FORECAST MODEL USING ARIMA FOR STOCK PRICES OF AUTOMOBILE SECTOR. Aloysius Edward. 1, JyothiManoj. 2

Transcription

1 FORECAST MODEL USING ARIMA FOR STOCK PRICES OF AUTOMOBILE SECTOR Aloysius Edward. 1, JyothiManoj. 2 Faculty, Kristu Jayanti College, Autonomous, Bengaluru. Abstract There has been a growing interest in modeling and forecasting stock prices over the past couple of decades. Auto Regressive Integrated Moving Average (ARIMA) models are one of the most important time series models used in financial forecasting over the past three decades. This paper attempts to address the forecasting of stock prices of Automobile sector. The forecasting models ARIMAs are applied to forecast the stock prices. Closer examination suggests that the stock prices are upward trends and could be considered as a worthy investment. Keywords : Forecasting, Stationary, Estimation, ARIMA, Time Series Modelling, Sectoral Stock Prices Introduction Sales forecast plays a prominent role in business strategy for generating revenue. Sales forecast depends on some of the factors as the market demand, promotion strategy used, living standard of the people, inflation rate, consumables price, public image of the company, market share, quality of service and so on. The inflation rate, petrol price, previous month sale are found to be more prominent parameters influencing the sales forecast of cars. The worldwide automotive industry has been enjoying a period of relatively strong growth and profitability, and annual sales have reached prerecession levels in some regions. Yet considerable uncertainty about the future remains. The most immediate challenge is the unevenness of global markets. Meanwhile, the Indian market s performance has been inconsistent. From the ground level, three powerful forces are roiling the auto industry: shifts in consumer demand, expanded regulatory requirements for safety and fuel economy, and the increased availability of data and information. There has been a growing interest in modeling and forecasting stock prices over the past couple of decades. One of the most important prevention of investors to invest in stock exchange is their unfamiliarity with the various methods and models for predicting the stock price. Price prediction process is possible using different method and models. A time series is a set of well- defined data items collected at successive pointsat uniform time intervals. Time series an analysis is an important part in statistics, which analyzes data set to study the characteristics of the data and helps in predicting future values of the series based on the characteristics. Forecasting is important in fields like finance, industry, etc. [1] Autoregressive and Moving Average (ARMA) model is an important method to study time series. The concept of autoregressive (AR) and moving average id: editorijrim@gmail.com 1

2 (MA) models was formulated by the works of Yule, Slutsky, Walker and Yaglom [1]. Autoregressive Integrated Moving Average (ARIMA) is based on ARMA Model. The difference is that ARIMA Model converts a non-stationary data to a stationary data before working on it. ARIMA model is widely used to predict linear time series data. [2] The ARIMA models are often referred to as Box-Jenkins models as ARIMA approach was first popularized by Box and Jenkins. The general transfer function model employed by the ARIMA procedure was discussed by Box and Tiao. (1975)[2] ARIMA model is often referred to as ARIMAX model when it includes other time series as input variables. [3] Pankratz (1991) refers to the ARIMAX model as dynamic regression. [2] The ARIMA procedure offers great flexibility in univariate time series model identification, parameter estimation, and forecasting Stock prices are not randomly generated values rather they can be treated as a discrete time series model and its trend can be analyzed accordingly, hence can also be forecasted. There are various motivations for stock forecastingone of them is financial gain. A system that can identify which companies are doing well and which companies are not in the dynamic stock market will make it easy for investors or market or finance professionals make decisions. Having an excellent knowledge about share price movement in the future helps the investors and finance personals significantly [4]. Since, itis necessary to identify a model to analyze trends of stock prices with relevant information for decision making, it recommends that transforming the time series using ARIMA is a better approach than forecasting directly, as it gives more accurate results [5]. But only predicting will not help if one cannot figure out the efficiency of the result. The paper is arranged in the following order: - Past studies by other researchers in the related field is expressed in the next part of the paper as Review of Literature which is followed by the Methodology adopted in this study and the Results obtained with their Discussion; finally Conclusion and references are attached. Review of literature ARIMA model is widely used to analyze the impact of past values in predicting future. Gerra (1959) presented a series of behaviour relations and identities which were believed to timulate the basic economic system for the egg industry. He indicated that in using the equations fitted (an econometric model) to forecast values of variables in the egg industry beyond the years for which equations were fitted, better estimates of the annual quantity variable (domestic egg consumption, egg production on farms, average number of layers on farms, and the number sold) were obtained from simultaneous equation approach, while better estimates for some variables like storage movement and price variables were obtained by least square method. Suits (1962) while presenting an econometric model of the U.S. economy demonstrated its use as a forecasting instrument and explored its implications for policy analysis. He divided the presentation in to two parts. Part - I deals with the general nature ofeconometric models using a highly simplified schematic example, illustrating how forecasts were made with a model, how a model could be modified to permit the introduction of additional information and judgment, and how short - run and long -run policy multipliers were derived from the inverse of the model. Pat-II presented 32 equation in economics. Lirby (1966) compared three different time-series methods viz., moving averages, exponential smoothing, and regression. He found that in terms of month-to-month forecasting, horizon was increased to six months. The regression models included was found to be the best method for longer-term forecasts of one year or more. id: editorijrim@gmail.com 2

