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It is easy enough to detect interdependence of two discrete-time white noise sequences - one just takes cross-covariance and compares it with variance. But what if both time series are "non-white", that is, there's long-range autocorrelation (like two Brownian random walks)? I assume one resorts to some kind of whitening, where one can read more about this?

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  • $\begingroup$ "Long-range autocorrelation" does not necessarily imply non-stationarity. Is therefore your question about two stationary, or two non-stationary, autocorrelated processes? $\endgroup$ Commented Oct 2, 2013 at 16:40
  • $\begingroup$ Yes, you're quite correct, - I meant that in the first case, subsequent samples are uncorrelated. $\endgroup$ Commented Oct 2, 2013 at 19:33
  • $\begingroup$ Welcome to the site @tristes (+1). Just curious: Are there three of you? ;-) $\endgroup$
    – cardinal
    Commented Oct 3, 2013 at 12:07
  • $\begingroup$ No wheat around, so far as I can see :-) $\endgroup$ Commented Oct 4, 2013 at 12:18

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When examining the possible cross-correlation between two white noises, we can separate and investigate distinctly whether they are "contemporaneously correlated" and/or whether they are "non-contemporaneously correlated". This is a useful distinction -knowing whether a variable "leads" or "lags" another, contrasted to them moving contemporaneously is important, especially for forecasting.
Now if we have two auto-correlated, but still covariance-stationary processes, then, as I have recently answered here, any continuous transformation/combination of them will be also stationary, and so the sample moments(time-averages) will be strongly consistent estimators of the ensemble moments(expected values):

$$\frac 1T\sum_{t=1}^{T}(X_t-\mu_x)(Y_t-\mu_y) \rightarrow_{a.s} E(X_t-\mu_x) (Y_t-\mu_y) = {Cov} (X_t,Y_t)$$

In other words, we do not need to "whiten" the processes to test for cross-correlation between them.

The problem we have here is that we cannot distinguish between contemporaneous and non-contemporaneous cross-correlation, since the two variables are functions each of their own past, but also of a current shock. To take the standard example, assume that both processes are zero-mean $AR(1)$. Then since they are both stationary, they have an $MA(\infty)$ representation, so

$${Cov} (X_t,Y_t) = E\Big (\psi(L)\varepsilon_t\cdot \phi(L)u_t\Big)$$

where $\varepsilon_t,\;u_t$ are the white noise building blocks of $X_t$ and $Y_t$ and $\psi(L),\; \phi(L)$ are polynomials in the lag operator. So this covariance, apparently between the contemporaneous realizations of the two processess, will include the term (coefficient of the lag polynomials ommited) $E(\varepsilon_tu_t)$ but also products of all lags and leads like $E(\varepsilon_tu_{t-1})$, $E(\varepsilon_{t-5}u_{t-2})$ etc, and it can be non-zero because any one of them is not zero. And the same will happen if we try ${Cov} (X_t,Y_s),\; t>s$. For applied forecasting matters, we can of course compare the actual values of, say, $\hat {Cov} (X_t,Y_t)$ and $\hat {Cov} (X_t,Y_{t-1})$ and if it so happens that the second is visibly larger than the first we could say that "$Y$ leads $X$", and use this relation for our purposes - but the underlying confounding of contemporaneous and non-contemporaneous cross-correlation does not go away.

ADDENDUM: A pre-whitening algorithm in the time domain (As given in Søren Bisgaard & Murat Kulahci (2011) book "TIME SERIES ANALYSIS AND FORECASTING BY EXAMPLE", chapter 8, p. 208)
The chapter is about "Transfer Function Models"

Step 1: Fit an ARIMA model to the input $X_t$

Step 2: Prewhiten the input series $X_t$ (i.e., compute the residuals for $X_t$)

Step 3: Prewhiten the output series $Y_t$ using the same model fitted to the input $X_t$

Step 4: Compute the cross correlation between the prewhitened $X_t$ ’s and $Y_t$ ’s

Note: you can also look up prewhitening using spectral analysis.

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  • $\begingroup$ We don't need to whiten processes to calculate their cross-correlation, - OK, maybe, but so computed cross-correlation no longer serves as a useful indicator of the presence of common component in those time series. $\endgroup$ Commented Oct 3, 2013 at 11:32
  • $\begingroup$ Why? It identifies common components alright - and as such it is useful in forecasting, as I wrote. I am also adding an algorithm for prewhitening in my answer, since it is of interest to you. $\endgroup$ Commented Oct 3, 2013 at 14:57
  • $\begingroup$ Brownian process is AR(1) with $\varphi=1$, and is not wide-sense stationary. Cross-correlation criterium does not work for detecting interdependence of two browninan walks. But if you compute cross-correlation of the difference between two successive values, then you easily detect interdependence. That is the kind of "whitening" I have in mind. $\endgroup$ Commented Oct 3, 2013 at 15:12
  • $\begingroup$ ...which refers to non-stationary processes, for which actually, "whiteninig" may be simpler, in the case the processes are Integrated (and they need to be also co-integrated). $\endgroup$ Commented Oct 3, 2013 at 15:37
  • $\begingroup$ OK, the reference in your answer update is the kind of reference I am looking for. Thank you. $\endgroup$ Commented Oct 3, 2013 at 16:15

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