The Buffer-Stock Model and the Marginal Propensity to

Consume. A Panel-Data Study of the U.S. States.

Marıa Jose Luengo-PradoNortheastern University

Bent E. SørensenUniversity 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 andlagged income using simulated data generated by the model. We calculate the predictedmarginal effects of changing state-specific persistence of income shocks, uncertainty of state-specific 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 thepredictions of the model and while the effect of state-level uncertainty cannot be determinedprecisely, indicators of individual-level uncertainty have strong effects consistent with themodel. Overall, the buffer-stock model clearly helps explain differences in consumer behavioracross states.

KEYWORDS: Buffer-stock, Consumption, Precautionary saving.JEL classifications: E21 - Consumption; Saving.

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 Hypoth-

esis (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 individual-

level 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 iden-

tically 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

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 state-

level 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 ap-

proaches 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 addi-

tion, 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. Fried-

man (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 expec-

tations 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,1Furthermore, compared to cross-country data, the data are collected in a consistent manner and most in-

stitutional 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 panel-data regressions with time-fixed effects will make the results more robust to potential biases that might obtainbecause the U.S. as whole is unable to borrow internationally at fixed interest rates.

2

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 closed-

form 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 uncer-

tainty 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).2For 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 hasbeen documented by Deaton (1987), Campbell and Deaton (1989), and Galı (1991).

3Numerous 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”. Kim-ball (1990) discusses the conditions under which one can expect to observe precautionary saving. Carroll andKimball (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. Underthe assumption that consumers have constant relative risk aversion utility functions, it is not possible to obtainclosed-form solutions and, therefore, numerical methods must be used to study the reaction of consumption andsaving 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) usingsurvey data.

3

Browning and Lusardi (1996) provide a comprehensive survey of the empirical literature. Gour-

inchas 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 in-

duced 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 unem-

ployment 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 coun-

tries 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 approxima-

tion of aggregate income uncertainty—find more precautionary saving in periods of high uncer-

tainty 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—

a 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 Gour-

inchas 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

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 uncer-

tainty 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 zt = azt−1 + ut; and we refer to higher values of the

parameter a as higher levels of persistence of the shocks. In order to explore the aggregate im-

plications 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. Let β < 1 be the discount factor and R the interest factor. Sjt 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 RSjt−1 plus Yjt units of labor income. The agent

5

chooses optimal consumption Cjt according to the maximization problem:

maxCjt

E0

{ ∞∑t=0

βt U(Cjt)

}

s.t. Sjt = RSjt−1 + Yjt − Cjt. (1)

The utility function is assumed to exhibit constant relative risk aversion: U(Cjt) =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 assump-

tions. Deaton (1991) explicitly imposes a no-borrowing constraint (Sjt > 0) but assumes agents

always receive positive income. Carroll (1992), on the other hand, endogenously generates a no-

borrowing 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 bor-

row 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:

Yjt = PjtVjtWst,

Pjt = Pjt−1 At Gst Njt. (2)

Labor income, Yjt, is the product of permanent income, Pjt, an idiosyncratic transitory shock,

Vjt, and a state-level transitory shock, Wst. At can be thought of as growth of permanent

income attributable to aggregate productivity growth in the economy, while Gst reflects growth

of permanent income specific to state s. Njt is a permanent idiosyncratic shock. log Njt, log Vjt

and log Wst are independent and identically normally distributed with means −σ2N/2, −σ2

V /2

−σ2W /2, and variances σ2

N , σ2V and σ2

W , respectively. log At is assumed to be an AR(1) process

with persistence aA, unconditional mean µA, and variance σ2A. log Gst is also an AR(1) process

6

with persistence as, mean 0, and variance σ2G.

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 Yjt = log At +

log Gst+log Njt+log Vjt−log Vjt−1+log Wst−log Ws,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 Yt = log At, while state-specific income growth is ∆ log Yst −∆ log Yt = log Gst + log Wst − log Ws,t−1.

2.2 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,

Xjt ≡ RSjt−1+Yjt.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 ′(cjt) = max{U ′(xjt), βREt[(At+1Gs,t+1Nj,t+1)−ρU ′(cj,t+1)]}, (3)

where cjt = Cjt/Pjt and xj,t+1 = (At+1Gs,t+1Nj,t+1)−1R(xjt − cjt) + Vj,t+1Ws,t+1.5 Individuals

distinguish aggregate from state-specific shocks and optimize accordingly. We use Euler equa-

tion 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 , and 5 points for A 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)4The budget constraint becomes Sjt = Xjt − Cjt and the liquidity constraint Cjt ≤ Xjt. Combining the

definition of cash-on-hand and the budget constraint, we can write an expression for the evolution of cash-on-hand: Xjt+1 = R(Xjt − Cjt) + Yj,t+1.

5A necessary condition for the individual Euler equation to define a contraction mapping isβREt[(At+1Gs,t+1Nj,t+1)

−ρ] < 1. This is the “impatience” condition common to buffer-stock models whichguarantees that borrowing is part of the unconstrained plan.

7

combination of the discrete approximations of the persistent permanent income shocks.6

2.3 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 Yt = At, obtaining

estimates of µA = 0.016, aA = 0.42 and σA = 0.02.7 (Data sources are described in Appen-

dix A). We estimate the more complicated process for state-specific income, ∆ log Yst−∆ log Yt =

log Gst + log Wst − log Ws,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 In the estimation of state-specific

processes, we demean the data prior to estimation.

In Table 1, we report the estimated values of as, σG, and σW for each of the 50 U.S. states.

The estimates of the standard deviation of the state-level transitory shock, σW , 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 as is –0.46 for Nebraska

while the largest value of 0.71 is found for South Carolina. The average value is 0.165—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 as = 0.165 and σG = 0.016—the

simple average across states. We choose σW = 0—no state-specific transitory shocks—since the

average σW across states is close to this value.6More details on how to solve this equation can be found, for example, in the appendix of Ludvigson and

Michaelides (2001).7If we allow for an aggregate transitory shock, we find a point estimate of 0 consistent with our specification.8Hahm (1999) explores the impact of growth on aggregate consumption using country-level data.

8

Idiosyncratic income shocks are taken from previous studies—see, for example, Carroll and

Samwick (1997) and Gourinchas and Parker (2001). In particular, we set σV = σN = 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. 0.12 is also the median net

financial wealth-to-income ratio for households with head under 55 in the Survey of Consumer

Finances, 1998.9 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 panel-

data 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 subject9Net financial wealth excludes nonfinancial assets (houses and vehicles) and mortgage debt. The median ratio

for all ages is 0.32. The median net worth to income ratio is 1.93 including all ages, and 1.11 for those youngerthan 55.

