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109 c2 = (7.17) T 1 / T \ Ao + Tl  Ci + X MYt] where the second hne uses the fact that Ai = Aq + Yi  Ci. We can rewrite the expectation as of period 2 of income over the remainder of life, Xj=2 EilYt], as the expectation of this quantity as of period 1, Xj=2 Ei\Yt], plus the information learned between period 1 and period 2, .1=22]  YJt=2\VYt\Thus we can rewrite (7.17) as c2 = "1 / T J \ Ac + y, ~ Q + X + X EzlYA  X t = 2 \t=2 f=2 (7.18) From (7.14), Ac + Yl + YJ=2 Ei[Yt\ equals TCi. Thus (7.18) becomes C2 = TCv  Ci + S£2[Tf]~ l£i[Yt] \f=2 t~2 = Q + / T T \ XE2\Yt]XEilYt] \t=2 f=2 (7.19) Equation (7.19) states that the change in consumption between period 1 and period 2 equals the change m the individuals estimate of his or her lifetime resources divided by the number of periods of life remaimng. Finally, note that the mdividuals behavior exhibits certainty equivalence: as (7.14) shows, the individual consumes the amount he or she would if his or her future incomes were certain to equal their means; that is, uncertainty about future income has no impact on consumption. To see the intuition for this certaintyequivalence behavior, consider the Euler equation relating consumption m periods 1 and 2. With a general instantaneous utility function, this condition is u(Q) = Ei[u{C2)]. (7.20) When utihty is quadratic, marginal utility is linear. Thus the expected marginal utility of consumption is the same as the marginal utility of expected consumption. That is, since E\[l ~ aCz] = 1  aEi[C2], for quadratic utihty (7.20) is equivalent to u(Ci) = u(£i[c2]). (7.21) This implies Q = Ei[c2].
•Although the specihc result that the change m consumption has a mean of zero and is unpredictable (equation [7.16]) depends on the assumpnon of quadratic utility (and on the assumption that the discount rate and the interest rate are equal), the result that departures of consumpnon growth from its average value are not predictable arises under more general assumpUons. See, for example, Problem 7.3. Indeed, it is said that when Hall hrst presented the paper deriving and testing the randomwalk result, one prominent macroeconomist told him that he must have been on drugs when he wrote the paper. The permanentincome hypothesis also makes predictions about how consumption responds to unexpected changes m income. In the model of Secnon 7.2, for example, the response to news is given by equation [7.19]. The hypothesis that consumption responds less than the permanentincome hypothesis predicts to unexpected changes m income is referred to as excess smoothness of consumption. Since excess sensitivity concerns expected changes m income and excess smoothness concerns unexpected changes, it is possible for consumption to be excessively sensitive and excessive!) smooth at the same time, for more on excess smoothness, see Campbell and Deaton (1989); West (1988); Flavin (1993); and Problem 7.4. This analysis shows that quadratic utility is the source of certaintyequivalence behavior: if utility is not quadratic, marginal utiUty is not linear, and so (7.21) does not follow from (7.20). We return to this point in Section 7.6. 7.3 Empirical Application: Two Tests of the RandomWalk Hypothesis Halls randomwalk result ran strongly counter to existing views about consumption.5 The traditional view of consumption over the business cycle implies that when output declines, consumption declines but is expected to recover; thus it implies that there are predictable movements in consumption. Halls extension of the permanentincome hypothesis, in contrast, predicts that when output declines unexpectedly, consumption declines only by the amount of the fall in permanent income; as a result, it is not expected to recover. Because of this divergence in the predictions of the two views, a great deal of effort has been devoted to testing whether predictable changes in income produce predictable changes in consumption. The hypothesis that consumption responds to predictable income movements is referred to as excess sensitivity oi consumption (Flavin, 1981). Campbell and Mankiws Test Using Aggregate Data The randomwalk hypothesis implies that the change in consumption is impredictable; thus it implies that no information available at time f 1 can
be used to forecast the change in consumption from f  1 to f. Thus one approach to testing the randomwallc hypothesis is to regress the change in consumption on variables that are known at f  1. If the randomwalk hypothesis is correct, the coefficients on the variables should not differ systematically from zero. This is the approach that Hall took in his original work. He was imable to reject the hypothesis that lagged values of either income or consumption cannot predict the change in consumption. He did find, however, that lagged stockprice movements have statistically significant predictive power for the change in consumption. The disadvantage of this approach is that the results are hard to interpret. Halls result that lagged income does not have strong predictive power for consumption, for example, could arise not because predictable changes in income do not produce predictable changes in consumption, but because lagged values of income are of little use in predicting income movements. Similarly, it is hard to gauge the importance of the rejection of the randomwalk prediction using stockprice data. Campbell and Mankiw (1989a) therefore use an instrumentalvariables approach to test Halls hypothesis against a specific alternative. The alternative they consider is that some fraction of consumers simply spend their current income, and the remainder behave according to Halls theory. This altemative implies that the change in consumption from period f  1 to period t equals the change in income between f ~ 1 and f for the first group of consumers, and equals the change in estimated permanent income between f ~ 1 and f for the second group. Thus if we let A denote the fraction of consumption that is done by consumers in the first group, the change in aggregate consumption is QCt ] = ( ,Yf ,) + (lA)Cf, (7.22) = \Zt + Vf, where Cf is the change in consumers estimate of then permanent income from f ~ 1 to t. Zf and Vf are almost surely correlated. Times when income increases greatly are usually also times when households receive favorable news about their total lifetime incomes. But this means that the righthandside variable in (7.22) is positively correlated with the error term. Thus estimating (7.22) by ordinary least squares (OLS) leads to estimates of A that are biased upward. The solution to correlation between the righthandside variable and the error term is to use instmmental variables (IV) rather than OLS. The intuition behind IV estimation is easiest to see using the twostage least squares interpretation of instrumental variables. What one needs are variables correlated with the righthandside variables but imcorrelated with the resid
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