Comparison of the Holt-Winters Exponential Smoothing Method with ARIMA Models: Forecasting of GDP per Capita in Five Balkan Countries Members of European Union (EU) Post COVID

Gross Domestic Product (GDP) is the most frequently used measure of total product in an economy while GDP per capita is used for comparing living conditions or for watching the convergence or divergence among member countries of European Union (EU). This paper presents how two techniques can be applied to the same data set and how their performance can be eva-luated and compared. We chose ARIMA model and Holt-Winters exponential smoothing method to forecast the GDP per capita of five Balkan coun-tries-members of EU and to find the model that provides more accurate prediction. To achieve this, we apply the Root Mean Square Error (RMSE), the Mean Actual Error (MAE), the Mean Actual Percentage Error (MAPE), the Symmetric Mean Absolute Percentage Error (SMAPE) criteria and Theil’s U statistics. Based on statistical metrics ARIMA is the best forecasting model and fits performance for the examined period in four out of five countries.


Introduction
Long-term economic growth corresponds to a constant increase of per capita national production. Μodels of economic growth are based on the production their geo-economic position and their comparative advantages.
Due to their geographical location, the rich natural resources, the combination of Mediterranean and continental climate and the power of their human capital, Balkans are located in a very favourable position with significant environmental comparative advantages. With a capable political initiative, intellectual potential, leadership for the society and eagerness to learn from the past, Balkans countries could build an economic development and become role-models for the European continent.
The economic challenges faced by the Balkan countries today, provide an opportunity for reform, which not only can place them into the right developmental pathway, but also can revitalise its young democracies. The people who live in the Balkans are seeking a development which will provide employment, a decent lifestyle as well as promoting a balanced and prudent use of natural resources. When it comes to partnership issues about a joint green economy, regions do not start from scratch. The Energy Community Treaty which was signed in Athens in 2005 represents a starting point, since it shows that a region can support treaties for the development of single markers based on common interests and solidarity. It proves that Balkan countries can prepare themselves, not only to obtain and make a proper use of European support, but also contribute essentially to its stability, growth and energy self-sufficiency of the EU.
Moreover, the relations of the EU with Western Balkans work under the frame of stability procedures and cooperation. They aim at the political stability of the countries and enabling their transition to a market economy, also at the boost of cooperation at a region level (due to their history) and the promotion of their inclusion in the EU. The security, the political and social stability, the economic growth as well as the possibility of their inclusion in the EU, contain the key to foreign direct investment in the region.

What Does the Paper Contribute to the Understanding of Economic Processes?
Human resources are a significant determinant of economic growth and progression as well as a source of competitive advantages. Balkan countries could benefit from an improved competitiveness in labour market together with increased investments at the sectors of education, science and technology, resulting to a skilled workforce. The steps followed in order to enhance human capital in combination with the increased labour mobility, can stimulate the general competitiveness, mitigate poverty and reinforce governance in all levels. Education is the cornerstone of economic region development and its integration to the European and global community. The close cooperation on issues of practical educational initiatives could benefit the life and perspectives of all civilians. Moreover, the Balkan region attracts the admiration because of its impressive coastlines, gastronomic heritage, cultural traditions and agricultural goods which attract thousands of tourists every year. Also, the economic power of tourism could potentially be expanded from regional tourist initiatives which praise the cultural heritage and diversity while strengthening the region infrastructure by connecting individual countries. Such initiative would have generated economic income to agricultural an urban societies in the whole region while strengthening the relation between people.

