Habit Formation in Consumption and Its Implications for Monetary-Policy Models

That candidate monetary policy models had not been tested against the dynamic behaviour of the data, and that the right test is likelihood-based, using the full vector autocovariance function of an unconstrained VAR that nests the structural model. The author argues that “it is not enough to match first and second unconditional moments, or a subset of conditional moments implied by the model,” as in early equilibrium business cycle studies, because “the working assumption among most economists is that monetary policy has only short-run effects on real variables. If so, it would be a major omission not to fully evaluate the short-run dynamic effects of monetary policy in a candidate model.” He is equally sceptical of impulse-response matching: relying on the response to a federal funds rate shock “may be quite misleading,” since Leeper, Sims and Zha show “the fraction of the variance of output, inflation, or interest rates accounted for by the unanticipated component of monetary policy is generally quite small.” The likelihood is preferred because it “incorporates all of the dynamic covariances among observable variables, weighted according to their contribution to the likelihood,” with the autocovariance function serving as a graphical companion that “may highlight visually a behavioral deficiency in the model that is more difficult to interpret from statistical evidence.”

Consumption and investment behave like jump variables, front-loading their entire response to shocks, against identified-VAR evidence of a gradual response peaking around a year. Summarising his earlier work, the author lists for consumption “extremely significant unexplained serial correlation in the consumption-income ratio, parameter estimates that indicate very little or no forward-looking behavior, and excessive sensitivity of consumption to current income arising from the rule-of-thumb behavior,” and for investment “very significant unexplained serial correlation in the investment-capital ratio, extreme sensitivity of model stability and uniqueness to small perturbations in parameter estimates, and a negative estimate of the capital share in income.” The dynamic failure is the one he pursues: disinflation simulations show “both consumption and investment act like ‘jump variables,’ completely front-loading or pulling forward in time their responses to shocks,” whereas identified VARs show “a gradual response over several years, with the peak response at one year or so.”

Because a model whose dynamics live in its error processes cannot claim those errors are policy-invariant. The author acknowledges that “in principle, one can augment a structural model that is dynamically deficient with an arbitrary error structure so as to exactly replicate the dynamic structure in the data,” and names Rotemberg and Woodford as taking that route. His objection is that “the error processes so identified cannot be considered ‘structural’ in any meaningful sense. We have no idea in what way the errors are linked to underlying behavior, and thus we can have no more confidence about their policy invariance than we have for reduced-form VARs or 1960s structural models sans explicit expectations. In essence, by putting the dynamic structure in the errors, the model becomes vulnerable to a Lucas critique of its errors.” He therefore assumes throughout that structural innovations are uncorrelated across time, though possibly correlated across equations, and states that “none of the dynamics in the results reported below may be attributed to across-time correlation in the error terms.”

Period utility depends on consumption relative to a reference level that is a geometric average of past consumption, with one parameter indexing how much the reference matters and another indexing how far back its memory reaches. The specification follows Carroll, Overland and Weil, and is “related in spirit to the pioneering work of Duesenberry.” Utility is no longer time-separable “because the consumption choice today influences the future habit reference level in next period’s and all future periods’ utility.” The author flags an admissibility restriction: the habit-importance parameter cannot exceed one, “because it implies that steady-state utility is falling in consumption.” The consumer’s problem is solved with a standard budget constraint and a time-varying real rate, where the ex ante real rate is a discounted weighted average of model-consistent forecasts of short real rates – the duration of the implied long-term real bond being “set to ten years for this paper” – and a separate discount rate for future income, following Campbell and Mankiw, indexes how far forward consumers look.

Numerical maximum likelihood on a system, nested inside an unconstrained VAR, using U.S. quarterly data for 1966:1-1995:4. The estimator follows Fuhrer and Moore (1995); its advantages are that “it allows estimation to proceed naturally from an unrestricted linear vector autoregression that nests all of the linear models considered to successively more-restricted linear models,” and that its finite-sample properties may be better than method-of-moments estimators. The acknowledged drawback is that “to the extent that any equation in the system is mis-specified, estimates of all the parameters in the system will (in principle) be affected.” The VAR contains log per capita nondurables and services consumption, log per capita disposable personal income, the federal funds rate, the price level, and log per capita GDP other than nondurables and services. Consumption and income are chain-weighted, per capita and detrended with the trend segmented in 1974; the price measure is the CPI excluding food and energy; the short rate is the quarterly average effective federal funds rate. In the first stage only the consumption function’s parameters are estimated, with income, funds rate, price and other-GDP equations left as unconstrained VAR equations.

