state. The growth rate of individual-level income is modeled as the sum of aggregate (country-wide) income growth, (aggregate) state-level income growth, and individual-level idiosyncratic income growth, where the latter is the sum of a random walk and a transitory shock. The standard deviation of idiosyncratic income is calibrated to have the value typically chosen in the literature, while the parameters of the aggregate and state-level components are estimated from the data. Because our empirical data are annual, we assume in our simulations that the frequency of consumption is twice that of the observations; i.e., we allow for time aggregation. We regress simulated consumption growth on the (simulated) state-specific current or lagged income growth. We repeat the simulations changing the baseline parameters one at a time in order to examine the predicted marginal effects in empirically relevant directions. Specifically, we add the risk of job loss to the baseline model; i.e., we add the possibility of disastrous, large, independently identically distributed (i.i.d.) transitory shocks which happen with low probability. Alternatively, we increase the persistence of state-level shocks, state-level uncertainty, or individual-specific risk. In particular, the effect of changing persistence of shocks has not been analyzed in the literature. Second, using the panel of U.S. states, we estimate the MPCs out of current and lagged 1

3 income by regressing consumption growth on current and lagged income growth, respectively. We do not attempt to identify the correct statistical model (data generating process) for statelevel consumption: we estimate the MPC out of current income, say, as a statistic calculated in the same way as the statistic calculated from the simulated data. We then can compare the statistic calculated from the data with the statistic calculated from the theory. As in the simulations, we allow the estimated MPCs to vary with persistence, with the unemployment rate, with state-level uncertainty, and with indicators for high (low) individual-level uncertainty; viz., the share of agriculture (government) employment in each state. Finally, we declare the model a success if the quantitative predictions for the MPCs match those found using the actual state-level data. The advantages of using state-level data are several. Compared with purely macro approaches the existence of 50 states with different income processes (some agricultural, some oil-based, etc.) vastly expands the relevant variation and the number of data points. In addition, by considering state-level variation that is orthogonal to aggregate variation, simultaneity problems are likely to be alleviated. Compared with studies using micro data, the state-level data are not plagued by the large amount of idiosyncratic variation in micro data that makes it hard to decide what patterns are likely to survive aggregation. Also, no sources of micro data (at least for the United States) have quite comprehensive data for income and consumption. For example, the much used Panel Study of Income Dynamics mainly records food consumption rather than the total retail sales available at the state-level. At the very least, studies based on state-level data complement micro-data based studies. 1 Marginal propensities to consume play a central role in Keynesian forecasting models. Friedman (1957) stresses that the MPC out of income shocks will depend on expectations regarding future income. Hall (1978), in the first rigorous implementation of this idea in a rational expectations framework, reach the, by then, astonishing conclusion that under simple assumptions consumption is a martingale; i.e., a regression of period t consumption growth on any variable known at period t 1 should return an estimate of zero. Regressions using aggregate data, 1 Furthermore, compared to cross-country data, the data are collected in a consistent manner and most institutional features do not vary across states, making our results less likely to suffer from left-out variable bias. Further, as argued by Ostergaard, Sørensen, and Yosha (2002) and Sørensen and Yosha (2000), the use of paneldata regressions with time-fixed effects will make the results more robust to potential biases that might obtain because the U.S. as whole is unable to borrow internationally at fixed interest rates. 2

4 however, consistently return an estimate significantly larger than zero when current growth in consumption is regressed on lagged aggregate income growth a phenomenon known as excess sensitivity (of current consumption to lagged income). The PIH-model also provides closedform solutions for the predicted growth in consumption as a function of innovations to income when income is described by a general Auto Regressive-Moving Average (ARMA) model. For example, if income is a random walk, consumption is predicted to move one-to-one with income. Empirical work using aggregate data consistently finds a smaller reaction of consumption to income shocks a phenomenon known as excess smoothness. 2 The buffer-stock model was designed to improve on the PIH-model while still preserving many of its insights stemming from rational forward-looking behavior. Roughly speaking, by assuming consumers cannot (or endogenously will not) borrow and allowing consumers to be more prudent and less patient than in Hall (1978), Deaton (1991) and Carroll (1992) demonstrate that the buffer-stock model has the potential to fit microeconomic and macroeconomic data better. In particular, the buffer-stock model predicts a high correlation of consumption with income regardless of expected future income, and that consumers will save more when they are subject to more uncertainty ( precautionary saving ). 3 Several papers have focused on identifying the presence of precautionary saving using micro data. The major challenge faced by this line of work is finding variables that reflect uncertainty but do not correlate with other characteristics that may increase saving. Commonly used uncertainty measures are household income variance, occupation, education, and unemployment risk. The literature offers diverse estimates ranging from no evidence of precautionary saving (e.g., Dynan 1993) to quite large amounts of precautionary saving (e.g., Carroll and Samwick 1997). 2 For studies of excess sensitivity, see Flavin (1981), Blinder and Deaton (1985), Campbell and Deaton (1989), and Attanasio and Weber (1993). Building on the results in Hansen and Sargent (1981), excess smoothness has been documented by Deaton (1987), Campbell and Deaton (1989), and Galí (1991). 3 Numerous authors have studied the effects of uncertainty with non-quadratic preferences. Leland (1968) names the difference between saving under uncertainty and saving under certainty precautionary saving. Kimball (1990) discusses the conditions under which one can expect to observe precautionary saving. Carroll and Kimball (1996) prove the concavity of the consumption function under precautionary saving. Caballero (1991) uses the exponential utility function and evaluates the effect precautionary saving may have empirically. Under the assumption that consumers have constant relative risk aversion utility functions, it is not possible to obtain closed-form solutions and, therefore, numerical methods must be used to study the reaction of consumption and saving to uncertainty. Early numerical studies of precautionary saving include Zeldes (1989b), Hubbard, Skinner, and Zeldes (1994), Skinner (1988) and Aiyagari (1994). Evidence of credit rationing has been provided by, e.g., Zeldes (1989a) using a sample splitting method, and by Jappelli (1990) and Perraudin and Sørensen (1992) using survey data. 3