9

to different shocks.10 We calculate marginal effects on the MPCs of changes in persistence, un-

employment, 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—Cst,

Yst—as the average of individual consumption and income over all consumers living in state s.

Then, we run the following panel regressions:

∆ log Cst = µs + vt + αc ∆ log Yst + εst, (4)

∆ log Cst = µs + vt + αl ∆ log Ys,t−1 + εst. (5)

µs are state-fixed effects and vt are time-fixed effects that control for aggregate effects. Thus,

αc is the estimated MPC out of current state-specific income shocks, and similarly αl 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 Cst = C1st + C2

st, where C1st and C2

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, zt = azt−1 + εt, and we only observe

it every hth period, then the observed process zτ , is also an AR(1) process with parameters10Thus, state-fixed effects are not necessary in the regressions with simulated data but are included for compa-

rability with the regressions using actual data.11Time aggregation generates excess sensitivity in a representative PIH model (see Working 1960).

10

aτ = ah and σ2ε,τ = σ2

ε(1 − a2h)(1 − a2)−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 state-

level 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 empir-

ical counterpart in our baseline calibration, while the simulated current MPC is larger. Lud-

vigson 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, Dıaz 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.12Since 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.13These are the average estimated standard errors of αc and αl in regressions (4) and (5), respectively, across

the 20 independent samples.

11

The baseline simulations have no state-specific transitory shocks and allow for the state-

specific permanent shocks to be persistent (as = 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 de-

crease 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.14For 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 0.15The 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 andlagged income growth as described in Appendix C.

12

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 0.05. 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 devi-

ation 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-to-

income ratio is 0.145 instead of 0.118), they can adjust consumption more promptly in response

to persistent positive permanent income shocks, which increases the current MPC. Also, con-

sumers 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 demon-

strate 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 change16The direction of the changes are chosen such that the model satisfies the convergence condition of footnote 5.

13

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 cst ≡ ∆ log Cst denote the growth rate of state-level consumption.17 In our implementation,

we regress cst on income growth, yst, and lagged income growth, ys,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 cst− c.t on yst− y.t and ys,t−1− y.,t−1, respectively,

17We tested the level of consumption (non-durable retail sales) for unit roots and could only reject the unit rootfor 4 and 0 states at the 5 and 1 percent level, respectively, for the 1976–1998 sample (the numbers are 2 and 1,respectively, for the 1964–1998 sample). Therefore, the growth rate of consumption is reasonably well modeledas a stationary variable.

18Empirically, it matters little if the data are adjusted by subtracting average values of the state-level variablesor if U.S.-wide aggregate values are subtracted. The method chosen here is the most straightforward in terms of

14

where c.t = 150Σ50

s=1cst 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):

zst = xst − x.t − xs. + x.., where xs. = 1T ΣT

t=1xst 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 zst equal

to 0. Using variables in this form is equivalent to including state-specific (and, as before, time-

specific) 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

cst = µs + vt + αc yst + εst , (6)

where the µs terms symbolize the inclusion of cross-sectional fixed effects and the vt terms

symbolize the inclusion of time-fixed effects. In the above regressions, αc 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

cst = µs + vt + αcst yst + εst , (7)

where

αcst = αc + ζc (Xst − X.t) .

In this regression, the current MPC is αc + ζc (Xst − X.t) where the time-specific average

of Xst is subtracted in order to remove U.S.-wide aggregate effects.19 We subtract the time-

implementation.19In order to estimate this model, we regress cst on yst, (Xst − X.t)(yst − y.t − ys. + y..), and time- and

state-specific dummy variables.

15

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 yt−1 for yt everywhere.

We use the sample period 1976–1998. 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 as 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 σG and the standard error of the innovation to the transitory component of aggregate

income σW . 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 σV 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 σV is higher. Based

16

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 individual-

level 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 σN 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: unem-

ployment, 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

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 state-

level 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 sta-

tistically 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 regres-

sion 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 0.26. 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 base-

line 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 state-

specific income is between 0.44 and 0.75. 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

is predicted to be about 0.02 while the MPC in Oklahoma (with persistence 0.42) is predicted

to be 0.45.20 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 σW = 0.02), is 0.22. The impact of

this parameter is not robust to the inclusion of the parameter of the variance of the permanent

component (σG), as can be seen from column (6), but (not tabulated) experimentation with the

specification reveals that the impact of σG is, in general, not robust or significant. Therefore,

our preferred specification is one that includes σW and not σG.

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 simi-

lar, 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 es-

timated 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-20The number for, e.g., Iowa, is obtained as 0.26 + 0.75 × (−0.16 − 0.165), where the term −0.165 corresponds

to the subtraction of the average value.

19

timated, we keep these three variables in columns (5)–(7) where we examine the impact of the

remaining regressors. Column (5) reports the effect of persistence. Surprisingly, given the strong

effect of persistence in the current MPC, the lagged MPC is not different in states with high vs.

low persistence of state-specific shocks. The last two columns examine the effect of aggregate

uncertainty. We find clearly insignificant effects of both aggregate standard deviations. These

regressors both have the expected negative sign but they are not robustly estimated.

4.4 Match of Theory and Data.

Match for the current MPC. Comparing the empirical findings of the panel-data estimations in

Table 6 with the predictions of the buffer-stock (and PIH) model in Table 3, the buffer-stock

model and the data strongly agree on a clear effect of persistency with the correct sign: the more

persistence, the higher the MPC. The coefficient found in the data is, in most columns, very

close to that predicted by a model with a 50-50 mix of buffer-stock and PIH consumers. We are

not able to significantly match the predicted effect of unemployment although the sign of the

estimated coefficient is as anticipated. The estimated coefficient to state-level aggregate transi-

tory uncertainty is significantly and has the predicted sign, although the estimated coefficient is

somewhat smaller than predicted by the model. The insignificant effect of state-level permanent

uncertainty is consistent with the relatively small impact found in our estimations. The in-

significant indicators of individual-level uncertainty (government and agricultural employment,

respectively) are consistent with the near-zero effects predicted by the model.21

Match for the lagged MPC. A comparison of the empirical results in Table 7 with the theo-

retical predictions for the lagged MPC reveals a very good fit. The model predicts a very small

effect of persistence if we consider a 50-50 split of PIH and buffer-stock consumers and we em-

pirically find no effect. State-level uncertainty is predicted to have a smaller effect on the MPC

from lagged income and we found none. Empirically, we found a significant negative effect of

unemployment in line with the model. The estimated effect is much larger than predicted, but

measured unemployment is not directly a measure of the probability of job loss so it is not really

meaningful to compare the orders of magnitude. This is also the case for the other indicators21The effect of individual-level uncertainty on the current MPC would, however, be larger in situations where

wealth holdings are higher. This can be seen from Table 9 in Appendix D.

