# Correlation of nonstationary time series (levels vs differences)

I wonder what the relationship between the empirical sample correlation of two time series in levels and the one of the differenced series is.

I know that for nonstationary variables, it makes little sense to calculate the correlation between them, so one should difference the series to make them stationary and calucate the correlation of the differenced series.

My questions:

How are the "valid" correlation (between 2 differenced stationary series) and the "wrong" correlation (between 2 series in levels) related?

If the "wrong" correlation strengthens over time, does that mean that the "valid" correlation does too, always?

While intuitively, this might make sense (the two variables are coming "closer together"), it confuses me that the correlation between stationary variables should be state-dependent.

• When you talk about correlation changing over time, do you mean the data generating process is changing? (I removed the differences tag as its definition is unclear. I added data-transformation instead.) Note that for integrated processes, theoretical correlation is ill-defined because variance is infinite, even though sample correlation can always be calculated (but it would not be a sensible estimate of the theoretical correlation just because the estimand is ill-defined). – Richard Hardy Aug 19 '17 at 19:25
• Yes! (as far as I understand your question). Basically, the input data change, e.g. corr(Xt,Yt) -> corr(Xt - Xt-1, Yt - Yt-1) – Kuma Aug 19 '17 at 19:32
• So you mean that the pair of random variables $(X_t,Y_t)$ have a different correlation than the pair $(X_{t+1},Y_{t+1})$? I am talking about random variables, not their realization which I would denote $(x_t,y_t)$ and $(x_{t+1},y_{t+1})$, respectively. Or are you talking about realizations and thus empirical correlations that for some reason happen to be higher in some windows within your sample and lower in other windows? (I have to go now, will be back only tomorrow.) – Richard Hardy Aug 19 '17 at 19:34
• I meant empirical correlation, yes! I am sorry, I did not clarify that. I understand that sample correlation is meaningless for two series that are characterized by unit roots. Yet, in empirical research, there are plenty of studies where researchers draw conclusions from calculating the correlation between two integrated series. Hence, I wondered how the sample correlation of differenced, i.e. stationary series is related to the sample correlation of non-stationary series (the one that I see as "wrong") and whether implications from the one carry over to the other. – Kuma Aug 19 '17 at 19:49
• But where does the part of correlation changing over time come? Or should we forget this over time thing and just focus on finding a relationship between correlation between $(x_t,y_t)$ and $(\Delta x_t,\Delta y_t)$? – Richard Hardy Aug 20 '17 at 7:26