Robustness of the Certainty-Equivalence Rejection in the Excess Sensitivity of Aggregate Consumption

Pozzi and Malengier (2007) estimated a two-type aggregate consumption model on a panel of 17 OECD countries (1981-2003) and rejected, through Lagrange-multiplier (score-type) tests computed within a nonlinear two-step panel GMM (generalized method of moments, an estimation method that matches model-implied moment conditions to their sample counterparts) estimator, the certainty-equivalence restriction on the expected consumption growth of forward-looking consumers. They concluded that the magnitude of the estimated excess sensitivity of private consumption to current income cannot be explained by a model based on certainty equivalence, so that certainty-equivalent readings overstate the share of rule-of-thumb (current-income) consumers. The problem is whether this rejection - and the precautionary-saving interpretation of excess sensitivity that it supports - is robust. It rests on a second-order Taylor approximation with constant conditional variance of consumption growth, on instrument validity under a possible MA(1) (first-order moving-average) serial error structure, on inference for nonlinear two-step GMM without a general finite-sample variance correction, and on a single annual macro panel.

Background

Under the life-cycle permanent income hypothesis with quadratic utility, Hall (1978) showed that consumption should follow a random walk: once lagged consumption is controlled for, no variable known at date $t$ should predict consumption growth. This result depends on certainty equivalence, the property of quadratic-utility expected-utility maximization that the optimal consumption plan depends on future income and interest rates only through their conditional means, so that income risk has no effect on the consumption decision. Empirically, aggregate consumption growth is strongly and robustly predictable by expected income growth, a violation of the random-walk prediction known as excess sensitivity (Flavin 1981). Campbell and Mankiw (1990) summarized this evidence in a widely used two-type model: a fraction $\lambda$ of aggregate income accrues to rule-of-thumb consumers who simply spend their current income, while the remaining consumers optimize intertemporally; estimates of $\lambda$ near 0.4 are often read as the population share of current-income consumers.

That reading has long been suspected to be contaminated by the certainty-equivalence assumption itself. With constant relative risk aversion (CRRA) utility $u(c)=c^{1-\gamma}/(1-\gamma)$, where $\gamma$ is the coefficient of relative risk aversion, income risk induces precautionary saving - consumers hold extra wealth as self-insurance, which lowers their marginal propensity to consume (MPC) out of wealth below its certainty-equivalent value and makes expected consumption growth respond positively to risk. If the MPC of optimizing consumers is lower than the certainty-equivalent benchmark, observed positive aggregate consumption growth need not indicate a large population of rule-of-thumb consumers: it can reflect the interaction of income risk with forward-looking behavior. Separating these two sources of predictable consumption growth is an identification problem that certainty-equivalent specifications cannot pose, let alone solve.

Pozzi and Malengier (2007) made the question operational for a macro panel. In their model, rule-of-thumb consumption is $c_{1t}=\lambda y_t$ with $y_t$ disposable labor income, while forward-looking consumers maximize CRRA utility subject to a budget constraint with interest rate $r$ and discount rate $\rho$. Writing $z=(1+r)/(1+\rho)$, a second-order Taylor expansion of their Euler equation (the first-order intertemporal optimality condition $E_t[(c_{2,t+1}/c_{2t})^{-\gamma}]=(1+\rho)/(1+r)$) around consumption growth equal to $z$, together with the assumption that the conditional variance $\sigma^2(z)=E_t[((c_{2,t+1}/c_{2t})-z)^2]$ is constant, yields the linear recursion

$$E_t[c_{2,t+1}]=\alpha\, c_{2t},\qquad \alpha=\frac{1+\gamma}{\gamma}z-\frac{1}{\gamma}z^{\gamma}+(1+\gamma)z^{-1}\frac{\sigma^2(z)}{2},$$

