# Testing for statistically significant difference in time series?

I have the time series of the prices of two securities, A and B, over the same period of time and sampled at the same frequency. I would like to test whether there is any statistically significant difference over time between the two prices (my null hypothesis would be that the difference is null). Specifically, I am using price differences as a proxy for market efficiency. Imagine A and B are a security and its synthetic equivalent (i.e. both are claims to exactly the same cash flows). If the market is efficient, both should have the exact same price (barring different transaction costs, etc.), or a zero price difference. This is what I would like to test for. What is the best way to do so?

I might have intuitively run a two-sided t-test on the "difference" time series, i.e. on the A-B time series, and tested for $\mu_0$=0. However, I have the suspicion that there might be more robust tests, that take into account things like potential homoskedastic errors or the presence of outliers. In general, are there things to watch out for when working with the prices of securities?

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I think to make this question answerable, we need a model of some type. In particular, what does it mean to ask if "there is a statistically significant difference over time between the two prices" unless there is some noise in observing the prices? There is no parameter here and no randomness. Perhaps you are wanting to make some assumption about some parameter of the price process over time. A "standard" formulation might look at the log-returns process $R_t = \log(X_t/X_{t-1})$ and assume that these are iid normal. (cont.) –  cardinal Apr 6 '12 at 19:01
(cont.) Then, one might want to test if the mean returns between the two processes are equal. But, that's getting a bit ahead of ourselves, perhaps, and also fixes rather strong (and, often, empirically false) assumptions on the price process. –  cardinal Apr 6 '12 at 19:02
@cardinal: I want to test the existence of ANY arbitrage strategy, to test for market efficiency. H0: market is efficient, therefore one is not able to make riskless profit with no investment of cash, using any imaginable strategy. –  lodhb Apr 7 '12 at 16:52
lodhb, that is interesting in that I did not interpret your question at all as having that as the main interest. This makes me think (i) the answer you have accepted has almost nothing to do with your comment, (ii) I'm not sure that @naught101, who has offered a bounty on your question, has read this as your intent and (iii) if this is really what you are looking to test, you might strongly consider updating your question to reflect this, though it might put naught101 in a bit of an awkward spot. –  cardinal Apr 7 '12 at 18:24
Doesn't bother me if the question changes. That's part of the risk of offering a bounty on someone else's question. Go for it. –  naught101 Apr 8 '12 at 2:29
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