Sales forecasting with limited information: Comparison between cumulative observations and rolling windows

Transcription

1 The Empirical Econometrics and Quantitative Economics Letters ISSN EEQEL all rights reserved Volume 2, Number 4 (December 2013), pp Sales forecasting with limited information: Comparison between cumulative observations and rolling windows Orakanya Kanjanatarakul 1, Komsan Suriya 2, Cheng Hsiao 3 and Christian Gourieroux 4 1 Faculty of Management Science, Chiang Mai Rajabhat University 2 Faculty of Economics, Chiang Mai University suriyakomsan@gmail.com 3 University of Southern California 4 CREST, University of Paris and University of Totonto ABSTRACT This analysis will forecast the sales of an innovative agro-industrial product, feta cheese produced by the Royal Project Foundation in Thailand. It will estimate a logistic model suggested by Stoneman (2010) that presents the S-curve following the product life cycle theory. By the first method, it will add a number of observations to the previous set of observations, which is called the process of cumulative observations. This process includes the cumulative knowledge from the past. By the second method, it will estimate the model by using the rolling windows method which deletes the oldest data after adding the newest ones. The analysis will compare the accuracy of both approaches in terms of mean absolute percentage error. The results show that the method of rolling windows outperforms the cumulative observations. The optimal window width is 15 months. Keywords: Sales forecasts, innovation, product life cycle, cumulative observations, rolling windows, JEL Classification: C53, O31, M31

2 12 EEQEL Vol. 3, No. 1 (March 2014) O. Kanjanatarakul et al 1. Introduction. Agro-industry produces significantly positive impact on nationwide economy of Thailand (Kanjanatarakul and Suriya, 2012). By this impact, the government supports the agro-industry especially via the Royal Project Foundation which extremely empowers the agency and the whole sector (Kanjanatarakul and Suriya, 2013 a). Sales forecasting of innovative agro-industrial product with limited information is a challenge for both scholars and practitioners. Traditional econometrics usually uses more than 30 observations to construct a time trend. However, practitioners cannot wait for 30 months to have the complete information for such a forecast. In this paper, we propose a method to use limited information from only 3 observations of the sales for the forecasting. The forecasts are adjusted according to the product life cycle theory, which suggests an S-shape evolution of sales over time that can be modeled using the logistic function as suggested by Stoneman (2010). Kanjanatarakul and Suriya (2013 b) propose the method of least squares with quadratic interpolation to estimate the S-curve. However, a drawback of their work is on the fixed parameter over time. This is because they use the method of cumulative observations which adds another new observation into the model but not delete any observation away. It assumes that from the beginning until the time that is evaluated by the model, the parameters of the model remain the same. This assumption may not be true especially in the long-run. A possible way to prevent this drawback is to use the method of rolling windows. The method will add a new observation into the model while delete the oldest observation off the model. It will keep the number of observations in the model constant with the updated information. Therefore, the parameters of the model are refreshed. This study will find out which method between cumulative observations and rolling windows is better. It will compare the results using mean absolute percentage error (MAPE) derived from the out-of-sample test. Moreover, it will figure out the optimal size of the windows if the technique sounds promising. The results of the study will enhance practitioners in agroindustry to accurately forecast the sales of their new products. 2. Literature review. Sales forecasting has its long history. Bass (1969) pioneers the quantitative method of the forecasting. His model is called Bass model. It captures the diffusion of the innovative products in two channels, the innovation and the imitation. This means that two reasons behind the diffusion are, first, people use it because they want to be different from other people and, second, people use it because they want to imitate other people. Bass model is popular among marketers until nowadays. Stoneman (2010) launches his book on soft innovation. In the book, he proposes another functional form for the sales forecasting. He simply uses the logistic function. This is challenging to Bass model. It does not capture the two dimensions of diffusion which are innovation and imitation like presented by Bass model. However, this simpler functional form may yield more accurate forecasting results. Kanjanatarakul and Suriya (2012) prove this curiousness by applying both Bass model and logistic function onto a set of empirical data of an innovative agro-industrial product. They find that the logistic function works better than Bass model. Therefore, this study chooses logistic function to be the functional form of the sales forecasts. The series of works of Kanjanatarakul and Suriya do not stop there. In 2013, they publish another paper regarding the sales forecasting under limited observations. They find that the sufficiency number of observations is between 7 to 24 observations of monthly data. However, the attacks on their works upon the strong assumption of fixed parameters over

