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.
Background In regression analysis, heteroscedasticity refers to unequal variances of the random error terms ϵ i {\displaystyle \epsilon _{i}} , such that
Var ( ϵ i ) = E ( ϵ i 2 ) − E ( ϵ i ) 2 = E ( ϵ i 2 ) = σ i 2 {\displaystyle \operatorname {Var} (\epsilon _{i})=E(\epsilon _{i}^{2})-E(\epsilon _{i})^{2}=E(\epsilon _{i}^{2})=\sigma _{i}^{2}} . It is assumed that E ( ϵ i ) = 0 {\displaystyle \operatorname {E} (\epsilon _{i})=0} . The above variance varies with i {\displaystyle i} , or the i t h {\displaystyle i^{th}} trial in an experiment or the i t h {\displaystyle i^{th}} case or observation in a dataset. Equivalently, heteroscedasticity refers to unequal conditional variances in the response variables Y i {\displaystyle Y_{i}} , such that
Var ( Y i | X i ) = σ i 2 {\displaystyle \operatorname {Var} (Y_{i}|X_{i})=\sigma _{i}^{2}} , again a value that depends on i {\displaystyle i} – or, more specifically, a value that is conditional on the values of one or more of the regressors X {\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
Var ( ϵ i ) = σ 2 {\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:
ln ( σ ϵ i 2 ) = ln ( σ 2 ) = γ ln ( X i ) + v i {\displaystyle \operatorname {ln} (\sigma _{\epsilon i}^{2})=\operatorname {ln} (\sigma ^{2})=\gamma \operatorname {ln} (X_{i})+v_{i}}
in which the error terms v i {\displaystyle 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 = β 0 + β 1 X i 1 + . . . + β p − 1 X i , p − 1 + ϵ i {\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{i1}+...+\beta _{p-1}X_{i,p-1}+\epsilon _{i}}
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