# If a DGP is ARIMA(P,D,0) does that imply no other variables affect this process?

Im currently working with some time series data and have ran a number of tests, presuming the need of a VAR or VECM for an optimal forecast.

However, upon going back to basics I noticed that the appropriate univariate models for the variables interested in are ARIMA(1,2,0) and ARIMA(1,1,0).

The reason why this is of interest to me is because there is no MA component in each of these models, only the process's previous values are influencing the determination of the DGP.

If a DGP is ARIMA(P,D,0) does that imply no other external variables affect this process?

• Can you explain why "no MA component" would suggest that no other variable is influencing the process? Either way, the answer is no: the fact that a process has a certain marginal distribution has no bearing on its joint distribution with any other process. Take two white noise processes with errors highly correlated at the same time across series. They are both ARIMA(0,0,0) but they are highly related. – Chris Haug Dec 1 '17 at 19:30
• @ChrisHaug edited the question – EconJohn Dec 1 '17 at 21:09
• @ChrisHaug Can you elaborate on how the white noise process, rejects the idea I suggested? i.e how an MA component suggests that there are other variables at play. – EconJohn Dec 3 '17 at 4:51

If we interpret your claim as "if an ARIMA process has no MA part, then it is independent of every other stochastic process", the answer is decidedly no. As I said above, just take two correlated white noise processes:

$$Y_t = \eta_t$$ $$X_t = \varepsilon_t$$ $$\text{corr}(\eta_t,\varepsilon_t) = \rho$$

The two processes will be more or less correlated depending on $\rho$. So, you have an ARIMA(0,0,0) process $Y_t$, but there exists another process related to it, $X_t$. Fundamentally, if you observe only $Y_t$, it is impossible to reject the existence of a related process $X_t$.

However, it sounds like your real question is closer to: "if I have found evidence that, marginally, my time series are well-modeled by an ARIMA process with no MA part, should I not bother using VAR/VECM?". The answer is also no, for the following reasons:

1. As I said above, there is no theoretical basis for this.
2. Since your goal is forecasting, you should be looking at out-of-sample forecast performance, not in-sample goodness of fit.
3. Most importantly, the fact that a univariate model fits a specific series well is a fact about its own distribution only. It says absolutely nothing about its relationship to other series.

you say " no MA component in each of these models, which would imply that some other variable is influencing this process." I say that a all ar structures can be re-presented as ma structures thus there is no magic (i.e. suggesting the proof that lurking/external/exogenous variables exist ) and have to be found. Where is your citation that an ma components suggests an unused series ? All ARIMA structure suggests that ....