# White's test interpretation

I am running a regression in python (a basic market model with just one index as regressor). After doing that I conduct the heteroscedasticity test on residuals using two tests, White and ARCH. I am having trouble understanding the situation when I only obtain significance in the White test. In other words, how should I proceed? Could using a GARCH model not be the appropriate solution?

• What is the model? Is it $y_t=\beta_0+\beta_1x_t+u_t$? (a basic market model with just one index as regressor does not tell me much.) Are you using time series data? Nov 17, 2023 at 11:30
• @RichardHardy yes it is! and yes Nov 17, 2023 at 12:16

I wouldn't base my choice of model on a test for heteroscedasticity. And I'm not the only one. Here is a quote from the great George Box:

To make the preliminary test on variances is rather like putting to sea in a rowing boat to find out whether conditions are sufficiently calm for an ocean liner to leave port!

And you can read more about this in a paper by the great Andrew Gelman which he titled: Hey, you. Yeah, you! Stop what you’re doing RIGHT NOW and read this Stigler article on the history of robust statistics .

If you want to check heteroscedasticity, I would do it visually and use judgement (data analysis requires judgement). Or, do the usual regression and a robust variant and compare results, and, again, use your judgement.

Using GARCH does not sound like a great option, because you have not established presence of autoregressive conditional heteroskedasticity (and you have used the ARCH-LM test that could have indicated that). Meanwhile, White's test assesses whether the variance of the residuals varies with the regressors (in your case only one). If you were to modify the model to account for that, you would want a model where, well, the variance of the residuals varies with $$x$$.

(This could still be done with a GARCH model in a dirty way. You would specify the conditional variance equation as GARCHX, including $$x$$ as an external regressor in the conditional variance equation. During estimation, you would restrict the parameters $$\alpha=0.001$$ and $$\beta=0.999$$, so that the estimated ARCH patterns are negligible. But there should be a more elegant alternative.)

• Thank you for explaining to me so well! What would be a more elegant way to approach this problem? Would using HAC estimators be a good idea? Nov 17, 2023 at 17:09
• @Mattia, you can use HAC, though you could possibly get more accurate point estimates by, say, using WLS that accounts for how error variance changes with $x$. Nov 17, 2023 at 20:39
• let me know if I got it. If there are ARCH effects, I use a GARCH models; if, instead, there are only heteroscedasticity issues (White test), I use WLS or OLS with HAC estimators. Sorry for my questions, I just started studying econometrics. Thank you again Nov 18, 2023 at 8:38
• @Mattia, that sounds right to me. Nov 18, 2023 at 8:49
• @Mattia, let me know if you need any further explanation. Otherwise, consider accepting the answer using the check mark to the left. This is how Cross Validated works. Nov 24, 2023 at 20:16