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I want to estimate a VAR-model with 6 variables, all of them are stationary. But when I analyse the time series by examining ACF, PACF and auto.arima in R. auto.arima confirms that two of the time series are ARMA(2,2), two AR(2) and two AR(1). Now I wonder if estimating a VAR-model is the correct method in this case? Are there any objections?

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Any method being the correct method is a strong claim that is hard if not impossible to prove in practice. Yet you can seek methods and models that are useful for your goals and justifiable from a practical perspective.*

Time series can be modelled using univariate models such as ARMA or multivariate models such as VAR. Depending on what you are interested in and what information you can condition upon, different models will be appropriate.

E.g. if at time $t$ you can only observe $x_{1,t}, x_{1,t-1}, \dots$ and you are interested in predicting $x_{1,t+1}$, then a univariate model such as ARMA is a natural choice. It models $x_{1,t+1}$ as a function of $x_{1,t}, x_{1,t-1}, \dots$.

If, on the other hand, you can observe $x_{1,t}, x_{1,t-1}, \dots, x_{2,t}, x_{2,t-1}, \dots, x_{6,t}, x_{6,t-1}, \dots$, then a multivariate model such as VAR is a natural choice. It models $x_{1,t+1}$ (and $x_{2,t+1}, \dots, x_{6,t+1}$) as a function of this larger set of values.

If you have the larger set of data, it should not be surprising that both a univariate and a multivariate model can be fit. There is no contradiction in that. Rather, you just have to make a choice of what is relevant for you.

*Your title says your data are ARMA or AR. This is unlikely to be exactly true unless you have generated the data yourself. More likely these are approximations. Nor is auto.arima giving you the model that has generated the data; it gives you an approximation that is expected to perform well out of sample (where well is to be understood in relation to the other models that were on auto.arima's search path).

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