^{1}

^{*}

^{2}

This work examines the impact of data transformation (for variance stabilization) and outlier adjustment (“linearization”) on the quality of univariate time series forecasts, considering each one separately, as well as in combination. Twenty of the most important time series of the Greek economy were used for this purpose. Empirical findings show a significant improvement in forecasts’ confidence intervals, but no substantial improvement in point forecasts. Furthermore, the combined transformation-linearization procedure improves substantially the non-normality problem encountered in many macroeconomic time series.

A univariate AutoRegressive Integrated Moving Average (henceforth ARIMA) model is a concise quantitative summary of the internal dynamics of a time series in a linear framework, as such, is useful for several reasons, amongst others for forecasting and model-based time series decomposition in unobserved components. This work will deal with the former and, in particular, with univariate forecasts, which usually serve either as short-term, or benchmark forecasts. However, economic time series from the real world are not usually “ready” to be used for forecasting purposes and they need to undergo some statistical preparation and pre-adjustment. This is because in time series of raw data variance non-stationarity may be present. Furthermore, very often there exist causes that disrupt the underlying stochastic process (existence of outliers, calendar effects, etc.). Their treatment is known as “linearization”.

Within that line of reasoning, statistical forecasts can be made after a series itself, or some variance stabilizing transformation of it, is “linearized” according to the general framework [

y t = w ′ t b + C ′ t η + ∑ j = 1 m α j μ j ( B ) I t ( t j ) + x t (1)

where: y t = f ( z t ) , f is some transformation of the raw series z_{t}, which may be necessary to stabilize the variance;

b = ( b 1 , ⋯ , b n ) is a vector of regression coefficients;

w ′ t = ( w 1 t , ⋯ , w n t ) denotes n regression or intervention variables;

C ′ t denotes the matrix with columns possible calendar effect variables (e.g. trading day) and η the vector of associated coefficients;

I t ( t j ) is an indicator variable for the possible presence of an outlier at period t j ;

μ j ( B ) captures the transmission of the j-th effect and α j denotes the coefficient of the outlier in the multiple regression model with m outliers;

x t follows in general a multiplicative seasonal ARIMA(p, d, q) (P, D, Q)s model:

φ ( B ) Φ ( B s ) ∇ d ∇ s D x t = θ ( B ) Θ ( Β s ) ε t (2)

where:

• φ ( B ) = 1 − ϕ 1 B − ⋯ − ϕ p B p is the so-called autoregressive polynomial of order p;

• θ ( B ) = 1 − θ 1 B − ⋯ − θ q B q is the so-called moving average polynomial of order q;

• ∇ d ≡ ( 1 − B ) d is the arithmetic difference operator of order d;

• ∇ s D ≡ ( 1 − B s ) D is the seasonal arithmetic difference operator of order D and seasonality s;

• Φ ( B s ) = 1 − Φ 1 B s − ⋯ − Φ P B P ⋅ s is the so-called seasonal autoregressive polynomial of order P and seasonality s;

• Θ ( B s ) = 1 − Θ 1 B s − ⋯ − Θ Q B Q ⋅ s is the so-called moving average polynomial of order Q and seasonality s;

• ε t is the stochastic disturbance.

As far as variance stabilization is concerned, if variance is somehow functionally related to the mean level it is possible to select a transformation to stabilize the variance. Widely used transformations to tackle this problem belong to the class of the power Box and Cox transformation [

z t λ if λ > 0

f ( z t ) = ln z t if λ = 0 (3)

− z t λ if λ < 0

On the other hand, outliers are major changes in values that especially stand out in a time series. In the TSW^{1} framework, of which use will be made in this work, three types of outliers are detected according to their effect in a time series: Additive outliers (AO), Transitory Change outliers (TC), and Level shifts (LS). In an additive outlier the value of only one observation is affected. In a transitory change the value of one observation is extremely high or low and then the size of the deviation reduces gradually. In the level shift the level of the time series is changed. As far as the detection of outliers is concerned within the TSW framework, outliers are automatically detected, classified and corrected using the Chen and Liu [

So, there are two effects with potential influence on forecasting: «linearization» and transformation, each of which separately, as well as in combination, may play an important role on time series forecasting.

At the empirical level, studies which have considered the merits of mathematical transformations on forecasting have demonstrated that transformation often does not have a positive effect on forecast accuracy [

On the other hand, at the theoretical level, Granger and Newbold [

Regarding time series linearization, such a procedure is utilized thus far mainly as a preadjustment task for seasonal adjustment [^{2} It is also remarked that even in studies coping with forecasting with transformed data the attention focuses almost exclusively on point forecasts, by and large disregarding interval forecasts.

Aiming at covering this research gap in the literature the objective of this work is in fact twofold: a) to examine the effect of «linearization» and transformation separately, as well as in combination, on both point forecasts and confidence interval forecasts; b) as a further application, we rank main economic indicators of the Greek economy in terms of statistical «forecastability». The intended approach will be practical.

The structure of the paper is as follows: In Section 2 details about the data to be used for the empirical analysis are given; Section 3 presents the empirical results and relevant comments; Section 4 summarizes and concludes the paper.

The data set comprises some of the most important macroeconomic time series for the Greek economy, which refer to: GDP; unemployment; prices of consumer goods and services; monetary aggregates; and balance of payments statistics. Particularly, in the balance of payments, a distinction is made between imports—exports of all goods and imports—exports of goods without fuels and ships, as according to a study by the Bank of Greece [

The monthly time series data cover the period from January 2004 to August 2018 and consist of one hundred and seventy-six (176) observations, except from Industrial Production Index, where available data existed from January 2010 to August 2018 (104 observations). The quarterly time series is that of Gross Domestic Product and covers the period from 1995 Quarter 1 to 2018 Quarter 3 (95 observations).

