What does the residual plot of a model against a term not in the model mean?

Question

Let's say I have a linear regression model $Y = \beta_0+\beta_1 X_1 + \beta_2X_2$.

I know that plotting the residuals against $X_1$, for example, will show that the model is a good fit if the residuals are randomly scattered.

But what if I took the residuals from that model and plot them against $X_1^2$, the quadratic term? If it looks nonrandom, does this mean that the quadratic term would be helpful to add to the model?

Answer 1

Not necessarily, it may point that you have a heteroscedasticity. Namely, a non equal variance. Actually, White's test for detecting non-equal variance is based exactly on this idea.