Quote from lolatency:
You're right. Improved F(X) would just give you a better estimate of the mean in the presence of correlation at various lags. IOW, you would know more readily whether price is trading in a region that could be considered far from the mean. In theory, your system would be more accurate in the presence of stationarity. [Recall, however, that the confidence interval for small n on the prediction of the mean even in ARMA is still ridiculously wide.]
But, again, you're right -- how do you even know if you've got stationarity? This is one of the reasons why they have terms like weak stationarity and strong stationarity. Strong stationarity implies the joint distribution of two disjoint, sequentially time-ordered subsets of a time series are the same. No serious market participant would ever bet on strong stationarity being present in the markets. Weak stationarity just assumes a constant mean and variance at every time-step, which we also KNOW is a bogus assumption.
But to address your question of knowing whether the regime changed, you could mathematically detect such a thing -- but it would take some time for the formal test to register. Whether this time is long enough for you to save yourself depends on your strategy.
I mean, one possible way to figure out whether your regime has changed is to take the two disjoint time-ordered subsets of the same time-series and do a non-parametric comparison of the CDFs of the two subsets. If the CDFs don't match, you've got a different underlying distribution and no longer have stationarity. In theory, you could do this all day long for every tick. You could also use something like Brown-Forsythe or Modified-Levine test on the residuals -- non-parametric tests to assess there is homoscedascity.
The real question is -- what is breaking that stationarity? It's almost certainly the volatility parameter of the stock price process that changes. ARMA models and the like assume homoscedascity. Your regime changes are going to come from the fact that you are almost assured that the volatility of the price process itself changes -- or, what they call like heteroscedascity.
The genius in pairs trading is that two correlated, stochastic processes will evolve in a similar fashion. The pair-hedge accounts for the simultaneous shift in paradigm, so your reversion happens within the context of whatever distribution currently determines the empirically determined probability distribution function, or what you're calling "the regime."