As mention in Section 1, the effect of transformation and the effect of linearization on forecasting will be examined at first each one separately and, subsequently, in combination. The aforementioned effects will be studied utilizing TSW.

Typical statistics to be used for the assessment of the quality of point forecasts are the following: 1) the Mean Absolute Percentage Error (MAPE) statistic given by: M A P E = 100 % n ∑ t = 1 n | A t − F t A t | , 2) the Mean Square Forecast Error (MSFE) statistic given by: M S F E = 1 n ∑ t = 1 n ( A t − F t ) 2 , and 3) the Mean Absolute Error (MAE) statistic M A E = 1 n ∑ t = 1 n | A t − F t | , where A t is the actual value and F t is the forecast value. Furthermore, as far as interval forecasts are concerned, the width of the forecast confidence interval (CI) will be considered.

Time Series | Observation frequency | Source |
---|---|---|

Gross Domestic Product (GDP) | Quarterly | ELSTAT |

Industrial Production Index (IPI) | Monthly | ELSTAT |

Consumer Price Index (CPI) | Monthly | ELSTAT |

Harmonised Index of Consumer Prices (HICP) | Monthly | ELSTAT |

Unemployment-thousands | Monthly | ELSTAT |

Unemployment-percentage | Monthly | ELSTAT |

Retail sales | Monthly | ELSTAT |

M1 | Monthly | BoG |

M2 | Monthly | BoG |

M3 | Monthly | BoG |

Balance of payments (BOP) - Transport - Payments | Monthly | BoG |

Balance of payments (BOP) - Transport - Receipts | Monthly | BoG |

Balance of payments (BOP) - Travelling - Payments | Monthly | BoG |

Balance of payments (BOP) – Travelling - Receipts | Monthly | BoG |

Balance of payments (BOP) - Sea transport - Payments | Monthly | BoG |

Balance of payments (BOP) - Sea transport - Receipts | Monthly | BoG |

Exports of Goods | Monthly | BoG |

Exports of Goods without fuels and ships | Monthly | BoG |

Imports of Goods | Monthly | BoG |

Imports of Goods without fuels and ships | Monthly | BoG |

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 adjustment^{3}.

From the results of

Point Forecast | With detected Outliers | Without Outliers |
---|---|---|

MAPE | 10/19 | 9/19 |

MSFE | 8/19 | 11/19 |

MAE | 9/19 | 10/19 |

Confidence Interval | With detected Outliers | Without Outliers |

Standard error (SE) | 19/19 | 0/19 |

Point Forecast | With detected Outliers | Without Outliers |
---|---|---|

MAPE | 9/19 | 10/19 |

MSFE | 11/19 | 8/19 |

MAE | 10/19 | 9/19 |

Confidence Interval | With detected Outliers | Without Outliers |

Standard error (SE) | 19/19 | 0/19 |

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) [

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

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

Point Forecast | All Outliers | Outliers without LS |
---|---|---|

MAPE | 6/15 | 9/15 |

MSFE | 5/15 | 10/15 |

MAE | 6/15 | 9/15 |

Confidence Interval | All Outliers | Outliers without LS |

Standard error (SE) | 14/15 | 1/15 |

Point Forecast | All Outliers | Outliers without LS |
---|---|---|

MAPE | 6/15 | 9/15 |

MSFE | 5/15 | 10/15 |

MAE | 6/15 | 9/15 |

Confidence Interval | All Outliers | Outliers without LS |

Standard error (SE) | 13/15 | 2/15 |

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 [

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.

The possible effect of transforming time series on forecasting quality is examined through

TIME SERIES | LOG-LEVEL PRETEST (Output from TSW) |
---|---|

Gross Domestic Product (GDP) | SSlevels/(SSlog*Gmean(levels)^2) = 1.1380170 LOGS ARE SELECTED |

Consumer Price Index (CPI) | SSlevels/(SSlog*Gmean(levels)^2) = 1.0781750 LOGS ARE SELECTED |

Harmonised Index of Consumer Prices (HICP) | SSlevels/(SSlog*Gmean(levels)^2) = 1.0954455 LOGS ARE SELECTED |

Industrial Production Index (IPI) | SSlevels/(SSlog*Gmean(levels)^2) = 1.0224433 LOGS ARE SELECTED |

Unemployment - thousands | SSlevels/(SSlog*Gmean(levels)^2) = 0.87725642 LEVELS ARE SELECTED |

Unemployment - percentage | SSlevels/(SSlog*Gmean(levels)^2) = 0.86356273 LEVELS ARE SELECTED |

Retail sales | SSlevels/(SSlog*Gmean(levels)^2) = 1.2755206 LOGS ARE SELECTED |

M1 | SSlevels/(SSlog*Gmean(levels)^2) = 0.98393639 LOGS ARE SELECTED |

M2 | SSlevels/(SSlog*Gmean(levels)^2) = 1.0714007 LOGS ARE SELECTED |

M3 | SSlevels/(SSlog*Gmean(levels)^2) = 1.0422806 LOGS ARE SELECTED |

Balance of payments (BOP) - Transport - Payments | SSlevels/(SSlog*Gmean(levels)^2) = 1.0351033 LOGS ARE SELECTED |

Balance of payments (BOP) - Transport - Receipts | SSlevels/(SSlog*Gmean(levels)^2) = 1.1641507 LOGS ARE SELECTED |

