We will answer these questions in the following sections. But first, we will consider an example to illustrate the problem.

Illustrative Example

Table 5.1 presents consumption expenditures (y) and income (x) for 20 families. Suppose that we estimate the equation by ordinary least squares. We get

= 0.847 + 0.899X

(0.703) (0.0253)

R- = 0.986 RSS = 31.074

(Figures in parentheses are standard errors.) We can compute the predicted values and the residuals. In Table 5.2 we present the residuals for the observations which are ordered by their x-values. As we can easily see they are larger (absolutely) for larger values of x. Thus there is some evidence that the error variances are not constant but increase with the value of x. In Figure 5.1 we show the plot of the residuals. This shows graphically (perhaps more than Table 5.2) that there is a heteroskedasticity problem.

Sometimes, the heteroskedasticity problem is solved by estimating the regression in a log-linear form. When we regress log on log x, the estimated equation is

log = 0.0757 + 0.9562 log x

(0.0574) (0.0183)

R = 0.9935 RSS = 0.03757

(Figures in parentheses are standard errors.) The i?-s are not comparable since the variance of the dependent variable is different. We discuss the problem of comparing Rs from linear versus log-linear form in Section 5.6. The residuals

Residual +2

-2 -

• 10

Figure 5.1. Example of heteroskedasticity.

Table 5.1 Consumption Expenditures (y) and Income (jc) for 20 Families (Thousands of Dollars)

Table 5.2 Residuals for the Consumption Function Estimated from Data in Table 5.1 (Ordered by Their jc-Values)"

"Residuals are rounded to two decimals.

from this equation are presented in Table 5.3. In this situation there is no perceptible increase in the magnitudes of the residuals as the value of : increases. Thus there does not appear to be a heteroskedasticity problem.

In any case, we need formal tests of the hypothesis of homoskedasticity. These are discussed in the following sections.

5.2 Detection of Heteroskedasticity

In the illustrative example in Section 5.1 we plotted the estimated residual , against x, to see whether we notice any systematic pattern in the residuals that suggests heteroskedasticity in the errors. Note, however, that by virtue of the

"Log x rounded to two decimals and residual multiplied by 10.

normal equation, , and x, are uncorrelated though ? could be correlated with X,. Thus if we are using a regression procedure to test for heteroskedasticity, we should use a regression of u, on x;, xJ, ... or a regression of , or on x„ X?, X?, .... In the case of multiple regression, we should use powers of S, the predicted value of y„ or powers of all the explanatory variables.

1. The test suggested by Anscombe and a test called RESET suggested by Ramsey both involve regressing , on y?, y?, . . . and testing whether or not the coefficients are significant.

2. The test suggested by White involves regressing uJ on all the explanatory variables and their squares and cross products. For instance, with three explanatory variables x,, , Xj, it involves regressing on x,, X2, Xj, X], xi, X3, XiXi, x,x„ and X3X,.

3. Glejser suggested estimating regressions of the type , = a + (3x„ , = a -I- p/Xj, , = a -I- Vx„ and so on, and testing the hypothesis

p = o.

The implicit assumption behind all these tests is that var(M,) = of = aiz) where z, is an unknown variable and the different tests use different proxies or surrogates for the unknown function /( ).

F. J. Anscombe, "Examination of Residuals," Proceedings of the Fourth Berlceley Symposium on Mathematical Statistics and Probability (Berkeley, Calif.: University of California Press, 1961), pp. 1-36.

J. B, Ramsey, "Tests for Specification Errors in Classical Linear Least Squares Regression Analysis," Journal of the Royal Statistical Society. Series B, Vol. 31, 1969, pp. 350-371. H. White, "A Heteroskedasticity Consistent Covariance Matrix Estimator and a Direct Test of Heteroskedasticity," Econometrica. Vol. 48, 1980, pp. 817-838.

H. Glejser, "A New Test for Homoscedasticity," Journal of the American Statistical Association. 1969, pp. 316-323.