while imposing certainty equivalence ($\sigma^2(z)=0$) gives the benchmark $\alpha^{CE}=\frac{1+\gamma}{\gamma}z-\frac{1}{\gamma}z^{\gamma}$. Since $\alpha>\alpha^{CE}$ exactly when risk depresses the MPC of optimizers, testing $\alpha=\alpha^{CE}$ separates the precautionary and rule-of-thumb explanations. Aggregating the two consumer types gives a nonlinear panel consumption equation that they estimate by two-step nonlinear system GMM (generalized method of moments, in which moment conditions are stacked and weighted by an estimated optimal weight matrix), combining first-differenced moments with levels moments in the spirit of Arellano and Bover (1995) and Blundell and Bond (1998) and using four linear-combination instruments at lag depth three to avoid instrument proliferation. On 17 OECD countries over 1981-2003 they obtain $\hat\alpha=1.024$ and $\hat\lambda=0.369$, and Lagrange-multiplier (LM) tests - score-type tests computed from the GMM criterion, used because conventional two-step GMM standard errors are downward biased in finite samples - reject $\alpha=\alpha^{CE}$ for $\gamma=1,3,5,7$ (p-values 0.003-0.005). Re-estimating under the constraint $\alpha=\alpha^{CE}$ drives $\lambda$ close to one: under certainty equivalence, all observed consumption growth is attributed to current-income behavior.

The paper is explicit that this conclusion rests on four maintained choices, each of which defines a research direction. First, inference: the analytical finite-sample correction for the two-step GMM variance estimator of Windmeijer (2005) exists only for linear moment conditions, so there is no general finite-sample-corrected variance for the nonlinear two-step estimator used here, and hypothesis testing leans on LM tests. Second, moment validity: the error may contain a first-order moving-average MA(1) component - for example from time aggregation of flow data or from durable goods purchases - under which some lagged instruments are invalid, and alternative instrument sets can lead to rejection of the overidentifying restrictions (the testable implication, checked by a Sargan-type test, that the instruments are uncorrelated with the error when there are more instruments than parameters). Third, specification of risk: the derivation of $\alpha$ assumes a constant conditional variance and a second-order expansion around $z$; whether the quantitative implication $\alpha>\alpha^{CE}$ survives time-varying conditional variances, higher-order terms, or alternative utility functions is not established. Fourth, external validity: the evidence covers one annual panel of 17 OECD countries over 1981-2003; generalization to other country samples, periods, or data frequencies is untested.

Problem Statement

Establish whether the rejection of certainty equivalence in the excess sensitivity of aggregate consumption is robust. Concretely, working in the framework of Pozzi and Malengier (2007), determine whether the finding that the estimated two-type consumption model satisfies $\alpha>\alpha^{CE}$ - that is, that the expected consumption growth of forward-looking consumers exceeds its certainty-equivalent benchmark because income risk depresses their marginal propensity to consume out of wealth - and the associated conclusion that certainty-equivalent specifications overstate the rule-of-thumb share $\lambda$ survive the four maintained choices on which they rest:

1. Specification of risk. Derive the aggregate consumption equation when the conditional variance of optimizing consumers' consumption growth is not constant (for example, under conditional heteroskedasticity of income or consumption risk, i.e. conditional variance that moves with the state of the economy), when terms beyond second order in the Euler-equation expansion are retained, or under alternative utility specifications; establish, for at least one of these relaxations, whether the relation $\alpha>\alpha^{CE}$ and the certainty-equivalent benchmark $\alpha^{CE}$ remain well defined and whether the empirical rejection of $\alpha=\alpha^{CE}$ persists.

2. Finite-sample inference. Provide finite-sample-valid inference for the nonlinear two-step panel GMM tests of $\alpha=\alpha^{CE}$ - that is, inference whose false-rejection rate under the null hypothesis is no larger than its nominal level - and check that the rejection is not an artifact of the downward bias of standard two-step standard errors.

3. Moment validity. Establish the robustness of the result to instrument choice and to the MA(1) serial correlation that time aggregation and durables may induce in the error, under which lagged instruments can be invalid.

4. External validity. Re-examine the test on at least one country sample, time period, or data frequency other than the 17-country annual OECD panel for 1981-2003.

The question is whether, with valid inference, the statement that the magnitude of the excess sensitivity of private consumption to current income cannot be explained by a certainty-equivalent model holds, and by how much the rule-of-thumb (excess-sensitivity) share estimated under certainty equivalence, $\lambda^{CE}$, exceeds the share $\lambda$ estimated once income risk and precautionary saving are admitted, that is, the magnitude of the fraction $(\lambda^{CE}-\lambda)/\lambda^{CE}$. The four parts above are the independent conditions on which the rejection rests: the statement is established only if the rejection survives every maintained choice and is overturned if any one of them removes it, while resolving a single part settles that robustness dimension and leaves the others open.

References

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