What does it mean in terms of regression if residuals are not white noise?

Question

I need help in answering this one, it is an exam question.

Answer 1

Strictly speaking, the residuals of a regression are not white noise. Since each residual is a function of the entire data set, the residuals are lightly correlated.

But ... there's correlation and there's correlation.

Residuals can fail to be "white noise" if:

• The regression model was not correctly specified. e.g $Y=a + bX + cX^2$ should have been chosen instead of a $Y=a+bX$.
• Additional covariates are needed
• the covariates are correct but the variance is not constant.
• The error structure was not normal to start with. e.g. if you fit a linear regression model to 0/1 count data, you will get weird residuals near the extremes. The solution, in this case, would be to fit a logistic model.

Bottom line: when the residuals fail to be white noise, a different model should be tried.

Answer 2

Short answer regarding time series regression: If they are not white noise (i.e. they are not normal, not have zero mean or serially autocorrelated), then your model is not fully adequate. Therefore, you should revise your model. Usually (but not always), this means that there is a significant autocorrelation (of some order) among the residuals so you should improve your model.