Balance of payments (BOP) - Travelling - Payments | SSlevels/(SSlog*Gmean(levels)^2) = 1.1509645 LOGS ARE SELECTED |

Balance of payments (BOP) – Travelling – Receipts | SSlevels/(SSlog*Gmean(levels)^2) = 4.3996100 LOGS ARE SELECTED |

Balance of payments (BOP) - Sea transport - Payments | SSlevels/(SSlog*Gmean(levels)^2)= 0.98863656 LOGS ARE SELECTED |

Balance of payments (BOP) - Sea transport - Receipts | SSlevels/(SSlog*Gmean(levels)^2) = 1.1948699 LOGS ARE SELECTED |

Exports of Goods | SSlevels/(SSlog*Gmean(levels)^2) = 0.95751942 LOGS ARE SELECTED |

Exports of Goods without fuels and ships | SSlevels/(SSlog*Gmean(levels)^2) = 0.96487436 LOGS ARE SELECTED |

Imports of Goods | SSlevels/(SSlog*Gmean(levels)^2) = 1.1118244 LOGS ARE SELECTED |

Imports of Goods without fuels and ships | SSlevels/(SSlog*Gmean(levels)^2) = 1.2957291 LOGS ARE SELECTED |

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

Point Forecast | TSW-no outliers | Levels-no outliers (Benchmark) |
---|---|---|

MAPE | 9/18 | 9/18 |

MSFE | 7/18 | 11/18 |

MAE | 9/18 | 9/18 |

Confidence interval | TSW - no outliers | Levels-no outliers (Benchmark) |

Standard error (SE) | 8/18 | 10/18 |

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

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

Point Forecast | TSW | Levels-no outliers (Benchmark) |
---|---|---|

MAPE | 8/18 | 10/18 |

MSFE | 8/18 | 10/18 |

MAE | 8/18 | 10/18 |

Confidence interval | TSW | Levels-no outliers (Benchmark) |

Standard error (SE) | 12/18 | 6/18 |

TSW for τ are presented in ^{5}. 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 [

The comparison of the results based on default critical τ values, as well as on Fischer – Planas recommendations are presented in ^{6}.

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 [

Observations | Default values for τ in TSW |
---|---|

164 | 0.358E+01 |

165 - 168 | 0.359E+01 |

169 - 172 | 0.360E+01 |

173 - 175 | 0.361E+01 |

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 ^{7} 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

Nelson and Granger [

Time series | Improvement of forecast quality | Same forecast quality | Deterioration of forecast quality |
---|---|---|---|

Gross Domestic Product (GDP) | MAPE, MSFE, MAE, SE (TSW) | MAPE, MSFE, MAE, SE (Levels - all outliers) | |

Consumer Price Index (CPI) | MAPE, MSFE, MAE (TSW) | MAPE, MSFE, MAE, SE (Levels –all outliers) | SE (TSW) |

Harmonised Index of Consumer Prices (HICP) | MAPE, MSFE, MAE (Levels –all outliers) MAPE, MSFE, MAE, SE (TSW), | SE (Levels - all outliers) | |

Industrial Production Index (IPI) | MAPE, MSFE, MAE (Levels –all outliers) | MAPE, MSFE, MAE, SE (TSW) | SE (Levels - all outliers) |

Unemployment - thousands | MSFE (Levels -all outliers, and TSW) | MAPE, MAE, SE (Levels - all outliers, and TSW) | |

Unemployment - percentage | MAPE, MSFE, MAE (Levels –all outliers, and TSW) | SE (Levels –all outliers, and TSW) | |

Retail sales | MAPE, MSFE, MAE (Levels – all outliers) | SE (Levels - all outliers) MAPE, MSFE, MAE, SE (TSW) | |

M1 | MAPE, MSFE, MAE, SE (Levels –all outliers, and TSW) | ||

M2 | MAPE, MAE (Levels –all outliers) | MSFE, SE (Levels - all outliers) MAPE, MSFE, MAE, SE (TSW) | |

M3 | MAPE, MSFE, MAE (Levels –all outliers) | SE (Levels - all outliers) MAPE, MSFE, MAE, SE (TSW), | |

Balance of payments (BOP) - Transport - Payments | MAPE, MSFE, MAE (TSW) | SE (TSW) MAPE, MSFE, MAE, SE (Levels - all outliers) | |

Balance of payments (BOP) - Transport - Receipts | MSFE (Levels – all outliers) MAPE, MSFE, MAE, SE (TSW) | MAPE, MAE, SE (Levels - all outliers) | |

Balance of payments (BOP) - Travelling - Payments | MAPE, MAE, SE (TSW) | MSFE (TSW) MAPE, MSFE, MAE, SE (Levels - all outliers) | |

Balance of payments (BOP) - Travelling - Receipts | SE (TSW) | MAPE, MSFE, MAE (TSW) MAPE, MSFE, MAE, SE (Levels - all outliers) | |

Balance of payments (BOP) - Sea transport - Payments | MAPE, MSFE, MAE (Levels: all outliers, and TSW) | SE (Levels - all outliers, and TSW) | |

Balance of payments (BOP) - Sea transport - Receipts | MAPE, MSFE, MAE (Levels: all outliers, and TSW) | SE (Levels - all outliers, and TSW) |

Exports of Goods | MAPE, MSFE, MAE (Levels –all outliers, and TSW) | SE (Levels- all outliers, and TSW) | |
---|---|---|---|

Exports of Goods without fuels and ships | MAPE, MSFE, MAE, SE (Levels - all outliers, and TSW) | ||

