1 Description of the Data Set

This is a data set of monthly beer production in Australia from January 1956 to December 1994. There are 468 observations with a monthly character variable and a numeric variable of the amount of beer produced. The plot below shows that there is both a seasonal and a positive trend.

2 Different Models

Holt’s smoothing model with both a damped and exponetnial trend an Holt-Winters model: additive (damped) and multiplicative (damped) will be fit to this data to determine the best model for predicition. The last 12 observations will be held off to test the accuracy of the different models.

The accuracy measures of various exponential smoothing models on beer production
ME RMSE MAE MPE MAPE MASE ACF1
SES 0.2726 18.7688 14.7629 -0.9279 10.8139 1.5717 0.0348
Holt Linear 0.0126 18.7702 14.7762 -1.1278 10.8405 1.5731 0.0352
Holt Add. Damped 0.2965 18.7707 14.7801 -0.9012 10.8314 1.5735 0.0348
Holt Exp. Damped 0.3267 18.7708 14.7866 -0.8651 10.8358 1.5742 0.0349
HW Add. -0.1640 9.9049 7.4503 -0.4598 5.4570 0.7932 -0.1110
HW Exp. -0.1260 9.6167 7.2178 -0.4534 5.3164 0.7684 -0.0936
HW Add. Damp 0.5484 9.8199 7.4546 0.1230 5.4793 0.7936 -0.1040
HW Exp. Damp 0.2789 9.5550 7.2152 -0.1562 5.3170 0.7681 -0.0900

None of the regular Holt’s smoothing models did very well. The HW Exponential Damped model appears to be the most appropriate.

From the above accuracy table that HW’s linear trend with an exponential damped seasonal model is the best of the eight smoothing models. This is consistent with the patterns in the original serial plot.

3 Final Model

Since we train the model with the training data and identify the best model using both training and testing data. Both methods yield the same results. To use the model for real-forecast, next we refit the model using the entire data to update the smoothing parameters in the final working model.

The accuracy measures of various exponential smoothing models on beer production
MSE MAPE
SES 1404.07455 18.621575
Holt.Add 1473.46171 18.815220
Holt.Add.Damp 1404.82287 18.624317
Holt.Exp 1405.17689 18.624961
HW.Add 112.56402 5.772278
HW.Exp 112.73479 5.620375
HW.Add.Damp 70.01085 4.641678
HW.Exp.Damp 83.44106 4.639548

From this we see that the HW additive damped model is the best with the HW exponential damped model being a close second. This slightly contradicts the model conclusions above but since they are so close, we will stick with the HW exponential damped model.

Now we will refit the model at the very end using the entire data to update the smoothing parameters in the final working model.

Estimated values of the smoothing parameters in Holt-Winters linear trend with Multiplicative Damped seasonality
x
alpha 0.0718686
beta 0.0089825
gamma 0.0001060

These are the final values of the model.