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What are the differences between VAR (vector auto regression) and MANOVA?

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Strictly speaking VAR has no 'explanatory' variables - everything is assumed to be endogenous. In VAR, a time series of multivariate dependent variables is assumed to be predictable on the basis of its joint past, back a certain number of time steps (the 'lag'). VARX, in contrast, is what a VAR model looks like when it also has a time series of explanatory variables. The X series that run parallel to the multivariate Y is typically just assumed to be exogenous.

Like a VARX model, MANOVA has multivariate dependent variable and also explanatory variables that are assumed to exogenous. However, there is no time series structure assumed between Y variables and therefore no lagged terms in the model.

MANOVA need not always be applied to experimental data, though it often is, and that makes the exogeneity assumption for X plausible. It is, underneath, simply a linear regression model with a multivariate dependent variable. Likewise, VAR is, underneath, a system of multivariate regressions predicting the present of one part of the dependent variable on the basis of its past and the pasts of the other parts of the dependent variable.

This leads to a second difference in practice. Often VAR models assume a diagonal covariance for the dependent variable, which means that model decomposes into a separately estimable sequence of linear regressions, one for each part of the dependent variable. MANOVA is typically applied when there is contemporaneous correlation between elements of the dependent variable that are not explainable by exogenous factors or the past.

Lütkepohl (2005) is a standard (updated) work VAR and related time series models.

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I like to think about the difference this way:

VAR is a system of regressions with lagged dependent variables and some other independent variables observed over time (observational data).

MANOVA is an advanced version of ANOVA, where there are more than one response is being measured (experimental data).

The response or the dependent variable for both is not univariate. It is a vector of dependent variables.

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