Imports of Goods | MAPE, MSFE, MAE, SE (Levels: all outliers) | MAPE, MSFE, MAE, SE (TSW) | |

Imports of Goods without fuels and ships | MAPE, MSFE, MAE (TSW) | MAPE, MSFE, MAE, SE (Levels: all outliers) | SE (TSW) |

Time series | Benchmark | Logs-no outliers | Levels-all outliers | TSW |
---|---|---|---|---|

Consumer Price Index (CPI) | 2 | 1 | 4 | 3 |

Harmonised Index of Consumer Prices (HICP) | 1 | 2 | 3 | 4 |

M3 | 1 | 2 | 3 | 4 |

M2 | 3 | 1 | 2 | 4 |

Gross Domestic Product (GDP) | 4 | 1 | 3 | 2 |

M1 | 4 | 2 | 3 | 1 |

Industrial Production Index (IPI) | 1 | 3.5 | 2 | 3.5 |

Retail sales | 1 | 3 | 2 | 4 |

Unemployment - thousands | 2 | 1 | 3.5 | 3.5 |

Balance of payments (BOP) - Transport - Receipts | 2 | 1 | 4 | 3 |

Balance of payments (BOP) - Sea transport - Receipts | 1 | 2 | 4 | 3 |

Unemployment - percentage | 3 | 1 | 3 | 3 |

Balance of payments (BOP) - Transport - Payments | 1 | 2 | 3 | 4 |

Imports of Goods without fuels and ships | 3 | 4 | 1 | 2 |

Exports of Goods without fuels and ships | 4 | 2 | 3 | 1 |

Exports of Goods | 3 | 2 | 4 | 1 |

Balance of payments (BOP) - Sea transport - Payments | 3 | 4 | 2 | 1 |

Imports of Goods | 3 | 4 | 2 | 1 |

Balance of payments (BOP) - Travelling - Receipts | 3 | 2 | 4 | 1 |

Balance of payments (BOP) - Travelling - Payments | 2 | 3 | 4 | 1 |

SUM | 47 | 43.5 | 59.5 | 50 |

Time series | Benchmark | Logs-no outliers | Levels –all outliers | TSW |
---|---|---|---|---|

Harmonised Index of Consumer Prices (HICP) | 3 | 4 | 1 | 2 |

Consumer Price Index (CPI) | 3 | 4 | 1 | 2 |

M1 | 3 | 4 | 1 | 2 |

M3 | 4 | 3 | 2 | 1 |

M2 | 4 | 3 | 2 | 1 |

Gross Domestic Product (GDP) | 4 | 3 | 2 | 1 |

Unemployment – percentage | 2 | 4 | 2 | 2 |

Industrial Production Index (IPI) | 2 | 3.5 | 1 | 3.5 |

Unemployment - thousands | 3 | 4 | 1.5 | 1.5 |

Exports of Goods without fuels and ships | 2 | 4 | 1 | 3 |

Retail sales | 4 | 2 | 3 | 1 |

Exports of Goods | 2 | 4 | 1 | 3 |

Balance of payments (BOP) - Transport - Receipts | 3 | 4 | 1 | 2 |

Balance of payments (BOP) - Transport - Payments | 2 | 4 | 1 | 3 |

Balance of payments (BOP) - Sea transport - Receipts | 4 | 3 | 2 | 1 |

Imports of Goods without fuels and ships | 4 | 3 | 2 | 1 |

Balance of payments (BOP) - Sea transport - Payments | 2 | 4 | 1 | 3 |

Imports of Goods | 4 | 3 | 2 | 1 |

Balance of payments (BOP) - Travelling - Payments | 4 | 2 | 3 | 1 |

Balance of payments (BOP) - Travelling - Receipts | 2 | 4 | 1 | 3 |

SUM | 61 | 69.5 | 31.5 | 38 |

that cost and effort are much lower nowadays than what they were at that time, we further note that Nelson and Granger [

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 [

Another serious concern expressed by Nelson and Granger [

The results of

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.

Time series | Benchmark | Logs-no outliers | Levels-all outliers | TSW |
---|---|---|---|---|

Consumer Price Index (CPI) | 2.889 | 1.931 | 0.423 | 0.999 |

Harmonised Index of Consumer Prices (HICP) | 6.289* | 3.923 | 8.263* | 5.850 |

M3 | 19.78* | 30.85* | 12.44* | 14.72* |

M2 | 16.71* | 19.27* | 16.31* | 7.519* |

Gross Domestic Product (GDP) | 14.17* | 2.967 | 3.699 | 0.541 |

M1 | 152.6* | 376.3* | 3.597 | 2.879 |

Industrial Production Index (IPI) | 1.118 | 0.996 | 1.118 | 0.996 |

Retail sales | 2.328 | 0.771 | 0.145 | 0.771 |

Unemployment - thousands | 9.745* | 12.58* | 7.613* | 7.613* |

Balance of payments (BOP) - Transport - Receipts | 5.526 | 5.788 | 0.7983 | 0.563 |

Balance of payments (BOP) - Sea transport - Receipts | 7.447* | 5.991* | 0.874 | 0.9231E−01 |

Unemployment - percentage | 7.584* | 11.30* | 7.584* | 7.584* |

Balance of payments (BOP) - Transport - Payments | 137.5* | 169.3* | 5.289 | 1.651 |

Imports of Goods without fuels and ships | 7.938* | 23.30* | 2.159 | 0.928 |

Exports of Goods without fuels and ships | 28.26* | 12.84* | 0.593 | 0.473 |

Exports of Goods | 0.404 | 0.282 | 0.180 | 0.380 |

Balance of payments (BOP) - Sea transport - Payments | 210.5* | 253.3* | 4.598 | 4.633 |