3 The Empirical Econometrics and Quantitative Economics Letters 13 time lead them to conduct this study to find out whether their traditional technique of cumulative observations or the method of rolling windows will yield more accurate forecasts. 3. Methodology 3.1 Logistic function Stoneman (2010) suggests scholars to use the logistic function for forecasting sales of new products as follows: V T = M 1+A exp ( βt) (1) where V T = Sales of innovative agro-industrial product; M = Maximum sales of innovative agro-industrial product; β = Growth parameter; A = Shift parameter; T = Time. 3.2 Least squares using quadratic interpolation algorithm The parameter estimation includes 8 steps as follows: Step 1: Set three initial values of parameter M. Transform the data by the Logistic transformation into a linear function. ln V T/M ln 1 V T /M 1 = βt (2) A Then, estimate parameter β using Ordinary Least Squares (OLS) Step 2: Take parameter M and β to forecast sales by this formula. V T = M 1 + A exp ( βt) The value of A is calculated by the following formula to fix the y-intercept at the first data of the series (Vo): A = M V 0 1 (3) Step 3: Calculate the Sum Squared Error (SSE). e 2 = V T V T N i=1 (4)

4 14 EEQEL Vol. 3, No. 1 (March 2014) O. Kanjanatarakul et al Step 4: Calculate the SSE at the three points using the three initial M values. Step 5: Search for a new M value by Quadratic Interpolation Step 6: Include the new M with other two previous M values which are located nearest to the new M. Then, estimate parameter β and calculate the SSE again. Step 7: Repeat steps 5 and 6 for 10,000 iterations. Step 8: Summarize the values of parameter M and β. It should be remarked that the estimation of β using OLS with logistic transformation may suffer from the heteroscedasticity problem. The problem may be avoided by using Estimated Generalized Least Squares or EGLS (Judge et al, 1986). However, Kanjanatarakul (2013) shows that OLS is better than EGLS in this case. Therefore, this study uses OLS rather than EGLS in the estimation of β. 4. Data The data are obtained from the Royal Project Foundation. They are monthly sales of feta cheese. The data cover the period during January 2010 to August Totally, the model has 32 observations. 5. Results We will first present the results from the method of rolling windows. The width of the windows was varied from 3 to 19. Due to the large size of the results, they are presented in the appendix section of this chapter. However, two of the best models with width of 15 and 16 observations, which demonstrate the lowest average MAPE in this class, are displayed here. Table 1: Estimation results for the Logistic function using OLS and quadratic interpolation with the method of rolling windows when the width of window is 15 Repeat Mstar beta SSE (Million) MAPE AIC BIC 1 1,156, , ,482, , ,806, , ,205, , ,729, , ,357, , Source: Calculation using own programming in Matlab.

5 The Empirical Econometrics and Quantitative Economics Letters 15 Table 2: Estimation results for the Logistic function using OLS and quadratic interpolation with the method of rolling windows when the width of window is 16 Repeat Mstar beta SSE (Million) MAPE AIC BIC 1 1,156, , ,483, , ,802, , ,206, , ,731, , Source: Calculation using own programming in Matlab. The study will find the best model with rolling windows using t-test. The results are shown in Tables 3.26 and Table 3: Descriptive statistics of MAPE, AIC and BIC from the estimation methods of rolling windows with widths15 and 16 Mean N Std. Deviation Std. Error Mean Pair 1 MAPE MAPE Pair 2 AIC AIC Pair 3 BIC BIC Source: Calculation using SPSS version The MAPE of the model with a width of 15 observations is slightly lower than that with 16 observations. The AIC and BIC of the former model are higher than those of the latter model. However, as shown in Table 3.26, these differences are not statistically significant. Therefore, the best model is the one with 15 observations because it uses fewer observations. Table 4: Comparison of MAPE, AIC and BIC for the estimation methods of rolling windows with widths 15 and 16 Paired 95% Confidence Std. Std. Error Sig. Differences Interval of the t df Deviation Mean (2-tailed) Mean Difference Lower Upper Pair 1 MAPE15 MAPE Pair 2 AIC15 AIC Pair 3 BIC15 BIC Source: Calculation using SPSS version In the next step, this study will compare the performances of the best model from the method of rolling windows to the best model with the method of cumulative observations. The results are shown in Tables 3.28 and 3.29.