20

of individual-level uncertainty. Agricultural and government shares in employment are strongly

significant in agreement with the model’s predictions. The marginal effect of a change in the

standard deviation of individual-level shocks is predicted to be of the same order of magnitude

as that of the aggregate shock, but typically the variance of individual-level income is an order

of magnitude larger than that of aggregate income, which is why the effect of individual-level

uncertainty is predicted to be stronger. Overall, the buffer-stock model is very successful in

predicting the differences in the lagged MPCs across states. The PIH-model, on the contrary,

has the implication that most of these are equal to zero!

5 Conclusion

The contributions of our paper are both theoretical and empirical. Based on simulating suitably

calibrated versions of the buffer-stock model, we find that the model predicts large effects on

the marginal propensities to consume out of current and lagged shocks to labor income from

persistence of aggregate shocks, aggregate uncertainty, and individual-level uncertainty. To the

best of our knowledge, a systematic quantification of the marginal impact of these economic

variables has not been done for this model.

Using panel-data regressions, we document varying propensities to consume across U.S.

states. More importantly, the propensities to consume vary in ways explained by a suitably

calibrated buffer-stock model. The good match of the positive predictions of the model with

aggregate data suggests that it is important to use this model to supplement the standard PIH-

model in macroeconomic forecasting. Our results are derived from cross-state differences in

consumer behavior but similar results will likely hold in aggregate data as uncertainty changes

over the business cycle. A challenging topic for future research is to document this conjecture—

this will be challenging because the limited number of aggregate business cycles provides few

degrees of freedom for econometricians.

21

Appendix A: The Data

We use state-level annual data from a variety of sources.

We construct state-level disposable labor income for the period 1964–1998 using data from

the Bureau of Economic Analysis (BEA). We define labor income as personal income minus

dividends, interest and rent, and social security contributions. We calculate after-tax labor

income by multiplying labor income by one minus the tax rate, where we approximate the tax

rate by total personal taxes divided by personal income for each state in each year. We refer to

the resulting series as disposable labor income or—for brevity—just as labor income or income.

The panel regressions in Section 4 use a shorter sample, 1976–1998, due to lack of availability

of other variables prior to 1976. However, in order to obtain more precise parameter estimates

we use the larger sample in the calibration described in Appendix B. (We also, for robustness,

used the BEA disposable personal income data by state and found qualitatively similar results.)

We perform state-by-state Augmented Dickey-Fuller (ADF) tests for unit roots in labor

income. These tests reject the unit root null hypothesis for only a few states at conventional

levels of significance.22 ADF tests provide somewhat weak evidence because they have low power

for samples as short as ours. The overall impression is, nevertheless, that U.S. state-level labor

income is well-described as an integrated process.23 We, therefore, treat labor income growth

as a stationary series.

We approximate state-level consumption by state-level retail sales published in the Survey

of Buying power, in Sales Management (after 1976, Sales and Marketing Management). Retail

sales are a somewhat noisy proxy for state-level private consumption but no better data seems to

exist. The retail sales data are available from 1963–1998. The correlation between annual growth

rates of aggregate U.S. total (nondurable) retail sales and aggregate U.S. total (nondurable and

services) private consumption from the National Income and Product Account, both measured22For the 1976–1998 sample, we can reject a unit root in labor income for only 7 states at the 5 percent level

of significance and for no states at the 1 percent level. Using the longer 1964–1998 sample, we can reject the unitroot for only 2 states at the 5 percent level and for none at the 1 percent level.

23Panel unit root tests are not attractive for these series since they are highly correlated across states. Os-tergaard, Sørensen, and Yosha (2002) show that panel unit root tests for disposable income—when aggregateincome is subtracted, making the data less correlated—provide little evidence against the unit root hypothesis.Disposable income is highly correlated with labor income state-by-state and the results using labor income would,therefore, be similar.

22

in real terms and per capita, is 0.84 (0.65). We transform the retail sales and labor income series

to per capita terms using population data from the BEA and deflate them using the Consumer

Price Index from the Bureau of Labor and Statistics (BLS).

We use state-level unemployment rates from the BLS available since 1976. In our implemen-

tation, we focus on the potential income loss from job separation and use unemployment rates

multiplied by 1 minus the average rate of income replacement. State-level income replacement

rates are from the Unemployment Insurance Financial Data Handbook published by the U.S.

Department of Labor. We also obtained the total number of employees (including proprietors

and partners) by state, as well as the number of employees in farming and government, from

the BEA.

Appendix B: Estimating the Process for State-Level DisposableLabor Income

Let yst be the growth rate of state-specific labor income. We apply a Kalman filter technique to

evaluate the likelihood function recursively, conditioning on the initial observation. The general

workings of Kalman filters are described in econometric textbooks. Concerning the technical

details, we parameterize the filter in terms of a transition equation for state s of the form

xst = Axs,t−1 + νst where

A =

as 0 0

0 0 −1

0 0 0

,

and ν ′st = (ust, log Wst, log Wst). The measurement equation is yst = (1, 1, 0)xst. as is the

persistence parameter, ust = σG εst, and log Wst = σW vst, where εst and vst are i.i.d. mean zero

shocks with variance 1. The remaining parts of the Kalman filter implementation are standard

and we leave out the details.24

The distribution of the estimated parameters is not well-known and we explored this issue

using Monte Carlo simulations. We do not tabulate the details but the results can be described24We hedged against local minima by performing grid-searches. Basically, we chose a grid for σW = 0 and

optimized over the remaining two parameters. The grid search did not reveal any local maxima.

23

as follows: the estimated standard errors are of the right magnitude and the estimated value

of σG is approximately unbiased. σW is also nearly unbiased, but a fraction of estimates will

accumulate exactly at 0. For small values of the persistence parameter, as, the estimated value

of as is unbiased when the true value of σW is 0 and approximately so for small values of this

parameter. For values of as above, say, 0.7 there will be some downward bias in this parameter

but not as much as in the plain AR(1) model (σW = 0), when the parameter is estimated by

Ordinary Least Squares (OLS). The estimated values of as are almost all much smaller than that.

When σW > 0, the persistence parameter displays downward bias, for example for σW = .01

and as = 0.30 the average value of as in simulations is about 0.25, and for very large values of

σW the bias in as becomes severe; but such large values of σW are not describing our data well

(the maximum estimated value is 0.027). Finally, we found that the fraction of estimated values

of σW exactly at 0 (for σG = 0.02 and as = 0.3) were about 35 percent when the true value is

σW = .01 and about 60 percent, when the true value is σW = 0.

