Park test
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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
In regression analysis, heteroscedasticity refers to unequal variances of the random error terms εi, such that
- var( εi ) = E[ ( εi )2 ] – [ E (εi ) ]2 = E[ ( εi )2 ] = ( σi )2.
It is assumed that E(εi) = 0. The above variance varies with i, or the ith trial in an experiment or the ith case or observation in a dataset. Equivalently, heteroscedasticity refers to unequal conditional variances in the response variables Yi, such that
- var( Yi | Xi ) = ( σi )2,
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
- var( εi ) = σ2, 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]
- ln[ ( σεi )2 ] = ln[ σ2 ] + γ ln[ Xi ] + vi
in which the error terms vi are considered well behaved.
This relationship is used as the basis for this test.
The modeler first runs the unadjusted regression
Yi = β0 + β1Xi1 + ∙∙∙ + βp−1Xi,p−1 + εi
where the latter contains p − 1 regressors, and then squares and takes the natural logarithm of each of the residuals (εi-hat), which serve as estimators of the εi. The squared residuals (εi-hat)2 in turn estimate (σεi)2.
If, then, in a regression of ln[ ( εi )2 ] on the natural logarithm of one or more of the regressors Xi, we arrive at statistical significance for non-zero values on one or more of the γi-hat, we reveal a connection between the residuals and the regressors. We reject the null hypothesis of homoscedasticity and conclude that heteroscedasticity is present.
Notes
The test has been discussed in the econometrics literature.[2] Stephen M. Goldfeld and Richard E. Quandt raise concerns about the assumed structure, cautioning that the vi may be heteroscedastic and otherwise violate assumptions of ordinary least squares regression.[3]
See also
Notes
- ^ a b Park, R. E. (1966). "Estimation with Heteroscedastic Error Terms". Econometrica. 34 (4): 888. JSTOR 1910108.
- ^ Gujarati, Damodar (1988) Basic Econometrics (2nd Edition), New York: McGraw–Hill. ISBN 0-07-100446-7 ( pp. 329–330)
- ^ 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.