Can a nonstationary ARMA always be made stationary after differencing?

I was wondering if a nonstationary ARMA can always be made stationary after differencing? The question arises from Metrics' comment on my previous question:

... you need to use ARIMA (which means you need to take the difference if ARMA is non-stationary). – Metrics yesterday

A non-stationary ARMA (2,3) means ARMA is say I(1), then it becomes ARIMA(2,1,3) which means if you difference y one time, then it becomes stationary ARMA. – Metrics yesterday

In the example he gave, why can a nonstationary ARMA(2.3) become ARIMA(2,1,3)? ARIMA is defined to be able to become stationary ARMA after differencing. How do we know a nonstationary ARMA(2.3) can become stationary ARMA after differencing?

Thanks and regards!

• @ Tim: I assume that you are the beginner in applied time series econometrics. So, I recommend you to have a book Applied Econometric Times Series by Walter and Enders. – Metrics Jul 25 '13 at 20:41
• @Metrics: Thanks! Does the book address my question and where in it? – Tim Jul 25 '13 at 21:08
• Yes, it will answer your question.It should be in chapter on ARMA. – Metrics Jul 26 '13 at 11:52
• @Metrics Walter Enders is precisely one author. – Nick Cox Jul 27 '13 at 15:59

A process which is integrated of order one, $I(1)$, is stationary after differencing once. A process which is integrated of order $d$, $I(d)$, is stationary after differencing $d$ times. There are tests for determining the integration order of a time series, like the Dickey-Fuller test or KPSS test. E.g. If you find from a test, that the process is not stationary, you may difference it and run the test on the differenced series to see if it's still non-stationary.

• Thanks! Does your reply mean that a nonstationary ARMA process may not be made stationary by differencing? – Tim Jul 24 '13 at 18:57
• No, not necessarily. Only a $I(1)$ process is stationary after differencing once. A non-stationary process is any $I(d)$ process where $d\neq0$. – fredrikhs Jul 24 '13 at 18:59
• Do you mean any nonstationary process can always be made stationary after some number of differencing? – Tim Jul 24 '13 at 19:02
• @whuber: Thanks! What you and I think is that the time series is continuous-time. But the actual time series is always discrete-time by discretizing the continuous time, and keeping differencing a time series with finite number of time values, will eventually leave only one point. But I am still confused: a nonstationary ARMA process have countably infinite number of points – Tim Jul 24 '13 at 19:59
• Please don't make assertions about what I think, Tim, because they might not be true. In particular, I haven't made any assumptions about what times are possible. You need to clarify what you're asking about, though: if you insist that your time series are actual data sets then, as you say, it's a triviality that $n-1$ differences will reduce them to nothing. But that's not what stationarity is about: it's a theoretical construct and applies, sensu strictu, only to mathematical models of time series, for which we may assume that $t \in \mathbb{N}$ for instance. – whuber Jul 24 '13 at 20:02

Fredrikhs is correct.

You can keep differencing any non-stationary time series until you arrive at a series which is stationary. One downside, however, is that you lose an observation with each difference.

• Thanks! why "you can keep differencing any non-stationary time series until you arrive at a series which is stationary"? What kind of observation do I lose with each difference? – Tim Jul 24 '13 at 19:22
• Shouldn't this be a comment? @Tim: If you have a series of three variables, $t_1$, $t_2$ and $t_3$, when you difference, you lose one. After differencing these three variables you get, $\Delta t_1$ which is $t_1-t_2$, and $\Delta t_2$ which is $t_2-t_3$, can you see that you have lost one observation? – fredrikhs Jul 24 '13 at 19:40
• @fredrikhs: Is the reason "why a nonsationary time series can always made stationary" that we can keep differencing till only one observation is left, and a single observation is stationary trivially? – Tim Jul 24 '13 at 19:53
• Not essentially. By differencing you decrease the integration order of the series by one. Most time series you will encounter is either $I(0)$, $I(1)$ or $I(2)$. But theoretically I guess they can be of even higher integration order. But the point is, the $d$'th difference of a $I(d)$ process is stationary. Except for the case @whuber mentioned. – fredrikhs Jul 24 '13 at 20:00
• Simply put, "stationary" just means that the statistical parameters (mean, variance, kurtosis, etc) don't change with time. You can see why your series would be stationary if you're down to your last observation...the mean of that one observation can't change because there's not other observation! This is an extreme example, but it illustrates the point. Hope that helps. – AOGSTA Jul 24 '13 at 20:00