Comparing the means of two time series I have data sets of the returns of two indexes in the same market (two different sets of stocks constituting each index), with 496 observations for each. I want to compare if the means are statistically different. I believe the variances are different, so I think I have to check if the variances are statistically different first. How would I do these things?
 A: The t test is primarily employed for data that is paired ( e.g. N independent pairs of readings before and after some activity) or for N independent readings on two characteristics (indices). You don't have independent observations since you have time series data that is most probably auto-correlated (within structure). The cross-correlation coefficient (among structure) also requires independent (within structure) draws as there is a requirement for joint normality which requires statistical independence of the draws. Again time series data is by it's very nature not usually independent. Please see "Why Do We Sometimes Get Nonsense Correlations between Time-series?" (1926), an investigation of a form of spurious correlation, in http://en.wikipedia.org/wiki/Udny_Yule AND http://empslocal.ex.ac.uk/people/staff/dbs202/cat/stats/corr.html for more. The best way to determine the relationship between two time series is to review How to identify transfer functions in a time series regression forecasting model?.
A: IrishStat makes a good point.  If your data are two time series you have to take correaltion into account when comparing means.  But the time series modeling is even more important because if the series are nonstationary because of a time-changing mean it may not make sense to just compare the averages over the length of the series. How the indices change with time would be more likely to be what you are interested in.
