Park test

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.