ACF vs. PACF

Explain the difference between ACF and PACF using regression slopes.

Answer

Under weak stationarity (the only setting where ACF and PACF are meaningful), both are simply regression slopes:

  • ACF at lag \(h\): slope \(\beta\) in

    \[ y_t = \alpha + \beta\,y_{t-h} + \varepsilon_t. \]

  • PACF at lag \(h\): slope \(\phi_{hh}\) in

    \[ y_t = \alpha + \phi_{h1}\,y_{t-1} + \dots + \phi_{h,h-1}\,y_{t-h+1} + \phi_{hh}\,y_{t-h} + \varepsilon_t. \]
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