Stock Price Prediction Using The ARIMA Model - IJSSST

2014 UKSim-AMSS 16th International Conference on Computer Modelling and SimulationStock Price Prediction Using the ARIMA Model1Ayodele A. Adebiyi., 2Aderemi O. Adewumi1,23Charles K. Ayo3School of Mathematic, Statistics & Computer ScienceUniversity of KwaZulu-NatalDurban, South AfricaDepartment of Computer & Information SciencesCovenant UniversityOta, Nigeriaemail: {adebiyi, adewumia}@ukzn.ac.zaemail: charles.ayo@covenantuniversity.edu.ngintelligence techniques [2]. ARIMA models are known to berobust and efficient in financial time series forecastingespecially short-term prediction than even the most popularANNs techniques ([8, 9, 10]. It has been extensively used infield of economics and finance. Other statistics models areregression method, exponential smoothing, generalizedautoregressive conditional heteroskedasticity (GARCH).Few related works that has engaged ARIMA model forforecasting includes [11, 12, 13, 14, 15, 16].In this paper extensive process of building ARIMAmodels for short-term stock price prediction is presented.The results obtained from real-life data demonstrated thepotential strength of ARIMA models to provide investorsshort-term prediction that could aid investment decisionmaking process.The rest of the paper is organized as follows. Section 2presents brief overview of ARIMA model. Section 3describes the methodology used while section 4 discussesthe experimental results obtained. The paper is concluded insection 5.Abstract— Stock price prediction is an important topic infinance and economics which has spurred the interest ofresearchers over the years to develop better predictive models.The autoregressive integrated moving average (ARIMA)models have been explored in literature for time seriesprediction. This paper presents extensive process of buildingstock price predictive model using the ARIMA model.Published stock data obtained from New York Stock Exchange(NYSE) and Nigeria Stock Exchange (NSE) are used with stockprice predictive model developed. Results obtained revealedthat the ARIMA model has a strong potential for short-termprediction and can compete favourably with existingtechniques for stock price prediction.Keywords- ARIMA model, Stock Price prediction, Stockmarket, Short-term prediction.I.INTRODUCTIONPrediction will continue to be an interesting area ofresearch making researchers in the domain field alwaysdesiring to improve existing predictive models. The reasonis that institutions and individuals are empowered to makeinvestment decisions and ability to plan and developeffective strategy about their daily and future endevours.Stock price prediction is regarded as one of most difficulttask to accomplish in financial forecasting due to complexnature of stock market [1, 2, 3]. The desire of manyinvestors is to lay hold of any forecasting method that couldguarantee easy profiting and minimize investment risk fromthe stock market. This remains a motivating factor forresearchers to evolve and develop new predictive models[4].In the past years several models and techniques had beendeveloped to stock price prediction. Among them areartificial neural networks (ANNs) model which are verypopular due to its ability to learn patterns from data andinfer solution from unknown data. Few related works thatengaged ANNs model to stock price prediction are [5, 6, 7].In recent time, hybrid approaches has also been engaged toimprove stock price predictive models by exploiting theunique strength of each of them [2]. ANNs is from artificialintelligence perspectives.ARIMA models are from statistical models perspectives.Generally, it is reported in literature that prediction can bedone from two perspectives: statistical and artificial978-1-4799-4923-6/14 31.00 2014 IEEEDOI 10.1109/UKSim.2014.67II.ARIMA MODELBox and Jenkins in 1970 introduced the ARIMA model.It also referred to as Box-Jenkins methodology composed ofset of activities for identifying, estimating and diagnosingARIMA models with time series data. The model is mostprominent methods in financial forecasting [1, 12, 9].ARIMA models have shown efficient capability to generateshort-term forecasts. It constantly outperformed complexstructural models in short-term prediction [17]. In ARIMAmodel, the future value of a variable is a linear combinationof past values and past errors, expressed as follows:Yt φ0 φ1Yt 1 φ2Yt 2 . φpYt p εt θ1εt 1 θ2εt 2 . θqεt q(1)where,Yt is the actual value andθ j areε t is the random error at t, φi andthe coefficients, p and q are integers that are oftenreferred to as autoregressive andrespectively.105moving average,

The steps in building ARIMA predictive model consist ofmodel identification, parameter estimation and diagnosticchecking [18].III.Figure 2 is the correlogram of Nokia time series. From thegraph, the ACF dies down extremely slowly which simplymeans that the time series is nonstationary. If the series isnot stationary, it is converted to a stationary series bydifferencing. After the first difference, the series“DCLOSE” of Nokia stock index becomes stationary asshown in figure 3 and figure 4 of the line graph andcorrelogram respectively.METHODOLOGYThe method used in this study to develop ARIMA modelfor stock price forecasting is explained in detail insubsections below. The tool used for implementation isEviews software version 5. Stock data used in this researchwork are historical daily stock prices obtained from twocountries stock exchanged. The data composed of fourelements, namely: open price, low price, high price andclose price respectively. In this research the closing price ischosen to represent the price of the index to be predicted.Closing price is chosen because it reflects all the activitiesof the index in a trading day.To determine the best ARIMA model among severalexperiments performed, the following criteria are used inthis study for each stock index. Relatively small of BIC (Bayesian or SchwarzInformation Criterion) Relatively small standard error of regression (S.E.of regression) Relatively high of adjusted R2 Q-statistics and correlogram show that there is nosignificant pattern left in the autocorrelationfunctions (ACFs) and partial autocorrelationfunctions (PACFs) of the residuals, it means theresidual of the selected model are white noise.The subsections below described the processes ofARIMA model-development.Figure 2: The correlogram of Nokia stock price indexA. ARIMA (p, d, q) Model for Nokia Stock IndexNokia stock data used in this study covers the periodfrom 25th April, 1995 to 25th February, 2011 having a totalnumber of 3990 observations. Figure 1 depicts the originalpattern of the series to have general overview whether thetime series is stationary or not. From the graph below thetime series have random walk pattern.Figure 3: Graphical representation of the Nokia stock price index afterdifferencing .Figure 4: The correlogram of Nokia stock price index after firstdifferencingFigure 1: Graphical representation of the Nokia stock closing price index106

