The partial autocorrelation function (PACF) measures the relationship between a time-series observation and a value k lags earlier after accounting for intervening lags. Unlike the autocorrelation function (ACF), which displays correlation at each lag, PACF helps distinguish lags with a direct relationship to the current observation. Its plot is useful when choosing the autoregressive order p of an ARIMA model. section 7.3.7 section 7.3.8
Use PACF as part of a stationarity and model-selection workflow:
- Separate training and test observations before preprocessing; inspect the training series with ACF and PACF plots. section 7.3.7
- Check stationarity. In the closing-price example, an augmented Dickey–Fuller test returned p = 0.576 for the training series; first-order differencing produced a series with p ≈ 1.25 × 10⁻¹⁶. section 7.3.7
- Compare ACF and PACF before and after differencing. On the differenced series, inspect significant nonzero lags when considering autoregressive terms; treat the resulting order as a candidate to evaluate, not a decision made by a plot alone. section 7.3.7 section 7.3.8
>>> fig, axes = plt.subplots(2, 2) #1
>>> plot_acf(train, ax = axes[0, 0]) #2
>>> axes[0, 0].set_title('ACF - Original') #3
>>> plot_pacf(train, ax = axes[0, 1]) #4
>>> axes[0, 1].set_title('PACF - Original') #5
>>> plot_acf(train_diff, ax = axes[1, 0]) #6
>>> axes[1, 0].set_title('ACF - Differenced') #7
>>> plot_pacf(train_diff, ax = axes[1, 1]) #8
>>> axes[1, 1].set_title('PACF - Differenced') #9
>>> plt.tight_layout() #10
>>> plt.show() #11
This Python listing uses statsmodels to place ACF and PACF plots for the original training series beside those for its first-order difference. Comparing the four plots makes the change in correlation structure visible. section 7.3.7
In a PACF plot, each vertical line is a partial autocorrelation coefficient; the shaded band is a confidence interval. Spikes outside it suggest significant partial autocorrelation. A sharp drop after the first few lags suggests that only those early lags have a substantial direct relationship with the current value. Read lag labels carefully: a lag-0 spike alone does not establish which past observation should enter an autoregressive model. section 7.3.7 section 7.3.8
PACF belongs to a model-identification approach that analyzes a series’ correlation structure. Exponential smoothing takes a different route: it applies exponentially decaying weights to past observations rather than explicitly using ACF and PACF to select model parameters. section 7.4.1
