Indicators of technical analysis on the basis of moving averages as prognostic methods in the food industry

Journal of Competitiveness 102 INDICATORS OF TECHNICAL ANALYSIS ON THE BASIS OF MOVING AVERAGES AS PROGNOSTIC METHODS IN THE FOOD INDUSTRY Andrea Kolkova Abstract Competitiveness is an important factor in a company’s ability to achieve success, and proper forecasting can be a fundamental source of competitive advantage for an enterprise. The aim of this study is to show the possibility of using technical analysis indicators in forecasting prices in the food industry in comparison with classical methods, namely exponential smoothing. In the food industry, competitiveness is also a key element of business. Competitiveness, however, requires not only a thorough historical analysis not only of but also forecasting. Forecasting methods are very complex and are often prevented from wider application to increase competitiveness. The indicators of technical analysis meet the criteria of simplicity and can therefore be a good way to increase competitiveness through proper forecasting. In this manuscript, the use of simple forecasting tools is confirmed for the period of 2009-2018. The analysis was completed using data on the main raw materials of the food industry, namely wheat food, wheat forage, malting barley, milk, apples and potatoes, for which monthly data from January 2009 to February 2018 was collected. The data file has been analyzed and modified, with an analysis of indicators based on rolling averages selected. The indicators were compared using exponential smoothing forecasting. Accuracy RMSE and MAPE criteria were selected. The results show that, while the use of indicators as a default setting is inappropriate in business economics, their accuracy is not as strong as the accuracy provided by exponential smoothing. In the following section, the models were optimized. With these optimized parameters, technical indicators seem to be an appropriate tool. Keywords: forecasting, technical indicator, exponential smoothing, simple average moving, exponential average moving, competitiveness JEL Classification: C53, G17, M21 Received: May, 2018 1st Revision: October, 2018 Accepted: November, 2018 1. INTRODUCTION Prognosis is an integral part of corporate governance. Prognostic practice is currently applied using a wide range of different approaches and methods. Forecasting methods can be classified in two ways. Qualitative methods include for example personal evaluation, panel match, the Delphi ▪ Kolkova, A. (2018). Indicators of Technical Analysis on the Basis of Moving Averages as Prognostic Methods in the Food Industry. Journal of Competitiveness, 10(4), 102–119. https://doi.org/10.7441/ joc.2018.04.07 joc4-2018-v2.indd 102 1.12.2018 11:18:02 103 method, historical comparison, and market research. The second group consists of quantitative methods, mostly reling on trending or causal models. In this paper, certain quantitative methods will be applied, namely trend design. The importance of using quantitative methods in business was evidenced in a research by Wisniewski (1996), with the proportion of enterprises using quantitative methods found to be 66 %. A rate of 24 % of companies indicated that the benefit of these methods is very high, while 7 % of respondents in this research claimed no benefit. At this time, most business managers in enterprises applying quantitative methods used them to establish basic and descriptive statistics, cash flow discounting, quality control and inventory. Approximately 67 % of companies used decision-making, compensation methods, with more than 50 % of such companies using simulations or regression analysis. Of course, it can be assumed that the use of quantitative methods in the corporate economy has increased even more with the development of computing. With the proliferation of this technology, the number and complexity of the methods and models used for the prognosis of business variables have also increased. We can now make prognoses-based predictions using fuzzy logic, artificial neural networks, genetic algorithms, as well as chaos theory. The aim of this study is to show the possibility of using technical analysis indicators, a method otherwise used predominantly for stocks, currencies and other financial assets, in predicting prices in the food industry in comparison with classical methods, namely exponential smoothing. This analysis examines accuracy based on ex-post forecasting. 