Testing for Heteroscedasticity

The REG Procedure

Testing for Heteroscedasticity

The regression model is specified as $y_i = x_i {\beta} + \epsilon_i$ , where the $\epsilon_i$ 's are identically and independently distributed: $E({\epsilon})=0$ and $E({\epsilon^' \epsilon}) = \sigma^2{I}$ .If the $\epsilon_i$ 's are not independent or their variances are not constant, the parameter estimates are unbiased, but the estimate of the covariance matrix is inconsistent. In the case of heteroscedasticity, the ACOV option provides a consistent estimate of the covariance matrix. If the regression data are from a simple random sample, the ACOV option produces the covariance matrix. This matrix is

(X'X)^-1 (X' diag(e_i²)X) (X'X)^-1

where

e_i = y_i - x_i b

The SPEC option performs a model specification test. The null hypothesis for this test maintains that the errors are homoscedastic, independent of the regressors and that several technical assumptions about the model specification are valid. For details, see theorem 2 and assumptions 1 -7 of White (1980). When the model is correctly specified and the errors are independent of the regressors, the rejection of this null hypothesis is evidence of heteroscedasticity. In implementing this test, an estimator of the average covariance matrix (White 1980, p. 822) is constructed and inverted. The nonsingularity of this matrix is one of the assumptions in the null hypothesis about the model specification. When PROC REG determines this matrix to be numerically singular, a generalized inverse is used and a note to this effect is written to the log. In such cases, care should be taken in interpreting the results of this test.

When you specify the SPEC option, tests listed in the TEST statement are performed with both the usual covariance matrix and the heteroscedasticity consistent covariance matrix. Tests performed with the consistent covariance matrix are asymptotic. For more information, refer to White (1980).

Both the ACOV and SPEC options can be specified in a MODEL or PRINT statement.

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