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I need some help in understanding the following:

I have a time series data (y) that I am using to run regression models. However, my dependent variable is log(y).

  1. Should I test for stationarity of y or log(y)?
  2. If say y or log(y) turn out to be non-stationary and I need to make the sure that my dependent variable is stationary, I need to difference it. Should I take first difference of y and then take log(first diff of y) as my dependent variable or take first diff of log(y)?
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  1. If your dependent variable in the regression is log($y$), then do the stationarity testing on log($y$) because typically stationarity requirement arises from statistics (assumption of a statistical model) rather than the subject-matter domain.
  2. First take logs, then difference. This transformation can be interpreted as (or indeed is) log-return of the original variable. It approximately equals percentage returns for small changes in $y$. Also, you cannot difference first and then take logs due to a technical reason; the differences will most likely include some negative values, and logs are defined on positive values only.
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