the buffer-stock model and the marginal propensity to
TRANSCRIPT
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
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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.
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