5 Browning and Lusardi (1996) provide a comprehensive survey of the empirical literature. Gourinchas and Parker (2001) estimate a structural model of optimal life-cycle consumption in the presence of realistic labor income uncertainty and find that households behave like buffer-stock consumers early in their working lives and like PIH households when they approach retirement. More recently, Carroll, Dynan, and Krane (2003) identify a precautionary effect on saving induced by unemployment risk for households with moderate levels of permanent income but not for households with low levels of income. Engen and Gruber (2001) utilize the variation in unemployment insurance programs across states and find lower precautionary saving for households with high unemployment risk in states with more generous unemployment insurance. Consistent with the buffer-stock model, McCarthy (1995) finds a higher marginal propensity to consume for households with relatively low-wealth. The amount of empirical evidence supporting the buffer-stock model with aggregate data is limited. Hahm (1999) finds higher saving rates and steeper consumption growth paths for countries with more earnings uncertainty using OECD data for 22 countries. Hahm and Steigerwald (1999) using U.S. aggregate data and a survey measure of GDP uncertainty as an approximation of aggregate income uncertainty find more precautionary saving in periods of high uncertainty of GDP. Ludvigson and Michaelides (2001) consider MPCs at the U.S. aggregate level and demonstrate that the buffer-stock model under certain conditions of incomplete information à la Pischke (1995) can explain at least part of the deviations from the predictions of the PIH-model at the aggregate level. In this paper, we find strong effects of income persistence on the marginal MPC out of current income in the direction predicted by the buffer-stock model. Persistence does not have a significant effect on the MPC out of lagged income but this result is quite consistent with our simulation results if we allow for a non-negligible proportion of the consumers to behave according to the PIH-model as suggested by Carroll (1997) and empirically verified by Gourinchas and Parker (2001). We find statistically insignificant evidence of unemployment risk affecting the current MPC, but significant effects of unemployment risk on the lagged MPC; the buffer-stock model would have predicted clear effects on both. The signs for both current and lagged MPCs are consistent with the predictions of our model. We find significant effects of aggregate transitory uncertainty on the current MPC, but not on the lagged. This is consistent 4

6 with our simulations showing stronger current than lagged effects. Finally, we find large effects of individual-level uncertainty on the aggregate MPC out of lagged income, consistent with the predictions of the model. Overall, the buffer-stock model is quite successful in explaining differences in aggregate consumer behavior across states. The remainder of the paper is organized as follows: Section 2 describes the buffer-stock model and its calibration. Section 3 describes the results from simulating the model. Section 4 estimates panel-data models for consumption and compares the estimated MPCs to the theoretical results found in Section 3. Section 5 summarizes. 2 The Model and its Calibration This section presents the buffer-stock model used to predict the effects of persistence and uncertainty on state-specific consumption. The specification allows for aggregate shocks, state-specific shocks, and individual-level shocks. We assume that aggregate and state-specific shocks show varying degrees of persistence. More precisely, we allow for components of income growth to follow AR(1) processes of the generic form z t = az t 1 + u t ; and we refer to higher values of the parameter a as higher levels of persistence of the shocks. In order to explore the aggregate implications of the buffer-stock model, we simulate income and consumption paths for individuals and find state-level income and consumption paths by averaging across consumers. We simulate the model for a number of states and estimate MPCs with panel-data regressions which include time-fixed effects using the simulated data. The inclusion of time-fixed effects removes aggregate effects and allows us to interpret the results as reflecting state-specific effects. 2.1 The model Consumer j s maximizes the present discounted value of expected utility from consumption of a nondurable good, C. Letβ<1bethe discount factor and R the interest factor. S jt is agent j s holding of a riskless financial asset at the end of period t. Each period, the funds available to agent j consist of the gross return on assets RS jt 1 plus Y jt units of labor income. The agent 5

