Park test

Background

In regression analysis, heteroscedasticity refers to unequal variances of the random error terms ε_i, such that

var( ε_i ) = E[ ( ε_i )² ] – [ E (ε_i ) ]² = E[ ( ε_i )² ] = ( σ_i )².

It is assumed that E(ε_i) = 0. The above variance varies with i, or the i^th trial in an experiment or the i^th case or observation in a dataset. Equivalently, heteroscedasticity refers to unequal conditional variances in the response variables Y_i, such that

var( Y_i | X_i ) = ( σ_i )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( ε_i ) = 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:

ln[ ( σ_εi )² ] = ln[ σ² ] + γ ln[ X_i ] + v_i  

in which the error terms 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 = β₀ + β₁X_i1 + ∙∙∙ + β_p−1X_i,p−1 + ε_i

where the latter contains p − 1 regressors, and then squares and takes the natural logarithm of each of the residuals (ε_i-hat), which serve as estimators of the ε_i. The squared residuals (ε_i-hat)² in turn estimate (σ_εi)².

If, then, in a regression of ln[ ( ε_i )² ] on the natural logarithm of one or more of the regressors X_i, we arrive at statistical significance for non-zero values on one or more of the γ_i-hat, we reveal a connection between the residuals and the regressors. We reject the null hypothesis of homoscedasticity and conclude that heteroscedasticity is present.