2. THEORETICAL BACKGROUND The history of prognosis is relatively short, dating only from the 1960s and early 1970s. The categorization as a separate scientific discipline is not unambiguous, and even the very definition of prognosis has varied considerably since its inception. For example, Holcr (1981) defines prognosis as a form of a forecast which meets certain requirements, and it must contain the time or space interval in which the predicted phenomenon is or will be discovered. The interval must be final, and there must be a principle possibility of an a priori estimation of the predicted phenomenon; the predicted phenomenon must be verifiable and, finally, the particular prognosis must be formulated completely accurately and unambiguously. Gál (1999) defines prognosis as a conditional statement about the future of an object or phenomenon based on scientific knowledge. According to Wishniewski (1996), the intention of prognosis is to reduce the uncertainty of knowledge about the future and provide additional information to allow managers to assess alternative options in the context of future conditions as well as to evaluate the future consequences of current decisions. More modern approaches to forecasting then include the definition of the prognosis as a method of transforming past experience into the expected future. To Vincur & Zajac, prognosis (2007, p. 12) is defined as a scientific discipline, the subject of joc4-2018-v2.indd 103 1.12.2018 11:18:02 Journal of Competitiveness 104 which is the study of the technical, scientific, economic and social factors and processes that act on the development of the world’s objective reality and which aims to create a vision - the prognosis of a future condition resulting from the interconnected effects of these factors and processes. Forecasting methods can be broken down into several categories, with the most well-known and most widely used divisions being within the general categories of qualitative and quantitative methods. Miller & Swinehart (2010) categorized methods into three different groups: exploratory or normative methods, evidence-based methods, and assumptions based on evidence. The third grouping is then a classical breakdown into qualitative and quantitative methods. Moro et al. (2015) classify methods as quantitative, semi-quantitative and qualitative methods. Kesten & Armstrong (2014) divide forecasting methods into simple and complex forecasting along with a whole range of other subdivisions, as depicted in Figure 1. Fig. 1 – Methodology Tree of Forecasting. Source: Armstrong & Green, 2014. In this paper, the breakdowns set forth in Esmaelian et al. (2017) on quantitative, semi-quantitative and qualitative methods will be used. 2.1 Qualitative forecasting Qualitative methods usually do not duplicate numerical evaluations of data, but the professional appreciation and verbal evaluation of the studied variables. These methods include, for example, an expert panel where a group of experts within a given organization study and discuss a given quantity from different points of view (Wisnivski, 1996). Another method is the relevant tree (Daim et al, 2006), a way of identifying the development phases, objectives and basic elements Index Data Mining/ Analytics Statistical Univariate Theorybased Databased Extrapolation models Multivariate Rule-based forecasting Unaided judgment Judgmental SelfOthers Simulated interaction (Role playing) Role No role Conjoint analysis Knowledge source Quantitative analogies Unstructured Structured Expert Forecasting Decomposition Structured analogies Neural networks Expert systems Intentions/ Expectations/ Experimentation Judgmental bootstrapping Segmentation Linear Classification Causal methods Regression analysis joc4-2018-v2.indd 104 1.12.2018 11:18:02 105 of a given enterprise quantity. A very similar method is the futures wheel, in which the event or quantity being investigated is considered the core of a wheel, and events or variables that can affect it are considered to be vanes. A very well-known and used technique is the SWOT analysis method, by which experts identify the strengths, weaknesses, opportunities and threats of the company or product. The literature review can also be considered another search method (Moro et al, 2015). 