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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?

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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.

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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.

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