7 chooses optimal consumption C jt according to the maximization problem: { } max C jt E 0 t=0 β t U(C jt ) s.t. S jt = RS jt 1 + Y jt C jt. (1) The utility function is assumed to exhibit constant relative risk aversion: U(C jt )= C1 ρ jt 1 ρ.with ρ>0 the agent is risk-averse and has a precautionary motive for saving. In the literature, buffer-stock saving behavior has been derived from two different assumptions. Deaton (1991) explicitly imposes a no-borrowing constraint (S jt > 0) but assumes agents always receive positive income. Carroll (1992), on the other hand, endogenously generates a noborrowing constraint by assuming individuals may receive zero income (a transitory disastrous state) with a very small probability, p. In this case, the agent will optimally never want to borrow to avoid U (0) =. We use Deaton s specification as our baseline with p, the probability of the disastrous state, set to zero. Following Michaelides (2003), we also consider the case where p is not zero and impose different lower bounds for the transitory shock. A positive lower bound may be interpreted as an income replacement program (unemployment benefits, welfare, health insurance, disability payments, etc.). We refer to the disastrous state throughout the paper as unemployment. Income is stochastic and the only source of uncertainty in the model. It is assumed to be exogenous to the agent. We assume the income of agent j in state s follows the model: Y jt = P jt V jt W st, P jt = P jt 1 A t G st N jt. (2) Labor income, Y jt, is the product of permanent income, P jt, an idiosyncratic transitory shock, V jt, and a state-level transitory shock, W st. A t can be thought of as growth of permanent income attributable to aggregate productivity growth in the economy, while G st reflects growth of permanent income specific to state s. N jt is a permanent idiosyncratic shock. log N jt,logv jt and log W st are independent and identically normally distributed with means σ 2 N /2, σ2 V /2 σ 2 W /2, and variances σ2 N, σ2 V and σ2 W, respectively. log A t is assumed to be an AR(1) process with persistence a A, unconditional mean µ A, and variance σ 2 A.logG st is also an AR(1) process 6

8 with persistence a s, mean 0, and variance σg 2. This income specification is particularly useful since it allows for consumers to share in aggregate and state-specific growth while the variance of their income can be calibrated to be dominated by idiosyncratic permanent or transitory components. The formulation implies that the growth rate of individual labor income follows an ARMA process, log Y jt =loga t + log G st +log N jt +log V jt log V jt 1 +log W st log W s,t 1, consistent with microeconomic evidence (e.g., MaCurdy 1982, Abowd and Card 1989). By the law of large numbers, aggregate income growth can be written as log Y t =loga t, while state-specific income growth is log Y st logy t =logg st +logw st log W s,t Solution Method A closed-form solution of the model does not exist and it must be solved by computational methods. Following Deaton (1991), the model is first reformulated in terms of cash-on-hand, X jt RS jt 1 +Y jt. 4 Given the homogeneity property of the utility function, all variables can be normalized by permanent income to deal with non-stationarity, as proposed by Carroll (1997). The first order condition of the problem becomes: U (c jt )=max{u (x jt ),βre t [(A t+1 G s,t+1 N j,t+1 ) ρ U (c j,t+1 )]}, (3) where c jt = C jt /P jt and x j,t+1 =(A t+1 G s,t+1 N j,t+1 ) 1 R(x jt c jt )+V j,t+1 W s,t+1. 5 Individuals distinguish aggregate from state-specific shocks and optimize accordingly. We use Euler equation iteration to solve Equation (3) numerically. x is discretized and the income shocks are approximated by discrete Markov processes following Tauchen (1986). We use 10 points for N, V and W,and5pointsforA and G. Interpolation is used between points in the x grid. The numerical technique delivers a consumption function c(x, A, G): normalized consumption as a function of normalized cash-on-hand and the aggregate and state-specific permanent states. In other words, our optimal policy function for consumption has 25 branches, one for each (A, G) 4 The budget constraint becomes S jt = X jt C jt and the liquidity constraint C jt X jt. Combining the definition of cash-on-hand and the budget constraint, we can write an expression for the evolution of cash-onhand: X jt+1 = R(X jt C jt)+y j,t+1. 5 A necessary condition for the individual Euler equation to define a contraction mapping is βre t[(a t+1g s,t+1n j,t+1) ρ ] < 1. This is the impatience condition common to buffer-stock models which guarantees that borrowing is part of the unconstrained plan. 7

