Plane Thermoelastic Waves in Infinite Half-Space Caused FACTA UNIVERSITATIS Series: Economics and Organization Vol. 12, No 4, 2015, pp. 323 - 332 EXAMINATION OF THE PERFORMANCES OF MAXIMUM LIKELIHOOD METHOD AND BAYESIAN APPROACH IN ESTIMATING SALES LEVEL  UDC 31:658.8 Nataša Papić-Blagojević 1 , Vinko Lepojević 2 , Sanja Lončar 3 1 Novi Sad Business School, Novi Sad, Serbia 2 University of Niš, Faculty of Economics, Niš, Serbia 3 Novi Sad Business School, Novi Sad, Serbia Abstract. The method of maximum likelihood and Bayesian method are widely used in data processing, not only in economics but also in other fields of research. In order to identify which approach has better performances, these methods are analyzed on the selected economic data. By comparing the estimated values obtained by applying the maximum likelihood method and Bayesian method on the data that was obtained from the company CaliVita Int., it was concluded that the Bayesian inference with informative priors gives more accurate estimates. Key words: method of maximum likelihood, Bayesian method, estimation, prior, sales level INTRODUCTION Historically, classical statistics have had the main role in researches compared to Bayesian methods, but in the near past, Bayesian approach has become very popular in creating statistical models for solving problems in different research fields. Bayesian inference is known as an analytical method that combines information obtained from the experiment with prior knowledge. Contrary to the maximum-likelihood approach, the Bayesian framework requires the explicit prescription of a prior probability distribution for the unknown signal parameters [15]. On the other side, the method of maximum likelihood, as a method of classical statistics, does not include any prior information that maybe exists from previous research. Very often, researchers are faced with the problem that is common for one data set and they have to estimate parameters in the moment of taking data. Received December 10, 2015 / Accepted December 29, 2015 Corresponding author: Nataša Papić-Blagojević Novi Sad Business School, Novi Sad, Serbia E-mail: npapic.blagojevic@gmail.com 324 N. PAPIĆ-BLAGOJEVIĆ, V.LEPOJEVIĆ, S. LONĈAR Many supporters of classical statistics do not accept the use of subjective prior information in objective economic science. The debate about the role of prior information in statistics is still going on. Actually, prior information is a controversial aspect of Bayesian methods, but Bayesians, as the final line of defence, have developed non- informative priors for many classes of model [9]. In order to get conclusions about the importance of prior information, the results of research with and without presence of prior information are compared in this paper. 1. LITERATURE REVIEW Even though Bayesian methods are powerful and can be used in a variety of analytic models, the strenuous programming and computational demands have discouraged many researchers from using them [19]. Today, the use of Bayesian methods in empirical researches is rapidly growing, because there is a lot of appropriate statistical software that can be successfully applied in different situations. According to the complexity, the models are very different, so the use of some is very simple and allows the researcher to easily reach the desired estimates and test statistics, while others require researchers to possess programming skills. The maximum likelihood method has broad and significant application in determining the statistical estimations with good characteristics. Its application, as well as application of Bayesian methods, is not only limited to economics, but it can be successfully implemented in psychology, medicine, biology, tourism, etc. Lemmon, Brown, Stanger-Hall and Lemmon [11] study the effect of ambiguous data, or missing values for research in biology using the method of maximum likelihood and Bayesian method. Ward [18] in his paper compared Bayesian and classical methods for estimation of ecological models, where the maximum likelihood criteria consistently favoured simpler population models when compared to Bayesian criteria. On the other hand, Flurry and Shepard [6] studied the wide application of Bayesian inference and likelihood methods in microeconomics, macroeconomics and financial econometrics. In doing so, they illustrate these methods on four problems in econometrics, producing rather generic methods. Taken together, these methods imply that if we can simulate from an economic model we can carry out likelihood based inference using its simulations [6]. Pitt, Silva, Giordani and Kohn [13] are concerned with developing a methodology for Bayesian inference for general time series state space models using Markov chain Monte Carlo (MCMC) simulation with the likelihood estimation. Fernández-Villaverde and