Habit formation matters a great deal, rule-of-thumb behaviour accounts for about a quarter of income, intertemporal substitution is small, and consumers are highly forward-looking – but the reference level is only one quarter deep. The baseline estimates are 0.80 (standard error 0.19) for the habit-importance parameter, 0.0015 (0.0039) for the memory parameter, 0.26 (0.13) for the rule-of-thumb income share, 6.11 (1.81) for the curvature parameter, 0.99 (0.01) for the forward-looking discount parameter, and 28.49 (5.17) for the coefficient on the expected real interest rate term; the log-likelihood is 2366.4. The author reads these as: “(1) habit formation is an economically important determinant in the utility function; (2) the habit formation reference level is essentially last period’s consumption level; (3) rule-of-thumb behavior is important, with about one-fourth of income accruing to rule-of-thumb consumers; (4) the intertemporal elasticity of substitution is quite small; (5) for those who look forward, the horizon is long; the parameter takes the estimated value .996 on a quarterly basis, .984 on an annual basis; and (6) the model explains most, but not all, of the autocorrelation in the consumption data,” the last shown by a Ljung-Box Q(12) of 83.7 for consumption with a p-value of essentially zero. (The Q statistics for income, the funds rate and inflation, at 6.1, 13.5 and 8.2, are all far from significant.)

Overwhelmingly, and the same is true of rule-of-thumb behaviour – while the structural model’s full set of restrictions is not rejected at all. “The hypothesis that habit formation is unimportant in this model – that the exponent on the reference level of consumption is zero – is overwhelmingly rejected. The chi-squared likelihood ratio test for this single restriction takes the value 21.4, with p-value of 4 times ten to the minus six. Similarly, the hypothesis that rule-of-thumb behavior is unimportant is strongly rejected. The chi-squared likelihood ratio test for the restriction takes the value 12.6, with p-value = 4 times ten to the minus four.” The third result is the one the author treats as notable: the likelihood ratio test for the constrained baseline model, “which incorporates the many zero restrictions and cross-equation restrictions implied by the structure of the consumption model and by rational expectations, takes the value 32.8, not significant at even the 10 percent level. This is one of relatively few cases in which the joint restrictions imposed by an optimization-based model with rational expectations cannot be rejected relative to the unconstrained model in which the constrained model is nested.”

The author says it is “lower than expected” and defends it in two ways rather than explaining it away. First, algebraically: rewriting period utility and setting the memory parameter to zero “shows that the essence of habit formation is that it mixes utility from the level of consumption with utility from the change in consumption. That is, the habit formation model with any normally shaped utility function will imply smoothing of both the level of consumption and its changes (provided [the habit-importance parameter] is not zero). Larger values of [the memory parameter] simply define the changes relative to a longer distributed lag of past consumption.” So “a single lag of consumption in the reference level is sufficient to impart the smoothness to changes in consumption expenditures that is absent in the standard life-cycle model.” Second, numerically: he repeats the disinflation simulation substituting 0.9 for the memory parameter, and reports that “the model’s behavior is altered only slightly by the change from one-quarter memory to more persistent memory in the reference level.” He adds a candid aside in a footnote: in this light “the habit formation may provide a reasonable approximation to a model with a standard utility function and costs of adjustment” in the change in consumption.

The consumption dynamics are recovered well, and the remaining discrepancies are inside the VAR’s own confidence bands. “The model has recovered the dynamic covariances for consumption expenditures quite well, capturing the persistence in the autocorrelation, as well as the persistent dynamic correlations between consumption and income, interest rates, and inflation.” Against 90% confidence intervals computed by Monte Carlo around the VAR’s autocovariance function, “the differences between the two autocorrelation functions are generally insignificant at the 10% level. Thus the correlations that the structural model cannot match are generally not precisely determined in the data.” Setting both the habit and rule-of-thumb parameters to zero, by contrast, leaves “the primary consumption dynamics in the model… almost totally missing. The simple PIH model cannot replicate the dynamics in the data.” When further restrictions are added – a simple Taylor rule for the funds rate, a Fuhrer-Moore contracting price specification, and an equation for non-consumption GDP – none “significantly deteriorate the likelihood,” though the author notes specific imperfections: the model makes the correlation of consumption with the lagged funds rate and lagged inflation “too strongly” negative, and makes the correlation between the funds rate and lagged consumption negative “while the VAR says it should be mildly positive.”

