The fact that Z is based on estimated coefficients causes two complications. First, the uncertainty about the estimated coefficients must be accounted for in finding the standard error of the estimate of A; this is done in the usual formulas for the standard errors of instrumental-variables estimates. Second, the fact that the first-stage coefficients are estimated introduces some correlation between Z and v in the same direction as the correlation between Z and v. This correlation disappears as the sample size becomes large; thus FV is consistent but not unbiased. If the instruments are only moderately correlated with the right-hand-side variable, however, the bias in finite samples can be substantial. See, for example. Nelson and Startz (1990).

ual. Once one has such instruments, the first-stage regression is a regression of the right-hand-side variable, Zf, on the instruments. The second-stage regression is then a regression of the left-hand-side variable, Q - Q-i, on the fitted value of Zf from the first-stage regression, Zf. That is, we estimate:

Cf - Cf-i == AZf + A(Zf - Zf) + Vf

(7.23)

= AZf + Vf.

The residual in (7.23), Vf, consists of two terms, Vf and A(Zf ~ Zf). By assumption, the instruments used to construct Z are not systematically correlated with Vf. And since Z is the fitted value from a regression, by construction it is uncorrelated with the residual from that regression, Z ~ Z. Thus regressing Cf - C 1 on Z yields a valid estimate of A.

The usual problem in using instrumental variables is finding valid instruments: it is often hard to find variables that one can be confident are uncorrelated with the residual. But in cases where the residual reflects new information between t - 1 and t, theory tells us that there are many candidate instruments: any variable that is known as of time t - 1 is uncorrelated with the residual.

We can now turn to the specifics of Campbell and Mankiws test. They measure consumption as real purchases of consumer nondurables and services per person, and income as real disposable income per person. The data are quarterly, and the sample period is 1953-1986. They consider various sets of instruments. They find that lagged changes in income have almost no predictive power for future changes. This suggests that Halls failure to find predictive power of lagged income movements for consumption is not strong evidence against the traditional view of consumption. As a base case, they therefore use lagged values of the change in consumption as instruments. When three lags are used, the estimate of A is 0.42, with a standard error of 0.16; when five lags are used, the estimate is 0.52, with a standard error of 0.13. Other specifications yield similar results.

Thus Campbell and Mankiws estimates suggest quantitatively large and statistically significant departures from the predictions of the random-walk model: consumption appears to increase by about fifty cents in response to an anticipated 1-dollar increase in income, and the null hypothesis of no effect is strongly rejected. At the same time, the estimates of A are far below

In addition, the instrumental-variables approach has overidentifying restrictions that can be tested. If the lagged changes in consumption are valid instruments, they are imcorrelated with v. This implies that once we have extracted all of the information in the instruments about income growth, they should have no additional predictive power for the left-hand-side variable: if they do, that means that they are correlated with v, and thus that they are not valid instruments. This implication can be tested by regressing the estimated residuals from (7.22) on the instruments and testing whether the instruments ha\e any explanatory power. Specifically, one can show that under the null hypothesis of valid instruments, the of this regression times the number of observations is asymptoticalh distributed with degrees of freedom equal to the number of overidentifying restrictions- that is, the number of instruments minus the number of endogenous variables.

In Campbell and Mankiws case, this TR statistic is distributed when three lags of the change in consumption are used, and 4 when hve lags are used. The values of the test statistic in the two cases are only 1.83 and 2.94; these are only in the 59th and 43rd percentiles of the relevant A" distributions. Thus the hypothesis that the instruments are valid cannot be rejected.

1. Thus the results also suggest that the permanent-income hypothesis is important to understanding consumption.

Sheas Test Using Household Data

Testing the random-walk hypothesis with aggregate data has several disadvantages. Most obviously, the number of observations is small. In addition, it is difficult to find variables withmuch predictive power for changes in income; it is therefore hard to test the key prediction of the random-walk hypothesis that predictable changes in income are not associated with predictable changes in consumption. Finally, the theory concerns individuals consumption, and additional assumptions are needed for the predictions of the model to apply to aggregate data. Entry and exit of households from the population, for example, can cause the predictions of the theory to fail tn the aggregate even if they hold for each household individually.

Because of these considerations, many investigators have examined consumption behavior using data on individual households. Shea (1995) takes particular care to identify predictable changes in income. He focuses on households in the Panel Study of Income Dynamics (commonly referred to as the PSID) with wage-earners covered by long-term union contracts. For these households, the wage increases and cost-of-living provisions in the contracts cause income growth to have an important predictable component.

Shea constructs a sample of 647 observations where the union contract provides clear information about the households future earnings. A regression of actual real wage growth on the estimate constructed from the union contract and some control variables produces a coefficient on the constructed measure of 0.86, with a standard error of 0.20. Thus the union contract has important predictive power for changes in eamings.

Shea then regresses consumption growth on this measure of expected wage growth; the permanent-income hypothesis predicts that the coeffi-

An altemative would be to follow Campbell and Mankiws approach and regress consumption growth on actual income growth by instrumental variables, using the constructed wage growth measure as an mstrument. Given the almost one-for-one relationship between actual and constructed eammgs growth, this approach would be likely to produce similar results.

Cient should be zero. The estimated coefficient is in fact 0.89, with a standard error of 0.46. Thus Shea also finds a quantitatively large (though only marginally statistically significant) departure from the random-walk prediction.

Recall that in our analysis in Sections 7.1 and 7.2, we assumed that households can borrow without limit as long as they eventually repay their debts. One reason that consumption might not follow a random walk is that this assumption might fail-that is, that households might face liquidity constraints. If households are unable to borrow and their current income is less than their permanent income, their consumption is determined by their current income. In this case, predictable changes in income produce predictable changes in consumption.

Shea tests for liquidity constraints in two ways. First, following Zeldes (1989) and others, he divides the households according to whether they have hquid assets. Households with hquid assets can smooth theh consumption by running down these assets rather than by borrowing. Thus if liquidity constraints are the reason that predictable wage changes affect consumption growth, the prediction of the permanent-income hypothesis will fail only among the households with no assets. Shea finds, however, that the estimated effect of expected wage growth on consumption is essentially the same in the two groups.

Second, foUowing Altonji and Slow (1987), Shea splits the low-wealth sample according to whether the expected change in the real wage is positive or negative. Individuals facing expected dechnes in income need to save rather than borrow to smooth their consumption. Thus if liquidity constraints are important, predictable wage increases produce predictable consumption mcreases, but predictable wage decreases do not produce predictable consumption decreases.

Sheas findings are the opposite of this. For the households with positive expected income growth, the estimated impact of the expected change in the real wage on consumption growth is 0.06 (with a standard error of 0.79); for the households with negative expected growth, the estimated effect is 2.24 (with a standard error of 0.95). Thus there is no evidence that liquidity constraints are the source of Sheas results.

7.4 The Interest Rate and Saving

An important issue concerning consumption involves its response to rates of return. For example, many economists have argued that more favorable