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.