# What are the three forms of the Park test for heteroskedasticity?

I understand the Park test for heteroskedasticity has three different forms. The best known one is in a log form: LN(Residual^2) = intercept + slope (LN(X)). The second one is in a linear form: Residual^2 = intercept + slope (X). In both cases if the regression coefficient of X (the independent variable you are testing for heteroskedasticity) then you have to reject the null hypothesis that residuals are homoskedastic relative to the levels of the tested independent variable X. Nevertheless, do you know how well established is the second form, the linear one? And, also what is the third form? For model variables I need to test, it is key that I can use the linear form because many of those variables are percent changes that can't be logged.

• Where did you see that the Park test has 3 forms? Can you provide a reference? – gung - Reinstate Monica Dec 10 '14 at 18:53
• The first form is a version of the spread vs level plot. IMHO it's an inferior choice because it does not use robust estimates of spread and it really shouldn't have an intercept, but for "nicely behaved" residuals and quick-and-dirty work it should be ok. As you note, it's applicable only when one intends to make Box-Cox transformations of a (necessarily) positive response variable. – whuber Dec 10 '14 at 19:20
• gung, the source I found mentioning the three forms of the test was not "well sourced." That's actually part of my question. Can one disclose what the 3d form is? And, based on good reference. – Sympa Dec 11 '14 at 22:41

## 1 Answer

I am less familiar with the Park test. The Wikipedia page only lists what you call the first form. What you are calling the second form is identical to the Breusch-Pagan test. It is very well established, for what that's worth. Regarding the distinction between using logged or non-logged predictors and response variables, it may help you to read this excellent CV thread: Interpretation of log transformed predictor. In general, the two-stage approach to modeling (i.e., test for assumptions, and fit standard model if non-significant or robust model if significant) is not recommended (see this excellent CV thread: A principled method for choosing between t test or non-parametric e.g. Wilcoxon in small samples). If you are worried about the possibility of heteroscedasticity, you would be better off just using robust methods, such as the Huber-White heteroscedasticity consistent 'sandwich' standard errors, by default. For some examples of various strategies that can be used with heteroscedastistic data, it may help to read my answer here: Alternatives to one-way ANOVA for heteroscedastic data.

• This is a very erudite paragraph. But, it does not answer the question. Also, as a clarification the Breusch-Pagan test does test for heteroskedasticity on the whole model (Residuals vs estimates). The Park test instead is customized to test for heteroskedasticity at the independent variable level (Residuals vs X1, or X2, etc...). Your comment brings out an interesting point. And, that is that the linear form of the Park test is identical in structure to the Breusch-Pagan test. Given that, I think this gives much legitimacy to Park's test linear form. – Sympa Dec 11 '14 at 22:26
• @GaetanLion, where do you see these 3 forms? Wikipedia only list 1, so far as I notice. You raise a good point that I had not read clearly enough & Wikipedia notes the Park test is run on "one or more of the regressors Xi". However, I would say that such a test on only 1 regressor would be potentially invalid in that it would be endogenous unless all regressors are perfectly uncorrelated (not only in the population, but also in your sample). – gung - Reinstate Monica Dec 11 '14 at 22:38
• gung, as mentioned I have seen 3 forms mentioned on the Internet. But, statement was not well sourced. I was hoping someone would know readily of those 3 forms and how well established they are. Thanks to you, we are comfortable with the linear form of Park test (similar to Breusch-Pagan). In Wikipedia, they mention you can use Park on one or more Xs. But, those Xs are tested separately unlike in the Breusch-Pagan test. Park test is designed to test a single variable just like Glejser test. Those are valid as structured even if the Xs are correlated to a certain degree. – Sympa Dec 11 '14 at 22:55
• @GaetanLion, who's to say there even are 3 forms? Can you link to where this was claimed on the internet? It may be totally spurious. As for your claim that the test would be valid even if variables are correlated, where are you getting that from? I demonstrate the problems w/ tests of heteroscedasticity on single, non-independent variables here. – gung - Reinstate Monica Dec 11 '14 at 23:07
• gung, I may let you argue your point regarding the invalidity of the Park test with Park himself. Hopefully, he is still alive. You could have the same debate regarding the Glejser test with Glejser himself constrained by the same condition (hopefully he is still alive). The URL below is where I did find the mentioning of three different forms of the Park tests. But, the Word document described only two of them. As you can tell this statement is not well sourced or referenced. go.owu.edu/~rjgitter/Heteroscedasticity%20Testing%20Homework.doc – Sympa Dec 11 '14 at 23:11