Park test explained

In econometrics, the Park test is a test for heteroscedasticity. The test is based on the method proposed by Rolla Edward Park for estimating linear regression parameters in the presence of heteroscedastic error terms.^[1]

Background

\epsilon _{i} , such that\operatorname{Var}(\epsilon _{i)=E(\epsilon} ^{2} ^{i} .

It is assumed that

\operatorname{E}(\epsilon _{i)=0} . The above variance varies with i , or the i ^{th} trial in an experiment or the i ^{th} case or observation in a dataset. Equivalently, heteroscedasticity refers to unequal conditional variances in the response variables Y _{i} , such that\operatorname{Var}(Y _{i|X} _{i)=\sigma} ^{2} ^{i} ,

again a value that depends on

i – or, more specifically, a value that is conditional on the values of one or more of the regressors X . Homoscedasticity, one of the basic Gauss–Markov assumptions of ordinary least squares linear regression modeling, refers to equal variance in the random error terms regardless of the trial or observation, such that\operatorname{Var}(\epsilon ^{2} _{i) = \sigma} , a constant.

Test description

Park, on noting a standard recommendation of assuming proportionality between error term variance and the square of the regressor, suggested instead that analysts 'assume a structure for the variance of the error term' and suggested one such structure:^[1]

\operatorname{ln}(\sigma ^{2 ) = \operatorname{ln}(\sigma} _{\epsilon i} ^{2 ) = \gamma\operatorname{ln}(X} ^{i) + v} _{i}

in which the error terms

v _{i} are considered well behaved.

This relationship is used as the basis for this test.

The modeler first runs the unadjusted regression

Y _{i=\beta} _{0+\beta} _{1X} _{i1} + ... + \beta _{p-1} X _{i,p-1} + \epsilon _{i}

where the latter contains p − 1 regressors, and then squares and takes the natural logarithm of each of the residuals (

\hat{\epsilon _{i}} ), which serve as estimators of the \epsilon _{i} . The squared residuals \hat{\epsilon ^{2} _{i}} in turn estimate \sigma ^{2} _{\epsilon i} .

If, then, in a regression of

ln {(\epsilon ^{2 )}} _{i} on the natural logarithm of one or more of the regressors X _{i} , we arrive at statistical significance for non-zero values on one or more of the \hat\gamma _{i} , we reveal a connection between the residuals and the regressors. We reject the null hypothesis of homoscedasticity and conclude that heteroscedasticity is present.

See also

• Breusch–Pagan test
• Glejser test
• Goldfeld–Quandt test
• White test

Notes

The test has been discussed in econometrics textbooks.^{[2] [3]} Stephen Goldfeld and Richard E. Quandt raise concerns about the assumed structure, cautioning that the v_i may be heteroscedastic and otherwise violate assumptions of ordinary least squares regression.^[4]

Notes and References

1. Park . R. E. . 1966 . Estimation with Heteroscedastic Error Terms . . 34 . 4 . 888 . 1910108 .
2. Book: Gujarati, Damodar . 1988 . Basic Econometrics . 2nd . New York . McGraw–Hill . 0-07-100446-7 . 329–330 .
3. Book: Studenmund, A. H. . Using Econometrics: A Practical Guide . limited . Fourth . Boston . Addison-Wesley . 2001 . 0-321-06481-X . 356–358 .
4. Goldfeld, Stephen M.; Quandt, Richard E. (1972) Nonlinear Methods in Econometrics, Amsterdam: North Holland Publishing Company, pp. 93–94. Referred to in: Gujarati, Damodar (1988) Basic Econometrics (2nd Edition), New York: McGraw-Hill, p. 329.