What Does the Paper Contribute to the GDP Forecasting
Methodology?
Forecasts combine expert judgement with a variety of existing and new information related to current and future developments. These include recent statistical results, combined with analyses based on econometric models and techniques. An important starting point in the forecasting process is the re-appraisal of economic climate of the countries and the general global economy. This index is measured in terms of growth compared to previous year. For the case of Eurozone and G7 economies, the short-term assessment includes indices which provide estimations of the short-run increase of GDP. These models combine information from "soft" indices, namely business climate, as well as "hard indices" such as industrial production and use different data frequencies as well as a variation of estimation techniques (see Sédillot & Pain, 2003;Mourougane, 2006). Nevertheless, statistical model indices are limited to their capacity to forecast quarterly increase of GDP, since the confidence intervals for the GDP estimations for the next quarter is 70% and is fluctuated between 0.4 and 0.8 percentage points depending on the country or region and uncertainty exists as long as the forecasting period is being extended. Forecasting error could also occur for various reasons such as revisions of the initial database, such as revisions of the original data.
A popular generation of forecasting tools being used to predict and control future values of phenomena, is the Box-Jenkins methodology known as the ARIMA methodology. Box and Jenkins have proved that their method is strong, especially when generating forecasts of short-run time series. In general, we could say that ARIMA models outweigh most sophisticated structural models in terms of its short forecasting ability. This methodology has been used extensively from many researchers in forecasting studies. The popularity of this method is due to the fact that it has proven its ability to accurately predict if all the conditions of its application are fulfilled.
The Holt-Winters method or algorithm allows users to smooth a time series and use data in order to forecast areas of interest. The exponential smoothing yields exponentially declining weights and values versus historical data to reduce the weight value for older data. In other words, we could say that more recent historic data are assigned more weight in forecasting than the older results. Box-Jenkins methodology, as well as Holt-Winters methodology, is considered dynamic linear model and part of the Bayesian approach.
In this paper, a comparative analysis is presented using two widely used linear models in order to forecast the per capita GDP in Balkan countries, member countries of EU (Bulgaria, Croatia, Greece, Romania, and Slovenia).The models  (Makridakis et al., 1982). The aim of this paper is to find the model that provides The rest of the paper is organized as follows: Section 2 describes the literature review while in Section 3 the theoretical background is given. Data and descriptive statistics gross domestic product at current prices per head of population provided in Section 4. In Section 5 the empirical results are presented. Ιn Section 6 we discuss the applicability of employing the Box-Jenkins and Holt-Winters methodologies on GDP forecasting and share some thoughts on GDP forecasting in general and finally, summary and conclusions are provided in Section 7.

Literature Review
The most suggested forecasting models, analyzed by many researchers, use the statistical method as a forecasting tool for future data. In the analysis of statistic- paper is that it was based on data collected by eight places while normally a model is based on data from one place. Dritsaki (2015) used the Box-Jenkins methodology, with an ARIMA (1, 1, 1) model, between 1980 and 2013 in order to forecast the real GDP in the case of Greece. The results showed that real GDP values outside the sample were improving steadily. Galadima (2016) attempted to model and forecast the per capita income in the case of Nigeria using yearly data from 1960 to 2015. Using the Box-Jenkins methodology, the study found out that the best model is the ARIMA (1, 1, 1). The forecasting based on the assessed model for 5 years shown impressive results in the case of the real per capita income for the sampling period with the minimum error in the model. Uwimana et al. (2018) used the ARIMA models in order to forecast the GDP of the 20 biggest economies in Africa. Based on the results of their assessments, they suggest that between 1990 and 2030, there will be an increasing trend of GDP whereas the mean economic growth of Africa would reach 5.52% and the GDP could reach between 2185.21 and 10,186.18 billion US dollars.
The aim of the paper of Oral (2019) was to compare the selected exponential smoothing methods for forecasting the indices of economic growth in Turkey and to determine the most suitable technique. The results of their paper showed that the additive Holt-Winters smoothing exponential model is the most appropriate method for seasonality forecasting of Turkey's economic growth indices. Nyoni and Muchingami (2019) use yearly data of the GDP per capita of Botswana from 1960 until 2017 as well as Box-Jenkins methodology and found out that the optimum model for the data under examination is the ARIMA (3, 2, 3) one. The study results show that Botswana's living standards will continue to improve during the next decade.
da Costa et al. (2020) investigate the effectiveness of time series classical models and the state space models, which are applied on Brazil's GDP. The models used were Seasonal Autoregressive Integrated Moving Average (SARIMA) and a Holt-Winters method, which are considered as classical time series models and the dynamic linear model particularly a state-space model. According to statistical measures on model comparisons, the dynamic linear model exhibited the best forecasting model and the most appropriate fit performance for the examined period incorporating the significance of growth rate structure.
Finally, Eissa (2020) applied yearly data in order to forecast the GDP per capita in the case of Egypt and Saudi Arabia. Using Box-Jenkins methodology, the results suggest that ARIMA (1, 1, 2) and ARIMA (1, 1, 1) models are the most appropriate to assess the GDP per capita for both countries. Also, the study outcomes suggest that there will be an increase of per capita GDP in both countries.
The per capita GDP is an important economic index which collects practical information helping policy makers in decision making. In this framework, the forecasting of per capita GDP is becoming a powerful tool for decision optimization in various sectors. The majority of studies dealt with GDP, used ARIMA models or Holt-Winters models and focused in one country. The present paper analyzes and compares the per capita GDP for five Balkan countries members of European Union using both ARIMA models as well as Holt-Winters model. Economic policy makers can take into consideration the forecasting per capita GDP after 2020 (pandemia year) in order to compare the growth levels and to cooperate more close with other Balkan countries leading them to prosperity.