Imports of Goods | 1.589 | 4.108 | 0.924 | 4.115 |

Balance of payments (BOP) - Travelling - Receipts | 15.31* | 6.740* | 26.78* | 4.696 |

Balance of payments (BOP) - Travelling - Payments | 2.286 | 0.326 | 2.013 | 1.978 |

effect (fourth column), and interval forecasts in

MAPE | ||||
---|---|---|---|---|

Time series | Benchmark | Logs-no outliers | Levels-all outliers | TSW |

Harmonised Index of Consumer Prices (HICP) | 0.241% | 0.238% | 0.252% | 0.257% |

Consumer Price Index (CPI) | 0.238% | 0.233% | 0.328% | 0.289% |

M2 | 0.625% | 0.645% | 0.697% | 0.650% |

M1 | 0.786% | 0.730% | 0.706% | 0.652% |

M3 | 0.561% | 0.627% | 0.661% | 0.653% |

Gross Domestic Product (GDP) | 0.760% | 0.704% | 0.745% | 0.729% |

Industrial Production Index (IPI) | 1.011% | 1.111% | 1.019% | 1.111% |

Retail sales | 1.424% | 1.718% | 1.594% | 1.666% |

Exports of Goods without fuels and ships | 3.718% | 2.793% | 3.208% | 2.517% |

Unemployment - thousands | 2.170% | 1.925% | 2.608% | 2.608% |

Balance of payments (BOP) - Sea transport - Receipts | 2.789% | 2.674% | 4.003% | 2.902% |

Unemployment - percentage | 2.917% | 2.265% | 2.917% | 2.917% |

Balance of payments (BOP) - Transport - Receipts | 2.748% | 2.717% | 2.868% | 2.922% |

Imports of Goods without fuels and ships | 3.032% | 3.412% | 2.582% | 3.026% |

Balance of payments (BOP) - Transport - Payments | 2.929% | 2.955% | 3.134% | 3.309% |

Balance of payments (BOP) - Sea transport - Payments | 5.515% | 5.918% | 5.077% | 3.883% |

Exports of Goods | 5.021% | 3.969% | 5.129% | 4.238% |

Imports of Goods | 6.027% | 6.093% | 5.705% | 5.750% |

Balance of payments (BOP) - Travelling - Receipts | 12.194% | 8.171% | 13.314% | 7.729% |

Balance of payments (BOP) – Travelling - Payments | 12.553% | 11.624% | 13.994% | 11.775% |

From the results of

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.

Percentage Standard Error | ||||
---|---|---|---|---|

Time series | Benchmark | Logs - no outliers | Levels-all outliers | TSW |

Harmonised Index of Consumer Prices (HICP) | 0.439% | 0.441% | 0.423% | 0.427% |

Consumer Price Index (CPI) | 0.454% | 0.464% | 0.419% | 0.443% |

M3 | 1.451% | 1.283% | 1.180% | 1.013% |

M2 | 1.455% | 1.321% | 1.219% | 1.090% |

M1 | 1.290% | 1.475% | 1.142% | 1.272% |

Gross Domestic Product (GDP) | 2.145% | 1.868% | 1.855% | 1.745% |

Unemployment - thousands | 2.809% | 3.865% | 2.572% | 2.572% |

Unemployment - percentage | 2.737% | 3.841% | 2.737% | 2.737% |

Industrial Production Index (IPI) | 2.808% | 2.890% | 2.805% | 2.890% |

Retail sales | 5.110% | 3.728% | 4.188% | 3.636% |

Imports of Goods without fuels and ships | 7.151% | 6.478% | 5.184% | 4.833% |

Balance of payments (BOP) - Transport - Receipts | 5.565% | 5.586% | 4.930% | 5.348% |

Balance of payments (BOP) - Sea transport - Receipts | 6.514% | 6.164% | 5.524% | 5.394% |

Exports of Goods without fuels and ships | 4.582% | 6.263% | 4.008% | 5.483% |

Balance of payments (BOP) - Transport - Payments | 5.817% | 7.217% | 4.469% | 6.101% |

Exports of Goods | 5.495% | 7.831% | 5.294% | 7.552% |

Imports of Goods | 8.170% | 7.875% | 7.700% | 7.559% |

Balance of payments (BOP) - Travelling - Receipts | 24.967% | 8.695% | 20.433% | 7.679% |

Balance of payments (BOP) - Sea transport - Payments | 7.234% | 9.847% | 5.807% | 7.779% |

Balance of payments (BOP) - Travelling - Payments | 17.607% | 14.935% | 16.151% | 14.157% |

Method | Correlation |
---|---|

Benchmark | 95.40% |

Logs-no outliers | 96.73% |

Levels-all outliers | 97.23% |

TSW | 93.05% |

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 [

This work dealt with the effect of data transformation for variance stabilization and linearization for outlier adjustment on the quality of univariate time series forecasts, following a practical approach.

There is clear evidence that linearization improves the forecasts’ confidence intervals, but not such evidence for the data transformation. Furthermore, no evidence was found that either transformation or linearization leads to better point forecasts. The combined effect of transformation-linearization leads to better forecast confidence intervals and improves substantially the non-normality problem encountered in many macroeconomic time series, but worsens point forecasts. There is also evidence that the overall forecasting performance using the TSW data transformation procedure is somewhat better than that of the other used models.

It must be remarked that the above results regarding the effect of data transformation were obtained within the restrictive framework of TSW, which allows the logarithmic transformation as the only alternative. Further research is needed on that mater using a larger dataset and the whole Box-Cox transformations framework.