3 Schmitz and watts (1970) used parametric modeling to forecast wheat yields in the United States, Canada, Australia and Argentina. The essence of this approach was that the data were used for identifying the estimation of the random components in the form of moving average and autoregressive process. It did not identify and measure the structural relationship as was attempted when forecasting with econometric models. They used exponential smoothing to forecast yields in United States and Canada. They also compared the forecasting accuracy between parametric modeling and exponential smoothing. Leuthold et.al (1970) In their study of forecasting daily hog price and quantities usedtheil s inequality coefficient for comparing the predicative accuracy of the different forecasting approaches. For price forecast to hog market they compared econometric model, random walk model, and mean model and for supply forecasts they compared econometric model, random walk model, mean model and time-series models. They concluded that the data required for time series modeling was the concerned data on the variable to be forecasted, whereas for econometric models data are needed on both the regressor and regressand. Therefore the forecasts using econometric model are slightly better than those using a stochastic non-casual frame-work. Further, the cost of making slightly greater error in using the latter will be less than the additional cost involved in setting up an econometric model and collecting the data. Lakshminarayan et al. (1977) developed the following form of Box-Jenkins model. Zt =Zt-I + at at at-13. to forecast the broiler chicken production for the year The mean absolute percentage of error was under 5 percent while the error in the total production for the year was 1.7 percent. The forecast followed the pattern of the actual data. The actual production was always within the 50 percent confidence limits of the forecast. Bessler (1982) reviewed the relationship between the adaptive expectation, the exponentially weighted moving average, and optional univariate statistical predators. He showed that the behavioural-based adaptive expectations were a sub class of both the exponentially weighted moving average and the ARIMA (0,1,1) model. The applicability of the adaptive expectations model to 25 empirical price and quantity series was investigated. The adaptive expectations behaviour and the optional statistical forecasts were equivalent to 13 series, 11 on yield and two on prices. Numerous price series while exhibiting the general form of the adaptive expectations (a ARIMA (0,1,1) process) did not have a coefficient of expectation within the originally hypothesized range. The behavior consistent with the model underlying these price series was trend extrapolation rather than averaging (averaging the most recent observation and its forecast). Series measured at monthly or quarterly intervals were not adequately modelled by adaptive expectations or as a ARIMA (0,1,1) process. Lee (1988) compared single equation price models of simple weighted, native, autoregressive moving average (ARMA) and future price lagged seventeen weeks (FPt-17) to determine the accuracy of price prediction for different market positions relative to futures market delivery. Simple weighted and native models exhibited four times less variability as measured by RMSE. FPt- 17 exhibited low Durbin Watson values and ARMA for RMSE model accurately reflected time trend changes (turning points). Bootstrapping confirmed the statistical accuracy of RMSE evaluation with histograms of MSE frequency distributions, widest for ARMA and narrow and simple weighted and native. The FPt-17 price expectation model improved in prediction accuracy when bootstrapped. Bootstrapping indicated that FPt-17 may be a more accurate source of outlook information for cash price than indicated by RMSE evaluation. id: editorijrim@gmail.com 3

