Example of ARIMA
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The ACF and PACF of the food employment data (see Example of autocorrelation and Example of partial autocorrelation) suggest an autoregressive model of order 1, or AR(1), after taking a difference of order 12. You fit that model here, examine diagnostic plots, and examine the goodness of fit. To take a seasonal difference of order 12, you specify the seasonal period to be 12, and the order of the difference to be 1. In the subsequent example, you perform forecasting.

1    Open the worksheet EMPLOY.MTW.

2    Choose Stat > Time Series > ARIMA.

3    In Series, enter Food.

4    Check Fit seasonal model. In Period, enter 12. Under Nonseasonal, enter 1 in Autoregressive. Under Seasonal, enter 1 in Difference.

5    Click Graphs. Check ACF of residuals and PACF of residuals.

6    Click OK in each dialog box.

Session window output

ARIMA Model: Food

 

 

Estimates at each iteration

 

Iteration      SSE   Parameters

        0  95.2343  0.100  0.847

        1  77.5568  0.250  0.702

        2  64.5317  0.400  0.556

        3  56.1578  0.550  0.410

        4  52.4345  0.700  0.261

        5  52.2226  0.733  0.216

        6  52.2100  0.741  0.203

        7  52.2092  0.743  0.201

        8  52.2092  0.743  0.200

        9  52.2092  0.743  0.200

 

Relative change in each estimate less than 0.0010

 

 

Final Estimates of Parameters

 

Type        Coef  SE Coef     T      P

AR   1    0.7434   0.1001  7.42  0.000

Constant  0.1996   0.1520  1.31  0.196

 

 

Differencing: 0 regular, 1 seasonal of order 12

Number of observations:  Original series 60, after differencing 48

Residuals:    SS =  51.0364 (backforecasts excluded)

              MS =  1.1095  DF = 46

 

 

Modified Box-Pierce (Ljung-Box) Chi-Square statistic

 

Lag            12     24     36  48

Chi-Square   11.3   19.1   27.7   *

DF             10     22     34   *

P-Value     0.338  0.641  0.768   *

Graph window output

 

Interpreting the results

The ARIMA model converged after nine iterations. The AR(1) parameter had a t-value of 7.42. As a rule of thumb, you can consider values over two as indicating that the associated parameter can be judged as significantly different from zero. The MSE (1.1095) can be used to compare fits of different ARIMA models.

The Ljung-Box statistics give nonsignificant p-values, indicating that the residuals appeared to uncorrelated. The ACF and PACF of the residuals corroborate this. You assume that the spikes in the ACF and PACF at lag 9 are the result of random events. The AR(1) model appears to fit well so you use it to forecast employment in the Example of Forecasting with ARIMA.