Is the absolute value of a stationary series also stationary?

I know that linear transformations of time series arising from (weakly) stationary processes are also stationary. Is this true, however, for a transformation of a series via taking the absolute value of each element as well? In other words, if {xi,iN} is stationary, then is {|xi|,iN} stationary as well?

Answer

In one particular case this is somewhat true:

If your time series is stationary with normally distributed error, then the absolute values of your original time series follow a stationary folded normal distribution. Since even weak stationarity means both the mean and variance are constant over time, the absolute values will also be stationary. For other distributions this means that the absolute values of the original time series are at least weakly stationary, as constant variance of the original values translates to a constant mean of the new values.

However, if your original time series only has a constant mean, the variance may change over time, which will affect the mean of the absolute values. Hence, the absolute values will not be (weakly) stationary themselves.

A more general answer would require some study of the moment generating function of the absolute value of a random variable. Perhaps someone with more mathematical background can answer that.

Attribution
Source : Link , Question Author : Arthur Campello , Answer Author : Frans Rodenburg

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