4 Seema (1990) in his study on structure of egg prices in Hyderabad (A.P.) applied linear trend model and worked out 12 months moving average and seasonal index for month-wise egg price date for the period 1973 to 89. He made projection by multiplying trend value and seasonal index. Sabur.S.A. et.al. (1993) in their paper used ARIMA models for forecasting the prices. They have shown that the ARIMA model has to be used only to short term forecasts. Khan S.A. et.al. (1995) have used multiple regression analysis to predict the yields of winter rice on the basis of the rainfall distribution. The R-Square for their model was more than 70%. Methodology An Auto regressive (AR) process is a series which is dependent on its own lagged values. The AR(p)model refers to the regression model where no repressors other than the current and previous values of p- lags of the variable are involved.it may be represented as Y t = α 0 + α 1Y t-1 + α 2Y t-2+ +α py t-p Moving average (MA) model is relevant if the AR process is not the only mechanism that generates Y, but it also involves the past values of the error terms. An MA (q) process represented as ε t=β 1ε t + β 2ε t-2 + β 3ε t-3 + +β qε t-q which is a linear combinations of white noise errors. When Y has both the characteristics of AR and MA, we refer to it as ARMA(p, q) process. [6] The objective of ARIMA model which are also known as Box-Jenkins model is to identify and estimate a statistical model which can be interpreted as having generated the sample data. Hence stationarity is an important pre-requisitemost of the financial time series are not stationary but integrated. Differencing the series will yield a stationary time series. If a series becomes stationary when differenced d times, we refer to the series as I(d). therefore, if we apply ARMA(p,q)to a series which is I(d), then the original time series is ARIMA(p, d, q). The Box Jenkins methodology suggests finding the valuesof p and q for AR and MA respectively by referring to the correlogram. The autocorrelation function graph indicates the value of q while the Partialautocorrelation function graph indicates the valueof p. For an MA (q) model, moving average of order q, ACF Dies Down or Cuts off after lag q while for AR (p), autoregressive of order p PACF Dies Down or Cuts off after lag p. [10] This is further confirmed by least values of Akaike s Information criterion (AIC) value; the least value of AIC is considered most suitable.[7] Model diagnosis can be carried out by the values of Root mean Square Error (RMSE) and Mean Absolute Percent Error (MAPE). Further, the prediction accuracy is measured by an accuracy measure defined as Accuracy percent = (1 residual/actual series value)*100 where residual is the absolute difference between actual and estimated values.[8] Result and Discussion The study deals with the closing price of Automobile Sector. 4 different companies pertaining to automobile sector is chosen for the study. The firms are Bajaj Auto, Tata Motors, Hero Motor Corp, Mahindra &Mahindra. Closing price from NSE is collected for 8 years; May 2008 to April 2015 for all id: editorijrim@gmail.com 4

5 the series. The general trend of all the series is to increase, which can be observed in Fig 1- line graph of the closing prices. Among the four Bajaj and Hero performs comparatively better than the rest. The descriptive Statistics of the four series is provided in Table 1. Fig 1. Graph showing the trend of the stock prices over the past 8 years. Company N Mean SD Skewn ess Kurtosi s Jarque- Bera Bajaj * Tata * Motors Hero * M & M * Table 2: Descriptive Statistics of closing price of each stock. *- significant at 5% level The highest average price is found to be for Hero Motor Corp with comparatively high standard deviation while the lowest is found for Mahindra & Mahindra at an average with SD All the series are asymmetric (skewness coefficient 0) and tend to have kurtosis very close to normal (kurtosis coefficient almost = 3). However Normality test suggests the presence of normality in the data. Test for Stationarity Stationarity of the series is the pre- requisite of any time-series to develop any forecasting model. In this study we have used Augmented Dickey- Fuller test to test for stationarity. The series and the first differenced data used for stationarity test. The null hypothesis for ADF test is that the series has Unit root. The results are displayed in table 2. id: editorijrim@gmail.com 5

6 ADF at Level t- Statistic p- value Conclusion ADF at First Difference t- Statistic p- value Conclusion Bajaj Not stationary Bajaj * 0.01 Stationary Tata Motors Not stationary Tata Motors * 0.01 Stationary Hero Not stationary Hero * 0.01 Stationary Mahindra Not stationary Mahindra * 0.01 Stationary Table 3: ADF test result at level and first difference. (level of significance 1%) All the four closing prices are non- stationary at level. Stationarity is attained at first difference. ADF test indicates that the series are stationary at first difference, i.e, the series are I(1).Once the stationarity is obtained, the parameters(p, q) for AR and MA models are located by inspecting the correlogram. 70 percent of the data is only selected for the construction of the model. For all the seriescorrelogram observed suggested, AR (1) to be best with no MA. This is confirmed by observing the AIC as the minimum with various combinations of (1, 1,0), (1,1,2),(2,1,0), (2,1,1),(2,1,2). No other values of p and q were used as Box- Jenkins method recommends total number of parameters to be less than 3.E-views software is used for data analysis in this paper. id: editorijrim@gmail.com 6

