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I hope everyone is doing well. I have two time series data sets (Time Series 1) and (Time Series 2). These two series are produced monthly (from 1-2020 to 12-2020) 12 data points. Thus, I took the difference (diff = Time Series 1 - Time Series 2) at each month. I would like to test whether the mean of this difference (diff data set) equals zero - a test that is similar to paired t-test. What is a valid statistical test in this case? I didn't think paired t-test is valid here because it is a time series. I include the plots to illustrate the data set.

Other posts suggested fitting ARIMA model. I don't think it's valid because I only got 12 data points.

enter image description here

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  • $\begingroup$ If you were to test the mean of the differences then you would be ignoring any time-related signal. Is that really what you are interested in? If so then the best approach will depend on the nature of the data and so you need to say what they are in some more detail. $\endgroup$ Commented Jul 11 at 21:16

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Perhaps you should try testing for stationarity of the difference time series. If that is the case, then proceed to fit a model to the difference, e.g. AR(p), and examine the test results for the constant term accordingly. In other words, if you can establish that the difference is a well behaved mean-zero stochastic process, then you would have want you want.

Although in this case, with such a small sample size, such analysis has little statistical power. I would recommend a resampling based approach for this situation.

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Time series 1 is always below Time series 2, so the difference should be negative. Is the gray line (diff) measured on a different scale?

Your time series is made of just 12 data points and can be thought of as a vector in $R^{12}$. If you knew or could estimate a (common) covariance matrix of the distributions generating these vectors, something similar to a Hotelling's $t$-test might be used. With just the information given, the problem does not seem solvable.

Let me just add that if the series were each white noise from a common distribution and not cross correlated, the probability of one being above the other on 12 consecutive months would just be $1/2^{12} \approx 0.000244$.

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