2.2 Quantitative forecasting These methods are usually based on mathematical-statistical techniques and numerical calculations, as indicated in Esmaelian et al. (2017). These include: trend analysis and trend extrapolation, which will be detailed in Chapter 3.1. Multi-stage analysis is a method that combines several models, as defined along with other concepts by Antonic et al. (2011). We can also include the lesser known Future Workshop method by Martino (2003), as well as system dynamics, a method that makes predictions based on dynamic tools such as neural networks, fuzzy logic, genetic algorithms, or chaos theory. In this paper, among the quantitative methods of forecasting, new methods of technical analysis will be included as possible tools of forecasting in the corporate economy. These will be presented along with the trend analysis and trend extrapolation method, which explained in greater detail in Chapters 3.1 and 3.2. 2.3 Semi-quantitative methods Semi-quantitative methods include, for example, monitoring. This method uses systematic loops to identify ideal conditions by means of feedback information. Another popular method is brainstorming, a process that collects a set of ideas about the future of an individual or a group of people. Morphological analysis, questionnaire/surveys, scenario planning can also be characterized as this type of method. The Delphi method (Esmaelian, 2017), which uses questionnaires in consecutive rounds to gather the views of as many experts as possible and to reach consensus, has also become popular. Also in wide use is stakeholder mapping (Saritas et al., 2013), (Vishnevskiy, 2015), a method which uses statistical techniques to predict who the stakeholders are, where they are and why they are interested in the product, bailout, etc. The text / data mining method used by, for example, Moro et al (2015), is one of the most recent techniques put into use. 3. RESEARCH OBJECTIVE AND METHODOLOGY In this paper, a prognosis regarding the evolution of selected prices in the food industry will be based on historical prices and the ex-post forecast will be tested. The high prediction capability of the ex-post model is a prerequisite for using the ex-ante prognosis model. The ex-post relationship and the ex-ante prognosis are shown in Figure 2. joc4-2018-v2.indd 105 1.12.2018 11:18:02 Fig. 2 – Time in Forecasting. Source: own according to Marček (2013), Vincúr (2007) Data for the main raw materials of the food industry, namely wheat food, wheat forage, malting barley, milk, apples and potatoes, has been analyzed. The data was obtained from the Czech Statistical Office from the monthly data collections from January 2009 to February 2018 in the Czech Republic. The data file has been analyzed and modified. Missing values were found regarding milk and potatoes and replaced by linear interpolation. Descriptive statistics of the data are defined in Table 1. Tab. 1 – Descriptive Statistic of the Analyzed Data. Source: own N Min Max Mean SD Variance Statistic SE Statistic wheat food 110 2612 6117 4214.04 87.305 915.662 838436.090 wheat forage 110 2400 5714 3838.84 76.713 804.575 647340.560 malting barley 110 3055 6029 4641.52 65.566 687.662 472878.894 cow’s milk 110 5921 9808 7860.31 98.128 1029.174 1059199.738 potatoes 111 2159.0 7314.0 4498.469 132.0651 1391.3926 1935973.272 apples 111 6931.0 14493.0 9895.765 126.8946 1336.9180 1787349.705 The statistical programs SPSS and R (with TTR and FORECAST packages) were used for the analysis. 