9 combination of the discrete approximations of the persistent permanent income shocks Calibration A good calibration of the income process is essential to obtain qualitative and quantitative predictions. We estimate the AR process for the aggregate shock, log Y t = A t, obtaining estimates of µ A =0.016,a A =0.42 and σ A = (Data sources are described in Appendix A). We estimate the more complicated process for state-specific income, log Y st logy t = log G st +logw st log W s,t 1, using a Kalman filter procedure described in Appendix B. In the data, the mean growth rate varies by state but in our calibration we do not explore the effects of state-varying growth rates. We make this choice because in-migration in some states, such as Nevada, is so large that mean growth rates over 20-odd years cannot easily be interpreted as reflecting the income growth prospects of individuals. 8 processes, we demean the data prior to estimation. In the estimation of state-specific In Table 1, we report the estimated values of a s,σ G, and σ W for each of the 50 U.S. states. The estimates of the standard deviation of the state-level transitory shock, ˆσ W, are non-zero for 24 states and significant at conventional levels for 11 states. The states with relatively large transitory shocks are typically agricultural (Iowa, Nebraska, etc.) consistent with our prior belief that agricultural states are subject to aggregate temporary shocks. The persistence of state-specific shocks varies widely across states. The point estimate of a s is 0.46 for Nebraska while the largest value of 0.71 is found for South Carolina. The average value is if the aggregate effect is not removed average persistence is significantly higher at 0.38, which reflects that the aggregate component of income growth displays much more persistence than the state-specific component (we do not otherwise display results from such estimations). This difference in persistence implies that forward looking consumers will react more to aggregate than to state-specific shocks. In our baseline calibration, we set a s =0.165 and σ G =0.016 the simple average across states. We choose σ W = 0 no state-specific transitory shocks since the average σ W across states is close to this value. 6 More details on how to solve this equation can be found, for example, in the appendix of Ludvigson and Michaelides (2001). 7 If we allow for an aggregate transitory shock, we find a point estimate of 0 consistent with our specification. 8 Hahm (1999) explores the impact of growth on aggregate consumption using country-level data. 8

10 Idiosyncratic income shocks are taken from previous studies see, for example, Carroll and Samwick (1997) and Gourinchas and Parker (2001). In particular, we set σ V = σ N =0.1. The probability of unemployment is 0 in our baseline simulation, although we explore a case with p =0.03 and two different replacement rates: 30 percent of income and 0. Finally, we choose risk aversion ρ = 2, an interest rate of 2 percent, and a discount rate of 3 percent. This combination delivers a median wealth-to-income ratio of 0.12 in the model is also the median net financial wealth-to-income ratio for households with head under 55 in the Survey of Consumer Finances, Since we do not model durables in this paper, it seems appropriate to match this ratio as opposed to the net worth to income ratio. Also, we focus on people under 55, who are more likely to behave as buffer-stock consumers (see Gourinchas and Parker 2001, Carroll 1997). 2.4 Aggregation Procedure It is well-known that consumption functions for a buffer-stock consumer are nonlinear, so explicit aggregation is needed to obtain implications for aggregate consumption. Our simulation exercise is similar in spirit to that of Ludvigson and Michaelides (2001), who calibrate their income process to match U.S. aggregate income and focus on explaining excess sensitivity and excess smoothness at the aggregate level. Our goal, however, is to assess how income persistence and income uncertainty impact the MPCs using state-level data, so we proceed in a different manner. Ideally, we would like our simulation exercise to be as close as possible to the empirical strategy in Section 4. Briefly, we would like to simulate 50 states with different persistence and uncertainty parameters and a common aggregate productivity shock. Then, we would run paneldata regressions with both state-fixed effects and time-fixed effects. The inclusion of time-fixed effects is important because this removes the first-order impact of the more persistent aggregate (U.S.-wide) shocks due to the non-linearity of the model, the results are not, however, identical to simulating the model without any U.S. wide component. Due to computational limitations, we simulate states, with state-specific shocks generated from a common distribution. In other words, our simulated states are ex-ante identical but ex-post different because they are subject 9 Net financial wealth excludes nonfinancial assets (houses and vehicles) and mortgage debt. The median ratio for all ages is The median net worth to income ratio is 1.93 including all ages, and 1.11 for those younger than 55. 9

