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?

**Answer**

I would not start by taking differences of stock prices, normalized for the same initial capital or not. Stock prices do not go below zero, so at best the differences between two stock prices (or accrued difference in initial capital outlay) would only be slightly more normal than the not normal distributions of price (or capital worth) of the stocks taken individually, and, not normal enough to justify a difference analysis.

However, as stock prices are approximately log-normal, I would start normalizing by taking the ratio of the two prices $\frac{\$A}{\$B}$, which obviates having to normalize to initial capital outlay. To be specific, what I am expecting is that stock prices vary as proportional data, that a change from a price of $\$1.00$ to $\$1.05$, discretization aside, is as expected as the change from $\$100.00$ to $\$105.00$. Then, all you have to worry about is whether the ratio of stock prices is increasing or decreasing in time. For that, I would suggest ARIMA or some other trending analysis.

**Attribution***Source : Link , Question Author : lodhb , Answer Author : Carl*