Testing for heteroskedasticity
Assuming that hi(a) is such that h;(0) = 1, tests for heteroskedasticity can be formulated in terms of the hypotheses
H0 : a = 0 H1 : a Ф 0.
We will describe the likelihood ratio, Wald, and Lagrange multiplier test statistics for these hypotheses, and then refer to other tests and evaluations that have appeared in the literature.
Using equation (4.27), the likelihood ratio (LR) test statistic is given by
Ylr = 2[L(a) - L(0)]
where e0 = y - Xb are the OLS residuals and e = y - X0(a) are the maximum likelihood residuals. When the null hypothesis of homoskedasticity holds, yLR has an approximate x(s) distribution.
The Wald (W) test statistic is given by
Yw = 7 V Уа,
where, applying partitioned-inverse results to equation (4.28), it can be shown that
This statistic is conveniently calculated as one-half of the regression sum-of - squares of e 0/60 on z, and a constant term. It has an approximate x 2s) distribution under H0 : а = 0. The Lagrange multiplier test statistic was derived by Breusch and Pagan (1979) and Godfrey (1978). To make it more robust to departures from normality, replacement of the denominator 204 by N-1 У N=1(e2 _ 60 )2 has been suggested (Koenker and Bassett, 1982).
Many more tests for heteroskedasticity have been suggested in the literature. See Pagan and Pak (1993) for a review and for details on how the various tests
can be classified as conditional moment tests. One popular test that we have not yet mentioned is the Goldfeld-Quandt (1965) test which uses the error variances from two separate least squares regressions to construct a finite sample F- statistic. Other classes of tests have been described by Szroeter (1978) and Fare - brother (1987). Lee (1992) suggests a test where the mean function is estimated nonparametrically and hence does not have to be precisely specified. Orme (1992) describes tests in the context of censored and truncated regression models. Also, tests for heteroskedasticity in these and other nonlinear models, such as discrete choice models and count data models, are reviewed by Pagan and Pak (1993). Numerous Monte Carlo studies have compared the finite sample size and power of existing and new test statistics. Typically, authors uncover problems with existing test statistics such as poor finite sample size or power, or lack of robustness to misspecification and nonnormality, and suggest alternatives to correct for such problems. A study by Godfrey and Orme (1999) suggests that bootstrapping leads to favorable outcomes. Other examples of Monte Carlo studies that have appeared are Evans and King (1988), Griffiths and Surekha (1986), Griffiths and Judge (1992) and Godfrey (1996). See Farebrother (1987) for some insightful comments on the results of Griffiths and Surekha (1986).
The work of White (1980) on testing for heteroskedasticity and testing hypotheses about в without specificing the precise form of the heteroskedasticity motivated others to seek estimators for в that did not require specification of the form of heteroskedasticity. Attempts have been made to specify estimators which are more efficient than OLS, while at the same time recognizing that the efficiency of GLS may not be achievable (Cragg, 1992; Amemiya, 1983). Carroll (1982) and Robinson (1987) develop adaptive estimators that assume no particular form of heteroskedasticity but nevertheless have the same asymptotic distribution as the generalized least squares estimator that uses a correct parametric specification. These adaptive estimators have been evaluated in terms of a second-order approximation by Linton (1996) and extended to time series models by Hidalgo (1992), to nonlinear multivariate models by Delgado (1992), and to panel data by Li and Stengos (1994). Szroeter (1994) suggests weighted least squares estimators that have better finite sample efficiency than OLS when the observations can be ordered according to increasing variances but no other information is available.