Theoretical Background
The procedures of time series consist of simple forecasts and smoothing methods, correlation analysis methods and ARIMA modeling. Simple forecasting methods and smoothing are based on the idea that reliable forecasts can be accomplished with modeling patterns on the data, which are usually visible on a time series plot, and afterwards with the extension of these patterns in the future. The choice of method should be based on whether the patterns are static (constant in time) or dynamic (changes in time), on the nature of trend and seasonal components and how far we want to forecast. The simple forecast and smoothing methods model data on a series and are quite easy with a time series diagram. This approach decomposes the data through its components' parts and afterwards extends the estimation of these components in the future for forecasting. Exponential smoothing is another category of data analysis during time. Data smoothing is considered either with some optimum weight produced by data estimation or is achieved with a given weight. The optimum weight of ARIMA model is gained by fitting the ARIMA (0, 1, 1) model which stores the fits. The smoothing values on ARIMA model fit but one time lag unit.

Testing for Stationarity
Advances on global economy during the last years, after the recession and pandemia, have shaken many economies worldwide. As the period that is examined in this paper contains these advances, it was advisable to use a unit root test containing these variations. Appropriate information concerning the unknown structural variations (breaks) help policymakers on the design of long run policy examining these structural breaks. Zivot and Andrews (1992) suggested three models for the examination of unit root which are the following: Model A allows for a one-time change in the intercept.

1ˆˆˆk
Model B is used to test for stationarity of the series around a broken trend.
Model C accommodates the possibility of a change in the intercept as well as a broken trend.
Null hypothesis in all three above models is ˆ0 α = which entails that y t time series contains a structural break but is not stationary. Alternative hypothesis is ˆ0 α < meaning that time series is a stationary procedure with an endogenous time structural break taking place in an unknown point in time TB.

