Autocorrelation (ACF/PACF) Plots

Just as correlation measures the extent of a linear relationship between two variables, autocorrelation measures the linear relationship between lagged values of data. A plot of the data vs. the same data at lag 𝑘 may show a positive or negative trend. If the slope is positive, the autocorrelation is positive; if there is a negative slope, the autocorrelation is negative. We will use the lag utility and scatterplots to demonstrate this for 3 lags using the Series A data.

ACF denotes the AutoCorrelation Function plot (sometimes called a Correlogram). PACF denotes the Partial AutoCorrelation Function plot.

We will use the ACF and PACF to get a general idea of what models should be used, but let the automatic algorithms do the heavy lifting of determining which model is optimal. Later, ACF on model residuals are used to assist in determining how well a model fits the autocorrelation structure, ideally with the residuals showing no statistically significant autocorrelation.

We will also examine how differencing results in stationarity with the ACF and PACF plots.

Open Monthly Airline Passengers - Series G.xlsx (Sheet 1 tab). This is the Series G data from Box and Jenkins, monthly total international airline passengers for January 1949 to December 1960.
Click SigmaXL > Time Series Forecasting > Autocorrelation (ACF/PACF) Plots. Ensure that the entire data table is selected. If not, Use Entire Data Table. Click Next.
Select Ln(Airline Passengers), click Numeric Time Series Data (Y) >>. Select Automatic Number of Lags. Specify Seasonal Frequency = 12 and Alpha Level = 0.05.

The automatic number of lags will be (a minimum of) Seasonal Frequency * 3 = 36 allowing us to see seasonal patterns in the ACF.
Click OK. The ACF and PACF plots are produced.

The slow decrease in the ACF as the lags increase is due to the trend, while the “scalloped” shape is due to the seasonality.

We will now compare to the differenced data for Ln(Airline Passengers). Select the Difference Data sheet (or recreate using SigmaXL > Time Series Forecasting > Utilities > Difference Data, with Nonseasonal Differencing (d) = 1 and Seasonal Differencing (D) = 1 with Seasonal Frequency = 12).
Redo the ACF/PACF Plots for the differenced data:

After nonseasonal and seasonal differencing, Lag 1 now shows a negative autocorrelation, but Lag 2 and following have mostly insignificant autocorrelations and partial autocorrelations. Lag 12 is due to the seasonality. This is in agreement with the results in Box and Jenkins (2016, Chapter 9, “Analysis of Seasonal Time Series”, p. 319) who also suggest that, after differencing, the model might be a moving average of order 1 and seasonal moving average of order 1. We will examine this later in ARIMA Forecasting.