In figure 5 the model checking was done withAugmented Dickey Fuller (ADF) unit root test on“DCLOSE” of Nokia stock index. The result confirms thatthe series becomes stationary after the first-difference of theseries.Figure 7 is the residual of the series. If the model isgood, the residuals (difference between actual and predictedvalues) of the model are series of random errors. Since thereare no significant spikes of ACFs and PACFs, it means thatthe residual of the selected ARIMA model are white noise,no other significant patterns left in the time series.Therefore, there is no need to consider any AR(p) andMA(q) further.Figure 7: Correlogram of residuals of the Nokia stock index.Figure 5: ADF unit root test for DCLOSE of Nokia stock index.In forecasting form, the best model selected can beexpressed as follows:Yt θ1Yt 1 θ 2Yt 2 ε t(2)Table 1 shows the different parameters of autoregressive(p) and moving average (q) among the several ARIMAmodel experimented upon . ARIMA (2, 1, 0) is consideredthe best for Nokia stock index as shown in figure 6. Themodel returned the smallest Bayesian or Schwarzinformation criterion of 5.3927 and relatively smalleststandard error of regression of 3.5808 as shown in figure 6. where,ε t Yt Yt(i.e., the difference between theactual value of the series and the forecast value)TABLE I: STATISTICAL RESULTS OF DIFFERENT ARIMAPARAMETERS FOR NOKIA STOCK INDEXARIMABICAdjusted R2(1, 0, 0)(1, 0, 1)(2, 0, 0)(0, 0, 1)(0, 0, 2)(1, 1, 0)(0, 1, 0)(0, 1, 1)(1, 1,2)(2, 1, 0)(2, 1, 00020.00000.00020.00350.00330.0031S.E. 3.58593.58543.58003.58083.5812The bold row represent the best ARIMA model among the severalexperiments.B. ARIMA (p, d, q) Model for Zenith Bank IndexThe stock data of Zenith bank used in this study coveredthe period from 3rd January, 2006 to 25th February, 2011 withtotal of 1296 observations. Figure 8 is the original pattern ofthe series. From the graph there was upward movement ofthe index from 2006 and downward movement is observedFigure 6: ARIMA (2, 1, 0) estimation output with DCLOSE of Nokiaindex.107

from 2008 possibly because of world financial crisisexperienced at that time.Figure 10: Graphical representation of the Zenith bank stock index firstdifferencingFigure 8: Graphical representation of the Zenith Bank stock index closingprice.Figure 9 is the correlogram of the time series of Zenithbank stock index. From the graph of the correlogram, theACF dies down extremely slowly which simply means thatthe time series is nonstationary. If the series is notstationary, there is need to convert to stationary series bydifferencing. After the first difference, the series“DCLOSE” of Zenith bank stock index becomes stationaryas shown in figure 10 and figure 11 of the line graph andcorrelogram of the series after first differencing.Figure 11: The correlogram of Zenith bank stock price index after firstdifferencing.Figure 12 is the ADF unit root test on “DCLOSE” of theseries which also indicates the first difference of the seriesbecomes stationary.Figure 9: The correlogram of Zenith Bank stock price index108

patterns left in the time series and there is no need forfurther consideration of another AR(p) and MA(q).Figure 12: ADF unit root test for DCLOSE of Zenith bank stock index.Table 2 shows the different parameters of autoregressive(p) and moveing average (q) of the ARIMA model in orderto get the best fitted model. ARIMA (1, 0, 1) is relativelythe best model as indicated in figure 13. The model returnedthe smallest Bayesian or Schwarz information criterion of2.3736 and relatively smallest standard error of regressionof 0.7872 as shown in figure 13.Figure 14: Correlogram of residuals of the Zenith bank stock index.In forecasting form, the best model selected can beexpressed as follows:Yt φ1Yt 1 θ1ε t 1 ε t(3)where, ε t Yt Yt(i.e., the difference between the actualvalue of the series and the forecast value)TABLE II: STATISTICAL RESULTS OF DIFFERENT ARIMAPARAMETERS FOR ZENITH BANK STOCK INDEXARIMABIC(1, 0, 0)(1, 0, 1)(2, 0, 0)(0, 0, 1)(0, 0, 2)(1, 1, 0)(0, 1, 0)(0, 1, 1)(1, 1,2)(2, 1, 0)(2, 1, 720.72280.07080.00000.06690.07010.00310.0036S.E. 81510.78730.78630.81440.8142The bold row represent the best ARIMA model among the severalexperimentsFigure 13: ARIMA (1, 0, 1) estimation output with DCLOSE of Zenithbank index.Figure 14 is the correlogram of residual of the seies. Fromthe figure it is obvious there is no significant spike of ACFsand PACFs. This means that the residual of this selectedARIMA model are white noise. There is no other significantIV.RESULTS AND DISCUSSIONThe experimental results of each of stock index arediscussed in the subsection below.109