Rubio-Ramírez [5] showed how to undertake likelihood-based inference in dynamic macroeconomic models. They also describe how to use the output to estimate the structural parameters of the model, those characterizing preferences and technology, and to compare different economies. Both tasks can be implemented from either a classical or a Bayesian perspective. Bayesian estimation may also be used for solving some problems that are commonly encountered in traditional statistics; for example, obtaining estimations for impossible parameters, identification of the model [8], and obtaining more precise parameters estimations [3]. Also, Bayesian methods are more plausible ways to analyze small sample data compared with the maximum likelihood method [19]. In this paper, the comparison of the maximum likelihood and Bayesian method is made on the data that was obtained from Examination of the Performances of Maximum Likelihood Method and Bayesian Approach... 325 the company CaliVita Int. Calculations were made for two types of prior information that can be applied in Bayesian inference, informative and non-informative, and the results are compared with maximum likelihood estimates. 2. RESEARCH METHODOLOGY AND RESULTS In order to make a comparison of the methods of classical and Bayesian statistics, the data were obtained from the company CaliVita Int. representative for the Republic of Serbia, Fitco LLC, Novi Sad. The selected data related to 252 products [12]. For research purposes, the products are sorted by: product name, product type, manufacturer, price and realized sales in the period from January to June 2014 and processed in statistical packages IBM SPSS and IBM SPSS Amos Version 21. IBM SPSS Amos applies a general approach to data analysis known as Structural Equation Modeling –SEM. It is also known as the analysis of covariance structures or causal modeling. A sample of 252 products is considered as sufficient for successful research tests because Bayesian statistics is not based on large samples [17]. Many articles also show the advantages of Bayesian statistics in terms of a small data set [19]; [10]. To perform the analysis, four variables were selected: Product type, Manufacturer, Sales level in May and Sales level in June. The variables Sales level in May and Sales level in June are defined as observed, endogenous variables. They are conditioned with two observed, exogenous variables: the Product type and the Manufacturer. Figure 1 shows the model for selected data. Fig. 1 Structural model Thus defined, the model indicates the need to test the impact of product types and manufacturers to sales level in May and June. At the same time, there is a need for examining the mutual influence and a correlation between the variables Product type and the Manufacturer. 2.1. Research results obtained by method of maximum likelihood The application of proper statistical package has enabled the estimation of the collected data using method of maximum likelihood and Bayesian method, and subsequently, drawing 326 N. PAPIĆ-BLAGOJEVIĆ, V.LEPOJEVIĆ, S. LONĈAR conclusions by comparing the results of the research. Table 1 represents the results obtained by applying maximum likelihood method on the selected data. Table 1 Maximum Likelihood Estimates Regression Weights: (Group number 1 - Default model) Estimate S.E. C.R. P Label SL_May <-- Manuf -3.730 0.792 -4.709 *** SL_May <-- PrType -0.941 1.502 -0.626 0.531 SL_June <-- Manuf -1.090 .465 -2.346 0.019 SL_June <-- SL_May 0.730 .035 20.581 *** SL_June <-- PrType -0.033 .845 -0.039 0.969 Standardized Regression Weights: (Group number 1 - Default model) Estimate SL_May <--- Manuf -0.308 SL_May <--- PrType -0.041 SL_June <--- Manuf -0.097 SL_June <--- SL_May 0.784 SL_June <--- PrType -0.002 Means: (Group number 1 - Default model) Estimate S.E. C.R. P Label PrType 6.996 0.142 49.276 *** Manuf 6.246 0.269 23.200 *** Intercepts: (Group number 1 - Default model) Estimate S.E. C.R. P Label SL_May 69.800 10.095 6.914 *** SL_June 11.827 6.191 1.910 0.056 Covariances: (Group number 1 - Default model) Estimate S.E. C.R. P Label Manuf <-> PrType 3.926 0.654 6.000 *** Correlations: (Group number 1 - Default model) Estimate Manuf <-> PrType 0.409 Variances: (Group number 1 - Default model) Estimate S.E. C.R. P Label Manuf 18.193 1.624 11.203 *** PrType 5.060 0.452 11.203 *** e1 2386.501 213.029 11.203 *** e2 754.074 67.312 11.203 *** Source: own calculations Examination of the Performances of Maximum Likelihood Method and Bayesian Approach... 