11 to different shocks. 10 We calculate marginal effects on the MPCs of changes in persistence, unemployment, and changes in transitory and permanent uncertainty by changing the parameters of our baseline calibration (for all states) one at a time. We simulate income paths for 30,000 individuals in 10 states 3,000 per state for a number of periods. All individuals share a common aggregate shock each period, and individuals living in the same state share state-specific shocks. Using the optimal consumption functions and the simulated income paths, we calculate state-level consumption, and state-level income C st, Y st as the average of individual consumption and income over all consumers living in state s. Then, we run the following panel regressions: logc st = µ s + v t + α c logy st + ε st, (4) logc st = µ s + v t + α l logy s,t 1 + ε st. (5) µ s are state-fixed effects and v t are time-fixed effects that control for aggregate effects. Thus, ˆα c is the estimated MPC out of current state-specific income shocks, and similarly ˆα l is the estimated MPC out of lagged state-specific income shocks for brevity, we refer to them as the current/lagged MPCs. We repeat this process 20 times and report, in Table 2, the average current and lagged MPCs across the 20 independent samples. In order to take into account that consumers decision intervals and data-sampling intervals may be different, we allow for temporal aggregation. In particular, we assume that while agents make decisions on a bi-annual basis, we only observe annual data. In our regressions with simulated data, state-level consumption in year t is C st = C 1 st + C 2 st, where C 1 st and C 2 st are calculated as the average of individual consumption in state s for the first and second half of the year; respectively, and analogously for income. 11 In Section 2.3, we present calibration parameters in annual terms. In our simulations, the parameters are adjusted to represent bi-annual periods. The mapping is straightforward for most parameters. Note that if z is an AR(1) process, z t = az t 1 + ε t, and we only observe it every h th period, then the observed process z τ, is also an AR(1) process with parameters 10 Thus, state-fixed effects are not necessary in the regressions with simulated data but are included for comparability with the regressions using actual data. 11 Time aggregation generates excess sensitivity in a representative PIH model (see Working 1960). 10

12 a τ = a h and σε,τ 2 = σε(1 2 a 2h )(1 a 2 ) 1 (see Bergstrom 1984). We simulate 250 bi-annual periods (125 years) but we use only the last 70 years of simulated data for our regressions. Using a different number of periods would change the standard error of the regressions but not the point estimates, at least not considerably. The purpose of our simulations is not to replicate the exact size of MPCs observed in statelevel data, but to determine how the current and lagged MPCs out of state-specific income shocks vary with persistence and uncertainty. The simulated lagged MPC is close to its empirical counterpart in our baseline calibration, while the simulated current MPC is larger. Ludvigson and Michaelides (2001) show that an explicitly aggregated buffer-stock model cannot replicate the excess smoothness and excess sensitivity observed in U.S. aggregate data and recur to incomplete information to generate some excesses. 12 Incorporating durables or habits into the model can also reduce the predicted current MPC substantially (see Carroll 2000, Luengo- Prado 2001, Michaelides 2001, Díaz and Luengo-Prado 2002). 3 Simulation Results Table 2 presents results comparing an explicitly aggregated buffer-stock model, as just described, to the closed-form predictions from a log-approximation to a representative-agent PIH-model (where the representative agent receives the state-level income process described in Appendix C. The table presents the MPCs out of current and lagged state-specific income for both models as well as the median wealth-to-income ratio for the buffer-stock model. Estimated standard errors are given in parentheses. 13 A complementary table, Table 3, reports the marginal effects of changing the parameters of our simulations on the MPCs. The marginal effects are computed as the change in the corresponding MPC relative to the baseline case divided by the change in the parameter being altered. Table 3 includes marginal effects for the aggregated buffer-stock model, for the log-linear approximation of the PIH-model, and for a combination of the two (a half-half combination of buffer-stock and PIH consumers). In Section 4, we compare the sign and size of these marginal effects to our empirical findings. 12 Since our model allows for individual-level, state-level and aggregate income shocks, we could introduce incomplete information several different ways. This is an interesting extension which we leave for future research. 13 These are the average estimated standard errors of ˆα c and ˆα l in regressions (4) and (5), respectively, across the 20 independent samples. 11

13 The baseline simulations have no state-specific transitory shocks and allow for the statespecific permanent shocks to be persistent (a s =0.165). In this case, the PIH predicts a current MPC higher than 1. Also, because of time aggregation, the PIH delivers a non-zero lagged MPC. For our baseline case, the current and lagged MPCs in the PIH model are 1.14 and 0.15 respectively. In the buffer-stock model, agents cannot borrow and even though they have some assets because of prudence, asset holdings are small due to impatience. Hence, individuals cannot increase consumption as much as PIH consumers would when facing a persistent positive permanent shock, resulting in a lower current MPC and a higher lagged MPC. For our baseline case, the current and lagged MPCs in the buffer-stock model are 1.01 and 0.17 respectively. 14 Decreasing persistence to 0 the case where state-specific permanent shocks are i.i.d. lowers the MPCs out of current and lagged income in the buffer-stock model (from 1.01 to 0.97 and from 0.17 to 0.15 respectively). In the PIH, the current MPC decreases as well (from 1.13 to 1), and the lagged MPC increases slightly (from 0.15 to 0.17). 15 Table 3 shows that the marginal effect of increasing persistence on the current MPC is 0.82 in the PIH-model and 0.24 in the buffer-stock model. The marginal effect of persistence on the lagged MPC is 0.13 in the PIH-model and 0.15 in the buffer-stock model. We add unemployment to our simulations using a annual probability of unemployment of 3 percent. We consider a case with an unemployment benefit that replaces 30 percent of average income and a case with no unemployment protection. With unemployment, the MPCs decrease greatly in the buffer-stock model due to higher precautionary saving; not surprisingly, the decrease in the current MPC is more substantial if no income replacement program is present. The median wealth-to-income ratio increases dramatically from 0.12 to 0.54 in the case with unemployment benefits and to 1.05 in the case with no income replacement. The marginal effects shown in Table 3 are quite large: a 1 percent change in the probability of job loss will lower the MPC out of current income by 0.03 in the case with no replacement changes of this magnitude are probably common over the business cycle. In the PIH-model, the MPCs are not affected. 14 For our baseline calibration and no time aggregation, the current and lagged MPCs for the buffer-stock model are 0.98 and 0.03 respectively. For the PIH model, 1.16 and The increase in the lagged MPC in the PIH model when decreasing persistence can be explained by a more sizable decrease in the variance on income growth relative to the covariance of current consumption growth and lagged income growth as described in Appendix C. 12

