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Testing for Heteroskedasticity

Heteroskedasticity refers to unequal variance in the regression errors.Heteroskedasticity can arise in a variety of waysand a number of tests have been proposed. Typically a test is designedto test the null hypothesis of homoskedasticity (equal error variance)against some specific alternative heteroskedasticity specification.

Examples

A Note on the Spelling of Heteroskedasticity

A 'c' is often used instead of a 'k' in the spelling ofheteroskedasticity. The research by McCulloch [1985] concludes that the word is derived from Greek roots and the proper English spelling is with a 'k'.
Reference:J. Huston McCulloch, 'On Heteros*edasticity', Econometrica, Vol. 53, 1985, p. 483.

Heteroskedasticity as a function of the explanatory variables

Test statistics are reported with the SHAZAM commands:

The DIAGNOS command uses the results from the immediately preceding OLS command to generate diagnostic tests.The HET option computes and reports tests forheteroskedasticity. These tests are obtained by using a function of the OLS residuals e_t as a dependent variable in an auxiliary regression. A number of alternative auxiliary regressions have been proposed as follows.

where X_t is a (K x 1) vector of observations on the explanatory variables (including the constant) for t=1,...,N. SSE is the sum of squared errors from the initial OLS regression. R² and SSR are the R-square and the regression sum of squares respectively from the auxiliary regression.

Note that the final two auxiliary regressions include cross-productsof the explanatory variables as regressors.Therefore, the application requires at least 2 explanatory variables.The final two test statistics are not reported for regressions that specify one explanatory variable.

In 'large samples' the test statistics have a chi-square distribution withdegrees of freedom as given in the D.F. column. This means that criticalvalues can be obtained from tables for the chi-square distribution,but the comparison is approximate only.

References for the various test statistics are given in theSHAZAM User's Reference Manual.

The ARCH (AutoRegressive Conditional Heteroskedasticity) test isin a different category from the others.This test has specific application to time series data and detectssuccessive periods of volatility followed by successive periods ofstability. This type of behaviour has been found in financial timeseries data.

Example

Heteroskedasticity has been found to be a feature of cross-section studieson household expenditure. This example, from Griffiths, Hill and Judge,uses a data set on household expenditure.The SHAZAM commands are:

The SHAZAM output can be inspected. The results from the DIAGNOS / HET command are:

The 5% critical value from a chi-square distribution with 1 degreeof freedom is 3.84. With the exception of the ARCH test,all test statistics exceed this value and so there is evidencefor heteroskedasticity in the estimated residuals.Of course, the ARCH test is of no relevance to this example sincethe data is cross-section data and the ARCH test has application totime series data.

Note that the first test statistic and the seventh test statisticare identical. As an exercise the user should verify that thesetests are always identical when the regression contains one explanatoryvariable.

SHAZAM output with tests for heteroskedasticity

The OLS estimation results are described in further detail inGriffiths, Hill and Judge [1993, Section 5.3.2].