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$R^2$ measures explained variance. In an autoregressive model like AR(k), we are carrying out a linear regression, and as such we would have an $R^2$ and an adjusted $R^2$. Why are they not used in practice?

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IMHO, using the R2 is irrelevant since it would just push you to use a larger regression order $k$ which would generally give you a smaller R2. The idea of fitting an AR (or any GLP) is to reproduce the underlying process with a model that is as simple as possible (since the idea is also to extract meaning out of the different coefficients)

This is why people generally look at information criterion such as the BIC or the AIC that englobe a penalty for the number of parameters in the model with a goodness of fit based on the likelihood of the fitted parameters (and hence of the model).

Now I guess you could consider the adjusted R2 but it would be somehow less general which I guess is the reason the AIC, BIC and other similar IC are popular.

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