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I learned ACF as correlation. For me, this is clearly seen in monthly weather data: where, I see the ACF between consequent months as 'high', meaning july temperature is going to be similar to august temperature. Also seeing high ACF for lag 11,12,13: meaning July 2018 temperature is going to be similar to July or August 2019.

Now, I don't understand what use can PACF bring in... While reading explanations like this one, residuals are often mentioned, and in my mind to have a residual you need to have a fit...

Why residuals are mentioned? and what would be PACF giving me in my 'weather' example I mentioned?

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Think of the ACF as unconditional auto-regression coefficients reflecting the importance of any 1 AND ONLY 1 particular lag. Now think of the PACF as a set of conditional auto-regression coefficients suggesting the importance of any specific lag GIVEN all previous lags.

The whole idea of jointly examining the ACF and the PACF is to tentatively identify the # of required lags in the model AND whether or not previous values are incorporated and/or previous errors in the regression model.

Box and Jenkins should have simply referred to these as unconditional and conditional lags but they decided to introduce ACF and PACF.

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