How to regress a stationary variable on a non stationary variable? Answer II

Demetris Koutsoyiannis responds:

I think that such questions should not be treated in an algorithmic manner and that it is important to formulate them in the clearest and most consistent manner.

So, let us assume that we have a nonstationary stochastic process X(t) and a stationary process Y(t); I have interpreted here “variable” as process because the notion of stationarity/nonstationarity is related to a (stochastic) process, not a variable. Is the question, how to establish a regression relationship between Y(t) and X(t)? For instance a relationship of the form Y(t) = a(t) X(t) + V(t), where a(t) is a deterministic function of time and V(t) a process independent or uncorrelated to X(t)? Without going into detailed analysis it seems to me that in such a relationship it is difficult to have a constant a(t) = a, i.e. independent of time. Also, V(t) should be nonstationary too. So, while we can consider a time series (observations) of the stationary Y(t) as a statistical sample, we cannot do the same for X(t) or V(t). So, I doubt if there is a statistical procedure to infer a(t) and the statistical properties of V(t) (mean, variance, etc.) which are functions of time too. In addition, I do not find such a relationship useful at all.

What I would propose is to construct a derivative process from X(t) which should be stationary. For example, if X(t) is a stationary process with stationary intervals, then we could construct Z(t) = X(t) – X(t – d) for some d and make the regression between Y(t) and Z(t), which are both stationary.

Here I would like to summarize my opinion that the notions of stationarity/nonstationarity are not statistical but theoretical ones. I mean that stationarity and nonstationarity are not properties of observed time series (series of numbers). In this respect, some widely used procedures to statistically detect stationarity in a time series are fundamentally incorrect. Stationarity and nonstationarity are properties of processes, i.e. mathematical models that we have devised to describe the time series. (More on this in my 2006 paper “Nonstationarity vs. scaling in hydrology”, Journal of Hydrology). And since models are our constructions (as opposed to time series which are constructed by nature for instance – here our role is just to observe) we know a priori if they are stationary or nonstationary. And if they are nonstationary, we always know how to construct a derivative process so as to be stationary.

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0 thoughts on “How to regress a stationary variable on a non stationary variable? Answer II

  1. Regression on errors, as in an ARIMA multi-dimension process ? There is more to be gained by further processing the residuals ? How do the, almost bell curve, errors drift ?

  2. Regression on errors, as in an ARIMA multi-dimension process ? There is more to be gained by further processing the residuals ? How do the, almost bell curve, errors drift ?

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