COMPARATIVE STUDY IN ESTIMATING VOLKSWAGEN S PRICE: ARIMA VERSUS ANN

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1 COMPARATIVE STUDY IN ESTIMATING VOLKSWAGEN S PRICE: ARIMA VERSUS ANN Florin Dan PIELEANU Academy of Economic Studies Bucharest Abstract The multiple techniques used for trying to predict the future prices of securities usually fall in two categories: statistical techniques and soft computing techniques. In the fi rst category one can fi nd ARIMA (autoregressive integrated moving average) or GARCH (generalized autoregressive conditional heteroskedasticity) models, and the former will be used in the present article. From the second category, the most important models are the artifi cial neural networks ANN, and such a model will be compaired to ARIMA in order to see which one performs better in the goal of estimating Volkswagen s future prices. It is widely known that this company was recently involved in a scandal which affected the company s shares. Data used is comprised of daily prices for a period of 4,5 years, and the article s main objective is to try and foresee the price for the next 6 months (the second semester of 2015). After this step, it can be observed which of the two models was more accurate, through comparison with the actual prices. The conclusion will confi rm or will refute the superiority of a model over the other, in the mentioned context. Keywords: ARIMA, ANN, estimation, comparative, technique Introduction The main techniques used in trying to predict the future prices of different securities from various markets fall usually in two categories: statistical techniques and soft computing techniques. The latter use inexact solutions to try to solve problems without a known algorithm. They tolerate approximations, uncertainty and partial truth. Some examples include: artificial neural networks, fuzzy logic, support vector machines and others. The solutions provided by them are unpredictable and are performing well when there is not enough information about the problem to be solved. Statistical techniques include ARIMA (autoregressive integrated moving average) models and GARCH (generalized autoregressive conditional heteroskedasticity) models. ARIMA (also known as Box-Jenkins model) is highly used in this direction, being considered as pretty effective for such purposes. The model relies on past values of the series and on its residual values, and it does not require previous relations like other models. So, ARIMA proves to be a sturdy model, with good results, mainly on short and medium terms. The artificial neural networks ANN - are soft computing techniques, which are also considered as really efficient for predicting the values of variables, regardless of the domain: social, finance, exchange rate, engineering or other issues [2, 4, 10, 19]. Some of their attributes, like the independence from certain assumptions, good 98 Romanian Statistical Review - Supplement nr. 2 / 2016

2 adjusting capacity and a fair generalization potential, make the ANN an attractive method for researchers and practitioners. In the same time, ANN can offer solutions for nonlinear problems (like the real ones), opposed to traditional models (like ARIMA), who are implying that the time series are generated from linear processes. The present article analyzes and compares the performances of ARIMA and ANN, in the particular case of future price estimation for the shares of Volkswagen company, in order to confirm or refute the reviews which assert the superiority of one model over the other. Recently, the german auto producer was accused of using a soft that permitted the manipulation of emission tests, in order to report smaller figures. As a consequence, the price of Volkswagen s shares plummeted from August 2015, after it was rising at the beginning of the year. The rest of the article is organized as follows: section 2 lists some related work, section 3 highlights the used methodology, section 4 summarizes and discusses the results, while section 5 draws the conclusions. Related Work The studies who purchased the same goal found proofs favoring both models, and below are listed only a few: - [11] compared ARIMA and ANN on Korean market, and the first was more accurate, especially on medium term - [12] considered the Indian market, and ARIMA was again more precise than ANN - [8] found both models as adequate for predicting water consumption, with ARIMA performing better for linear data, and ANN for nonlinear data - [16] analyzed market in Malaysia, and ANN was better than ARIMA - [15] made a comparison on the market in Indonesia, where ANN predicted more exactly - Trying to forecast the maximum ozone concentrations, [17] proved the priority of ANN More studies in the same direction are [1, 3, 5, 6, 7, 9, 13, 14, 18]. Due to split results, this article wants to clarify the matter and to find a result obviously valid only in the presented context. METHODOLOGY Used data EViews software and Matlab Neural Network were used for ARIMA and ANN models, respectively. The data used in this research are historical daily stock prices for Volkswagen, consisting of open price, lowest price, highest price, close price and volume traded, on a 4,5 years period, starting from until , with 1177 total observations. The open price is the opening price of the shares at the Revista Română de Statistică - Supliment nr. 2 /

