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So I am quite confused about stationarity in ARIMA processes. For example, a Random Walk is an ARIMA process with order (1,0,0). Does this mean that a Random walk is stationary?

Stationarity implies that the covariance matrix is constant, is this the case for a random walk? Are all ARIMA processes stationary (I believe so)?

Thank you!

EDIT: I am just going to leave this link here, because it clarified a wrong assumption I made in the question above regarding the formulation of an random walk.

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2 Answers 2

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First of all random walk is a (0,1,0) ARIMA series. Random walk is a non stationary process. That is why we need this one differentiation. Try it in R plot(cumsum(rnorm(100))) although this plot(diff(cumsum(rnorm(100)))) is a stationary process. I think that you think about white noise as a stationary process (0,0,0).

I am predicting that you thought about Box_Jenkins transformation: https://en.wikipedia.org/wiki/Box%E2%80%93Jenkins_method where is always a step od differencing (,d,) to achieve stationarity. Remember that sometimes one differentiation is not satisfactory to brings it.

Summing up if you have some time series it might be non stationary although by differentiation you could bring it to world of stationary processes. If you want to stay with original time series you could use exponential smoothing theory which not demanding such step. https://en.wikipedia.org/wiki/Exponential_smoothing for example Holt-Winters. However ARIMA covers all space of exponential smooting models.

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  • $\begingroup$ My question was not about how to make a time series series stationary, but rather asking if a random walk is a stationary process, given that it is an ARIMA process (maybe my question was not phrased very clearly I realize now). My confusion arrives from the fact that the mean of a random walk can change or not? Thank you! $\endgroup$
    – user272585
    Jul 24, 2020 at 10:53
  • $\begingroup$ Random walk is a non stationary process. That is why we need this one differentiation. Try it in R plot(cumsum(rnorm(100))) although this plot(diff(cumsum(rnorm(100)))) is stationary. $\endgroup$
    – polkas
    Jul 24, 2020 at 10:59
  • $\begingroup$ Yeah so my confusion is clear now. For some reason I was thinking that an ARIMA process needs to be stationary, but that is not true. An ARMA process is stationary. Thank you! $\endgroup$
    – user272585
    Jul 24, 2020 at 11:02
  • $\begingroup$ I would rather say that ARMA model is made to analyze a stationary processes. stats.stackexchange.com/questions/65193/… $\endgroup$
    – polkas
    Jul 24, 2020 at 11:10
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The I in ARIMA is for Integrated. The difference between ARMA and ARIMA is specifically that an ARIMA process does not have to initially be stationary (although you have to difference it to be stationary).

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