Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence

Hall argues that empirical work on the life cycle-permanent income hypothesis has been “seriously weakened by failing to take proper account of the endogeneity of income when it is the major independent variable in the consumption function” (Introduction, p. 971). He notes that the standard fix – simultaneous-equations estimation using instrumental variables – rests on an “uneasy compromise” between instruments that are plausibly exogenous and instruments that meaningfully explain income, and that “the hypothesis of exogeneity is untestable” (p. 971). His alternative is to ask what can be learned from a consumption regression “where it is conceded from the outset that none of the right-hand variables is exogenous,” derived from the stochastic implications of the theory itself rather than from an assumption about which variables are exogenous (p. 971).

For a consumer who maximizes expected discounted lifetime utility subject to an intertemporal budget constraint, with stochastic labor earnings as the only source of uncertainty and a constant, known real interest rate r no less than the subjective discount rate, Hall proves (Appendix) that E_t[u’(c_{t+1})] = [(1+delta)/(1+r)] u’(c_t) – next period’s expected marginal utility of consumption is simply today’s marginal utility scaled by a constant factor reflecting the gap between the interest rate and time preference (Section I, “Theory,” p. 972). Earnings need not be independent over time or stationary; the only requirement is that their conditional expectation exists (p. 972).

“No information available in period t apart from the level of consumption, c_t, helps predict future consumption, c_{t+1}, in the sense of affecting the expected value of marginal utility. In particular, income or wealth in periods t or earlier are irrelevant, once c_t is known” (Corollary 1, p. 973). This is the paper’s strong, sharply falsifiable implication: because current consumption already summarizes everything the household knows about its lifetime resources, no other variable dated t or earlier – not lagged income, not lagged wealth – should have any additional power to predict consumption once lagged consumption is included in a regression.

Corollary 3 shows that for quadratic utility, consumption obeys the exact linear regression c_{t+1} = beta_0 + lambda*c_t + epsilon_{t+1}; Corollary 4 gives the analogous nonlinear statistical model for constant-elasticity-of-substitution utility; and Corollary 5 shows that whenever period-to-period changes in marginal utility are small – because the interest rate is close to the discount rate and the underlying shocks are modest – “consumption itself obeys a random walk, apart from trend” (pp. 973-975). Hall also traces through the implied stochastic behavior of human capital and total wealth, showing that optimizing consumer behavior makes both consumption and (separately) wealth evolve as random walks with trend, cautioning explicitly that this “is not accurate” to summarize merely as “consumption is proportional to wealth, wealth is a random walk, and so consumption is a random walk” – rather, the underlying optimizing behavior independently makes each variable a random walk (p. 976).

The test regresses consumption on its own lagged value and a vector of other variables dated t-1 or earlier, then applies an F-test for whether those other lagged variables can be excluded (Section II, pp. 976-977). Because the theory itself – not an assumption about which variables are exogenous – predicts that lagged variables other than consumption should have zero coefficients, “valid tests can be performed with any variable that is known in period t-1 or earlier,” and the test “can be tested rigorously without any assumptions about exogeneity” (pp. 971, 977) – in contrast to structural consumption-function regressions on contemporaneous income, which the paper explicitly does not attempt to estimate or interpret as structural relations (p. 986).

First, the “excess sensitivity” view associated with Tobin and Dolde (1971) and Mishkin (1976), that liquidity-constrained consumers cannot smooth transitory income fluctuations and so consume too much of current income; Hall shows algebraically that this alternative implies lagged income retains predictive power for consumption unless the liquidity-constrained group’s income happens to follow exactly the same stochastic process as aggregate consumption (Section II, pp. 977-978). Second, the ad hoc distributed-lag view, originating with Friedman (1957, 1963) and used by Modigliani (1971) and others, that permanent income is approximated by a fixed (e.g., Koyck/geometric) distributed lag of past actual income – Hall shows that if this distributed lag is “nonoptimal” (not the one an optimizing consumer would actually use), lagged income again retains predictive power for consumption beyond lagged consumption alone (pp. 978-979).