3.1 Forecasts based on exponential equalization For this article, quantitative methods of forecasting based on exponential alignment were selected. Exponential smoothing is used for short-term forecasting in various modifications. Prognoses based on exponential smoothing consist of weighted averages of past values, with scales exponentially decreasing with the age of the data used (Hyndman, 2018). Exponential alignment methods include simple exponential smoothing, Holt’s exponential smoothing and Winter’s exponential smoothing. As Bergmeir et al (2016) states, “the general idea of exponential smoothing is that recent observations are more relevant to forecasting than older observations, meaning that they should be weighted more highly.” Parametr Estimation Period Forecasting ex post Forecasting ex ante Present time Period of Forecasting Period of Quantification Time joc4-2018-v2.indd 106 1.12.2018 11:18:02 107 Simple exponential smoothing defines the prognosis as an exponential average and is used only for non-periodic time series. The relationship of the extended equation has the Shape, ܵ௧=ߙσ(1െߙ)௜ݕ௧ିଵ+(1െߙ)௧ܵ଴ ௧ିଵ ௜ୀ଴ , where (1) ߚ=ߚ(ܵ௧െܵ௧ିଵ)+ (1 െߚ)ߚଵ,௧ିଵ ෣, where (2) ܵ௧െܵ௧ିଵ =ߚ଴,௧ ෢െߚ଴,௧ିଵ ෣ with T being the length of the time series, yt-1 the value of the time series, α ∈ (0, 1) the equalization constant, and S0 the initial equalization value Brown’s multiple exponential smoothing defines the prognosis of polynomial trends with multiple exponential averages, which are obtained by another exponential equalization of already obtained exponential averages. Holt extended Brown’s exponential smoothing by an adaptive estimation of the trend component with the new balancing constant β (Vincur & Zajac 2007). The equalization constant can be defined thusly, ܵ௧=ߙσ(1െߙ)௜ݕ௧ିଵ+(1െߙ)௧ܵ଴ ௧ିଵ ௜ୀ଴ , where (1) ߚ=ߚ(ܵ௧െܵ௧ିଵ)+ (1 െߚ)ߚଵ,௧ିଵ ෣, where (2) ܵ௧െܵ௧ିଵ =ߚ଴,௧ ෢െߚ଴,௧ିଵ ෣ ܵ௧=ߙσ(1െߙ)௜ݕ௧ିଵ+(1െߙ)௧ܵ଴ ௧ିଵ ௜ୀ଴ , where (1) ߚ=ߚ(ܵ௧െܵ௧ିଵ)+ (1 െߚ)ߚଵ,௧ିଵ ෣, where (2) ܵ௧െܵ௧ିଵ =ߚ଴,௧ ෢െߚ଴,௧ିଵ ෣ is the current state of trend and β1, t-1 is the adaptive estimate of the trend directive over time. β is then the equalization constant. Holt’s double parametric linear exponential smoothing is a modification for the stochastic trend series. Damped trend methods have emerged as a response to the drawbacks of Holt linear methods that show a continuous trend. Empirical evidence, however, suggests that this can lead to excessive forecasts, especially in the longer forecast horizon. Methods of damped trends then include a parameter that dampens the trend on a straight line (Hyndman, 2018). There are currently several other methods summarized by Taylor (2003) as an additive damped trend method, multiplicative damped trend method, additive Holt-Winters method, multiplicative Holt-Winters method, Holt-Winters damped method. 3.2 Forecasts based on technical analysis indicators The objective of the technical analysis is to anticipate the future development of assets based on an analysis of their past developments. Techniques based on technical indicators are always based on mathematical statistics. The technical analysis uses not only technical indicators, but also graphical methods, with a more modern name of price action which are known even from the 18th century, when the Japanese applied their first candle charts to their rice deals. Today, they are published slightly less than technical indicators such as Lee & Jo (1999), and are the subject of research rather based on programming. There are a lot of technical indicators. Back in 1988, Colby published an encyclopedia of technical market indicators (Colby, 2003). George Lane published his Lane’s stochastic oscillator more than three decades ago (Lane, 1984), or even in the 1970s, the relative strength developed by Wilder (1978). In the 80s-90s of the 20th century, indicators belonging to a group of channel systems were published, for Bollinger bands, John Bollinger (Bollinger, 1992), or Kaufman (1987). Of the newer indicators, for example, the Chaikin oscillator is known (Achelis, 2001) or today the most widely used MACD indicator introduced by Appel (2005). In 2007 (Cheung & Kaymak, 2007), a concept combining technical indicators and fuzzy logic was introduced. Abbasi joc4-2018-v2.indd 107 1.12.2018 11:18:02 Journal of Competitiveness 108 and Abouec also used a system derived from neuro-fuzzy logic (Abbasi & Abouec, 2008). In 2009, Chavarnakul & Enke, (2009) developed a hybrid exchange trading model using the Neurofuzzy concept called the Genetic Algorithm (NF-GA). In 2015, technical indicators (specifically MACD and the lesser-known Gann-Hilo indicator) and fuzzy logic were used again (Chourmouziadis & Chatzoglou, 2015). Currently, there are still new indicators based on both fuzzy modeling and a combination of individual statistical and mathematical indicators, and so the list of indicators is far from complete. The existing