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Suppose that the expectational error of the Euler equation is described by €,t+1 “ allPt+\ + where \j/t+! is an aggregate shock and Tju+j an idiosyncratic shock assumed to be i.i.d. across individuals and over time and independent of at and щ+1 (notice that the aggregate shock is allowed to affect consumers in different fashion). This representation of the error term encompasses several models. Given the assumed structure of the error term, var(^if+i)= y/f+x var(«•) + var(^.;+)). If r\ulAi is indeed i.i.d., then var( 77,-,,+/) is constant over time. If у/, is also i.i.d., so that constant over time, cov[var(£//+,),var(c,•_,)] = () and var(c,>/) is obviously a valid instrument for var(c,->f). If instead щ is heteroskedastic and/or y/2t+] exhibits a considerable amount of persistence, then var(c,>/) may not be a valid instrument for var(c;f/). For instance, if the conditional time variance of y/t+I follows an AtfCtf(l) process, cov[var(£. (+1), var(c, f = 0 and our instrument is still a valid one; but if y/2t+x evolves according to an ARCH of order 2 or higher, or as any GARCH process, it is not. Similar arguments apply to r}iit+l and to its cross sectional variance.

Although we develop the previous example with quadratic utility, the same considerations apply to the more general model that relaxes certainty equivalence. In particular, when the innovations to the Euler equation can be decomposed into an aggregate shock and an individual-specific shock, the period t-1 variance of the marginal utility of consumption is a valid instrument for its period t variance if: (a) the aggregate shock is part of the agents’ information set; (b) its second moments do not exhibit persistence over time; (c) the evolution of the cross sectional variance of the idiosyncratic shocks is not correlated with the cross sectional variance of consumption. These are not predictions of the permanent income- life cycle model. Thus, our test that 7F= I is a joint test of the model and of these additional assumptions. A finding that да 1 cannot therefore be taken necessarily as a contradiction of the model; but a finding that tt= 1 implies that the model and our additional assumptions are consistent with the data.

We also experiment with an alternative instrument. Theory suggests that in the case of certainty equivalence the cross sectional variance of consumption is an increasing function of age, so that age is likely to be correlated with the dependent variable even relaxing certainty equivalence. However, for age to be a valid instrument, one must assume that it is uncorrelated with the cross sectional variance of 77,,f and with the time variance of щ . The latter assumption is quite plausible. The former, however, is questionable, especially when one considers periods of the life cycle during which uncertainty changes in a dramatic and yet systematic way. The most obvious example is retirement. For this reason in the empirical section we use age both in addition to and in place of the lagged variance as an instrument for the current variance. We also check if our estimates of /rare sensitive to the inclusion of the retired in our samples.

This post was written by , posted on May 24, 2014 Saturday at 2:36 pm