6 16 EEQEL Vol. 3, No. 1 (March 2014) O. Kanjanatarakul et al Table 5: Estimation results for the Logistic function using OLS and quadratic interpolation No. of obs Mstar Beta SSE (Million) MAPE AIC BIC 3 1,801, ,479, ,481, ,803, , ,154, , ,154, , ,170, , ,154, , ,152, , ,152, , ,151, , ,147, , ,156, , ,156, , ,153, , ,471, , ,474, , ,159, , ,481, , ,173, , ,253, , ,188, , ,100, , ,805, , ,807, , ,814, , ,227, , ,837, , ,407, , ,157, , Source: Calculation using own programming in Matlab. Table 6: Descriptive statistics of MAPE, AIC and BIC from the estimation methods of rolling windows with width 15 and the model with the method of cumulative observations (OLS with quadratic interpolation) Mean N Std. Deviation Std. Error Mean Pair 1 MAPE CUMMAPE Pair 2 AIC CUMAIC Pair 3 BIC CUMBIC Source: Calculation using SPSS version 11.0

7 The Empirical Econometrics and Quantitative Economics Letters 17 Table 7: Comparison of MAPE, AIC and BIC between the estimation methods of rolling window with width 15 and the model with the method of cumulative observations (OLS with quadratic interpolation ) Paired 95% Confidence Std. Std. Error Sig. Differences Interval of the t df Deviation Mean (2-tailed) Mean Difference Lower Upper Pair 1 MAPE15 CUMMAPE Pair 2 AIC15 CUMAIC Pair 3 BIC15 CUMBIC Source: Calculation using SPSS version The results show that the MAPE of the model with rolling windows is much lower than that of the model with cumulative observations. However, the AIC and BIC of the model with rolling windows are higher than those of the model with cumulative observations. These results are clearly significant at the confidence level of 99%. To judge which one is better, the purpose of the prediction should be the first priority. The model with smaller MAPE calculated from the out-of-sample test should be considered better. Therefore, the model using rolling windows should be preferred to the model using cumulative observations. 6. Conclusions and further studies. This study answers the research question whether the method of rolling windows will give more accurate forecasting results than the method of cumulative observations. It will also find out, if the rolling window is better, the optimal size of the windows. It uses a set of empirical data of an innovative agro-industrial product, feta cheese produced by the Royal Project Foundation in Thailand to prove this curiousness. It measures the accuracy of the models by mean absolute percentage error (MAPE). It is clear from the results that the method of rolling windows outperforms the method of cumulative observations. The optimal window width is 15 months. This confirms that the assumption of floating parameters over time is more suitable than the fixed parameters over time. It is because the rolling windows can adaptively capture the change of the sales situation which evolves continuously along the product life cycle. The challenge of forecasting with limited information is still going on. Even though the big data becomes more and more important and may threaten social research dramatically (Suriya, 2013) the limited data cannot be ignored by social researchers. A further technique on sales forecast of innovative agro-industrial product is the forecasting using belief functions. This is the development of Kanjanatarakul, Sriboonchitta and Denoeux (2014). Ultimately, we hope that these developments of the use of limited information to accurately forecast the sales or to analyze them in efficient and creative ways will enhance economists to work on limited resources that may not be avoided in conducting field research. The study should not be restricted to only innovative agro-industrial products (Kanjanatarakul and Suriya, 2013 c). The techniques should be extended to other fields such as innovative tourism products (Suriya, 2012; Srichoochart and Suriya, 2012) and new transportation services (Suriya et al, 2012) that practitioners usually collect only limited data of their sales in the introductory stage and would like to know whether the new products or services should be continuously supported or terminated. This is crucial for business to make the decision.

8 18 EEQEL Vol. 3, No. 1 (March 2014) O. Kanjanatarakul et al References Bass, Frank A New Product Growth Model for Consumer Durables, Management Science 15 (January): pp Judge, George, et al Introduction to the Theory and Practice of Econometrics. 2 nd ed. New York: John Wiley and Sons. Kanjanatarakul, Orakanya, An Economic Analysis of the Royal Project Foundation: Sales Forecast of Feta Cheese and Impact on Thai Economy. Doctoral dissertation, Faculty of Economics, Chiang Mai University. Kanjanatarakul, Orakanya and Komsan Suriya Economic impact of agro-industrial sector on nationwide economy of Thailand: A general equilibrium approach, The Empirical Econometrics and Quantitative Economics Letters 1, 4 (December 2012): pp Kanjanatarakul, Orakanya and Komsan Suriya Comparison of sales forecasting models for an innovative agro-industrial product: Bass model versus logistic function, The Empirical Econometrics and Quantitative Economics Letters 1, 4 (December 2012): pp Kanjanatarakul, Orakanya and Komsan Suriya a. Role of government funding on the empowerment of the Royal Project Foundation: A general equilibrium approach, The Empirical Econometrics and Quantitative Economics Letters 2, 1 (March 2013): pp Kanjanatarakul, Orakanya and Komsan Suriya b. Sufficient Number of Observations for Sales Forecasts with Limited Information: A Case of an Innovative Agro- Industrial Product, International Journal of Intelligent Technologies and Applied Statistics 6, 2 (June 2013), pp Kanjanatarakul, Orakanya and Komsan Suriya c. Forecasting the sales of an innovative agro-industrial product with limited information: A case of feta cheese from buffalo milk in Thailand, Proceedings in the Ecomod2013 conference in Prague. Kanjanatarakul, Orakanya, Songsak Sriboonchitta and Thierry Denoeux Forecasting using belief functions: An application to marketing econometrics, International Journal of Approximate Reasoning. In press (February 2014). Stoneman, Paul Soft Innovation: Economics, Product Aesthetics and the Creative Industries. Oxford: Oxford University Press. Srichoochart, Siriporn and Komsan Suriya Factors influencing the intention to purchase an innovative tourism service for learning Northern Thai intellectualities using a conditional Logit model, In Peter Calkins et al (eds.) Unifying Econometrics: Bridging the Gap between Theory and Practice. Chiang Mai: CMSE Press, pp Suriya, Komsan Big data, The Empirical Econometrics and Quantitative Economics Letters 2, 4 (December 2013): pp (Book review) Suriya, Komsan An economic analysis of tourism demand and innovative tourism products: A case study of Lampang province in Thailand, The Empirical Econometrics and Quantitative Economics Letters 1, 1 (March 2012): pp Suriya, Komsan et al New services development and pricing strategy of rail transporter in Thailand, The Empirical Econometrics and Quantitative Economics Letters 1, 1 (March 2012): pp

