PACFPlots

PACF plots, short for partial autocorrelation function plots, are graphs used in time series analysis to show the direct relationship between observations at different lags. Each value at lag k represents the correlation between X_t and X_{t-k} after removing the influence of intervening observations up to lag k-1.

They are commonly used in ARIMA model identification. For an autoregressive process of order p, the PACF tends to cut off after lag p, whereas for a moving-average process the ACF tends to cut off after lag q. In practice, patterns are approximate and influenced by sample size and noise.

Reading the plot: bars indicate partial autocorrelation estimates; dashed lines show approximate 95% significance bounds. Bars extending beyond the bounds are considered statistically significant, suggesting a direct lag effect not explained by shorter lags. The sign indicates positive or negative association.

Computation and usage: PACF values are obtained by regressing X_t on X_{t-1}...X_{t-k} for each k, or by fitting an autoregressive model. Software implementations include Python's statsmodels.plot_pacf and R's pacf function. Before plotting, series should be stationary; differencing or deseasonalizing may be required.

Limitations: PACF interpretation depends on stationarity and data quality. Outliers and seasonal patterns can distort the plot, and cutoffs are heuristic in finite samples. It is usually considered alongside the ACF and model diagnostics.