3 start of the training day, the lowest price represents the minimum price during the trading day, the highest price represents the maximum price of the day, and the closing price shows the shares price when the market closes. For this article, the closing price is chosen to represent the predicted value, because it reflects all the activities of the day. The main goal of the study is to forecast the future price of Volkswagen s shares on a 6 months horizon (the second semester of year 2015), based on available data, and to observe which one of the ARIMA and ANN models behaved better, by comparison with the real values for the period. ARIMA model in Volkswagen s case In order to see if the time series is stationary or not, its graphic, correlogram and Dickey-Fuller test are studied. The following situation occurs: Graph of Volkswagen s price time series Figure 1 Correlogram of Volkswagen s price time series Figure Romanian Statistical Review - Supplement nr. 2 / 2016

4 DF test for Volkswagen s price time series Figure 3, which proves that the series isn t stationary. For making it so, the first difference will be approached. After it, the changes show: Graph of Volkswagen s price time series (first difference) Figure 4 Correlogram of Volkswagen s price time series (first difference) Figure 5 l fv lk i i i (fi Revista Română de Statistică - Supliment nr. 2 /

5 DF test for Volkswagen s price time series (first difference) Figure 6 And the stationarity of the series can now be observed. Next step is to determine the ARIMA s orders p and q. The criteria for this are: relatively small Akaike, Schwarz and S.E (standard error) values, relatively high adjusted R 2 value. The following table highlights the obtained results for various p and q parameters, criteria-wise: Different ARIMA models and their values for chose criteria Table 1 ARIMA Akaike Schwarz Adjusted R 2 S.E (1,0,0)* 5,495 5,500-0,0002 3,775 (1,0,1) 5,494 5,503 0,0016 3,771 (0,0,1)* 5,496 5,501-0,0002 3,777 (2,0,0)* 5,492 5,501-0,0008 3,767 (0,0,2)* 5,498 5,506-0,0006 3,778 (1,1,0)* 5,497 5,505-0,0007 3,776 (0,1,1)* 5,498 5,506-0,0007 3,778 (2,0,1)* 5,494 5,507-0,017 3,769 (0,1,2)* 5,499 5,512-0,011 3,778 (2,1,0)* 5,493 5,507-0,0012 3,768 (1,0,2) 5,495 5,508 0,0019 3,770 * one or more parameters are not statistically significant So the choice will be made between the ARIMA(1,0,1) and ARIMA(1,0,2) models, on the basis of above-mentioned criteria. For both, the Q-statistics show no pattern left in the autocorrelation (ACF) and partial autocorrelation (PACF) functions of the residuals, which implies that the residuals represent only white noise. Also the comparison of theoretical and empirical autocorrelation and partial autocorrelation values shows adequacy: 102 Romanian Statistical Review - Supplement nr. 2 / 2016

6 Q-statistics Figure 7 Actual vs. theoretical ACF and PACF Figure 8, for ARIMA(1,0,1) and: Q-statistics Figure 9 Revista Română de Statistică - Supliment nr. 2 /

7 Actual vs. theoretical ACF and PACF Figure 10,for ARIMA(1,0,2). Additional, it can be seen that the second model has a higher adjusted R 2 value, and the first one has smaller Akaike and Schwarz values, with S.E being almost equal. As a result, the ARIMA(1,0,1) model will be chosen, being expressed, in the forecast form, as: Ŷt = ϕ1yt-1 - θ1et-1+ξt ANN model in Volkswagen s case It has the below-presented form: Yt = w0 + Σwj*g(woj + Σwij*yt-1) + εt with j=1.q, and i=1.p, where: p = the number of input nodes; q = the number of hidden nodes; wj and wij = connection weights Six input variables were supplied into the model: opening price, lowest price, highest price, closing price, adjusted closing price and trading volume. The creation of the ANN predictive model with Matlab requires the following steps: - selecting model type: one that can forecast the future values of a variable, based on past values of it - creating the network topology: the number of input variables, hidden neurons and output neurons must be selected - selecting training method for the network (in this case the LM method), selecting training, validation and testing percentages, selecting performance function (in this case the MSE - mean squared error function) 104 Romanian Statistical Review - Supplement nr. 2 / 2016

