We will investigate how time series linearization affects the quality of both point forecasts and confidence interval forecast. Here linearization will not be considered in its generality, as described in Section 1, but will be confined to outliers’ detection and adjustment³. Table 2 presents the number of best forecasts, in terms of minimization of the corresponding statistic, with data in levels. Auxiliary Table 3 presents the number of best forecasts with log-transformed data indistinguishably for all time series, as it is often the case to use log-transformed data in econometric analyses. It is noted that in one time series with levels (that of unemployment expressed in percentages) and one time series in logs (that of industrial production index) no outliers were detected, hence, the total number of time series considered reduced to nineteen for each case.

From the results of Table 2 and Table 3 it is apparent that, when outliers are considered, forecasts are better in every single case in terms of the width of the forecast confidence interval. In contrast, there is no obvious improvement in

point forecasts. One point that should be stressed is that such results are in general dependent upon the specific characteristics of each time series, especially upon whether an outlier lays among the first, the middle or the last observations. For this reason, it would be desirable to use a large number of time series, so as to draw conclusions of indisputable confidence. Although the number of time series used in this work is relatively small (though comparable to that of other similar works, see for instance Nelson and Granger, 1976) [5] the evidence that lead to the above conclusions, in particular regarding the width of the forecast confidence interval, is so convincing that it really stands far and beyond any concern related to micronumerosity.

After a level shift outlier, all observations subsequent to the outlier move to a new level. In contrast to additive and transitory outliers a level shift outlier reflects a major change in the stochastic process and affects many observations, as it has a permanent effect. For this reason, the case with only additive and transitory outliers (i.e. excluding level shifts) was considered, and their effect on forecasts was examined separately, performing the same analysis as in Section 3.1. It is noted that this time only fifteen time series were considered, i.e. those including all types of outliers. The results are presented in Table 4 and Table 5.

From the results below it is obvious that there is a trade-off: confidence interval forecasts are better with level shift outliers included and, conversely, point forecasts are better excluding level shifts. Given the influence of the level shift

outliers it would be desirable to possibly consider stricter identification criteria for them relative to the other two types of outliers. It is noted that in existing statistical software specializing on time series analysis there is no such an option and a purpose-built routine should be created by the researcher.

As far as the effect of data transformation is concerned, at first it is important to note that the effect of a transformation is meant in two ways: 1) direct and 2) indirect (through its influence on outlier detection). Indeed, regarding the later, it has been shown that data transformation affects the number and the character of outliers in a time series [12] [13].

The possible need for a data transformation of the original time series data will be examined using the TSW routine. Once a decision about the proper data transformation is made, TSW will be used for further analysis on statistical forecasting.

Table 6 presents the results on the decision about, transforming or not, the original time series data. The twenty series were analyzed following the standard TSW procedure. It is noted that the only alternatives available with TSW are either the log-transformation, or no transformation. Using the TSW routine for these twenty cases, TSW suggested the logarithmic transformation of the original data for eighteen cases. It is remarkable that only for the two series of unemployment TSW suggests no transformation.

The possible effect of transforming time series on forecasting quality is examined through Table 7. From the results below it is concluded that point forecasts with TSW transformation method are the same in comparison with that of no transformation in terms of MAPE and MAE, and slightly worse in terms of

MSFE. As already explained, forecasts on transformed variables are not optimal in terms of MSFE. Similarly, confidence interval forecasts are shorter in only eight out of the eighteen cases using transformations with the TSW approach. Thus, data transformations using the TSW routine do not seem to improve

either point forecasts or forecasts confidence intervals. We note, however, that in this case a larger data set is needed for more solid conclusions. Moreover, the outcome may be a result of the restriction of TSW to use only the logarithmic transformation. Further research is needed on that matter allowing for a wider range of transformations.

The results of the examination of the forecasting performance combining both linearization and data transformation are presented in Table 8. The conclusion that is derived is that, by and large, the combined effect does not lead to better point forecasts but leads to improved confidence interval forecasts. The conclusion about the forecast confidence interval is reasonable and, to a large extent, expected, as with the adjustment for outliers the process variance is reduced. It is possible to exploit this reduction in obtaining forecasts with increased confidence.

