Example of Univariate Box-Jenkins Analysis

Series F
We analyze the series F data set in Box, Jenkins, and Reinsel, 1994. A plot of the 70 raw data points is shown below.

The data do not appear to have a seasonal component or a noticeable trend. (The stationarity of the series was verified by fitting a straight line to the data versus time period. The slope was not found to be significantly different from zero (p-value = 0.2).)

Model Identification
We compute the autocorrelation function (ACF) of the data for the first 35 lags to determine the type of model to fit to the data. We list the numeric results and plot the ACF (along with 95 % confidence limits) versus the lag number.

The ACF values alternate in sign and decay quickly after lag 2, indicating that an AR(2) model is appropriate for the data.

Model Fitting

We fit an AR(2) model to the data.
Xt=δ+ϕ1X(t−1)+ϕ2X(t−2)+At

The model fitting results are shown below.

δ = 51.1286
Residual standard deviation = 10.9599

Test randomness of residuals:
Standardized Runs Statistic Z = 0.4887, p-value = 0.625

Forecasting

Using our AR(2) model, we forcast values six time periods into the future.

The “historical” data and forecasted values (with 90 % confidence limits) are shown in the graph below.