Chapter 6: Results of the HWES and ARIMA forecasting models
6.2 Holt Winter's Exponential Smoothing (HWES)
6.2.2 Results and discussion of the application of the HWES model
6.2.2.3 Predictions based on the superior HWES Models
Daily NZX 50 Index data from 2009 to 2017 is used for sample period one, having 2,173 observations in total. I have used 1,521 observations from sample period one for algorithm training, and the remainder (652) is used for testing. Sample period two is from 2007 to 2017, consisting of 2,672 total observations, of which 1,870 are used for model training, and the rest (802) is used for validation purposes. Sample period three contains daily NZX 50 Index data from 2007 to 2020, including the COVID-19 pandemic data. This data set consists of 3,401 total observations, of which 2,381 are used for model training, and the balance of 1,020 is used for testing purposes.
Both reformulated HWES models are fitted for each sample period tested, and the prediction results are presented in the appendices in the following order. Table B.1 and Table B.4 in Appendix B show the forecast results of HWEM Model 1 and HWES Model 2, derived from their application to the NZX 50 Index from 2009 to 2017 (sample period one). Table C.1 and Table C.4 in Appendix C represent the prediction results of HWES Model 1 and HWES Model 2 derived from HWES models’ application to the NZX 50 Index from 2007 to 2017 (sample period two). Similarly, Table D.1 and Table D.4 in Appendix D provide the prediction results of HWES Model 1 and HWES Model 2 applied to NZX 50 Index from 2007 to 2020 (sample period three). Upon examining the actual and predicted series in Tables B.1, B.4, C.1, C.4, D.1, and D.4, it is evident that both HWES models are efficient in predicting the NZX 50 Index. However, a closer inspection of the actual and predicted series reveals that HWES Model 1 is a more effective model for Period 1 (2009 – 2017) and Period 3 (2007 – 2020) as the deviations of actual values from the predictions (residuals) of the HWES Model 1 are marginally lower than the residuals of HWES Model 2. These findings are consistent with my conclusions based on the MAE, MAPE, and RMSE comparisons presented in Table 6.4.
Figures 6.7 – 6.12 portray the comparative time plots displaying the actual and the predicted series from HWES. More precisely, Figure 6.7 and Figure 6.8 represent the time plots showing observed values and the forecast values obtained from HWES Models 1 and 2, respectively, when the models are applied to sample period one (2009 to 2017). Similarly, Figure 6.9 and Figure 6.10 portray time plots displaying actual series against the predicted series derived from HWES Models 1 and 2, respectively, when the models are applied to sample period two (2007 to 2017). Likewise, Figure 6.11 and Figure 6.12 represent time plots displaying actual values against the predicted values extracted from HWES Models 1 and 2, respectively, when the models are applied to period three (2007 to 2020). A careful close
inspection of each of these figures confirms the predictive precision of the devised HWES models.
Subsequently, the residuals, which are the deviations of the actual observations from the estimations, are derived from the applications of the HWES models tested in three samples.
The residuals are then further analysed. The descriptive summary statistics of the residuals are presented in Table B.2 and Table B.5 in Appendix B, Table C.2 and Table C.5 in Appendix C and Table D.2 and Table D.5 in Appendix D. More precisely, B.2 (B.5), C.2 (C.5) and D.2 (D.5) display the residuals of HWES Model 1 (HWES Model 2) applied to sample periods one, two, and three, respectively. For example, during sample period one (2009 – 2017), the mean (average) of the residuals for the HWES Model 1 is 0.3166, which is marginally lower than the mean of the residuals (0.3183) for the HWES Model 2. Similarly, the centre measures of the residuals are clustered around 0, which confirms the prediction accuracies of the tested HWES models.
In my study, I use Ordinary Least Squares regression (OLS) regression to investigate the simple linear relationship between "prediction results from each technique" as a function of the actual observations. OLS method is a standard technique to estimate the linear relationship between independent variables and a dependent variable. This method estimates the parameters in the regression by minimising the sum of squared residuals. The coefficient of determination (R2,[304]–[306]), which measures the explanatory power of the estimated regression, is used to determine the model's goodness of fit.