ARIMA Models
An Autoregressive Integrated Moving Average (ARIMA) model is a generalization of an Autoregressive Moving Average (ARMA) used in econometrics. ARIMA is one of the types of models in the Box-Jenkins methodology (Box & Jenkins, 1976) for analysis and forecasting a time series (see Dritsaki & Dritsaki, 2020).
The ARIMA (p, d, q) can be expressed as: , , , p ϕ ϕ ϕ and 1 2 , , , q ϑ ϑ ϑ are the parameters of autoregressive and moving average terms with order p and q respectively. L is the difference operator defined as If 1 ρ = , the model is a random walk.  Random walk without drift: A random walk without drift is a procedure where the dependent variable can be estimated on a lagged period by itself together with an error term which is a white noise, known as random shock. The formula of a random walk without drift excludes the intercept. The mean is constant during time in a random walk featuring a non-stationary stochastic process (Gujarati & Porter, 2008).  Random walk with drift: A random walk with drift is a process where the variable is dependent from its own lagged values and a random shock. However, the model which can estimate a random walk with drift consists of an intercept known as drift which is denoted as β. This parameter indicates if time series tend upwards or downwards, according to positive or negative sign of β. A random walk with drift is a non-stationary stochastic procedure where mean and variance increase in the course of time (Gujarati & Porter, 2008).
where the constant is the average period-to-period change (the long run shift) on y t . This model could be fit as regression model without a constant in which the first difference of y t is the dependent variable. Given that it consists only a non seasonal difference and a constant term, it is classified as ARIMA (0, 1, 0) model without constant. A random walk model without drift could be an ARIMA (0, 1, 0) model without constant. The ARIMA (0, 1, 0) model is a non stationary time series, simply a random walk, a cumulative sum innovations or shocks. Considering that the expected value of an innovation is zero, the expected cumulative sum on period ahead is just the current value of cumulative sum. Thus, a forecasting is equal with the last observation. Meanwhile, a forecasting for a stationary time series will almost never be equal with the last observation.

The Box-Jenkins Methodology
Box-Jenkins procedure (Box & Jenkins, 1976) in the time series analysis is a me- which satisfactorily presents the stochastic procedure where the sample derived.
The univariate Box-Jenkins methodology is purely a tool for forecasting and is used for short time forecasting. The Box-Jenkins approach consists of the following steps:  Preparation of the data for a constant variance (series stationarity).
 Identification of the model.
 Estimation of the model.
 Diagnostic checking of the model.

Methods Exponential Smoothing (Holt-Winters Models)
When data appear a random variation during time, smoothing methods can be used to reduce or eliminate the result of these variations. So, when we discuss about time series smoothing we refer to the reduction or elimination of the variance on this series. Exponential smoothing was proposed in the late 1950s by Brown (1959), Holt (1957) and Winters (1960)  Forecasting function: Level function: Trend function: trend and without seasonal variances. This method is similar with the double smoothing method as both of them create forecasting with linear trend but without seasonal component. While the double smoothing method uses one parameter, this method uses two. The Holt-Winters model without seasonality is presented in the following equations: Level equation: Trend equation: Forecasting equation: The first equation computes the level at time t as a weighted average of the last observation of time series t y and the level for forecasting. It is based on 1 t L − at 1 t − period and on 1 t b − slope at time 1 t − . Slope expresses the linear increase of level, in a time unit, meaning that is represents the "discrete derivative" of level. The basic idea is that level moves though a straight line but due to shocks or implicit errors, deviates from the straight line. This is revealed by the fact that the last observation of time series t y is different from Finally, the third equation refers to the way that a new observation carries on h steps ahead in the future, based on past time series. In other words, it features the last level along with the last slope h steps in the future.

Forecasting
One of the main reasons in the analysis of time series model is forecasting. Forecasting is distinguished into static and dynamic. Static forecasting, known as one-step ahead forecast, is always using the lagged values of the Y time series for the forecasts. Dynamic forecasting on the other hand, known as the multi-step forecast, applies the real lagged variable Y in order to estimate the first forecasting value. Then it uses the first forecasting value in order to forecast the second forecasting value and so on (Dritsaki, 2015). If s is the first sample observation to be forecasted, then we have the following

Model Evaluation
The precision of the forecasts depends on the forecasting error. Moreover, statistic measures are being used for this purpose, such as: The Root Mean Square Error (RMSE) The Mean Actual Error measures the mean actual deviation of the prediction values from the actual ones.
The Mean Actual Percentage Error is (MAPE) We should mention that the smaller the values of the above indices, the better the fitting of the predicted time series to the actual ones, in other words, the bet- investigation. If both U 1 and U 2 Theil statistics are equal to zero then we are talking about a perfect prediction. Generally the U 1 Theil statistic lies between zero and one, whereas the U 2 Theil statistic has no boundaries.