Appendix 1 presents the ARIMA models for the benchmark model and the combination of TSW variance stabilizing method linearization. It is noted that the differences in the ARIMA models for the two time series where no transformation was needed (that of unemployment expressed in percentages and thousands) should be attributed to the existence of outliers adjusted by linearization.

Let Y ^ T + 1 / Φ T denote the optimal one step ahead linear prediction of Y T + 1 given the information set Φ T , which includes information up to time T, e T + 1 = Y T + 1 − Y ^ T + 1 / Φ T denote the associated forecast error, and

σ T + 1 2 = [ Y T + 1 − Y ^ T + 1 / Φ T ] 2 denote the associated variance. The observation Y T + 1 is considered as an outlier if the null Hypothesis: Ho: e T + 1 = 0 is rejected. The appropriate statistic to test Ho is τ = e T + 1 σ T + 1 .

However, theory cannot predict the critical value of τ above which the corresponding observation can be considered as an outlier. Α usual practice is to relate the critical value of τ with the length of a time series. The default values of

TSW for τ are presented in Table 9⁵. In the course of our experimentation it was observed that outlier detection (as well as ARIMA models for the linearized-transformed series), were very sensitive to the value of parameter τ. In order to examine, whether or not, the critical τ values could have any noticeable effect on our final conclusions, as an alternative set of critical values for τ we used those suggested by Fischer and Planas [14], who examined a very large number of time series. Their critical values for τ were set at 3.5, 3.7 and 4.0 for series lengths of less than 130 observations, between 131 and 180, and more than 180 observations, respectively.

The comparison of the results based on default critical τ values, as well as on Fischer – Planas recommendations are presented in Table 10, while the detected outliers for each time series and each set of values for the parameter τ are presented in Appendix 2. Looking at Appendix 2 it is observed that the detection of outliers is indeed sensitive even to the examined small changes in the value of τ. On the other hand, however, from the results of Table 10, it is apparent that using the Fisher and Planas critical values for τ leads to mixed results regarding the effect on forecast quality. By and large, there is only very weak evidence of improvement using the Fischer-Planas recommendations⁶.

The skill of a forecast can be assessed by comparing the relative proximity of both the forecast and a benchmark to the observations. The presence of a benchmark makes it easier to compare approaches and for this reason a benchmark is proposed to establish a common ground for comparison. In the present case an obvious benchmark is to use the twenty-time series described in section 2, non-linearized and non-transformed. Although there exist established formal tests for forecast evaluation [15] [16] [17] in this work, in line with its practical

character, it suffices to use a very simple and transparent ad-hoc forecasting evaluation approach based on point and interval forecasts.

More specifically, for the point forecasts for each time series and for each model an arithmetic value is assigned in ascending order based on the corresponding value of the MSFE statistic (i.e. 1 for the minimum MSFE value, 2 and 3 for the second and third lower MSFE value respectively, 4 for the maximum MSFE value). Then, adding up the arithmetic values for all series for a particular model their sum will represent the performance of the model. Models will be ranked according to the value of the corresponding sum. Apparently, the model with the lowest sum will be considered as the best one. For interval forecasts the same procedure will be followed replacing the value of the MSFE statistic with the value of the corresponding standard error around the point forecasts. The results are shown in Table 11 and Table 12⁷ and more detailed results are quoted in Appendix 3. It is clarified that TSW transformation approach is coupled with the outlier detection-adjustment approach.

From the results of Table 11 and Table 12 it is evident that the performance of TSW approach for point forecasts is not better than that of the benchmark model (as a matter of fact is slightly worse). On the other hand, for the forecast confidence intervals the Levels—all outliers method has a better performance than TSW and the benchmark model. Furthermore, TSW outperforms the benchmark model. A rather crude way to procced to an overall evaluation of the four models is to add up their performances in the two categories (i.e. point and interval forecasts). The addition gives the values of 108, 113, 91 and 88 for the benchmark model, Logs—no outliers, Levels—all outliers and TSW method respectively, which means that TSW method performs clearly better that the benchmark model and further the overall performance of the TSW method is slightly better than that of the “levels-all outliers model” and clearly better than that of the other two models.

