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I'm currently looking at a paper involving panel data and intend to run a similar test on a different data set. However, I'm having a difficult time understanding exactly what test the authors performed to get their results. I have 12 years of panel data and want to compare the average of the first X years to the average of the last X years for each individual in the panel. I then plan on summarizing the data to show how many individuals fell above/below the relevant test statistic. The authors simply said they used a dependent t-test, but could anyone provide more detail on what test is appropriate here and what the relevant test statistic would be to compare against?

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Any introductory text book to statistical data analysis is a good starting point here (see e.g. Mathematical Statistics and Data Analysis by John Rice).

A dependent (sometimes also called paired) t-test seems to be appropriate here, as the pairs can consist of a before-after comparison, as in your case. The relevant test statistic is the t statistic, which follows a t distribution. So you could test the null hypothesis that there are no differences between both means.

Note however that the t-test is a parametric test, as it assumes that the observed differences between the mean of the first $x$ years and the last $x$ years are a sample from normal distribution. The approximate validity of the confidence interval and hpypothesis test follows from the Central Limit Theorem. If the sample size is small and/or the true distribution of the differences is far from normal, the test may not be very reliable.

A non-parametric alternative would be (Wilcoxon) signed rank test. This test does not depent on the normality assumption and since differences are replaced by ranks it is insensitive to outliers. It is also almost as powerful as the t-test, even when the normality assumption holds and thus may be a good second check in your case.

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