327 All values that are explained in Table 1 also appear on graphs; first, after calculating standardized estimates (Fig. 2) and second, after calculating unstandardized estimates (Fig. 3). Fig. 2 Structural model with standardized estimates The value 0.41 is correlation value between Product type and Manufacturer. Values - 0.04; -0.31; -0.10; 0.00 (according to Table 1 this value is -0.002) and 0.78 are Standardized Regression Weights. If we calculate unstandardized estimation, the results will be as follows: Fig. 3 Structural model with unstandardized estimates The values from Figure 3 presented in Table 1 are sorted according to their meaning. It can be concluded, that the covariance between Product type and Manufacturer is estimated to be 3.93. Right next to the covariance in the S.E.column is presented the estimated standard error of about 0.654. The estimate 3.93 indicates that the observation of approximately normally distributed random variables is centred around population covariance with a standard deviation of 0.654. The values of critical ratios are represented in the C.R. column. Dividing the covariance estimate by the estimate of its standard error gives z = 3.926/0.654 = 6.000. In other words, the covariance estimate is 6 standard errors above zero. The probability of getting a critical ratio as large as 6 in absolute value is less than 0.001, so the covariance 328 N. PAPIĆ-BLAGOJEVIĆ, V.LEPOJEVIĆ, S. LONĈAR between Product type and Manufacturer is significantly different from zero at the 0.001 level (two-tailed). The estimated covariance between Product type and Manufacturer (3.93) is the value that will be compared with the Bayesian estimates. In our example, we used non- informative and informative priors in order to get more precise estimates. 2.2. Research results obtained by Bayesian method The Bayesian paradigm is characterized by several advantages relative to the classical one, like the coherence of the whole paradigm, which is derived from the systematical applying of the Bayes law, the concept of subjective probability, the general character of the Bayesian methods which do not ask for special regularity conditions, the sounder definition of the concepts of confidence interval as well as testing [1]; [14]. For carrying out the research using Bayesian estimation it is necessary to select the appropriate prior distribution. In many cases, chosen prior will contain very little information, so the conclusions will be based only on data. Such information is called non-informative prior [12]. However, there is no prior distribution that is completely non-informative, even uniform distribution that IBM SPSS Amos used as default for each parameter, because each prior distribution carries some information. The results obtained from the Bayesian analysis will change if prior distribution changes. In an analysis conducted by the author, it will be seen that changing prior distribution affects the results of research in terms of improving their accuracy. 2.2.1. Application of non-informative uniform prior In many cases, our prior beliefs are vague and thus difficult to translate into an informative prior. We therefore want to reflect our uncertainty about the model parameter(s) without substantially influencing the posterior parameter inference. The so-called non-informative priors, also called vague or diffuse priors, are employed to that end [16]. A non-informative prior might be used in the genuine absence of prior information, or if there is disagreement about the likely values of hypotheses or parameters. It may also be used in comparison with more informative priors as one aspect of a sensitivity analysis regarding posterior inferences according to the prior [2]. In literature, non-informative priors are also called objective priors and they are part of objective Bayesian analysis [7]. In our example, initially selected prior distribution is uniform prior distribution and it has the character of non-informative distribution. The results for estimated covariance between Product type and Manufacturer with the applied uniform distribution are shown in Table 2. Table 2 Bayesian estimates with non-informative prior Mean S.E. S.D. C.S. Covariances PrType<->Manuf 4.036 0.005 0.683 1.000 Source: own calculations Examination of the Performances of Maximum Likelihood Method and Bayesian Approach... 329 If the prior distribution is uniform then the posterior mean will be close to the estimate obtained by method of maximum likelihood. It is confirmed in our example where the estimated posterior mean for covariance between Product type and Manufacturer is 4.036 and the estimated covariance obtained by the method of maximum likelihood is 3.926. In Fig. 4 we can see that the posterior distribution is centered close to 4 that corresponds to the posterior mean (4.036). Fig. 4 Posterior distribution with non-informative prior In this way, it was confirmed that in the case where the prior information is diffuse or non-informative, the results of classical and Bayesian statistics differ very little. In order to find more precise estimates in the next step we choose informative prior. 