Figure 6.7: HWES (Model 1) Actual versus Prediction - Period one (2009 – 2017)
Figure 6.8: HWES (Model 2) Actual versus Prediction - Period one (2009 – 2017)
Figure 6.9: HWES (Model 1) Actual versus Prediction - Period two (2007 – 2017)
Figure 6.10: HWES (Model 2) Actual versus Prediction - Period two (2007 – 2017)
Figure 6.11: HWES (Model 1) Actual versus Prediction - Period three (2007 – 2020)
Figure 6.12 HWES (Model 2) Actual versus Prediction - Period three (2007 – 2020)
Tables B.3, B.6, C.3, C.6, D.3, and D.6 provide the simple OLS regression results showing the predictions from the HWES models applied to different sample periods (periods one, two, and three) as a function of the actual observations. In these OLS regressions, I have used forecasts from each HWES model as the dependent variable and the observed (actual) values as the independent variable. For example, Table B.3 provides the simple OLS regression results showing the linear relationship between the prediction from the HWES Model 1 applied to sample period one and the observed values. More precisely, the R2 in Table B.3 shows that 99.55% of the total variation in the predictions of HWES Model 1 is explained by the actual observations, confirming the explanatory power of the regression of predictions on HWES Model 1 and the observed values. The slope coefficient suggests that if the variable “actual observation” increased by one, HWES Model 1 based predictions increased by 0.9989. This evidence confirms the proximity of the forecasts from HWES model 1 and the actual values and validates the prediction efficacy of HWES Model 1.
Finally, three widely used hypothesis tests are performed to determine the effectiveness of the OLS regressions presented in Tables B.3, B.6, C.3, C.6, D.3, and D.6. These investigations further validate the prediction effectiveness of HWES Models tested in my study. For example, Table B.3 show simple regression results of the predictions from HWES Model 1 as a function of the actual observations. Firstly, it is possible to assess the overall joint significance of each of the estimated regressions. For example, an overall joint significance test is conducted to ascertain any explanatory power of the regression between the predictions from HWES Model 1 and the actual observations. A joint explanatory power confirms that the two variables, namely the predictions from HWES Model 1 and the actual observations, are closely adjoining to one another. For this investigation, using the F-test, a null hypothesis of no collective explanatory power of the estimated regression (all regression parameters are zero) is tested at a 5 percent significance level. For example, the P-value of the F-statistic for the
estimated OLS regression presented in Table B.3 is 0.0000. It is statistically significant at 5 percent significance, thus, leading to a conclusion that estimated OLS regression has a joint explanatory power. This finding is consistent with all the other OLS regressions presented in Tables B.6, C.3, C.6, D.3, and D.6. These findings confirm the actual observations and the predictions are in proximity to each other and confirm the predictive efficacy of the HWES models. Secondly, a hypothesis test for the estimated slope is conducted to determine whether the independent variable is a significant (linear) predictor of the dependent variable. For example, the P-value of the slope coefficient in Table B.3 is 0.0000, which is less than the 5%
significance level. This evidence confirms the rejection of the null hypothesis that the actual slope coefficient is zero. This finding confirms the independent variable “actual observations”
is a significant predictor of the dependent variable “predictions from HWES Model 1”. This finding is consistent with the rest of the OLS regressions presented in Tables B.6, C.3, C.6, D.3, and D.6, and confirm the rejection of the null hypothesis that the actual slope coefficient is zero. This finding justifies that the variable named “actual” has a statistically significant effect on the “predicted” series from HWES in OLS regressions presented in Tables B.3, B.6, C.3, C.6, D.3, and D.6. Thirdly, the Coefficient of Determination (R2,[304]–[306]), which measures the explanatory power of the estimated regressions, is used to assess the goodness of fit of the estimated OLS models related to HWES models. R2 has limiting values of 1 (fits perfectly) and 0 (not better than the mean of the dependent variable). For example, the R2 of Table B.3 is 0.9955, which confirms the superior explanatory power of the OLS regression of HWES Model 1 for period one (2009 – 2017). This finding is consistent with the rest of the OLS results stated in Tables B.6, C.3, C.6, D.3, and D.6.
A comparative discussion on the predictive efficacies of all the tested models will be performed in Chapter 8.