7 Date: 01/07/16 Time: 14:02 Sample: Included observations: 1680 Autocorrelation Partial Correlation AC Autocorrelation PAC Q-Stat Partial Prob Correlation AC PAC IJRFM Volume 6, Issue 4 (April, 2016) (ISSN ) Fig 2(a)Bajaj Auto. Date: 01/07/16 Time: 14:00 Sample: Included observations: Fig 2(c) Tata motors Date: 01/07/16 Time: 13:56 Sample: Included observations: 1966 Autocorrelation Partial Correlation AC PAC Q-Stat Prob Fig 2(b) Hero Motor Corp Fig 2(d) Mahindra & Mahindra Figure2: Correlogram of Closing price of Bajaj Auto, Tata Motors, Hero Motor Corp, Mahindra &Mahindra The correlograms of all the series more or less resembled each other and the the figure suggests AR with lag 1 and no MA. Still the model accuracy was confirmed with different combinations of (1, 1,0), (1,1,2),(2,1,0), (2,1,1),(2,1,2) The model is appropriateness is also confirmed by Root Mean Square Error (RMSE) and Mean Absolute Percent Error (MAPE). This is reassured by analysingthe percent of accuracy which is determined as Accuracy percent is (1 residual/actual series value)*100. id: editorijrim@gmail.com 7

8 Sector Firm p d α 1 AIC RMSE MAPE Percent of Accuracy Sectorwise % of Accuracy Automobiles Bajaj Tata Motors Hero Mahindra Table 4: Parameter estimation for ARIMA model in the 70% of test data The coefficient of the AR model implies a slow convergence of the series. Moreover it is interesting to notice that the prediction to a great extent can be dependent on only the variable value with unit lag and is not significantly influenced by the error terms. This mentions the immediate future prediction power the values of the stock prices. Fig 3: Forecast of the series The prediction accuracy is checked for the remaining 30% of test data; the result displayed in Table 5. Sector Firm RMSE Accuracy% Sector-wise % of Accuracy Automobiles Bajaj Tata Motors Hero Mahindra Table 5: Result of accuracy check on the 30% of test data. id: editorijrim@gmail.com 8

9 The test data also provides a high degree of accuracy (91.88%). The results are confirming the appropriateness of the ARIMA models developed. Conclusion The present study has the objective to develop an appropriate ARIMA model for analysis and forecast of stock prices of automobile sector. Fourautomobile companies and their closing stock price of 8 years were usedfor the study. The data is partitioned into two- 70% of observations were utilized for model development while the remaining 30% for confirmation of the accuracy of the model developed. The developed models all have common characteristic that they are all integrated at first order and are Autoregressive models with lag 1 having no MA characteristics. The prediction accuracy is also highly acceptable (> 75% accuracy). Since the series are highly correlated to the immediate past values forecast accuracy will be more. This phenomenon of the stock prices uniformly across various companies can be made use of for prediction and investment decisions. References [1]Chen, S., et al. "The time series forecasting: from the aspect of network." arxiv preprint arxiv: (2014). [2] Box, George EP, and George C. Tiao. "Intervention analysis with applications to economic and environmental problems." Journal of the American Statistical Association (1975): [3] A. Pankratz, Forecasting with Dynamic Regression model s, Wiley Interscience, [4] Devi, B. Uma, D. Sundar, and P. Alli. "An Effective Time Series Analysis for Stock Trend Prediction Using ARIMA Model for Nifty Midcap - 50." [5] Al Wadia, MohdTahir Ismail S, Selecting Wavelet Transforms Model in Forecasting Financial Time Series Data Based on ARIMA Model, Applied Mathematical Sciences, Vol. 5, 20 11, no. 7, [6]Gujarathi, Porter, Gunasekar. (2012). Basic Econometrics, McGraw Hill Pvt. Ltd. [7]David Raymond Anderson. (2008). Model based inference in the life sciences: a primer on evidence, New York, Springer. [8]Mondal, Shit, Goswami. (2014).Study of effectiveness of time series modelling (ARIMA)in forecasting stock prices; International Journal of Computer Science, Engineering and Applications (IJCSEA)Vol.4, No.2. id: editorijrim@gmail.com 9