All in all, our estimation method has the potential for highly biased parameters in samples

with 35 years of observations. However, the estimated parameter values are all in the range (low

persistence, small transitory errors) where the bias is “small” (no larger than typical for OLS

time series regressions on similar data).

Appendix C: The MPC in the PIH case

This appendix describes how to calculate the approximate current and lagged MPCs for the

PIH-model. First, we show how consumption reacts to income innovations and then we discuss

time-aggregation issues.

Consumption Growth and Innovations to Income

Given our assumptions about the income process, state-specific income growth is ∆(log Yst−log Yt) = log Gst + log Wst − log Ws,t−1. For notational simplification, let gt = log Gst, wt =

log Wst and log Yst − log Yt = yt. By assumption, gt = asgt−1 + εt. We drop the superscript in s

hereafter. Thus, ∆yt = gt + wt − wt−1 = a∆yt−1 + wt − (1 + a)wt−1 + awt−2.

Because the PIH is formulated in levels rather than logs, we must assume the process for the

24

first difference of state-specific income can be approximately described by the process for the log

difference: ∆(log Yst − log Yt) � ∆(Yst − Yt). Let ut = εt + wt. Hansen and Sargent (1981) and

Deaton (1992) show that if income can be represented by the ARMA process a(L)yt = b(L)ut,

the PIH predicts that: ∆Ct = r1+rut × b

(1

1+r

)/a

(1

1+r

).

The ARMA representation for our state-specific income process, (1 − aL)(1 − L)yt = εt +

wt − (1 + a)wt−1 + awt−2, is going to be of the form (1 − aL)(1 − L)yt = (1 + b1L + b2L2)ut.

We can easily solve for b1 and b2. Define Zt ≡ (1 − aL)(1 − L)yt. E(ZtZt−2) = aσ2w = b2σ

2u,

which implies b2 = aσ2w

σ2u

. E(ZtZt−1) = [−(1 + a) − (1 + a)a]σ2w = (b1 + b1b2)σ2

u. Plugging in for

b2, b1 = −(1+a)2σ2w

σ2u+aσ2

w, where σ2

u = σ2ε + σ2

w.

The consumption change in period t can be written as:

∆Ct =1 + r

1 + r − a

[1 +

b1

1 + r+

b2

(1 + r)2

]ut = Hut. (8)

If there are only permanent shocks (i.e, σw = 0), H = 1+r1+r−a . Also, H equals 1 when a = 0

and increases with persistence. If there are only transitory shocks (i.e., σε = 0), H = r1+r .

Time Aggregation

We assume that agents make consumption decisions bi-annually but we only observe data

at annual frequencies (i.e., we observe Ct = C1t + C2

t , where Cit is consumption in the ith half

of year t, i = 1, 2).

Using equation (8)—valid for the relevant frequency in the agent’s optimization problem—we

can derive an expression for C1t and C2

t :

C1t = C2

t−1 + Hu1t = C1

t−1 + H(u1t + u2

t−1),

C2t = C1

t + Hu2t = C2

t−1 + H(u2t + u1

t ), (9)

where the last part follows from recursive backward substitution. Thus, the annual consumption

change in period t is simply:

∆Ct = (C1t + C2

t − C1t−1 − C2

t−1) = H(u2t + 2u1

t + u2t−1). (10)

25

Given our state-specific income specification, we can write:

y1t = y2

t−1 + g1t + w1

t − w2t−1 = y1

t−1 + g2t−1 + g1

t + w1t − w1

t−1

y2t = y1

t + g2t + w2

t − w1t = y2

t−1 + g1t + g2

t + w2t − w2

t−1. (11)

Then,

∆yt = (y1t + y2

t − y1t−1 − y2

t−1) = g2t−1 + 2g1

t + g2t + w2

t + w1t − w2

t−1 − w1t−1.

Using recursive backward substitution, we obtain:

∆yt = d0g1t−1 + d1ε

2t + d2ε

1t + d3ε

2t−1 + w2

t + w1t − w2

t−1 − w1t−1, (12)

where d0 = (a + 2a2 + a3), d1 = 1, d2 = a + 2 and d3 = (a2 + 2a + 1).

We can calculate the MPC out of current and lagged state-specific income as cov(∆Ct,∆yt)var(∆yt)

and cov(∆Ct,∆yt−1)var(∆yt−1) respectively. Using equations (10) and (12), it is easy to show that:

var(∆yt) =(

d20

1−a2 + d21 + d2

2 + d23

)σ2

ε + 4σ2w,

cov(∆Ct, ∆yt) = H[(d1 + 2d2 + d3)σ2ε + 2σ2

w],

cov(∆Ct, ∆yt−1) = H(d1σ2ε + σ2

w).

If a = 0 and σw = 0 (i.e, permanent shocks are i.i.d. and there are no transitory shocks),

the MPC out of current income is 1 and the MPC out of lagged income is 1/6. Therefore, time

aggregation produces robust excess sensitivity in the PIH model (see Working 1960).

Furthermore, both cov(∆Ct, ∆yt) and var(∆yt) increase with a. For small values of a, the

covariance increases faster and the current MPC increases in a initially, but could eventually

decrease. Finally, increasing σw relative to σε decreases the current MPC since the actual

persistence of the overall income shock decreases.

26

Appendix D: Further Simulation Results

Tables 8 presents results for a higher discount rate, 5 percent. The results are remarkably

similar to the case with a lower risk aversion presented in Section 3. The median wealth-to-

income ratio falls considerably, resulting in a lower current MPC and a higher lagged MPC for

the baseline case. Similarly, persistence and unemployment have stronger effects on the lagged

MPCs. More idiosyncratic uncertainty, either permanent or transitory, still produces a lower

lagged MPC, while there is no significant effect on the current MPC. More aggregate permanent

uncertainty results in a higher current MPC and a lower lagged MPC because of the increase

in wealth. Finally, more aggregate transitory uncertainty clearly results in lower current and

lagged MPCs.

Increasing risk aversion to 3 produces interesting results. We can see in Table 9 that the

median wealth-to-income ratio increases to almost 0.5 from 0.12 in our original simulations. In

this situation, the buffer-stock consumer is seldom liquidity constrained and the consumers’s

behavior is close to that of a PIH consumer. The current (lagged) MPC is 1.06 (0.12) versus

1.14 (0.15) in the PIH case. The marginal effects of persistence are close to those in the PIH

case as well: increasing persistence increases the current MPC but lowers the lagged MPC.

Unlike the PIH, changes in uncertainty still have important effects on the MPCs for the buffer-

stock model. However, in this case, the effects are particularly strong on the current MPCs.

