something like a paried t-test for dependent variables

can anyone recommend a test for the following problems.
1) Say I have repeated measurements \(x_1,...,x_n\) and \(y_1,...,y_n\) such that each measurement may depend on the past. How can I test whether the means in both tests are equal, i.e. mean of \(x_1,...,x_n\) equal to mean of \(y_1,...,y_n\) ? I have learned that the t-test is not applicable in this situation since it allows only for dependence between \(x_i\) and \(y_i\) but not for serial dependence as indicated here. Note that I assume \(x_i\) and \(y_i\) to be independent.

2) Same situation as above but in addition let also \(x_i\) and \(y_i\) be dependent. So searial dependence and inter-goup dependence. Again the means of the groups should be compared.

Btw: Normality can be assumed but I am also happy for non-parametric alternatives.
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Ambassador to the humans
Do you have a specific form for the dependence in mind? Do you think that the dependence structure is the same for the x and the y series?
Thanks for that important question. Indeed the dependence should be equal. But it would take another test to confirm this. Are there alternatives for both cases, as in equal and possibly unequal dependence structure?


TS Contributor
1. The first one seems like a classical time-series problem. If you assume that the random sample inside the group jointly follows a multivariate normal, the only problem left is how to do specify/model the covariance structure among the observations. Since you only have \( n \) observations, you must have a certain assumption on the covariance matrix; otherwise there will be too many parameters.

2. The second one can be substantially complex if you consider cointegrated time series model or other. Again I guess if you want to extend the theory you will need some assumptions here. I am not a time series expert so better seek other advices.


Ambassador to the humans
Well I was just trying to think of how to formalize the model. It would be fairly easy if we're willing to assume a simple AR(1) type structure for the dependence. We wouldn't need to assume the AR parameters are equal for both series but it would be useful to know if we could reasonably assume that both follow some type of AR structure (or any other type of dependence structure - as long as we can give a general form for it)

(And I'm only focusing on the first problem right now)
Not quite. If it was AR(1) I'd have the idea to take differences but this is not the case and the dependence structure might reach futher back into the past. Say it's at least some kind of ARMA(p,q) model with p,q >= 1, or equally complex.