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COPYRIGHT 2001 MIT Press Journals
I. Introduction
THE estimation of key parameters in the household's utility function has been a longstanding goal of empirical research on consumer behavior. These parameters are of interest because they furnish a quantitative gauge of theoretically important concepts, such the strength of the precautionary saving motive, the degree of prudence and risk aversion, and the willingness of consumers to substitute consumption over time.
In order to estimate parameters from the utility function, a suitable empirical specification must first be chosen. A wide range of applications employ specifications that rely on linear approximations to dynamic Euler equations, and a virtual cottage industry has been created from the estimation of linearized consumption Euler equations.(1) Despite the popularity of this approach, an important unanswered question is whether linear approximations of Euler equations produce accurate estimates of key parameters in the consumer's utility function.
In this paper, we investigate the properties of linearized Euler equations within the context of one type of empirical application: the growing literature on precautionary saving that examines how consumption growth and saving behavior are affected by uncertainty. This application often involves regressing consumption growth on measures of uncertainty in expected consumption growth (the idea being that, with precautionary motives, future uncertainty will depress current consumption and raise consumption growth). The estimating equation is derived from a second-order Taylor expansion of the Euler equation, which is the first-order condition for optimal consumption choice and relates marginal utility today to expected marginal utility tomorrow. The parameter estimates can be used to measure the strength of precautionary saving motives, where, in the absence of precautionary motives, future uncertainty should not affect consumption growth.
The empirical work on precautionary saving has produced some anomalous results, in part due to differences in empirical strategies. One set of studies has investigated the effect of income risk on the level of consumption or wealth accumulation, and suggests that precautionary motives may explain a significant fraction of wealth accumulation (Lusardi, 1993; Carroll, 1994; Carroll & Samwick, 1998). Other studies produce more mixed results. Guiso, Jappelli, and Terlizzese (1992) find that precautionary saving can explain only a small fraction of saving using a self-reported measure of earnings uncertainty from Italian household data. One potential shortcoming of these studies is that income (or earnings) variability may be a poor proxy for income risk if most of the variation in income is predictable. Skinner (1988) shows that precautionary saving can comprise a large fraction of aggregate savings in a life-cycle model, but comparisons of savings rates across occupation groups provide little support for the theory.
In this paper, we focus on tests of precautionary saving motives based on linear approximations to consumption Euler equations. Studies that use this approach typically find that the estimated effects of consumption uncertainty on consumption growth are small, indicating that precautionary motives are weak or nonexistent (Dynan, 1993; Kuehlwein, 1991). For utility functions characterized by decreasing absolute risk aversion, these latter results also imply implausibly low levels of relative risk aversion. More specifically, given that the within-period utility function is isoelastic, such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], this literature yields estimates of [Rho] that are generally below 1.3, and are often insignificantly different from 0.
We investigate one possible reason for these small estimates of [Rho]. Specifically, the method of estimating [Rho] (which utilizes a second-order Taylor expansion of the Euler equation) relies on linear or log-linear approximations of the Euler equation. If the Euler equation is sufficiently nonlinear, these approximations will be poor, implying that estimates of p may differ from their true values simply because the Euler equation is approximated. We refer to this divergence between the estimated and true values as approximation bias.
In this paper, we study how approximation bias due to linear approximation influences parameter estimates. We take a two-step approach. First, we examine nonlinearities in the consumption Euler equation. We start with a standard intertemporal optimization problem, in which a finite-lived consumer with isoelastic preferences chooses consumption and saving given current wealth and the current shock to income. For several sets of assumptions about the parameters that govern preferences and the income process, we solve numerically for the function that relates consumption to wealth and the income state in each year of "life." We use these consumption functions to solve for the relationship between expected consumption growth and uncertainty in consumption growth, and contrast this with the relationship implied by the second-order approximation. We find substantial departures from the linear relationship implied by the second-order approximation.
How might these nonlinearities influence actual parameter estimates? The second step in our analysis investigates this issue. In particular, we ask how nonlinearities are likely to be translated into approximation bias in estimates of [Rho]. Typical estimates based on linearized Euler equations involve using household-level data to regress consumption growth over a given time period on a measure of variability in consumption growth, such as average squared consumption growth, using either ordinary least-squares or instrumental-variables techniques. In the latter case, squared consumption growth is often instrumented using variables such as education and occupation indicators; the assumption is that these factors predict variability in consumption growth but are uncorrelated with the error term. The second step in our analysis allows us to investigate the extent to which this assumption is likely to be true.
We examine the extent of bias using both ordinary least-squares (OLS) and instrumental-variables (IV) estimators by performing a Monte Carlo study in which we choose a utility function (that is, a value of p) and simulate panel data on optimal consumption for a large number of "households." For our results to be informative about the extent of bias in actual parameter estimates, it is important that the heterogeneity across our simulated households mimics that found in true data. To do this, we use income data from the Panel Study of Income Dynamics (PSID) to estimate the parameters of the income process for households in sixteen different education/occupation groups. We then solve for the appropriate consumption functions using each of the sixteen sets of parameter estimates. With these consumption functions in hand, it is straightforward to simulate data for households from the different education/occupation groups (with the fraction of households from each group in the simulated data equal to the fraction of households in each group in the PSID). These data are used to investigate the properties of OLS estimates of [Rho], as well as IV estimates that use occupation and education indicators as instruments.
