Park test

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 ${\displaystyle \epsilon _{i}}$, such that

${\displaystyle \operatorname {Var} (\epsilon _{i})=E(\epsilon _{i}^{2})-E(\epsilon _{i})^{2}=E(\epsilon _{i}^{2})=\sigma _{i}^{2}}$.

It is assumed that ${\displaystyle \operatorname {E} (\epsilon _{i})=0}$. The above variance varies with ${\displaystyle i}$, or the ${\displaystyle i^{th}}$ trial in an experiment or the ${\displaystyle i^{th}}$case or observation in a dataset. Equivalently, heteroscedasticity refers to unequal conditional variances in the response variables ${\displaystyle Y_{i}}$, such that

${\displaystyle \operatorname {Var} (Y_{i}|X_{i})=\sigma _{i}^{2}}$,

again a value that depends on ${\displaystyle i}$ – or, more specifically, a value that is conditional on the values of one or more of the regressors ${\displaystyle X}$. 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

${\displaystyle \operatorname {Var} (\epsilon _{i})=\sigma ^{2}}$, 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]

${\displaystyle \operatorname {ln} (\sigma _{\epsilon i}^{2})=\operatorname {ln} (\sigma ^{2})=\gamma \operatorname {ln} (X_{i})+v_{i}}$

in which the error terms ${\displaystyle v_{i}}$ are considered well behaved.

This relationship is used as the basis for this test.

The modeler first runs the unadjusted regression

${\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{i1}+...+\beta _{p-1}X_{i,p-1}+\epsilon _{i}}$

where the latter contains p − 1 regressors, and then squares and takes the natural logarithm of each of the residuals (${\displaystyle {\hat {\epsilon _{i}}}}$), which serve as estimators of the ${\displaystyle \epsilon _{i}}$. The squared residuals ${\displaystyle {\hat {\epsilon _{i}}}^{2}}$ in turn estimate ${\displaystyle \sigma _{\epsilon i}^{2}}$.

If, then, in a regression of ${\displaystyle \ln {(\epsilon _{i}^{2})}}$ on the natural logarithm of one or more of the regressors ${\displaystyle X_{i}}$, we arrive at statistical significance for non-zero values on one or more of the ${\displaystyle {\hat {\gamma }}_{i}}$, 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

1. Park, R. E. (1966). "Estimation with Heteroscedastic Error Terms". Econometrica. 34 (4): 888. JSTOR 1910108.
2. Gujarati, Damodar (1988). Basic Econometrics (2nd ed.). New York: McGraw–Hill. pp. 329–330. ISBN 0-07-100446-7.
3. Studenmund, A. H. (2001). Using Econometrics: A Practical Guide (Fourth ed.). Boston: Addison-Wesley. pp. 356–358. ISBN 0-321-06481-X.
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.