Allowing for unemployment decreases the current MPC significantly, and especially when no

income replacement program is in place. Reducing idiosyncratic permanent uncertainty decrease

wealth holdings dramatically (the median wealth-to-income ratio falls to just 0.07) resulting in

a lower current MPC and a much higher lagged MPC (i.e., consumers are often constrained).

Decreasing idiosyncratic transitory uncertainty has a less dramatic effect on wealth: there is a

small increase in the current MPC, since less saving is necessary, but no effect in the lagged

MPC. Increasing aggregate uncertainty more than doubles the median wealth-to-income ratio,

which translates into a slightly higher current MPC and a slightly lower lagged MPC. Increasing

transitory aggregate uncertainty still reduces the MPCs.

27

Appendix E: Estimation of Standard Errors

Because the parameters for the state-level income processes are estimated in an initial regression

they are random variables. This “generated regressors” problem leads to bias in the standard

errors reported by OLS. We, therefore, use a “parametric bootstrap” procedure to calculate

standard errors for all the coefficients.

Our approach is as follows. We regress consumption growth on the non-generated regressors

Xst (including fixed effects) and the estimated regressors Ys (which do not vary over time) using

OLS (after weighting the variables with state- and time-specific estimated standard errors):

cst = Xstγ + Ysδ + es ,

where s = 1, . . . , 50 is an index of the states, t = 1, ..., T is an index of time, and γ and δ

are OLS-coefficients. From this regression, we retrieve the estimated values γ and δ and the

estimated standard error se of the residuals es. We proceed to estimating the standard errors of

γ and δ from the following Monte Carlo experiment. In each iteration l we draw from an i.i.d.

N(0, se) distribution a vector of variables, e(l)st (s = 1, . . . , 50 ; t = 1, . . . , T ). We generate the

variable

c(l)st = Xstγ + Ysδ + e

(l)st .

Then, for each state s and time-period t, we generate Y(l)s by drawing from an N(Ys, ΣY s)

distribution where ΣY s is the variance matrix with the estimated standard errors of Ys reported

in Table 1 in the diagonal (for example, 0.017 for persistence in Alabama).

We then perform a panel-data regression of c(l)st on Xst and Y

(l)s and record the estimated

coefficients γ(l) and δ(l). We repeat this for l = 1, ..., 25000 and then calculate the standard

errors of γ(l) and δ(l). These are the standard errors reported in the tables.

28

References

Abowd, J., and D. Card (1989): “On the Covariance Structure of Earnings and HoursChanges,” Econometrica, 57, 411–445.

Aiyagari, S. R. (1994): “Uninsured Idiosyncratic Risk and Aggregate Saving,” QuarterlyJournal of Economics, 109, 659–684.

Attanasio, O., and G. Weber (1993): “Consumption Growth, the Interest Rate and Aggre-gation,” Review of Economic Studies, 60, 631–649.

Bergstrom, A. (1984): “Continuous Time Stochastic Models and Issues of Aggregation overTime,” in Handbook of Econometrics, Vol.2, chap. 20, pp. 1146–1210. North-Holland, NewYork.

Blinder, A., and A. Deaton (1985): “The Time Series Consumption Function Revisited,”Brookings Papers on Economic Activity, 1985:2, 465–521.

Browning, M., and A. Lusardi (1996): “Household Saving: Micro Theories and MicroFacts,” Journal of Economic Literature, 34, 1797–1855.

Caballero, R. (1991): “Earnings Uncertainty and Aggregate Wealth Accumulation,” Ameri-can Economic Review, 81, 859–871.

Campbell, J. Y., and A. Deaton (1989): “Why is Consumption so Smooth?” Review ofEconomic Studies, 56, 357–374.

Carroll, C. D. (1992): “The Buffer-Stock Theory of Saving: Some Macroeconomic Evidence,”Brookings Papers on Economic Activity, 2, 61–156.

(1997): “Buffer-Stock Saving and the Life Cycle/Permanent Income Hypothesis,” Quar-terly Journal of Economics, 112, 1–55.

(2000): “‘Risky Habits’ and the Marginal Propensity to Consume out of PermanentIncome,” International Economic Journal, 4, 1–40.

Carroll, C. D., K. Dynan, and S. D. Krane (2003): “Unemployment Risk and Precaution-ary Wealth: Evidence from Households’ Balance Sheets,” Review of Economic and Statistics,85, 586–604.

Carroll, C. D., and M. S. Kimball (1996): “On the Concavity of the Consumption Func-tion,” Econometrica, 64, 981–992.

Carroll, C. D., and A. A. Samwick (1997): “The Nature of Precautionary Wealth,” Journalof Monetary Economics, 40, 41–71.

Deaton, A. (1987): “Life-Cycle Models of Consumption: Is the Evidence Consistent with theTheory?” in Advances in Econometrics. Fifth World Congress, Vol. II, ed. by Truman F.Bewley. Cambridge University Press, New York, NY.

(1991): “Saving and Liquidity Constraints,” Econometrica, 59, 1121–1248.

29

(1992): Understanding Consumption. Oxford University Press, New York, NY.

Dıaz, A., and M. J. Luengo-Prado (2002): “Precautionary Savings and Wealth Distributionwith Durable Goods,” Mimeo.

Dynan, K. (1993): “How Prudent are Consumers?” Journal of Political Economy, 101, 1104–1113.

Engen, E. M., and J. Gruber (2001): “Unemployment Insurance and Precuationary Saving,”Journal of Monetary Economics, 47, 545–579.

Flavin, M. (1981): “The Adjustment of Consumption to Changing Expectations about FutureIncome,” Journal of Political Economy, 89, 974–1009.

Friedman, M. (1957): A Theory of the Consumption Function. Princeton University Press,Princeton, NJ.

Galı, J. (1991): “Budget Constraints and Time Series Evidence on Consumption,” AmericanEconomic Review, 81, 1238–1253.

Gourinchas, P.-O., and J. Parker (2001): “Consumption over the Life Cyle,” Econometrica,70, 47–91.

Hahm, J.-H. (1999): “Consumption Growth, Income Growth and Earnings Uncertainty: SimpleCross-Country Evidence,” International Economic Journal, 13, 39–58.

Hahm, J.-H., and D. Steigerwald (1999): “Consumption Adjustment under Time-VaryingUncertainty,” Review of Economic and Statistics, 81, 32–40.

Hall, R. E. (1978): “Stochastic Implications of the Life Cycle-Permanent Income Hypothesis:Theory and Evidence,” Journal of Political Economy, 86, 971–987.

Hansen, L. P., and T. Sargent (1981): “A Note on Wiener-Kolmogorov Predition Formulasfor Rational Expectations Models,” Economics Letters, 8, 255–260.

