# Definition of $I(0)$ and its relation to stationarity

I thought that $$I(0)$$ time series were simply stationary time series. But the definition on the Wikipedia page says:

A time series is integrated of order 0 if it admits a moving average representation with: $$\sum_{k=0}^\infty |b_k|^2 < \infty$$ Therefore, all stationary processes are $$I(0)$$, but not all $$I(0)$$ processes are stationary.

Is the definition on the page correct?

The concept of order of integration is not uniquely defined, and there exist multiple definitions that to a large degree overlap. Some of the more hand-wavy definitions rather imprecisely define it as "the number of differences needed to achieve stationarity". Engle and Granger (1987) and Johansen (1995) define the concept more precisely:

A series with no deterministic component which has a stationary, invertible ARMA representation after differencing $$d$$ times is said to be integrated of order $$d$$ (Engle and Granger, 1987)

A stochastic process $$Y_t$$ which satisfies $$Y_t-E(Y_t)=\sum_{i=0}^\infty C_i\epsilon_{t-i}$$ is called $$I(0)$$ if $$\sum_{i=0}^\infty C_iz^i$$ converges for $$|z_i|<1$$ and $$\sum_{i=0}^\infty C_i\neq 0$$. (Johansen, 1995)

One important implication stemming from these definitions is that an MA(1) process with $$\theta=1$$ is not $$I(1)$$ (because it is not invertible, and because $$\sum_{i=0}^\infty C_i=1-1=0$$). So according to these definitions, not all weakly stationary processes are stationary.

The definition on Wikipedia is therefore neither correct nor incorrect; it is merely one definition. For more discussion, see Davidson (2009).

Davidson, James (2009) "When is a time series I(0)?". Chapter 13 of The Methodology and Practice of Econometrics, a festschrift for David F. Hendry edited by Jennifer Castle and Neil Shepherd, Oxford University Press. https://people.exeter.ac.uk/jehd201/WhenisI0.pdf

Engle, Robert & Granger, Clive (1987) Co-Integration and Error Correction: Representation, Estimation, and Testing. Econometrica, 55(2), 251-276. doi:10.2307/1913236

Johansen, Soren (1995) "Likelihood-Based Inference in Cointegrated Vector Autoregressive Models", Oxford University Press. https://academic.oup.com/book/27916

It's correct because of the subtle difference between stationarity and wide-sense stationarity (WSS). All $$I(0)$$ are wide-sense (week-sense) stationary, or in other terms, covariance stationary, as implied in the first sentence of the wiki entry of Order of Integration. This doesn't mean that the series is stationary, (i.e. strongly stationary). But, a stationary series with finite mean and covariance is by definition WSS. The WSS requires mean to be constant, and covariance to depend on only the time difference between the samples. But, the stationarity is a far stronger requirement. You might see that the term stationarity is being used in some sources to refer to WSS for simplicity.

• But, a stationary series is by definition WSS. A counterexample: a series of i.i.d. Cauchy random variables are strictly but not weakly stationary. Also, I think using the terms stationary and strongly/strictly stationary interchangeably is misleading. Neither form of stationarity is "the genuine" or "the true" form. In my understanding, we have two equally important forms: weak and strict. Commented Apr 13, 2023 at 18:07
• Thanks on the Cauchy note. I understand the confusion, but using stationarity and strict/strong stationarity interchangeably is not something I chose to say, but something which is already present in random processes literature (e.g. last wiki link). Commented Apr 13, 2023 at 22:42
• Thank you. I am not persuaded by the Wiki link. I have been dealing with stationarity for quite a while, and my impression on the use of the two terms (based on the many sources I have encountered) is different. Being explicit about the form of stationarity also prevents ambiguity, and I find that useful. Commented Apr 14, 2023 at 7:00
• I think the question should clarify what sort of stationarity it's talking about if it's not strict-sense stationarity. I can't seem to find the referenced sentence in wikipedia. I've mostly seen stationarity = strict stationarity in books about random processes in electronics engineering (which is my background). I vaguely remember one of them was from Leon-Garcia, Probability, Statistics, and Random Processes For Electrical Engineering. It is pretty common to call strict-sense stationarity as stationarity. Commented Apr 14, 2023 at 16:45
• Thank you. I am more familiar with the statistics literature and time series textbooks, papers and lecture notes by statisticians. There it seems quite common to fully spell out which type of stationarity it is. Since this is a statistics site, I think using statistical terminology makes sense (similarly to how we use a sample of [size] 100 instead of 100 samples which would be the common terminology in biology). And since spelling the terms fully also removes ambiguity, I can only see benefits in such a choice. Commented Apr 14, 2023 at 17:31