2.2.2. Application of informative normal prior The prior that contains the most amount of certainty about the population parameter is an informative prior. Informative priors contain strict numerical information that is crucial to the estimation of the model and can have a large impact on final estimates [17]. The problem with using an informative prior is that people might use different background information (or interpret it differently). Thus, informative priors often seem subjective [4]. In order to prove previous claims, instead of initially selected uniform distribution, we now choose normal distribution as prior distribution. Normal distribution has the character of informative prior distribution. The results for estimated covariance between Product type and Manufacturer with the applied normal distribution are shown in Table 3. Table 3 Bayesian estimates with informative prior Mean S.E. S.D. C.S. Covariances PrType<->Manuf 1.887 0 0.279 1.000 Source: own calculations In Fig. 5 we can see that the posterior distribution is now centered close to 1.9 that correspond to the posterior mean (1,887). 330 N. PAPIĆ-BLAGOJEVIĆ, V.LEPOJEVIĆ, S. LONĈAR Fig. 5 Posterior distribution with informative prior If we want to make final conclusions, we should compare results of applied methods. First, it is necessary to compare the posterior standard deviation of Bayesian statistics, (denoted S.D.) with a standard error of classical statistics (denoted S.E.) which is a useful measure of uncertainty. The value of S.D. is 0.279 and of S.E. is 0.654. Lower value of S.D. indicates that the results of Bayesian methods are more precise. Second, we have to compare estimated covariance between Product type and Manufacturer. Unlike the case where the uniform distribution was a prior and where the results were not much different compared to classical estimates, in the case where the normal distribution is chosen for a prior, the situation is changing. Now, estimated posterior mean for covariance between Product type and Manufacturer is 1.887 and the estimated covariance obtained by the method of maximum likelihood is 3.926. So, we can conclude that we can get more accurate estimates if we use normal informative distribution as a prior distribution. CONCLUSION For many years, classical statistics had objective advantage compared to Bayesian approach. Supporters of Bayesian statistics did not have an opportunity to emphasize the possibility of applying Bayesian methods in data processing, because there was a real inability to perform complex methods to handle large amounts of data. The emergence of adequate software solutions has enabled intensive application of Bayesian methods in different research areas. Comparative analysis of maximum likelihood method and Bayesian method was obtained on 252 products and their sales level from January to June 2014. By using the statistical package IBM SPSS Amos and constructing the structural model, the base for further analysis was made. The example has shown the importance of prior information in the Bayesian estimation, and thereby confirmed that the Bayesian approach is more complex because there is an obvious need for estimating a prior probability and examining its sensitivity. Including normal prior distribution, as informative prior, it was concluded that the estimates obtained from Bayesian approach represent an improvement of estimates obtained by classical method in terms of accuracy. Examination of the Performances of Maximum Likelihood Method and Bayesian Approach... 331 REFERENCES 1. Caraiani, P. (2010) Forecasting Romanian GDP using a BVAR Model, Romanian Journal of Economic Forecasting, 4/2010, 76-87. 2. Congdon, P. 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(2007) Bayesian analysis of longitudinal data using growth curve models, International Journal of Behavioral Development, 31 (4) , 374-383. 332 N. PAPIĆ-BLAGOJEVIĆ, V.LEPOJEVIĆ, S. LONĈAR ISPITIVANJE PERFORMANSI METODA MAKSIMALNE VERODOSTOJNOSTI I BAJESOVOG PRISTUPA U OCENJIVANJU NIVOA PRODAJE Metod maksimalne verodostojnosti i Bajesov metod nalaze široku primenu u obradi podataka, ne samo u oblasti ekonomije već i u drugim poljima istraživanja. Cilj rada je da se kroz komparativnu analizu ova dva metoda na odabranim ekonomskim podacima identifikuje koji od pristupa ima bolje karakteristike u datim okolnostima. Poređenjem ocenjenih vrednosti dobijenih primenom metoda maksimalne verodostojnosti i Bajesovog metoda na podacima o ostvarenoj prodaji kompanije CaliVita Int. izvedeni su zaključci o performansama oba pristupa. U radu je pokazano da odabirom informativnih apriornih informacija, Bajesov pristup daje preciznije ocene u odnosu na rezultate dobijene primenom metoda maksimalne verodostojnosti. Kljuĉne reĉi: metod maksimalne verodostojnosti, Bajesov metod, ocenjivanje, apriorna informacija, nivo prodaje