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While studying for a time series paper I came across the terms 'partial autocorrelation function' (PACF) and 'autocorrelation function' (ACF) in conjugation to $AR$ and $MA$ processes, why is it such that PACF cuts down for $AR$ process and not for $MA$ process?

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In AR(p) models, PACF cuts off after lag $p$. Assuming invertibility, an MA model of some finite order corresponds to AR($\infty$), so, its PACF doesn’t cut off.

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