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What is the exact difference between an autoregressive (AR) and vector autoregressive model (VAR)?

I always thought that VAR would just be for more than two variables, until I learned that AR can also have more than two variables. I.e. I had a misunderstanding(?) that AR models are not always bivariate.

Before I write too much potentially false assumptions about the difference between AR and VAR, it would rather be interested to see if anyone can make provide a quick and good summary.

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    $\begingroup$ Isn't a VAR simply an AR with vectors? $\endgroup$ – Matthew Gunn Oct 24 '16 at 12:10
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VAR (vector autoregression) is a generalization of AR (autoregressive model) for multiple time series, identifying the linear relationship between them. The AR can be seen as a particular case of VAR for only one serie.

The hypothesis necessary to apply the VAR is the series is just that one influence other in a intertemporal way.

About AR with more than one serie, you may have seen something like AR models of different series that have any relationship between them, as for example simultaneous equations, where there is some equilibrium condition.

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I think we can distinguish these types of models:

We can conclude that:

  • AR(1) is a simple linear regression because it is linear in parameters, has one variable and one equation. Please note that $y_t$ is different from $y_{t-1}$, although we might assume stationarity or etc.

  • AR(p) for p>1 is in fact a multiple linear regression model, because it has one equation and more than one variables ($y_t,\; y_{t-1},\; ...,\; y_{t-p}$). It might have other exogenous variables.

  • VAR(p) model is a multivariate linear regression model. It has more than one equation and variable. It might also have different exogenous variables.

I am open to comments.

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