Hubbard, R. G., J. Skinner, and S. P. Zeldes (1994): “The Importance of Precaution-ary Motives in Explaining Individual and Aggregate Saving,” Carnegie-Rochester ConferenceSeries on Public Policy, 40, 59–125.

Jappelli, T. (1990): “Who is Credit Constrained in the U.S. Economy?” Quarterly Journalof Economics, 105, 219–235.

Kimball, M. S. (1990): “Precautionary Saving in the Small and in the Large,” Econometrica,58, 53–73.

Leland, H. (1968): “Saving and Uncertainty: The Precautionary Demand for Saving,” Quar-terly Journal of Economics, 82, 465–473.

Ludvigson, S., and A. Michaelides (2001): “Can Buffer Stock Saving Explain the Con-sumption Excesses?” American Economic Review, 91, 631–647.

30

Luengo-Prado, M. J. (2001): “Durables, Nondurables, Down Payments, and ConsumptionExcesses,” Mimeo.

MaCurdy, T. E. (1982): “The Use of Time-Series Processes to Model the Error Structure ofEarnings in Longitudinal Data analysis,” Journal of Econometrics, 18, 83–114.

McCarthy, J. (1995): “Imperfect Insurance and Differing Propensities to Consume AcrossHouseholds,” Journal of Monetary Economics, 36, 301–327.

Michaelides, A. (2001): “Buffer Stock Saving and Habit Formation,” Mimeo.

(2003): “A Reconciliation of Two Alternative Approaches towards Buffer Stock Saving,”Economics Letters, 79, 137–143.

Ostergaard, C., B. E. Sørensen, and O. Yosha (2002): “Consumption and AggregateConstraints: Evidence from U.S. States and Canadian Provinces,” Journal of Political Econ-omy, 110, 634–645.

Perraudin, W., and B. E. Sørensen (1992): “The Credit-Constrained Consumer: An Em-pirical Study of Demand and Supply in the Loan Market,” Journal of Business & EconomicStatistics, 10, 179–193.

Pischke, J.-S. (1995): “Individual Income, Incomplete Information and Aggregate Consump-tion,” Econometrica, 63, 805–840.

Skinner, J. (1988): “Risky Income, Life Cycle Consumption, and Precautionary Savings,”Journal of Monetary Economics, 22, 237–255.

Sørensen, B. E., and O. Yosha (2000): “Intranational and International Credit MarketIntegration: Evidence from Regional Income and Consumption Patterns,” in IntranationalMacroeconomics, ed. by Gregory Hess and Eric van Wincoop. Cambridge University Press,New York, NY.

Tauchen, G. (1986): “Finite State Markov Chain Approximations to Univariate and VectorAutoregressions,” Economics Letters, 20, 177–181.

Working, H. (1960): “Note on the Correlation of First Differences of Averages in a RandomChain,” Econometrica, 28, 916–918.

Zeldes, S. (1989a): “Consumption and Liquidity Constraints: An Empirical Investigation,”Journal of Political Economy, 142, 305–346.

(1989b): “Optimal Consumption with Stochastic Income: Deviations from CertaintyEquivalence,” Quarterly Journal of Economics, 104, 275–298.

31

Table 1: Parameters of Time Series Process for State-Level Disposable Labor Income.

(1) (2) (3)Persistence σG σW

Alabama 0.16 (0.17) 0.74 (0.09) 0.00 (2.80)Alaska 0.37 (0.16) 4.57 (0.54) 0.00 (1.02)Arizona 0.36 (0.28) 1.29 (0.31) 0.13 (0.87)Arkansas –0.14 (0.17) 1.70 (0.20) 0.00 (1.76)California 0.17 (0.19) 1.17 (0.14) 0.00 (0.87)Colorado 0.59 (0.20) 1.03 (0.25) 0.38 (0.18)Connecticut 0.39 (0.16) 1.56 (0.18) 0.00 (2.17)Delaware –0.04 (0.17) 1.46 (0.17) 0.00 (1.31)Florida 0.09 (0.17) 1.24 (0.15) 0.00 (0.69)Georgia 0.57 (0.20) 0.65 (0.15) 0.28 (0.11)Hawaii 0.32 (0.30) 2.35 (0.70) 0.75 (0.76)Idaho –0.53 (0.15) 2.49 (0.29) 0.00 (5.08)Illinois 0.59 (0.21) 0.55 (0.16) 0.43 (0.09)Indiana 0.06 (0.64) 1.31 (0.87) 0.54 (1.00)Iowa –0.16 (0.60) 1.67 (1.08) 2.24 (0.59)Kansas –0.04 (0.35) 1.10 (0.50) 0.43 (0.66)Kentucky –0.24 (0.15) 1.21 (0.14) 0.00 (1.61)Louisiana 0.50 (0.14) 1.73 (0.20) 0.00 (0.47)Maine –0.16 (0.16) 1.61 (0.19) 0.00 (2.21)Maryland 0.20 (0.17) 1.38 (0.16) 0.00 (0.86)Massachusetts 0.45 (0.15) 1.48 (0.17) 0.00 (0.48)Michigan 0.10 (0.27) 1.69 (0.61) 0.39 (1.17)Minnesota –0.18 (0.30) 1.26 (0.55) 0.98 (0.47)Mississippi 0.02 (0.18) 1.55 (0.18) 0.00 (1.41)Missouri –0.25 (0.16) 1.22 (0.14) 0.00 (0.55)Montana –0.23 (0.16) 2.78 (0.33) 0.00 (2.41)Nebraska –0.46 (0.34) 1.25 (0.67) 1.99 (0.46)Nevada 0.48 (0.18) 1.43 (0.29) 0.37 (0.30)New Hampshire 0.60 (0.24) 1.40 (0.47) 0.82 (0.29)New Jersey 0.25 (0.76) 1.31 (0.96) 0.31 (1.56)New Mexico 0.57 (0.24) 0.78 (0.29) 0.69 (0.17)New York 0.10 (0.17) 1.55 (0.18) 0.00 (1.36)North Carolina –0.06 (0.17) 1.04 (0.12) 0.00 (0.45)North Dakota –0.09 (0.30) 7.71 (3.34) 3.91 (3.74)Ohio –0.28 (0.16) 1.07 (0.13) 0.00 (1.15)Oklahoma 0.42 (0.27) 1.36 (0.38) 0.47 (0.34)Oregon 0.43 (0.32) 1.06 (0.38) 0.67 (0.24)Pennsylvania 0.07 (0.17) 0.94 (0.11) 0.00 (0.50)Rhode Island 0.36 (0.37) 1.50 (0.58) 0.65 (0.48)South Carolina 0.71 (0.16) 0.53 (0.18) 0.55 (0.11)South Dakota –0.23 (0.29) 3.94 (1.88) 2.67 (1.85)Tennessee –0.01 (0.14) 0.97 (0.11) 0.00 (2.87)Texas 0.48 (0.15) 1.28 (0.15) 0.00 (0.31)Utah 0.48 (0.23) 1.05 (0.25) 0.32 (0.24)Vermont 0.37 (0.33) 1.10 (0.38) 0.64 (0.25)Virginia 0.14 (0.16) 1.14 (0.13) 0.00 (0.29)Washington 0.40 (0.15) 1.45 (0.17) 0.00 (0.50)West Virginia 0.15 (0.16) 1.52 (0.18) 0.00 (2.70)Wisconsin –0.08 (0.16) 1.01 (0.12) 0.00 (0.60)Wyoming 0.48 (0.25) 2.20 (0.62) 0.80 (0.52)