Nelson and Granger [5] utilized the Box-Cox transformations, amongst others, for forecasting purposes (point forecasts) using twenty-one actual economic time series. As they failed in getting superior forecasts, they reached to the rather pessimistic conclusion that it is not worthwhile to make use of these transformations bearing in mind the extra inconvenience, effort and cost. Their point of view was subsequently adopted by other researchers as well, as already mentioned in the introductory section. Lest to get too disappointed, despite the fact

that cost and effort are much lower nowadays than what they were at that time, we further note that Nelson and Granger [5] did not associate forecasts on transformed time series with an outlier detection-adjustment approach. Furthermore, their conclusion was based only on point forecasts, disregarding forecast confidence intervals. The latter are of much importance especially in cases where the focus is on best-worst forecast scenarios. For instance, such is the case with actuarial time series on mortality rates, which may be used further for the construction of pension plans. As shown above, the combination of transformation-linearization leads to shorter forecast confidence intervals.

It should also be stressed that neither in the existing research works thus far, nor in the present one, the treatment of the effect of data transformation on time series forecasting is complete for the simple reason that no work extends the analysis in a bivariate (in general multivariate) framework. Indeed, the existence of variance non-stationarity in time series will contaminate the pre-whitening process (for details about the pre-whitening process see Box and Jenkins, 1976 [18] ), consequently, the sample cross correlation function, so it will mask the true dynamic relationship between two series, one of which is supposed to be the leading indicator, thus affecting negatively the conditional (in this case) forecasts.

Another serious concern expressed by Nelson and Granger [5] was the fact that the problem of acute non-normal distributions they found in most macroeconomic time series they analyzed was restored only very little by their use of data transformations. Table 13 presents the results for the Jarque-Bera statistic for normality [19]. This statistic is distributed as chi-square with two degrees of freedom. An asterisk right next to an arithmetic value of Table 13 indicates a rejection of the null hypothesis of normality at the 5% significance level (critical value = 5.99).

The results of Table 13 allow, again, for a more optimistic view, inasmuch as it is evident that there is a general shift towards normality from the benchmark model to TSW transformation-linearization procedure. The phenomenon on some occasions is really very pronounced indeed (e.g. in the series of M1 and Balance of Payments–transport-payments). This allows for computational algorithms such as maximum likelihood estimation, as well as standard statistical tests, to be legitimately employed with transformed-linearized data.

Seizing the opportunity of the above analysis, it is useful to assess the forecastability of the twenty time series of the Greek economy. Here forecastability will be perceived in both point and confidence interval forecasts. For the former the MAPE statistic will be employed. For the latter the percentage standard error statistic will be introduced as the mean average of the ratio of the forecasts’ standard error over the corresponding actual value, so as to make forecasts of the various series mutually comparable. In all cases one-step-ahead forecasts will be performed. It is stressed that although these forecasts are technically perfectly acceptable, nevertheless they are purely statistical, hence, a-theoretical, and they can only serve as benchmark forecasts in order to evaluate the merit of more structural econometric forecasts. Table 14, Table 15 shows the results in terms of statistical forecastability, according to the combined transformation-linearization effect (denoted as TSW). More specifically, point forecasts in Table 14 are presented in descending ordered according to the value of the Mean Absolute Percentage Error (MAPE) statistic for the combined transformation-linearization

effect (fourth column), and interval forecasts in Table 15 are presented in descending order according to the value of the Percentage Standard Error statistic for the combined transformation-linearization effect (fourth column).

From the results of Table 14, Table 15, it is observed that although there are many similarities in the two Tables, the ordering is not exactly the same. For this reason, the linear correlation coefficient between orderings based on MSFE and the percentage standard error was used. In all cases there is a strong positive correlation (see Table 16). The method of Levels –all outliers has the highest correlation, while TSW has the lowest.

From Tables 14-15 it is also noticeable that the BOP series are the least forecastable in both Tables. Regarding imports-exports it is noted that the former are less forecastable than the latter. Furthermore, imports-exports excluding fuels and ships are clearly more forecastable than imports-exports including them.

This justifies, here from the statistics point of view, the separate recording and usage of the imports-exports without the inclusion of fuels and ships, as presented in the official BOP statistics for Greece [20].