14 Next, we examine the marginal effects of uncertainty by changing the standard deviation of the different income shocks one at a time. Contrary to the PIH case, these changes have large effects on the MPCs in the buffer-stock model. We start with the idiosyncratic shocks by reducing their standard deviations by half (one at a time). Because of less uncertainty, agents save less, which might be expected to lead to higher current MPCs. The median wealth-to-income ratio falls from 0.12 to 0.05 (0.06) when reducing the standard deviation of the idiosyncratic permanent (transitory) shock from 0.10 to However, we do not observe significant differences in the current MPC from the baseline case. This is because with liquidity constraints, less saving implies agents cannot increase consumption more than income in response to a persistent positive permanent shock. This effect would tend to lower the MPCs out of current income and offset the former effect. Moreover, because of lower saving agents are liquidity constrained more often, resulting in higher lagged MPCs. The marginal effects in Table 3 look quite large, but a unit increase in the standard deviation of any of the income shocks corresponds to quite a massive increase in uncertainty. Finally, more state-level aggregate uncertainty is introduced by changing the standard deviation of the state-level shocks one at a time. We increase the standard deviation in this case. 16 More state-level aggregate permanent uncertainty results in a higher current MPC in these simulations. Because agents hold more assets due to higher uncertainty (the median wealth-toincome ratio is instead of 0.118), they can adjust consumption more promptly in response to persistent positive permanent income shocks, which increases the current MPC. Also, consumers are constrained less often and the lagged MPC decreases slightly. Introducing state-level aggregate transitory uncertainty lowers both MPCs due to higher precautionary saving in the buffer-stock model. In this case, the MPCs decrease in the PIH-model as well. Intuitively, the current MPC depends only on the persistence of shocks in the PIH-model and the effect of higher state-level transitory uncertainty comes from the temporary shocks getting larger relative to the persistent shocks, thereby lowering the average persistence of shocks. Our findings are potentially important for macroeconomic forecasting because they demonstrate that small changes in uncertainty result in substantial changes in the behavior of aggregate consumption. In particular, a small change in the probability of job loss can significantly change 16 The direction of the changes are chosen such that the model satisfies the convergence condition of footnote 5. 13

15 the way consumption reacts to income growth. We explore the robustness of the results to variations in the baseline parameters. For brevity, we discuss results for a lower risk aversion parameter here (Table 4) and present alternative scenarios in Appendix D. The main determining factor for the results is how the wealth-to-income ratio changes. Lowering risk aversion from 2 to 1 decreases the median wealth-to-income ratio considerably. Consumers now have less resources with which to react to persistent permanent income shocks, resulting in a lower current MPC and a higher lagged MPC. In general, the lower the median wealth-to-income ratio in the model, the stronger the effect of parameter changes on the lagged MPC and the weaker the effect on the current MPC. The effects of persistence and unemployment are robust: the higher persistence, the higher the MPCs; the higher unemployment, the lower the MPCs. Also, more aggregate transitory uncertainty clearly results in lower current and lagged MPCs. As before, more idiosyncratic uncertainty either permanent or transitory produces a lower lagged MPC, but not much of an effect on the current MPC. More aggregate permanent uncertainty results in a higher current MPC because the change leads to a slight increase in wealth; however, now there is no significant effect on the lagged MPC because wealth holdings are not sufficiently high to alleviate the liquidity constraint. 4 Panel-data Estimation of the MPCs 4.1 Econometric Implementation Let c st logc st denote the growth rate of state-level consumption. 17 In our implementation, we regress c st on income growth, y st, and lagged income growth, y s,t 1, respectively. Aggregate policy and aggregate interest rates affect consumption. It is not obvious how to best capture such aggregate effects using exogenous regressors, so we follow Ostergaard, Sørensen, and Yosha (2002) and perform all regressions in terms of the deviations from the average value across states in each time period. 18 In symbols, we regress c st c.t on y st ȳ.t and y s,t 1 ȳ.,t 1, respectively, 17 We tested the level of consumption (non-durable retail sales) for unit roots and could only reject the unit root for 4 and 0 states at the 5 and 1 percent level, respectively, for the sample (the numbers are 2 and 1, respectively, for the sample). Therefore, the growth rate of consumption is reasonably well modeled as a stationary variable. 18 Empirically, it matters little if the data are adjusted by subtracting average values of the state-level variables or if U.S.-wide aggregate values are subtracted. The method chosen here is the most straightforward in terms of 14