Data
For the analysis of our paper, annual data are used for GDP per capita at 2015 reference levels (ECU/EUR) for all examined countries from 1990 until 2020. Based on this data, the ARIMA and Holt-Winters-no seasonal models are developed for each country and then are applied for forecasting per capita GDP. Annual data for GDP per capita are downloaded from the World Bank's Development Indicators. In Table 1

 Time series plots
On the following figure, the graphs of GDP per capita for each country are featured.
From Figure 1 we can see that GDP per capita in four countries (except Greece) have an upward trend in a large time span until 2019 and afterwards a downfall because of COVID-19 pandemia. Nevertheless, focusing on the movement of GDP per capita for all countries we can argue that it is a random walk model.  Estimation of linear series trend.
In the following diagram, the linear trend model and trend analysis plot for the examined countries are presented ( Figure 2).
The results show that there is a linear trend of GDP per capita for the examined countries. So, the GDP per capita for all countries present a linear trend and is regarded as a random walk model.

Testing for Stationarity
In Table 2, the corresponding timing of the structural break of GDP per capita for each country is presented.

Estimation and Diagnostic Tests of the Models
From the moment that the models are ARIMΑ (0, 1, 0) random walk models, the estimation will be employed using the ordinary least squares methodology. The diagnostic tests of the models consist of the specification model (Ramsey, 1969) RESET test, residuals' normality (Jarque & Bera, 1980), residuals' autocorrelation (Breusch & Godfrey, 1981), heteroscedasticity (White, 1980) and autoregressive conditional heteroscedasticity (ARCH model (Engle, 1982)). Table 3 features the results of all models' estimation.
The results from the above table indicate that coefficients' from all models are statistical significant in 1% level of significance. Diagnostic tests in most of the models and for all countries face slight problems, so we can proceed on with forecasting. Notes: 1. *, ** and *** show significant at 1%, 5% and 10% levels respectively. 2. Critical values intercept: −5.34 (1%), −4.93 (5%), −4.58 (10%), trend: −4.80 (1%), −4.42 (5%), −4.11 (10%), both: −5.57 (1%), −5.08 (5%), −4.82 (10%). 3. In brackets we note down the time lags for the corresponding equations. 4. The optimal lag length is selected using t-sig, with the maximum lag set to 4. Notes: 1. *, ** and *** show significant at 1%, 5% and 10% levels respectively. Table 4 presents the forecasting accuracy of the three exponential smoothing models which are computed on the basis of statistics measures. The results in Table 4, with all forecasting indices, indicate that Holt-Winters model is the best for all countries in obtaining a much accurate short-term out-of-sample of GDP per capita. So, this model can be used for forecasting. Table 5 presents the smoothing parameters, the level, trend and the sum of squares residuals for the non seasonal Holt-Winters model on the examined countries. The smoothing parameters, alpha and beta, are used for the components estimation on the level and on trend. Smoothing parameters get values between 0 and 1. Values near zero denote a relatively little weight on recent observations when making forecasts of future values. Conversely, values closer to 1 denote that a great weight falls on the observations of the distant past in order to obtain future forecast values. Table 6 presents the indices of forecasting accuracy on ARIMA (0, 1, 0) random walk model and Holt-Winters-non seasonal for the examined countries. For evaluation sample we define the period 2020-2024 giving four years prediction and as training sample for the least-square, mean square error methods and MSE ranks we define the years 2015-2019.