9 The Empirical Econometrics and Quantitative Economics Letters 19 Annex The estimation results of the Logistic function using rolling windows are presented here, with the width of window ranges from 3 to 20. Table A-1 The estimation results of the Logistic function using OLS and quadratic interpolation with the method of rolling window when the width of window is ,801, ,752, ,196, ,608,479, ,694, ,135,837, ,332, ,686,089, ,103, ,774,806, ,073, ,186,289, ,264, ,118,112, ,707, ,522,025, ,496, ,831,558, ,747, ,007, ,465, ,231,801, ,834, ,601,992, ,001, ,551,572, ,112, ,623,759, ,446, ,564,367, ,267, ,288,384, ,901, ,218,213, ,807, ,530,561, Table A-2 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,479, E ,814, E ,220, E ,752, E ,387, E ,167, E ,138, E ,329, E ,800, E ,627, E ,854, E ,625, E ,035, E ,220, E ,422, E ,821, E ,726, E

10 20 EEQEL Vol. 3, No. 1 (March 2014) O. Kanjanatarakul et al Table A-3 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,481, E ,853, E ,240, E ,786, E ,414, E ,206, E ,185, E ,390, E ,870, E ,697, E ,962, E ,750, E ,188, E ,418, E ,664, E ,122, E Table A-4 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,803, E ,211, E ,746, E ,373, E ,152, E ,120, E ,309, E ,777, E ,579, E ,815, E ,570, E ,967, E ,155, E ,328, E ,712, E Table A-5 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,154, E ,493, E ,816, E ,221, E

11 The Empirical Econometrics and Quantitative Economics Letters ,750, E ,382, E ,162, E ,123, E ,313, E ,785, E ,597, E ,831, E ,588, E ,003, E Table A-6 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,154, E ,485, E ,809, E ,214, E ,744, E ,372, E ,134, E ,105, E ,287, E ,754, E ,554, E ,781, E ,532, E Table A-7 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,170, E ,493, E ,816, E ,226, E ,757, E ,376, E ,157, E ,128, E ,317, E ,783, E ,597, E ,835, E

12 22 EEQEL Vol. 3, No. 1 (March 2014) O. Kanjanatarakul et al Table A-8 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,154, E ,484, E ,810, E ,214, E ,729, E ,357, E ,135, E ,098, E ,279, E ,739, E ,547, E Table A-9 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,152, E ,484, E ,809, E ,202, E ,726, E ,352, E ,130, E ,088, E ,266, E ,731, E Table A-10 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,152, E ,484, E ,797, E ,200, E ,722, E ,349, E ,120, E ,080, E ,255, E

13 The Empirical Econometrics and Quantitative Economics Letters 23 Table A-11 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,151, E ,470, E ,794, E ,195, E ,717, E ,337, E ,105, E ,067, E Table A-12 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,147, E ,473, E ,798, E ,199, E ,720, E ,343, E ,113, E Table A-13 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,156, E ,482, E ,806, E ,205, E ,729, E ,357, E Table A-14 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,156, E ,483, E ,802, E ,206, E ,731, E

14 24 EEQEL Vol. 3, No. 1 (March 2014) O. Kanjanatarakul et al Table A-15 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,153, E ,478, E ,801, E ,206, E Table A-16 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,471, E ,795, E ,197, E Table A-17 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,474, E ,801, E Table A-18 The estimation results of the Logistic Model using OLS and quadratic interpolation with the method of rolling window when the width of window is ,159, E