16 where c.t = 1 50 Σ50 s=1 c st is the time-specific mean of consumption growth and similarly for the other variables. Removing time-specific means is equivalent to including a dummy variable for each time-period. Such time-specific dummy variables are referred to as time-fixed effects in the panel-data literature. Including time-fixed effects implies that we are measuring the effect on state-specific consumption of state-specific changes in income. We also want our results to be robust to permanent differences between the states. For instance, some states may have higher consumption growth due to demographic factors that are hard to control for. We, therefore, also remove state-specific averages; i.e., we use data in the form (for a generic variable x): z st = x st x.t x s. + x.., where x s. = 1 T ΣT t=1 x st is the state-specific mean of x and the last term is the overall average across states and time; this is added to keep the mean of z st equal to 0. Using variables in this form is equivalent to including state-specific (and, as before, timespecific) dummy variables. In the language of panel-data econometrics, we include a state-fixed effect (also referred to as a cross-sectional fixed effect ). We will use the shorter panel-data econometric notation and write our regressions as c st = µ s + v t + α c y st + ε st, (6) where the µ s terms symbolize the inclusion of cross-sectional fixed effects and the v t terms symbolize the inclusion of time-fixed effects. In the above regressions, α c is measuring the current MPC. The main focus of our empirical work is to examine if the MPC changes in the way predicted by the theoretical model. If X is a variable that might affect the MPC, we allow the MPC to change with X by estimating the regression c st = µ s + v t + α c st y st + ε st, (7) where α c st = α c + ζ c (X st X.t ). In this regression, the current MPC is α c + ζ c (X st X.t ) where the time-specific average of X st is subtracted in order to remove U.S.-wide aggregate effects. 19 We subtract the timeimplementation. 19 In order to estimate this model, we regress c st on y st, (X st X.t)(y st ȳ.t ȳ s. +ȳ..), and time- and state-specific dummy variables. 15

17 specific average X.t from the X variable so the ζ-coefficient will not pick up variations in the average (across states) MPC over time. We do not subtract the state-specific average from the X variable. The whole point of the exercise is to gauge if the MPC varies across states and, indeed, many of the X-variables we utilize are constant over time and would become trivially zero if the state-specific average was subtracted. In our implementation, we will often include more than one interaction variable and each of them will be treated as explained here. Our regressions using lagged income are done in the exact same fashion, substituting y t 1 for y t everywhere. We use the sample period State-level disposable labor income is constructed as described in Appendix A. We approximate state-level consumption by state-level retail sales. We transform retail sales and labor income to per capita terms and deflate them using the Consumer Price Index. See Appendix A for further details. 4.2 Selection of Regressors We turn to the empirical estimation of the MPCs as functions of state-level variables. As interaction terms, we use variables that approximate the parameters of the theoretical model. Persistence of aggregate shocks. Our measure of the persistence of aggregate shocks in state s is the estimated parameter â s shown in Table 1, column (1). Aggregate state-level uncertainty. We have two estimated parameters of state-level aggregate uncertainty: the standard error of the innovation to the persistent component of aggregate income ˆσ G and the standard error of the innovation to the transitory component of aggregate income ˆσ W. These are the numbers shown in Table 1, column (2) and column (3), respectively. Individual-level income volatility. We use the share of farmers in a state. Farmers are subject to substantially higher transitory income uncertainty than other income groups as documented in table 4 in Carroll and Samwick (1997). In other words, farmers may be particularly subject to the type of uncertainty that is captured by the parameter σ V in the model. In our simulations, all agents at the disaggregated level are subject to the same stochastic process for uncertainty and the simulations reveal that the aggregate lagged MPC will be lower when σ V is higher. Based 16