8 Results and Discussions ARIMA model s results The ARIMA(1,0,1) model offers results for estimating Volkswagen s shares prices on a 6-month period, results that are presented below: Date Estimated Real price price (Ep) (Rp) Difference Date Estimated Real price price (Ep) (Rp) Difference ,03 140,2-0, ,3 118,9-1, ,28 139,15-0, ,99 118,9-0, ,03 134,9-0, ,99 111,2-1, ,03 134,3-0, ,6 133,7-0, ,03 131,7-0, ,8 161,35-0, ,53 137,6-0, ,65 167,4-0, ,53 139,6-0, ,95 167,5-0, ,53 140,4-0, ,55 166,85-0, ,53 132,5-0, ,87 165,5-0, ,53 136,05-0, ,15 166,25-0, ,53 137,3-0, ,75 166,9-0, ,53 136,75-0, ,7 169,6-0, ,78 137,85-0, ,44 165,4-0, ,78 139,65-0, ,3 161,15-0, ,53 141,3-0, ,55 159,95-0, ,53 134,25-0, ,8 164,4-0, ,53 134,55-0, ,6 159,95-0, ,53 130,45-0, ,1 164,4-0, ,53 127,15-0, ,8 159,5-0, ,03 124,65-0, ,65 161,95-0, ,78 123,9-0, ,55 166,7-0, ,53 125,3-0, ,3 170,5-0, , , ,8 171,15-0, , , ,8 165,95-0, ,53 116,8-1, ,5 167,8-0, ,54 118,45-1, ,55 158,55-0, ,3 117,2-1, , , ,5 118,15-1, ,25 168,05-0, ,3 119,75-0, , , ,05 120,1-0, ,8 175,2-0, ,55 121,9-0, ,55 178,05-0, ,55 120,1-0, , , ,55 117,15-0, ,55 180,45-0, ,8 124,1-0, ,55 177,75-0, ,8 126,75-0, ,29 186,2-0, ,75 126,1-0, ,59 194,15-0, ,05 125,4-0, ,3 191,55-0, ,05 124,05-0, , , ,54 121,8-0, ,81 192,4-0, , , , , ,55 121,85-0, ,8 185,75-0, ,55 123,9-0, ,68 184,05-0, ,55 119,95-0, ,66 185,55-0, ,05 116,25-0, ,15 185,9-0, ,05 118,6-0, ,55 190,6-0, ,6 121,2-0, ,8 185,45-0, ,55 123,8-0, ,8 189,55-0, ,25 128,6-0, ,82 198,5-0, ,3 130,6-0, ,24 196,1-0, ,56 132,45-0, ,15 198,9-0, ,55 125,9-0, , , ,55 116,2-1, ,2 202,7-0, ,91 114,9-1, ,75 203,15 0, ,3 106,9-1, ,75 199,15-0, ,8 102,8-1, ,75 201,5 0, ,34 101,15-1, ,03 205,75 0, ,8 105,05-1, ,95 203,4 0, ,3 104,95-1, ,3 203,45 0, ,8 103,3-1, ,3 197,5-0, ,8 107,1-1, ,05 202,95-0, ,3 115,55-1, ,3 208,25-0, Revista Română de Statistică - Supliment nr. 2 /

9 The common graphic for both series, the predicted and the real price, looks like this: Actual vs. predicted price - ARIMA Figure 11 ARIMA forecasted a rise in the shares price, compared to the real price, which actually plummeted. Surely that the scandal in which the company was involved was responsible for the drop, but this event could not be anticipated by our model. ARIMA just estimated a continuous increase, based on historical data, which indicated such an evolution over the last years. ANN model s results After several trials, using different network architectures based on the ANN algorithm, the chosen structure was the one that gave the best accuracy in prediction and the smallest MSE, based on testing data. Below are the ANN results, as predicted daily prices for the second semester of year 2015: 106 Romanian Statistical Review - Supplement nr. 2 / 2016

