The Partial Autocorrelation Function (PACF) is a key tool in time series analysis. It measures correlation between observations and their lags, controlling for intermediate lags. This helps identify the order of autoregressive (AR) terms in ARIMA models.

PACF plots display partial autocorrelation values for different lags. Significant spikes suggest including corresponding lags in the AR model. Used with ACF plots, PACF helps determine appropriate ARIMA models for forecasting and analysis.

Partial Autocorrelation Function (PACF)

Partial autocorrelation function definition

• Measures the correlation between an observation and its lag, while controlling for the effects of intermediate lags
  • Removes the influence of shorter lags when calculating the correlation for longer lags (lag 1, lag 2)
• Identifies the extent of the lag in an autoregressive (AR) model
  • Helps determine the order of the AR term in ARIMA models (AR(1), AR(2))
• Calculated as the correlation that results after removing the effect of any correlation due to terms at shorter lags
  • PACF at lag $k$ is the correlation after removing the effect of lags 1, 2, ..., $k-1$

PACF plot interpretation

• Displays the partial autocorrelation values for different lags (lag 1, lag 2, lag 3)
  • Significant spikes suggest the inclusion of the corresponding lag in the AR model
• For an AR($p$) model, the PACF will have significant spikes at lags 1, 2, ..., $p$, and cuts off to zero afterwards
  • The lag at which the PACF cuts off indicates the order of the AR term (AR(1), AR(2))
• Gradual decay in the PACF suggests a higher-order AR term or the presence of a moving average (MA) term
  • Indicates the need for further investigation using the ACF plot

ACF vs PACF comparison

• ACF (Autocorrelation Function) measures the correlation between an observation and its lagged values
  • Helps identify the presence of AR and MA terms in a time series model (ARMA, ARIMA)
• PACF focuses on identifying the order of the AR term in ARIMA models
  • Controls for the influence of shorter lags when calculating correlations
• ACF and PACF plots are used together to determine the appropriate ARIMA model
  • ACF identifies the presence of AR and MA terms (ARMA(1,1), ARIMA(1,1,1))
  • PACF determines the order of the AR term (AR(1), AR(2))

PACF for moving average models

• Examine the PACF plot to identify significant spikes at different lags (lag 1, lag 2, lag 3)
• The lag at which the PACF cuts off or the last significant spike occurs indicates the appropriate order of the AR model
  • Example: PACF with significant spikes at lags 1 and 2, and cuts off afterwards, suggests an AR(2) model
• Gradual decay in the PACF without a clear cut-off suggests considering higher-order AR models or the presence of MA terms
  • Use the ACF plot in conjunction with the PACF to determine the appropriate ARIMA model (ARIMA(1,1,1), ARIMA(2,1,2))