The authors declare no conflicts of interest regarding the publication of this paper.

Milionis, A.E. and Galanopoulos, N.G. (2019) Forecasting Economic Time Series in the Presence of Variance Instability and Outliers. Theoretical Economics Letters, 9, 2940-2964. https://doi.org/10.4236/tel.2019.98182

Time series | Benchmark | TSW |
---|---|---|

Gross Domestic Product (GDP) | ARIMA (0,1,1) (0,1,1)_{4} | ARIMA (1,1,0) (0,1,1)_{4} |

∇ ∇ 4 Y t = ( 1 + 0.118 Β ) ( 1 + 0.425 Β 4 ) ε t | ( 1 − 0.160 Β ) ∇ ∇ 4 ln Y t = ( 1 + 0.267 Β 4 ) ε t | |

Industrial Production Index (IPI) | ARIMA (2,0,0) (0,1,1)_{12} | ARIMA (1,1,0) (0,1,1)_{12} |

( 1 + 0.379 Β + 0.547 Β 2 ) ∇ 12 Y t = ( 1 + 0.950 Β 12 ) ε t | ( 1 − 0.554 Β ) ∇ ∇ 12 ln Y t = ( 1 + 0.831 Β 12 ) ε t | |

Consumer Price Index (CPI)_{ } | ARIMA (0,1,0) (0,1,1)_{12} | ARIMA (1,1,0) (0,1,0)_{12} |

∇ ∇ 12 Y t = ( 1 + 0.260 Β 12 ) ε t | ( 1 + 0.146 Β ) ∇ ∇ 12 ln Y t = ε t | |

Harmonised Index of Consumer Prices (HICP) | ARIMA (0,1,0) (0,1,1)_{12} | ARIMA (0,1,0) (0,1,1)_{12} |

∇ ∇ 12 Y t = ( 1 + 0.347 Β 12 ) ε t | ∇ ∇ 12 ln Y t = ( 1 + 0.326 Β 12 ) ε t | |

Unemployment – thousands | ARIMA (3,2,1) (0,1,1)_{12} | ARIMA (3,2,1) (0,1,1)_{12} |

( 1 − 0.681 Β − 0.674 Β 2 + 0.062 Β 3 ) ∇ 2 ∇ 12 Y t = ( 1 + 0.758 Β ) ( 1 + 0.938 Β 12 ) ε t | ( 1 − 1.153 Β − 1.123 Β 2 − 0.340 Β 3 ) ∇ 2 ∇ 12 Y t = ( 1 + 0.614 Β ) ( 1 + 0.907 Β 12 ) ε t | |

Unemployment – percentage | ARIMA (2,2,1) (0,1,1)_{12} | ARIMA (2,2,1) (0,1,1)_{12} |

( 1 − 0.726 Β − 0.715 Β 2 ) ∇ 2 ∇ 12 Y t = ( 1 + 0.734 Β ) ( 1 + 0.816 Β 12 ) ε t | ( 1 − 0.726 Β − 0.715 Β 2 ) ∇ 2 ∇ 12 Y t = ( 1 + 0.734 Β ) ( 1 + 0.816 Β 12 ) ε t | |

Retail sales | ARIMA (0,1,1) (0,1,1)_{12} | ARIMA (0,1,1) (0,1,1)_{12} |

∇ ∇ 12 Y t = ( 1 + 0.364 Β ) ( 1 + 0.566 Β 12 ) ε t | ∇ ∇ 12 ln Y t = ( 1 + 0.334 Β ) ( 1 + 0.631 Β 12 ) ε t | |

M1 | ARIMA (0,2,1) (0,1,1)_{12} | ARIMA (3,1,0) (0,1,1)_{12} |

∇ 2 ∇ 12 Y t = ( 1 + 0.838 Β ) ( 1 + 0.682 Β 12 ) ε t | ( 1 + 0.0001 Β + 0.178 Β 2 + 0.422 Β 3 ) ∇ ∇ 12 ln Y t = ( 1 + 0.685 Β 12 ) ε t | |

M2 | ARIMA (3,1,0) (1,0,1)_{12} | ARIMA (1,1,1) (0,1,1)_{12} |

( 1 + 0.328 Β + 0.040 Β 2 + 0.307 Β 3 ) ( 1 + 0.868 Β 12 ) ∇ Y t = ( 1 + 0.656 Β 12 ) ε t | ( 1 + 0.933 Β ) ∇ ∇ 12 ln Y t = ( 1 + 0.627 Β ) ( 1 + 0.800 Β 12 ) ε t | |

M3 | ARIMA (0,2,1) (0,1,1)_{12} | ARIMA (0,2,1) (0,1,1)_{12} |

∇ 2 ∇ 12 Y t = ( 1 + 0.695 Β ) ( 1 + 0.824 Β 12 ) ε t | ∇ 2 ∇ 12 ln Y t = ( 1 + 0.660 Β ) ( 1 + 0.838 Β 12 ) ε t | |

Balance of payments (BOP) – Transport – Payments | ARIMA (0,1,1) (0,1,1)_{12} | ARIMA (0,1,1) (0,1,1)_{12} |

∇ ∇ 12 Y t = ( 1 + 0.188 Β ) ( 1 + 0.847 Β 12 ) ε t | ∇ 2 ∇ 12 ln Y t = ( 1 + 0.440 Β ) ( 1 + 0.841 Β 12 ) ε t | |

