pacf

Partial autocorrelation function (PACF) is a tool in time series analysis that measures the direct association between observations separated by k time steps after removing the influence of the intervening observations. For a stationary process X_t, the partial autocorrelation at lag k is the coefficient on X_{t−k} in the linear regression of X_t on X_{t−1}, ..., X_{t−k}. Equivalently, it is the correlation between X_t and X_{t−k} after accounting for the linear effects of X_{t−1} through X_{t−k+1}.

Computation and interpretation: The PACF can be obtained from the Durbin–Levinson algorithm or by fitting an autoregression of order k and taking the coefficient on X_{t−k}. In practice, software returns sample PACF values with confidence intervals under a white-noise assumption. The PACF is often plotted alongside the ACF (autocorrelation function) to help identify appropriate model structure.

Role in model identification: In ARIMA modeling, the PACF is used to inform the order of the autoregressive part. For an ideal AR(p) process, the PACF exhibits a sharp cutoff after lag p, being nonzero up to p and effectively zero thereafter, while the ACF does not cut off. For an MA(q) process, the ACF cuts off after q while the PACF decays gradually. For ARMA models, neither function typically shows a perfect cutoff, but patterns in the PACF and ACF guides model selection and diagnostics.

Notes: Interpretation assumes stationarity; nonstationarity, seasonality, or outliers can distort PACF estimates. Modern tools routinely provide standard errors and confidence bands for the estimated PACF values.