Notes: The table displays the estimated parameters of a time series model for real disposable labor income. Let yst be thelog of per capita disposable labor income (deflated by the CPI) in state s. The model is: yst = µs +log Gst +σWs (log Wst−log Ws,t−1), where µs is a constant for each state, log Wst (an i.i.d. innovation to the level of yit) is normally distributedand independent of Gst, and log Gst = as log Gs,t−1 +σGs εst where εst are i.i.d. normal innovations. as is persistence, σGs

is the standard deviation of the permanent component and σWs is the standard deviation of the transitory component.The table reports the estimates of as in column (1), 100 times σGs in column (2), and 100 times σWs in column (3). Themodel is estimated by Maximum Likelihood using a Kalman Filter. Standard errors in parentheses. Sample 1964–1998.

Table 2: Sensitivity to Current and Lagged Income in Simulated Data

Buffer-Stock PIHMedian

MPC Wealth-Income MPCCurrent Lagged Ratio Current Lagged

Baseline 1.007 0.170 0.118 1.136 0.146(0.011) (0.041) (0.011)

No Persistence 0.967 0.146 0.114 1.000 0.167as = 0 (0.005) (0.038) (0.010)

UnemploymentReplacement 0.970 0.144 0.544 1.136 0.146

(0.013) (0.040) (0.024)

No replacement 0.916 0.142 1.052(0.013) (0.038) (0.037)

Less idiosyncraticuncertainty

σV = 0.05 1.008 0.194 0.050 1.136 0.146(0.010) (0.041) (0.007)

σN = 0.05 0.988 0.225 0.058 1.136 0.146(0.009) (0.039) (0.004)

More aggregateuncertainty

σG = 0.032 1.024 0.162 0.145 1.136 0.146(0.012) (0.042) (0.015)

σW = 0.01 0.883 0.148 0.119 0.647 0.133(0.016) (0.038) (0.011)

Notes: The columns labelled “current” report the estimated value of the parameter αc from the regression: ∆ log Cst =µs + vt + αc∆ log Yst + εst. The columns labelled “lagged” report the estimated value of the parameter αl for laggedincome from the regression ∆ log Cst = µs + vt + αl∆ log Ys,t−1 + εst. For the columns “buffer-stock”, consumption issimulated as described in the text. The simulated data are based on the individual-level income process ∆ log Yjt =log At + log Gst + log Wst − log Ws,t−1 + log Njt + log Vjt − log Vj,t−1, where log At and log Gst are AR(1) processeswith persistence aA and as respectively, unconditional means µA and 0, and standard deviation σA and σG. log Njt,log Vjt and log Wst are independent and identically distributed with standard deviations σN , σV and σW , and means−σ2

N/2, −σ2V /2 and −σ2

W /2, respectively. Baseline parameters in annual terms: µG = 0.0165, σA = 0.02, aA = 0.42.σG = 0.016, as = 0.165, σW = 0; and σN = σV = 0.1. The interest rate is 2 percent, the discount rate is 3 percent. Thecoefficient of risk aversion ρ = 2 and the unemployment probability is 0. In the case with unemployment, p = 0.03 andthe replacement rate is 30 percent when present. State-level consumption and income (Cst, Yst) are averages over 3,000individuals in each state. There are 10 states and 70 periods. We report average MPCs for 20 independent simulations.Average estimated standard errors in parentheses. For the columns labelled “PIH” the numbers are calculated for alog-linear approximation of a representative-agent PIH-model, with the interest rate equal to the discount rate and therepresentative agent receiving the aggregate income process.

33

Table 3: Sensitivity to Current and Lagged Income in Simulated Data. Marginal

Effects

Buffer-Stock PIH Combined

Current Lagged Current Lagged Current Lagged

Persistence 0.24 0.15 0.82 -0.13 0.53 0.01

UnemploymentReplacement –1.21 –0.88 0.00 0.00 –0.60 –0.44

No replacement –3.03 –0.94 0.00 0.00 –1.52 –0.47

Idiosyncratic uncertaintyσV –0.03 –0.49 0.00 0.00 –0.01 –0.24

σN 0.37 –1.10 0.00 0.00 0.19 –0.55

Aggregate Uncertainty

σG 1.08 –0.50 0.00 0.00 0.54 –0.25

σW –12.41 –2.21 –48.90 –1.38 –28.70 –1.75

Notes: The columns present marginal effects calculated using the MPCs in Table 2. Each marginal effect iscalculated as the change in the corresponding MPC relative to the baseline case divided by the change in theparameter being altered. The table includes marginal effects for the simulated buffer-stock model, the log-linearapproximation of a representative-agent PIH-model, and a combination of the two. For the combined case, it isassumed that half of the agents are PIH consumers and half the agents are buffer-stock consumers.

34

Table 4: Sensitivity to Current and Lagged Income in Simulated Data. ρ = 1

Buffer-Stock

Marginal MedianMPC effect Wealth-Income

Current Lagged Current Lagged Ratio

Baseline 0.984 0.221 – – 0.057(0.009) (0.039) (0.004)

No persistence 0.973 0.148 0.07 0.44 0.056(0.004) (0.038) (0.004)

Unemploymentreplacement 0.967 0.156 –0.37 –2.17 0.334

(0.012) (0.040) (0.013)no replacement 0.918 0.160 –2.20 –2.03 0.663

(0.011) (0.038) (0.018)Less idiosyncraticuncertaintyσN = 0.05 0.982 0.249 0.04 –0.56 0.04

(0.008) (0.037) (0.003)σV = 0.05 0.985 0.255 –0.02 –0.68 0.018

(0.007) (0.039) (0.002)More aggregateuncertaintyσG = 0.032 0.992 0.221 0.50 0.00 0.062

(0.009) (0.040) (0.005)σW = 0.01 0.888 0.185 –9.60 –3.60 0.057

(0.013) (0.037) (0.005)

Notes: All parameters as in Table 2, except ρ = 1.

