White test with large model and few samples impossible?

Question

I have a linear model with 12 regressors and sample size 42, which I want to test for heteroscedasticity. Hence, I applied a white test in the following way

• regress $y=X\beta + e$
• regress residuals $\hat e^2=\beta_1+\beta_2x_2+\cdots+\beta_{13}x_2^2+...+\beta_{25}x_{1}x_2+\cdots+u $
• reject $H_0: \sigma_i^2=\sigma^2$ if $N\cdot R^2_{\hat e^2} > \chi^{2, 1-\alpha}_{s}$

Since I have $N=42$ and $s=90$* the test statistic is maximum $42$ and can thus never exceed $\chi^{2, 1-\alpha}_{s}$, which is $112$ for $\alpha=5%$.

What does that mean? Can there be no Heteroscedasticity in this case? This feels counterintuitive since I actually get an $R^2=1$ in step 2 and thus the $x$ terms have much explanatory power on the residuals. So am I doing something wrong?

Thanks in advance for your thoughts!

$*$ ($12$ variables, $12$ squared terms and $\frac{n*(n-1)}{2}=66$ crossterms)

Answer 1

In your step 2 you are getting a perfect fit because you have more predictors than observations so $R^2=1$ is inevitable. I think most people would start the process here by plotting the residuals against the fitted values and looking at the spread of the residuals in that plot. Alternatively plot $\sqrt(| e |)$ against the fitted where $e$ is the residual in your notation, the so-called scale-location plot.