Our results indicate that regressions of consumption growth on consumption growth squared produce estimates of [Rho] that are biased down from the true value. OLS estimates produce the most bias, with estimates of [Rho] that are between 12% and 30% of the true value. The IV estimates fare somewhat better, but are still biased down, with estimates of [Rho] that are typically around 60% of the true value. The reason for the bias is that the instruments (the occupation and education indicators that determine the income process parameters) are correlated with the higher-order moments of consumption growth that are in the error term of the linearized equation. Another feature of our results is that the extent of bias varies with wealth: when the sample is split between "low-wealth" and "high-wealth" households, the estimates of [Rho] are biased down most for the low-wealth group, and for some estimation techniques are actually biased up for the high-wealth group. Differences in estimates of [Rho] across wealth groups may be interpreted as evidence that poorer households are more liquidity constrained than are wealthier households (Zeldes, 1989; Dynan, 1993). For example, if poorer households face binding borrowing constraints, the usual Euler equation will not hold for those households. Binding constraints will force these agents to consume their current income instead of their permanent income, and those households will have faster consumption growth because the binding constraint forces them to defer consumption. If current income is also less volatile than permanent income, the reasoning goes, estimates of prudence for constrained households will be biased down.(2) Our results indicate that these differences may also arise as the result of approximation bias.
To document the difference between the approximate and true relationship between expected consumption growth and uncertainty in consumption growth, our investigation requires us to first compute the "true" consumption function. Because there is no known analytical solution to this problem, we must seek a numerical solution that is itself an approximation of the true solution. Obviously, our approach to investigating linearization bias in Euler equation estimation will not be very informative unless the approximate numerical solution we compute is far closer to reality than that of the linearized Euler equation. We provide the results of two robustness checks for our numerical solution in appendix A. First, we give the solutions to the model with double the number of grid points for wealth used in our benchmark model; because the numerical accuracy can be made ever more accurate by using finer and finer grids, this check is one way of demonstrating that the number of grid points we have used is sufficient to make the numerical solution very accurate. Second, we report the results of a test for Euler equation accuracy recommended by Den Haan and Marcet (1994). The focus of this paper is not to document the superiority of numerical solution techniques over linearized Euler equation approaches. Indeed, this question has been investigated extensively in other recent work that typically finds that the linearized Euler equation fares much worse than numerical-solution techniques (Judd, 1998; Taylor & Uhlig, 1990). Instead, we focus here on investigating the substantive issue of potential bias in Euler equation estimation of consumption functions, an important problem in many empirical applications.
Section II discusses in more detail how approximations to Euler equations have been used in previous literature, and why estimates based on these approximations may be biased. Section III describes our methods for computing consumption functions, and examines how the approximate relationship between consumption growth and the variance of consumption growth differs from the relationship implied by our numerical solutions. Section IV describes our estimation of income processes for each occupation/education group using household data, and discusses the Monte Carlo results. Section V concludes.
II. Approximations to Euler Equations
We start with a simple but general model of consumption. Individuals choose consumption and saving in each period so as to maximize expected lifetime utility. We assume that there is one asset, At, and that assets held between t and t + 1 earn a gross return of [R.sub.t+1]. Decisions are made conditional on current resources (cash on hand) held at the beginning of the time period, and on information about future incomes and interest rates. Utility is additively separable and is discounted across periods at rate [Delta]. Subutility functions in each period are identical and isoelastic-elastic. The maximization problem is summarized as
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Cash on hand ([x.sub.t] [equivalent] [A.sub.t] + [y.sub.t]) evolves according to
(2) [x.sub.t+1] = [R.sub.t+1][[x.sub.t] - [C.sub.t]] + [y.sub.t+1],
where [y.sub.t+1] is labor income earned in time t + 1.
Following most of the literature on precautionary saving, we assume that the real interest rate in not stochastic and is fixed at r = R - 1. The only uncertainty consumers face is in labor income, which fluctuates from period to period. In this case, the Euler equation associated with utility maximization is
(3) u'([C.sub.t]) = (1 + r/1 + [Delta])[E.sub.t][u'([C.sub.t+1])],
or, in the specific case of isoelastic-elastic utility:
(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Because marginal utility is not linear in consumption in equation (4), it is not possible to derive an equation that relates expected future consumption to current consumption. Instead, researchers commonly linearize the right side of equation (4) and derive an equation that relates the expected growth in consumption to the expected squared growth in consumption. Specifically, taking a second-order Taylor approximation of marginal utility in t + 1 around the point [C.sub.t], inserting into equation (4), and rearranging yields
(5) [E.sub.t][[C.sub.t+1] - [C.sub.t]/[C.sub.t]] = 1/[Rho][r - [Delta]/1 + r] + (1 + [Rho]/2) x [E.sub.t] [([C.sub.t+1] - [C.sub.t]/[C.sub.t]).sup.2]] + [v.sub.t],
where 1 + [Rho] equals the coefficient of relative prudence...
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