Balance of payments (BOP) – Transport – Receipts | ARIMA (3,1,1) (0,1,1)_{12} | ARIMA (3,1,0) (0,1,1)_{12} |

( 1 − 0.393 Β − 0.050 Β 2 + 0.264 Β 3 ) ∇ ∇ 12 Y t = ( 1 − 0.288 Β ) ( 1 + 0.950 Β 12 ) ε t | ( 1 − 0.208 Β − 0.110 Β 2 + 0.229 Β 3 ) ∇ ∇ 12 ln Y t = ( 1 + 0.895 Β 12 ) ε t | |

Balance of payments (BOP) – Travelling – Payments | ARIMA (1,0,0) (0,1,1)_{12} | ARIMA (1,0,0) (1,0,0)_{12} |

( 1 + 0.339 Β ) ∇ 12 Y t = ( 1 + 0.506 Β 12 ) ε t | ( 1 + 0.328 Β ) ( 1 + 0.608 Β 12 ) ln Y t = ε t | |

Balance of payments (BOP) – Travelling – Receipts | ARIMA (1,0,0) (1,1,0)_{12} | ARIMA (1,0,0) (1,1,0)_{12} |

( 1 + 0.731 Β ) ( 1 − 0.371 Β 12 ) ∇ 12 Y t = ε t | ( 1 + 0.598 Β ) ( 1 − 0.422 Β 12 ) ∇ 12 ln Y t = ε t | |

Balance of payments (BOP) – Sea transport – Payments | ARIMA (0,1,1) (0,0,0)_{12} | ARIMA (0,1,1) (0,1,1)_{12} |

∇ Y t = ( 1 + 0202 Β ) ε t | ∇ ∇ 12 ln Y t = ( 1 + 0.334 Β ) ( 1 + 0.824 Β 12 ) ε t |

Balance of payments (BOP) – Sea transport – Receipts | ARIMA (3,1,1) (0,1,1)_{12} | ARIMA (3,1,1) (0,1,1)_{12} |
---|---|---|

( 1 − 0.388 Β − 0.020 Β 2 + 0.281 Β 3 ) ∇ ∇ 12 Y t = ( 1 − 0.262 Β ) ( 1 + 0.848 Β 12 ) ε t | ( 1 − 0.676 Β − 0.187 Β 2 + 0.170 Β 3 ) ∇ ∇ 12 ln Y t = ( 1 − 0.515 Β ) ( 1 + 0.867 Β 12 ) ε t | |

Exports of Goods | ARIMA (0,1,1) (0,1,1)_{12} | ARIMA (0,1,2) (0,1,1)_{12} |

∇ ∇ 12 Y t = ( 1 + 0.414 Β ) ( 1 + 0.950 Β 12 ) ε t | ∇ ∇ 12 ln Y t = ( 1 + 0.349 Β + 0.111 Β 2 ) ( 1 + 0.950 Β 12 ) ε t | |

Exports of Goods without fuels and ships | ARIMA (0,1,1) (0,1,1)_{12} | ARIMA (0,1,1) (0,1,1)_{12} |

∇ ∇ 12 Y t = ( 1 + 0.485 Β ) ( 1 + 0.922 Β 12 ) ε t | ||

Imports of Goods | ARIMA (0,1,1) (0,1,1)_{12} | ARIMA (0,1,1) (1,1,1)_{12} |

Imports of Goods without fuels and ships | ARIMA (0,1,1) (0,1,1)_{12} | ARIMA (0,1,1) (0,1,1)_{12} |

Time series | τ -default TSW critical values | τ –Fisher-Planas |
---|---|---|

Gross Domestic Product (GDP) | OUTLIERS:57 AO (1 2009) | OUTLIERS:57 AO (1 2009) |

Industrial Production Index (IPI) | OUTLIERS: NO OUTLIERS DETECTED | OUTLIERS: NO OUTLIERS DETECTED |

Consumer Price Index (CPI) | OUTLIERS: 93 LS ( 9 2011), 119 AO (11 2013) | OUTLIERS: 119 AO (11 2013) |

Harmonised Index of Consumer Prices (HICP) | OUTLIERS: 119 AO (11 2013) | OUTLIERS: 119 AO (11 2013) |

Unemployment – thousands | OUTLIERS: 60 LS (12 2008), 95 LS (11 2011), 98 TC (2 2012), 126 LS (6 2014), 148 TC (4 2016), 156 TC (12 2016) | OUTLIERS: 60 LS (12 2008), 95 LS (11 2011), 98 TC (2 2012), 126 LS (6 2014), 148 TC (4 2016), 156 TC (12 2016) |

Unemployment – percentage | OUTLIERS: NO OUTLIERS DETECTED | OUTLIERS: NO OUTLIERS DETECTED |

Unemployment – thousands | OUTLIERS: NO OUTLIERS DETECTED | OUTLIERS: 113 AO (5 2013), 139 AO (7 2015) |

M1 | OUTLIERS: 139 LS (7 2015) | OUTLIERS: 139 LS (7 2015) |

M2 | OUTLIERS: 100 AO (4 2012), 102 AO (6 2012), 133 LS (1 2015), 138 TC (6 2015) | OUTLIERS: 100 AO (4 2012), 102 AO (6 2012), 138 TC (6 2015) |

M3 | OUTLIERS: 100 AO (4 2012), 102 AO (6 2012), 133 LS (1 2015), 138 TC (6 2015) | OUTLIERS: 100 AO (4 2012), 102 AO (6 2012), 133 LS (1 2015), 138 TC (6 2015) |

Balance of payments (BOP) – Transport – Payments | OUTLIERS: 60 LS (12 2008), 133 LS (1 2015) | OUTLIERS: 60 LS (12 2008), 133 LS (1 2015) |

