Parameters estimation and diagnostic checking

4.3.3.1 Classic

The Expert Modeler in SPSS gives us the following parameters estimation for the Classic model ARIMA(0, 0, 1)(1, 0, 0):

Table 3. Model parameters for the Classic chairs.

All of the five predictors turned out to be significant in the model. In addition, the model contains a constant.

Table 4. Model statistics for the Classic chairs.

Three outliers are detected and accounted for. Those are two additive outliers; one in year 2013 week 48 and one in year 2014 week 28, and a local trend in year 2013 week 6. We see from the model statistics that the P-value from Ljung-Box test is 0,310. The P-value is larger than 0,05 and we fail to reject H0, that the residuals are random, with 95%

confidence. This means that the model is a good fit. Though, the MAPE-value indicates are 25% error in the forecasting values. The actual values in the data are not very small, so that is probably not the reason why the error is a bit large. This has to be taken into

account when using the forecasted values. The stationary R-square value is the upper layer, with a value of 0,771. This also indicates that the model is a good fit.

Figure 24. ACF and PACF residuals from the model for Classic chairs.

Figure 25. Noise residuals from the model of Classic chairs.

The residuals show the differences between expected and actual values for the model.

When we look at the remaining residuals in the ACF and PACF we can see from lag 27 that there is a very small amount of autocorrelation left. This suggests that the model still can be improved, but the amount is so small (-0,146) that it looks like the model has captured the patterns in the data very well. From the graph of the residuals, there are some peaks that should have been accounted for in order to make the model even better. How well this model fits is illustrated in figure 26. We can see that the fitted line is very close to the observed values.

4.3.3.2 City Starbase

The Expert Modeler in SPSS gives us the following parameters estimation for the City Starbase model ARIMA(0, 0, 0):

Table 5. Model parameters for the City Starbase chairs.

All of the five predictors turned out to be highly significant in the model. In addition, the model contains a constant.

Table 6. Model statistics for the City Starbase chairs.

Three outliers are detected and accounted for in the model; two transient outliers in year 2014 (week 2 and 44) and one additive outlier in year 2015 (week 32). We see from Ljung-Box test that the P-value (0,791) is larger than 0,05, so we fail to reject H0. This

means that the residuals are random, with 95% confidence, and that the model is a good fit.

Though, the MAPE-value indicates are 75% error in the forecasting values. I conclude that MAPE is not a good estimate of fit to this model, since the actual data set contains zero values, which gives a high MAPE. The stationary R-square value is the upper layer, with a value of 0,731. This also indicates that the model is a good fit.

Figure 27. Residual ACF and PACF from the model of City Starbase chairs.

Figure 28. Noise residuals from the model of City Starbase chairs.

Figure 29. Plots of the actual sales versus the fitted values and forecasted values from the model of City Starbase chairs.

Since this model do not contain any AR or MA terms, we know that the residuals are just deviations from the mean. The model is a pretty good fit, as we can see from figure 29.

The lowest points are fitted, but many of the peaks are not accounted for. In addition, from the graph of noise residuals it looks like there are two outliers that are not detected in year 2014 (around week 22 and 28). It is reasonable to say that the model can be improved.

4.3.3.3 Signature

The Expert Modeler in SPSS gives us the following parameters estimation for the Signature model ARIMA(1, 0, 0):

Table 7. Model parameters for the Signature chairs.

Only four of the predictors turned out to be significant in this model. Week 1 was not significant and hence, not included. In addition, the model does not contain any constant.

Table 8. Model statistics for the Signature chairs.

Four outliers are detected and accounted for in the model. Those are three additive outliers in year 2015 (week 28, 32 and 49), and one transient outlier in year 2015 week 10.

We see from Ljung-Box test that the P-value (0,755) is larger than 0,05, so we fail to reject H0. This means that the residuals are random, with 95% confidence, and that the model is a good fit. Though, the MAPE-value indicates a 211% error in the forecasting values. I conclude that MAPE is not a good estimate of fit to this model as well, since the actual data set contains zero values, which gives a high MAPE. In addition, the actual data values are quite low and contribute to the high MAPE value. The stationary R-square value is the upper layer, with a value of 0,875. This indicates that the model is an excellent fit.

Figure 30. ACF and PACF residuals from the model of Signature chairs.

Figure 31. Noise residuals from the model of Signature chairs.

Figure 32. Observed values versus fitting values, and forecasted values from the model of Signature chairs.

When we look at the remaining residuals in the ACF and PACF it looks like the model has captured the patterns in the data very well. How well this model fits is illustrated in figure 32. We can see that the fitted line is very close to the observed values, and that all peaks are accounted for. In conclusion, the model is a very good fit to the data.