10 Date Estimated price (Ep) Real price (Rp) Difference Date Estimated price (Ep) Real price (Rp) Difference ,02 140,2 0, ,4 118,9-0, ,96 139,15 0, ,68 118,9-0, ,46 134,9 0, ,65 111,2-0, ,75 134,3 0, ,01 133,7 0, ,88 131,7 0, ,95 161,35 0, ,68 137,6 0, ,18 167,4 0, ,98 139,6 0, ,17 167,5 0, ,42 140,4 0, ,23 166,85 0, ,31 132,5 0, ,79 165,5 0, ,24 136,05 0, ,68 166,25 0, ,88 137,3 0, ,43 166,9 0, ,34 136,75 0, ,05 169,6 0, ,66 137,85 0, ,27 165,4 0, ,7 139,65 0, ,31 161,15 0, ,37 141,3 0, ,67 159,95 0, ,39 134,25-0, ,69 164,4 0, ,55 0, ,54 159,95 0, ,01 130,45-0, ,75 164,4 0, ,62 127,15-0, ,8 159,5 0, ,7 124,65-0, ,24 161,95 0, ,19 123,9-0, ,87 166,7 0, ,54 125,3 0, ,7 170,5 0, , , ,88 171,15 0, , , ,15 165,95 0, ,43 116,8-0, ,59 167,8 0, ,45 118,45-0, ,76 158,55 0, ,31 117,2-0, , , ,11 118,15-0, ,16 168,05 0, ,65 119,75-0, , , ,58 120,1-0, ,52 175,2 0, ,92 121,9-0, ,29 178,05 0, ,18 120,1-0, , , ,95 117,15-0, ,58 180,45 0, ,25 124,1-0, ,63 177,75 0, ,14 126,75 0, ,17 186,2 0, ,24 126,1-0, ,32 194,15 0, ,82 125,4-0, ,75 191,55 0, ,05 124,05-0, , , ,54 121,8-0, ,61 192,4 0, , , , , ,89 121,85-0, ,29 178,05 0, ,43 123,9-0, , , ,45 119,95-0, ,58 180,45 0, ,29 116,25-0, ,63 177,75 0, ,72 118,6-0, ,17 186,2 0, ,99 121,2-0, ,32 194,15 0, ,25 123,8-0, ,75 191,55 0, ,62 128,6 0, , , ,04 130,6 0, ,61 192,4 0, ,4 132,45 0, , , ,54 125,9 0, , , ,4 116,2-0, ,14 202,7 0, ,71 114,9-0, ,07 203,15 0, ,23 106,9-0, ,13 199,15 0, ,93 102,8-0, ,08 201,5 0, ,68 101,15-0, ,42 205,75 0, ,64 105,05-0, ,78 203,4 0, ,3 104,95-0, ,49 203,45 0, ,24 103,3-0, ,33 197,5 0, ,18 107,1-0, ,54 202,95 0, ,41 115,55-0, ,17 208,25 0, The commom plot for predicted and real price series looks like below: Revista Română de Statistică - Supliment nr. 2 /

11 Actual vs. predicted price - ANN Figure 12 ANN proposes a fairly stable price for the studied period, and this tends much towards the real one in the second half of the interval. For the first half, ANN is not accurate in forecasting, but then performs quite good, by closing in to the real values triggered by the Volkswagen scandal. Conclusions The present study comprehends the empirical results of compairing ARIMA and ANN models for estimating the future price of Volkswagen shares. This headto-head was employed in order to highlight which one is more accurate in the chosen context. The conclusions show that ANN looks more appropriate for this goal, since it performed better, being quite close to the real values in half of the studied time period, while ARIMA proposed estimations that are far from them in the whole interval. But this fact happens because of the dispute that involved Volkswagen. So the results can t be considered as absolute benchmarks, being valid only in the presented conditions. Future research can surely clarify the problem in more depth. *** This work was presented at the Seminar of Statistics' Octav Onicescu "held in the month of January REFERENCES 1. H. Al-Qaheri, A. E. Hassanien, A. Abraham - Discovering stock price prediction rules using rough sets, Neural Network World, vol. 18, no. 3, pag , G.Ardali, M. Khashei, M. Bijari Improvement of auto-regressive integrated moving average models using fuzzy logic and artificial neural networks (ANNs), Neurocomputing, vol. 72, no. 4 6, pag , S. Atsalakis and K. P. Valavanis - Forecasting stock market short-term trends using a neuro-fuzzy based methodology, Expert Systems with Applications, vol. 108 Romanian Statistical Review - Supplement nr. 2 / 2016

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