Balance of payments (BOP) – Transport – Receipts | OUTLIERS: 59 LS (11 2008) | OUTLIERS: 36 TC (12 2006), 59 LS (11 2008) |

Balance of payments (BOP) – Travelling – Payments | OUTLIERS: 92 AO ( 8 2011) | OUTLIERS: 92 AO ( 8 2011) |

Balance of payments (BOP) – Travelling – Receipts | OUTLIERS: 2 AO (2 2004), 113 LS (5 2013) | OUTLIERS: 2 AO (2 2004), 113 LS (5 2013) |

Balance of payments (BOP) – Sea transport – Payments | OUTLIERS: 60 LS (12 2008), 113 LS (5 2013), 133 LS (1 2015) | OUTLIERS: 59 LS (11 2008), 113 LS (5 2013), 133 LS (1 2015) |

Balance of payments (BOP) – Sea transport – Receipts | OUTLIERS: 36 TC (12 2006), 59 LS (11 2008), 129 AO (9 2014) | OUTLIERS: 36 TC (12 2006), 59 LS (11 2008), 129 AO (9 2014) |

Exports of Goods | OUTLIERS: 81 AO ( 9 2010) | NO OUTLIERS DETECTED |

Exports of Goods without fuels and ships | OUTLIERS: 60 LS (12 2008), 81 AO (9 2010) | OUTLIERS: 60 LS (12 2008), 81 AO (9 2010) |

Imports of Goods | OUTLIERS: NO OUTLIERS DETECTED | OUTLIERS: NO OUTLIERS DETECTED |

Imports of Goods without fuels and ships | OUTLIERS: 39 AO (3 2007), 59 LS (11 2008), 75 AO (3 2010), 82 AO (10 2010), 139 TC (7 2015) | OUTLIERS: 39 AO (3 2007), 59 LS (11 2008), 75 AO (3 2010), 82 AO (10 2010), 139 TC (7 2015) |

Time series | Benchmark | Logs – no outliers | Levels – all outliers | TSW |
---|---|---|---|---|

Consumer Price Index (CPI) | 0.074 0.241 0.461 | 0.069 0.236 0.471 | 0.163 0.332 0.426 | 0.123 0.293 0.450 |

Harmonised Index of Consumer Prices (HICP) | 0.100 0.255 0.466 | 0.102 0.252 0.467 | 0.107 0.267 0.448 | 0.114 0.272 0.452 |

M3 | 1,551,599 947 2448 | 1,939,771 1060 2166 | 1,947,577 1116 1989 | 2,166,840 1100 1709 |

M2 | 2,410,091 1048 2440 | 2,014,253 1085 2220 | 2,224,942 1165 2046 | 2,479,304 1094 1831 |

Gross Domestic Product (GDP) | 252,244 371 1004 | 208,470 342 878 | 230,028 363 869 | 212,606 354 819 |

M1 | 1,318,053 908 1490 | 997,812 844 1704 | 1,138,385 815 1319 | 849,764 752 1470 |

Industrial Production Index (IPI) | 1.618 0.955 2.665 | 1.639 1.049 2.751 | 1.619 0.963 2.663 | 1.639 1.049 2.751 |

Retail sales | 3.159 1.423 5.111 | 4.366 1.731 3.740 | 3.815 1.591 4.194 | 4.389 1.671 3.646 |

Unemployment – thousands | 546.2 20.8 26.6 | 469.1 18.4 36.9 | 819.2 24.8 24.4 | 819.2 24.8 24.4 |

Balance of payments (BOP) – Transport – Receipts | 1919 36.0 70.3 | 1789 35.7 71.2 | 2585 38.9 62.2 | 2225 38.5 68.0 |

Balance of payments (BOP) – Sea transport – Receipts | 1215 31.2 70.2 | 1259 30.2 66.9 | 2803 45.1 59.4 | 1574 32.6 58.5 |

Unemployment – percentage | 0.399 0.584 0.544 | 0.285 0.454 0.773 | 0.399 0.584 0.544 | 0.399 0.584 0.544 |

Balance of payments (BOP) – Transport – Payments | 1002 25.4 49.7 | 1083 25.6 61.9 | 1106 27.3 37.9 | 1217 29.0 51.8 |

Imports of Goods without fuels and ships | 12,479 98.1 224.7 | 14,750 108.9 203.5 | 11,069 83.7 163.0 | 12,246 96.5 152.1 |

Exports of Goods without fuels and ships | 6020 67.3 81.3 | 3476 50.1 111.6 | 4520 58.4 71.0 | 2793 45.5 97.7 |

Exports of Goods | 20,174 130.6 138.8 | 17,024 101.8 199.6 | 20,562 133.4 133.9 | 16,877 108.5 192.8 |
---|---|---|---|---|

Balance of payments (BOP) – Sea transport – Payments | 1276 31.0 39.8 | 1418 33.4 54.5 | 1095 28.7 31.8 | 711.8 21.8 42.9 |

Imports of Goods | 97,620 263.4 345.1 | 99,541 266.2 333.7 | 93,509 250.3 324.0 | 93,330 252.6 319.3 |

Balance of payments (BOP) – Travelling – Receipts | 19,885 87.7 96.6 | 16,863 84.6 113.7 | 34,684 135.3 79.2 | 13,120 78.6 98.9 |

Balance of payments (BOP) – Travelling – Payments | 1563 24.4 28.8 | 1564 23.2 24.3 | 1687 26.9 25.7 | 1560 23.4 23.0 |