ones are then subjected to various tests (da Costa, 2015; Kolkova, 2017; Kresta, 2015). In this study, an innovative attempt is made to apply technical indicators to business economy phenomena as well. Technical analysis indicators have not yet been used to predict the business economy and are not yet part of any research work, so their use can be a tool to significantly increase the competitiveness of the business. For the sake of scale, only some technical indicators have been selected, namely indicators on the basis of rolling averages, which are also one of the most used in the practice of financial transactions. Since the exponential equalization method is also based on the methodological basis of moving averages, it can be assumed that these indicators may also be an appropriate tool for predicting business phenomena. Sliding averages calculate the average value of the data in the width of its timeframe. For example, a 7-day moving average means the average value of the last week, 14 days in the last two weeks. After joining the rolling average of all days, we create a rolling average curve. The moving average is now a whole range. The basis is Simple Moving Average (SMA) and can be defined by the relationship, ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) N is the number of days for which the SMA is numbered. Moving averages are used to smooth the data in an array to help eliminate noise and identify trends. The simple moving average is literally the simplest form of a moving average. Each output value is the average of the previous n values. In a simple moving average, each value in the time period carries equal weight, and values outside of the time period are not included in the average. This makes it less responsive to recent changes in the data, which can be useful for filtering out those changes. Exponential moving average (EMA) is considered to be a better tool than a simple moving average (Elder, 2006) because it attaches greater weight to current data and changes in price correspond faster than with the simple one. It is used in countless technical indicators. It can be expressed by the relationship, ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) N is the number of days to quantify the EMA. Double exponential moving average (hereafter DEMA), as reported by FM Labs (2016), is a smoothing indicator less lag than straight EMA. It is more complex than just moving average. DEMA was developed by Mulloy (1994). It can be defined by the relationship, joc4-2018-v2.indd 108 1.12.2018 11:18:03 109 ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) The Zero-Lag exponential moving average is a variation of the EMA. This indicator was created by Ehlers & Way (2010) and keeps the benefit of the heavier weighting of recent values but attempts to remove lag by subtracting older data to minimize the cumulative effect. It is expressed by the relationship, ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) Hull moving average (hereafter only HMA), developed by Alan Hull (2012), is an improved variant of the moving average, which shows the moment of trend reversal quite accurately. It is defined by the relationship, ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) WMA is weighted moving average. Arnaud Legoux moving average (hereafter ALMA) by the authors Legoux & Kouzis-Loukas uses the curve of the normal (Gauss) distribution which can be placed by offset parameter from 0 to 1. This parameter allows regulating the smoothness and high sensitivity of the moving average. Sigma is another parameter that is responsible for the shape of the curve coefficients. This moving average reduces a lag of the information but still being smooth to reduce noises. ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) size is the window size. 3.3 Forecasting accuracy If it is possible to predict the values by multiple methods or models, it is advisable to choose the one that provides the smallest errors. The error rate should be evaluated at the time of known values in the ex-post forecasting period. For the evaluation then, if the chosen error estimating variable is 0 then the prognosis is flawless. In the case of a positive error, the model underestimates the fact, and vice versa, in the case of a negative model error, the fact overestimates the fact. R-squared, root mean square error (hereafter RMSE), mean absolute percentage error (hereafter MAPE), maximum absolute perceived error (hereafter MaxAPE), mean absolute error (hereafter MAE), maximum absolute error (hereafter MaxAE) and the normalized Bayesian information criterion (hereafter Normalized BIC) are used as prognostic accuracy measures. The formulas in this article were drawn mainly from (Vincur & Zajac, 2007) and (Marček, 2013). R-squared is usually called the coefficient of determination. It is the proportion of variation in variable explained by the model, joc4-2018-v2.indd 109 1.12.2018 11:18:04 Journal of Competitiveness 110 ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) R-squared statistics, however, are generally considered to have relatively poor predictive abilities. Armstrong (2001, p. 461) identified 6 studies on the use of R-Squared and found a relatively small relationship with the precision forecast. Therefore, other statistics are introduced, which are simpler, more useful, but also easier to understand than R-squared (Hyndman & Koehler, 2006). Newer methods are discussed in Chen et al (2017). When defining the RMSE variable, it is necessary to first describe the Mean square error (MSE), which expresses an average square error by the relationship, ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) However, it is preferable to use the standard deviation, which is RMSE by the relationship, ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) MSE and RMSE have the same unit as the original time series. In the same units, the MAE is also declared, this being the average deviation of the actual values from the forecasts. It can be described by the relationship, ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) Other indicators are referred to as relative forecast accuracy rates and are expressed as a percentage. These indicators do not depend on the time series units of measure, and therefore we can use them when comparing the forecast accuracy of different variables. One of these is, for example, MAPE, which represents an average error of forecasts compared to actual values by the relationship, ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . (23) The MaxAPE value is also expressed as a percentage and represents the largest predicted error. On this basis, we can get an idea of the worst possible scenario of our forecast. Max AE is then the largest forecasted error, expressed in the same units as the dependent series. Like MaxAPE, it is useful for imagining the worst-case scenario of your forecasts. Maximum absolute error and maximum absolute percentage error may occur at different series points, for example when the absolute error for a large series value is slightly larger than the absolute error for a small series value. In that case, the maximum absolute error will occur at the larger series value and the maximum absolute percentage error will occur at the smaller series value. Normalized Bayesian Information Criterion (hereafter Normalized BIC) can be defined, a general measure of the overall fit of a model that attempts to account for model complexity. It is a score based upon the mean square error and includes a penalty for the number of parameters in the model and the length of the series. The penalty removes the advantage of models with more parameters, making the statistics easy to compare across different models for the same series. ܵܯܣ=σ௜௡௣௨௧ ಿ భ௡, where (3) ܧܯܣ=ܧܯܣିଵ+ܭή(݅݊݌ݑݐെܧܯܣିଵ), or (4) ܧܯܣ=ܭή݅݊݌ݑݐ+ (1 െܭ)ήܧܯܣିଵ , where (5) 1 2  N K , where (6) ܦܧܯܣ= 2 ήܧܯܣ(݅݊݌ݑݐ)െܧܯܣ(ܧܯܣ(݅݊݌ݑݐ)) (7) ܼܮܧܯܣ=ܭή൫2ή݅݊݌ݑݐ଴െ݅݊݌ݑݐି௟௔௚൯+ (1 െܭ)ήܼܮܧܯܣିଵ, where (8) ݈ܽ݃=௡ିଵ ଶ. (9) ܪܯܣ௧=ଵ σ௜ ೞ ೔సభ ήσ(ݏെ݅)(݂݂݀݅௧ି௜) ௦ିଵ ௜ୀ଴ , where (10) ݉=ቔ௡ ଶቕ, (11) ݏ=උξ݊ඏ, (12) ݂݅ݎݏݐ௧=ଵ σ௜ ೞ ೔సభ ήσ(݉െ݅)(݅݊݌ݑݐ௧ି௜) ௠ିଵ ௜ୀ଴ , (13) ݏ݁ܿ݋݊݀௧=ଵ σ௜ ೞ ೔సభ ήσ(݊െ݅)(݅݊݌ݑݐ௧ି௜) ௡ିଵ ௜ୀ଴ , (14) ݂݂݀݅௧= 2 ή݂݅ݎݏݐ௧െݏ݁ܿ݋݊݀௧, or (15) ܪܯܣ=ܹܯܣ(2 ήܹܯܣ௡ ଶെܹܯܣ(݊),ݏݍݎݐ(݊)), where (16) ܣܮܯܣ=ଵ ேைோெσ݌(݅)݁ି(೔ష೚೑೑ೞ೐೟)మ ഑మ ௦௜௭௘ ௜ୀଵ , where (17) ܴଶ=σ൫௬೛ି௬೛ ෞ൯మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ , or (18) ܴଶ= 1 െσ௘೛మ ಿ ೛సభ σ൫௬೛ି௬ ത൯మ ಿ ೛సభ . (19) ܯܵܧ=ଵ ெσ൫ݕ௣െݕ௣ ෞ൯ଶ ே ௣ୀ௡ାଵ . (20) ܴܯܵܧ=ξܯܵܧ. (21) ܯܣܧ=ଵ ெσหݕ௣െݕ௣ ෞห ே ௣ୀ௡ାଵ . (22) ܯܣܲܧ=ଵ ெσห௬೛ି௬೛ ෞห ௬೛ ଶ ே ௣ୀ௡ାଵ . 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