18 on this result, we will examine if the lagged MPC is lower in states where a relatively large number of consumers can be expected to have high variance of transitory idiosyncratic income. In our implementation, we use the interaction variable farm share (number of employed including proprietors in farming divided by total employment in the state). As discussed earlier, agricultural states may also have a high level of aggregate uncertainty and one might question if the share of agricultural employees can then be interpreted as a measure of individuallevel uncertainty. The interpretation is, however, valid in a multiple regression that also includes a separate measure of aggregate uncertainty in our case, the measure discussed earlier. Government sector jobs are less subject to the vagaries of nature and to the state-level business cycle implying that the share of government employees may be a good proxy for states with a low value of individual-level transitory uncertainty see table 4 in Carroll and Samwick (1997). One may also hypothesize that government employees are less likely to be subject to permanent idiosyncratic shocks (captured by σ N in the model) but since the predicted impact of transitory and permanent idiosyncratic shocks on the aggregate MPCs are similar, it is not important for our current purpose to sort this out. Individual-level probability of job loss. Our final indicator of individual-level uncertainty is the unemployment rate. When the unemployment rate is high, the risk of job loss is higher this is almost true by definition and we assume that this is associated with a higher risk of catastrophic income loss. In practice, the risk of job loss is quite unevenly distributed across the population and not necessarily well captured by the unemployment rate. On the other hand, casual empiricism suggests that periods of high unemployment are periods of high income uncertainty, so it is well-motivated to examine if high unemployment affects the MPCs in the direction suggested by our model. Because we focus on the potential income loss from job separation, we multiply the state-level unemployment rate by one minus the state s average unemployment insurance replacement rate; for brevity, we use the term unemployment. Correlation matrix for regressors. Table 5 presents the correlations of our regressors: unemployment, the share of farmers in total employment, the share of government employment, and the estimated persistence and the standard deviations of aggregate permanent and transitory shocks found in Table 1. Unemployment has low correlation with the other regressors, while 17

19 the share of agriculture is strongly correlated with persistence of income as well as with both parameters for aggregate uncertainty. The share of government is not highly correlated with other regressors. Finally, we observe a high positive correlation between the standard deviations of permanent and transitory state-level shocks. 4.3 Empirical results of the panel-data estimations We estimate the model from the empirical state-level data in the same manner as the statelevel regressions using simulated data, except that we also allow for state- and time-specific variances. In specifications that involve the state-level persistence and uncertainty parameters, the standard errors will be biased because these parameters are measured with error (they are generated regressors ). We calculate correct standard errors using a Monte Carlo method described in Appendix D. We report a selection of specifications. We chose to report the specifications with the statistically significant interaction effects included in most regressions. We prefer to include these one-by-one in order to convey to the reader if the coefficients to these interactions are robustly estimated. The interaction terms that are insignificant are included to show the point estimates, but these variables are included one at a time in order to not clutter the table. Current MPC. Column (1) of Table 6 reports the MPC out of current income from a regression of consumption growth on current income growth and persistence of state-specific shocks. We find that the MPC for a state with average persistence is The coefficient to the income variable can be interpreted as the MPC for an average state because the interactions have all been demeaned. This is clearly lower than the coefficients near 1 found in Table 2; our baseline version of the buffer-stock model is, therefore, not the full truth. This is, of course, the well-known excess smoothness result. Our main focus is on the interaction effects. The main conclusions are quite clear. The effect of persistence is estimated robustly with very large t-statistics and this variable is, therefore, included in all of columns (1) (6). The estimated value of the coefficient to persistence of statespecific income is between 0.44 and This corresponds to a large economic impact. For example, using the estimated values in column (1), the MPC in Iowa (with persistence 0.16) 18

20 is predicted to be about 0.02 while the MPC in Oklahoma (with persistence 0.42) is predicted to be The effect of persistence is estimated to be slightly smaller when other interaction terms are included. The other interaction term that is estimated to have a significant impact is the aggregate state-specific transitory variance. The predicted partial impact on the MPC of moving from no transitory aggregate uncertainty, as in Maryland, to an agricultural state with high transitory uncertainty, such as Nebraska (with ˆσ W = 0.02), is The impact of this parameter is not robust to the inclusion of the parameter of the variance of the permanent component (ˆσ G ), as can be seen from column (6), but (not tabulated) experimentation with the specification reveals that the impact of ˆσ G is, in general, not robust or significant. Therefore, our preferred specification is one that includes ˆσ W and not ˆσ G. No other regressors are robustly or significantly estimated at the conventional 5 percent level. We illustrate this by including each of them one-by-one. While not statistically significant, the coefficients to these all have the expected signs. Lagged MPC. Table 7 examines the same specification in terms of the MPC out of lagged income ( excess sensitivity ). For the average state, the MPC is estimated to be between 0.22 in column (2), and 0.29 in column (3). The sensitivity of consumption to lagged income is clearly and robustly lower in agricultural states. The estimated value is around 5, implying that a 10 percent increase in agricultural employment can be expected to lower the lagged MPC by 0.5. The order of magnitude and the level of significance is very robustly estimated and we include the share of agriculture in all specifications. In column (2), we add the share of government employees which is also robustly significant with the expected positive sign. The effect is similar, if slightly lower, in absolute magnitude to that found for agriculture. Column (3) includes unemployment but not the share of government employees, and column (4) includes interaction terms for both unemployment and government employees. Unemployment is nearly significant with the expected negative sign. When other interaction terms are included, it is robustly estimated with a coefficient of around 12 which implies that an increase in the unemployment rate of 1 percentage point will lower the MPC by around 0.1. Because agricultural employment share, government employment share, and unemployment are all robustly and significantly es- 20 The number for, e.g., Iowa, is obtained as ( ), where the term corresponds to the subtraction of the average value. 19