35

Table 5: Correlation Matrix of Regressors

(1) (2) (3) (4) (5) (6)

(1) Unemployment Rate 1.00 –0.28 0.20 0.16 –0.06 –0.38(2) Farm Share 1.00 0.04 –0.58 0.47 0.64(3) Govt. Share 1.00 0.17 0.35 0.05(4) Persistence 1.00 –0.23 –0.17(5) σG 1.00 0.62(6) σW 1.00

Notes: The table shows how the variables correlate across the 50 U.S. states over the years 1976–1998. “UnemploymentRate” is the average unemployment rate for each state multiplied by one minus the state’s average unemployment insurancereplacement rate over the period 1976–1998. “Farm Share” is the ratio of employees (including proprietors) in farming tothe total number of employees in each state over the same sample. “Govt. Share” is defined similarly. “Persistence” refers,for each state, to the number in the column “Persistence” in Table 1, while the standard deviations in rows (5) and (6) arethe numbers given in columns (2) and (3) of Table 1, respectively. Time-specific averages are removed before calculatingthe correlations.

36

Table 6: Sensitivity to Current Income: Non-Durable Retail Sales

(1) (2) (3) (4) (5) (6)

State-fixed effects yes yes yes yes yes yesTime-fixed effect yes yes yes yes yes yes

yst 0.26 0.35 0.35 0.35 0.34 0.34(4.24) (4.93) (5.75) (5.40) (4.71) (5.12)

Interaction terms:

Persistence 0.75 0.55 0.58 0.56 0.48 0.44(4.84) (3.29) (3.65) (3.44) (2.80) (2.50)

σW - –6.71 –7.96 –7.07 –7.73 –21.27- (1.62) (2.32) (1.80) (1.74) (4.49)

Farm share - - 0.64 - - -- - (0.40) - - -

Unemployment rate - - - –0.90 - -- - - (0.19) - -

Govt. share - - - 1.86 -- - - (1.16) -

σG - - - - 8.52- - - - (1.65)

Notes: Model: cst = µs +vt +αcyst +ζc (Xst−X.t)(yst− y.t− ys. + y..)+εst. yst is state s’s labor incomegrowth (real and per capita). cst is state s’s nondurable consumption growth (real and per capita). µs

is a cross-sectional fixed effect and vt is a time-fixed effect. X is one of the variables that may affect theMPC—see Table 5 for definitions. t-statistics in parentheses. Sample 1976–1998.

37

Table 7: Sensitivity to Lagged Income: Non-Durable Retail Sales

(1) (2) (3) (4) (5) (6) (7)

State-fixed effects yes yes yes yes yes yes yesTime-fixed effects yes yes yes yes yes yes yes

ys,t−1 0.28 0.22 0.29 0.22 0.23 0.22 0.22(5.67) (4.21) (5.77) (4.09) (4.13) (4.16) (4.17)

Interaction terms:

Farm share –4.21 –3.71 –5.65 –5.68 –5.33 –4.85 –5.14(4.95) (4.36) (4.89) (4.96) (4.30) (3.24) (3.63)

Govt. share - 3.13 - 3.79 3.62 4.37 3.90- (3.14) - (3.81) (3.49) (3.36) (3.73)

Unemployment rate - - –7.92 –11.66 –12.03 –11.86 –11.99- - (1.76) (2.61) (2.69) (2.63) (2.64)

Persistence - - - - 0.11 - -- - - - (0.91) - -

σG - - - - - –1.66 -- - - - - (0.63) -

σW - - - - - - –1.74- - - - - - (0.65)

Notes: Model: cst = µs + vt + αlys,t−1 + ζl (Xst − X.t)(ys,t−1 − y.,t−1 − ys. + y..) + εst. ys,t−1 is state s’slagged labor income growth (real and per capita). cst is state s’s nondurable consumption growth (realand per capita). µs is a cross-sectional fixed effect and vt is a time-fixed effect. X is one of the variablesthat may affect the MPC—see Table 5 for definitions. t-statistics in parentheses. Sample 1976–1998.

38

Table 8: (Appendix) Sensitivity to Current and Lagged Income in Simulated Data. 5

percent discount rate

Buffer-Stock

Marginal MedianMPC effect Wealth-Income

Current Lagged Current Lagged Ratio

Baseline 0.984 0.221 – – 0.058(0.009) (0.039) (0.004)

No persistence 0.973 0.148 0.07 0.44 0.057(0.004) (0.039) (0.004)

Unemploymentreplacement 0.954 0.168 –1.00 –1.77 0.379

(0.012) (0.039) (0.012)no replacement 0.895 0.169 –2.97 –1.73 0.813

(0.011) (0.037) (0.019)Less idiosyncraticuncertaintyσN = 0.05 0.981 0.257 0.06 –0.72 0.035

(0.007) (0.038) (0.002)σV = 0.05 0.985 0.255 –0.02 –0.68 0.018

(0.007) (0.039) (0.002)More aggregateuncertaintyσG = 0.032 0.992 0.219 0.50 –0.13 0.064

(0.005) (0.040) (0.004)σW = 0.01 0.887 0.186 –9.70 –3.50 0.058

(0.013) (0.037) (0.005)

Notes: All parameters as in Table 2, except the discount rate is 5 percent.

39

Table 9: (Appendix) Sensitivity to Current and Lagged Income in Simulated Data. ρ = 3

Buffer-Stock

Marginal MedianMPC effect Wealth-Income

Current Lagged Current Lagged Ratio

Baseline 1.060 0.121 – – 0.481(0.014) (0.044) (0.080)

No persistence 0.963 0.141 0.59 –0.12 0.415(0.005) (0.038) (0.067)

Unemploymentreplacement 0.998 0.121 –2.07 –0.00 1.012

(0.014) (0.042) (0.079)no replacement 0.928 0.125 –4.40 0.13 1.621

(0.014) (0.039) (0.084)Less idiosyncraticuncertaintyσN = 0.05 0.993 0.209 1.34 –1.76 0.072

(0.009) (0.040) (0.005)σV = 0.05 1.080 0.121 –0.40 –0.00 0.383

(0.014) (0.045) (0.081)More aggregateuncertaintyσG = 0.032 1.067 0.118 0.44 –0.19 1.388

(0.014) (0.045) (0.181)σW = 0.01 0.903 0.107 –15.70 –1.40 0.483

(0.018) (0.041) (0.079)

Notes: All parameters as in Table 2, except ρ = 3.

40