Forecasting Accuracy: Seasonal-ARIMA vs. Holt-Winters
From Table 6, the out-of-sample forecast performances of ARIMA models and Holt-Winters model were ranked for all countries using accuracy measure statistics: RMSE, MAE, MAPE, SMAPE, and Theil U 1 . The ARIMA model showed the smallest forecasting errors in four out of five examined countries whereas for Romania, the Holt-Winters model of non seasonal smoothing was the most suitable for obtaining a much accurate short-term out-of-sample GDP per capita. Forecasting evaluation criteria denote that composite forecasts are superior compared with the individual models. Table 7 presents the forecasted values for ARIMA (Bulgaria, Croatia, Greece Slovenia) and Holt-Winters (Romania) models for four periods ahead starting from 2021 until 2024.
The forecasted values in all countries for GDP per capita and the corresponding models are presented in Figure 3.
The results in Figure 3 show an increase of forecasted values on GDP per capita for four countries (Bulgaria, Croatia, Romania and Slovenia) for the years 2021 until 2024. Specifically, Bulgaria will have the largest increase on GDP per    capita for the next four years and Romania will follow. Croatia and Slovenia seems from Figure 3 to have the same increase rates on GDP per capita. Greece is the only country which will have the smallest increase on GDP per capita for the forecasted years (in green colour). The fact that Greek economy is affected by the global financial crisis, the Greek debt crisis and pandemia resulted in the shrinkage of GDP per capita which will continue in the years ahead.
The forecasted values of GDP per capita in percentages for the following years are showed in Table 8.
From Table 8, it seems that the most important percentage increase of GDP per capita in the next four years is found in Bulgaria and the lowest increase in Greece.

Discussion
The significance of per capita GDP as a measure of economic growth could be examined in three different aspects. First of all, per capita GDP reflects the magnitude of economic growth in industrial countries. Second, in order to achieve a fair measurement of the GDP per capita, the individual income should not differ from one country to another. Therefore, countries that emphasize improving per capita income should also pay attention to social justice and equality. Third, it has been proved that per capita GDP relates to the level of social stability in a country (Zhang, 2013).
Given the importance of GDP per capita as a measure of economic growth, its forecast is a useful tool for conducting economic policy. The future is uncertain and forecasting the future is thus inherently difficult. Those responsible for policy-making decisions need to understand the economic condition of a country in order to make the best possible policy decisions. Given that GDP is considered a very important index for every government, its forecast could become a very useful tool for policymakers not only in the case of delivering an economic development plan but also dealing with possible recessions in advance. These decisions are made frequently under uncertainty not only as far as future economic conditions are concerned, but also under the current economic situation. Based M. Dritsaki, C. Dritsaki

Summary and Conclusion
The current paper aims at forecasting the GDP per capita for the period between Finally, the results of the analysis showed an increase in the predicted prices of the GDP per capita for all countries between 2021 and 2024 for both models under examination. The largest GDP per capita increase for this period was detected in Bulgaria and was more than 5% for all years, while for the case of Greece it was the lowest. While both models provided predictions with great accuracy, the unexpected drop of the GDP per capita in all countries during the pandemic period of 2020 seems to have affected Greece because of the touristic product.
In addition, we should highlight that the sudden COVID-19 outburst, had a huge impact on the world economy, disturbing the economic activity of every country, corporate income as well as household income. Pandemia drove many countries to suspension, closing down business activity. As a result, GDP per capita had a prompt decline in all countries and also Balkan countries. Specifically, the decrease of GDP per capita in Bulgaria was 3.41%, in Croatia 9.77%, in Greece 9.94%, in Romania 3.48% and in Slovenia 5.6% in 2020. However, the results of the paper show an increase in the forecasted values on GDP per capita for all examined countries from 2021 until 2024.
The forecasting models have been created based on time series regression, although in recent years progress in forecasting macroeconomic variables using neural networks and machine learning algorithms, has been occurred. Neural networks and ML algorithms though require access to big data set in order to produce sufficient results, to understand complex patterns and relationships between variables. Thus, the current paper uses Box-Jenkins and Holt-Winters methodologies which can use a small number of data, as the most appropriate for forecasting GDP increase.