The consumption series is real per capita consumption of nondurables and services (1972 dollars, quarterly, U.S. National Income and Product Accounts), chosen over total consumption specifically to avoid conflating the results with the imputation procedure for durable goods’ service flows (Section III, p. 979). Regressing consumption on its own lag (for quadratic utility, and for CES utility with two elasticity values) shows extremely high predictive power of lagged consumption for current consumption in all three specifications, with residuals showing no obvious systematic pattern – six residuals exceed two standard errors in the expected proportion, clustering around recognizable recession episodes such as 1974:4 and the Korean War quarter of 1950:4 (Section III, pp. 979-981).

No: adding consumption lagged two, three, and four quarters to the basic regression (c_{t-2}, c_{t-3}, c_{t-4}) improves the forecast of current consumption by only “about 10 cents per person per year,” with an F-statistic of 1.7 against a 5 percent critical value of 2.7 (Section IV, pp. 981-982). Hall notes this as “only very weak evidence against the pure life cycle-permanent income hypothesis” and highlights that, unlike most other aggregate economic time series, consumption shows no sign of following a second-order difference equation capable of generating stochastic cycles (p. 982).

A single lag of real disposable income has “essentially no predictive value at all” (F-statistic of 0.1 against a critical value of 3.9); a four-quarter unconstrained distributed lag has a small, marginally significant F-statistic of 2.0 (critical value 2.4 at 5 percent) with a long-run “marginal propensity to consume” that is actually slightly negative; and a 12-quarter Almon-lag specification gives similarly marginal results (Section V, “Can Consumption Be Predicted from Disposable Income?”, pp. 982-984). Hall summarizes: “There is a statistically marginal and numerically small relation between consumption and very recent levels of disposable income… there is no evidence at all supporting the view that a long distributed lag covering several years helps to predict consumption” (p. 984) – casting only “a little doubt” on the pure hypothesis.

Lagged real Standard & Poor’s stock price index values do have a statistically unambiguous relationship with consumption – an F-statistic of 6.5 against a critical value of 2.4, with each individual lag coefficient significant – “which in a formal sense refutes the simple random-walk hypothesis,” though the improvement in forecasting power is numerically small (the regression standard error falls only from about $14.60 to $14.40 per person per year) (Section VI, “Wealth and Consumption,” pp. 984-985). Hall reconciles this by proposing a modified hypothesis in which “some part of consumption takes time to adjust to a change in permanent income,” so a variable correlated with a recent change in permanent income – such as a stock price move, since stock prices are themselves close to a random walk – can have genuine, if modest, predictive value without contradicting the theory’s central content (Section VII, “Implications of the Empirical Evidence,” pp. 985-986).

“Beyond the next few quarters consumption should be treated as an exogenous variable” in forecasting models, since any information available today about future income is already reflected in today’s permanent income and hence in current consumption; forecasts of consumption can be improved only slightly, and only one quarter ahead, using current stock prices (Section VIII, “Implications for Forecasting and Policy Analysis,” pp. 986-987). For policy, “the findings of this paper go no further than supporting the view that policy affects consumption only as much as it affects permanent income” and only through genuinely new information about policy; Hall stresses explicitly that this does not mean policies affecting income have no effect on consumption – “a permanent tax reduction generates an immediate increase in permanent income and thus an immediate increase in consumption” – but that “policies that have a transitory effect on income are incapable of having a transitory effect on consumption,” which “certainly complicates the problem of formulating countercyclical policies that act through consumption” (p. 987).

All the theoretical results assume a known, constant real interest rate; Hall notes that known, pre-announced variation in the interest rate over time would require only minor amendment (mainly to the trend factor lambda_t), with the practical importance of this depending on the elasticity of intertemporal substitution, but that if the future real interest rate is genuinely uncertain at the time a consumption decision is made, “the theoretical results no longer apply,” though he sees “no strong reason for this to bias the results of the statistical tests in one direction or another” (Section I, p. 976). He also cautions against reading any of the paper’s regressions as structural: “it is important not to treat any of the equations of this paper as structural relations between consumption and the variables used to predict it” – for example, the negative coefficients on lagged income in Table 3 should not be read as income having a negative causal effect on consumption (Section VII, p. 986).