# Simple Linear Regression, Serially Correlated Residuals, and Interpretation

## Question

Can I draw any proper conclusions about the linearity and strength of a relationship between two non-stationary time series (I'm considering two series of interest rates, series $$A$$ and series $$B$$) using simple linear regression when the residuals are serially correlated and heteroskedastic?

## Background

I see that computing a naive Pearson correlation coefficient (PCC) doesn't help since this calculation yields a random variable regardless of sample size. Since the $$\beta$$ in simple linear regression is related to the PCC, I think this would also yield a random variable for the coefficient in the model.

## Interpretation

If I run a simple linear regression, and despite non-IID residuals, have an $$R^2$$ of $$0.75$$, can I say that "series $$A$$ explains $$75%$$ of the variability in series $$B$$"? Or do I have to qualify that statement by saying "series $$A$$ explains $$75%$$ of the variability in series $$B$$ in this particular sample"? The model is still consistent (p.19), but not unbiased?

## 2 Answers

Without stationarity (and therefore ergodicity) you can't assume that the linear regression is a consistent estimator. Basically your observations are not iid.

A linear regression at that point is just a geometric construct, the line that minimizes squared distance, but it doesn't really have statistical meaning.

I think you need to say "series A explains 75 of the variabilitiy in series B in this particular sample" and if you want to generalize this you need to do cross validation.