The paper tests that interpretation directly and rejects it. “An alternative interpretation of the results of this paper and Fuhrer (1996) is that the restrictions imposed on the price specification and the funds rate reaction function are invalid, and are interfering with the real-side dynamics of consumption and output. To test this possibility, I estimate a model with reduced-form processes for consumption and income, so that only the restrictions from the price and interest rate specifications constrain the model.” The comparison “suggests that this interpretation is invalid. The Fuhrer-Moore price specification and the simple reaction function capture the dynamics in these variables without distorting their dynamic interactions with consumption and income (or vice versa).” The converse test, reported in a footnote, runs the other way: “the model with restrictions on consumption, but without restrictions on prices and interest rates, requires rule-of-thumb and habit formation behavior to match the moments in the data.”

Habit formation produces the gradual, hump-shaped consumption response the data show, and the absence of it corrupts the model’s inflation dynamics as well. Starting from steady state, the long-run inflation target is cut unexpectedly “from about 5% (the estimate from the data) to 2%.” In the habit model, “inflation falls gradually from its old steady state to the new, lower equilibrium. Interestingly, consumption also responds gradually, with its peak response at a year or so; the full response takes three to four years.” Against this, the no-habit model gives “an example of a model in which the mis-specification of the real side compromises the behavior of the nominal side. The persistence of inflation in this model… is significantly decreased by the rapid (and counterfactual) response of real variables to a disinflationary shock.” The author is careful to note that the problem is not solved by rule-of-thumb consumers alone: “the specification including rule-of-thumb consumers still exhibited rapid response to shocks.”

The paper checks and finds it is. Solving the nonlinear model with the parameters estimated from the linear one, for the same disinflation simulation, gives “nearly identical results.” Substituting the linear model’s solutions for consumption, income and real rates into the nonlinear first-order conditions, “the maximum absolute error in the nonlinear Euler equations is about .01, compared to steady-state marginal utility of about -1.” And “the estimate of lifetime utility for the disinflation simulation is very similar whether computed using the solution paths from the linear model or from the nonlinear model.” The author does flag the method used: the nonlinear solutions use “a ‘certainty equivalence’ solution technique that does not compute the stochastic distribution of the endogenous variables via value-function programming,” because “the state dimension of the model would make the computation time for such a method prohibitive.”

Three reasons, all about what the model cannot support rather than about computation. First, “given the empirical significance of consumers who appear to follow a rule of thumb, it is difficult to know whose utility we should maximize in evaluating policy alternatives. One could maximize the utility of the (forward-looking, rational) habit-formation consumers, but one could not know the welfare implications for the rule-of-thumbers.” Second, “the model as it stands includes no explicit cost of inflation” – inflation reaches consumers only indirectly, through the real disruptions the Fed causes in bringing it back to target – so “without any direct cost of inflation, the optimal policy from the consumers’ point of view is one that minimizes fluctuations in consumption. This is not a satisfying or interesting policy conclusion.” Third, “because the bulk of the welfare cost arguably arises from discrete shifts in employment status for a small fraction of the population, the representative agent model may not provide an accurate measure of the relative welfare costs of pursuing different monetary policies.” He suggests the usual weighted output-and-inflation loss function “may be a reasonable approximation to use until the thornier issues… are tackled,” and “leave[s] the computation of ‘optimal’ policy responses in this model for future work.”

It claims that only specifications imposing smoothness on the change in consumption will succeed empirically, and that the hump-shaped response is a statistically significant feature of the data; it declines to claim that habit formation is the unique or regime-invariant explanation. The conclusion states that habit formation “allows the model to replicate key dynamic correlations among consumption, output, interest rates, and inflation to a degree that standard models cannot,” specifically the hump-shaped response “to income, interest rate, and inflation shocks,” and that it does so “because it imparts a motive for consumers to smooth the change, as well as the level of consumption.” But immediately: “Other specifications may also afford improvements in the empirical performance of the standard model. This paper suggests, however, that only specifications that impose some smoothness on the change in consumption will be successful empirically.” And on stability: “The specification set forth in this paper might not be robust across shifts in monetary or other policy regimes. But only through rigorous econometric testing of this and alternative specifications across regime shifts can observational equivalence (or empirical dominance) of alternative specifications and stability of any one specification across policy shifts be determined.” The author’s own summary of the contribution is deliberately modest – “a small step,” “a modest improvement.”