1 Journal of Statistical and Econometric Methods, vol.3, no.3, 2014, ISSN: (print), (online) Scienpress Ltd, 2014 Forecasting Stock Market Series with ARIMA Model Fatai Adewole Adebayo 1, Ramysamy Sivasamy 2 Dahud Kehinde Shangodoyin 3 and Abstract Forecasting financial time series such as stock market has drawn considerable attention among applied researchers because of the vital role which stock market play on the economy of any nation. To date, autoregressive integrated moving average (ARIMA) model has remained the mostly widely used time series model for forecasting stock market series the problem of selecting the best ARIMA model for stock market prediction has attracted a huge literature in empirical analysis in view of its implication for national economics. In this paper, we consider the problem of selecting best ARIMA models for stock market prediction for Botswana and Nigeria. Using the standard model selection criteria such as AIC, BIC, HQC, RMSE and MAE we evaluate the forecasting performance of various candidate ARIMA models with a view to determining the best ARIMA model for predicting stock market in each country under investigation. The outcome of the 1 University of Botswana. 2 University of Botswana. 3 University of Botswana. Article Info: Received : May 18, Revised : June 28, Published online : August 15, 2014.
2 66 Forecasting Stock Market Series with ARIMA Model empirical analysis indicated that ARIMA (3,1,1) and ARIMA (1,1,4) models are the best forecast models for Botswana and Nigeria stock market respectively. Keywords: ARIMA; Information Criteria; Stock market; Forecasting; Unit root tests 1 Introduction Stock market forecasters focus on developing a successful approach for forecasting or predicting index value of stock prices. The vital idea to successful stock market prediction is achieving best result and minimizes the inaccurate forecast of the stock price. Indisputably, forecasting stock indices is very difficult because the market indices are highly fluctuating as result of increase or decrease that characterize the stock price. Fluctuations are affecting the investor s belief. Determining more effective ways of stock market index prediction is important for stock market investors in order to make more informed and accurate investment decisions. Hsien-Lun et al.  (2010) used ARIMA model and vector ARMA model with fuzzy time series method for forecasting. Fuzzy time series method especially heuristic model performs better forecasting ability in short-term period prediction. The ARIMA model creates small forecasting errors in longer experiment time period. In this paper, the author investigated whether the length of the interval will influence the forecasting ability of the models or not. Contreras, et al. (2003)  used ARIMA models to predict next day electricity prices; they have found two ARIMA models to predict hourly price in the electricity market of Spain and California. The Spanish model needs 5 hours to predict future prices as proposed to the 2 hours needs by the California model. Kumar et al. (2004)  used ARIMA model to forecast daily maximum surface ozone concentration in
3 F.A. Adebayo, R. Sivasamy and D.K. Shangodoyin 67 Brunei Darussalam. They have found that ARIMA (1, 0,1) was suitable for the surface 0 3 data collected at the airport in Brunei Darussalam. Tsitsika, et al. (2007)  used ARIMA model to forecast pelagic fish production. The final model selected was of the form ARIMA (1, 0, 1) and ARIMA (0, 1, 1).Azad et al. (2011)  used ARIMA model in forecasting Exchange rates of Bangladesh. By using Box Jenkins methodology they tried to find out best model for forecasting. They have found that ERNN (exchange rate neural network) model shows better performance than ARIMA. Merh, (2011)  used ANN and ARIMA model in next day stock market forecasting. They used ANN (4-4-1) and ARIMA (1, 1,1) for forecasting the future index value of sensex (BSE 30). The forecasting accuracy obtained for ARIMA (1,1,1) is better than ANN(4-4-1). Liv et al (2011)  used ARIMA model in forecasting incidence of hemorrhagic fever with renal syndrome in china. The goodness of fit test of the optimum ARIMA (0,3,1) model showed non significant autocorrelation in the residuals of the model. Similarly, Datta (2011) used ARIMA model in forecasting inflation in Bangladesh Economy. He showed that ARIMA (1,0,1) model fits the inflation data of Bangladesh satisfactorily. Al-Zeaud (2011) used ARIMA model in modeling and forecasting volatility. The result showed that best ARIMA models at 95% confidence interval for banks sector is ARIMA (2,0,2) model. Uko et al. (2012) examined the relative predictive power of ARIMA, VAR and ECM model in forecasting inflation in Nigeria. The result shows that ARIMA is good predictor of inflation in Nigeria and serves as a benchmark model in inflation forecasting. Following the introduction, the paper is organized as follows. Section two discusses the methodology to be used to analysis the data. Section three examines ARIMA model selection. Section four describes the data use, econometric methodology and the model with empirical investigation with result reported. Section five ends with conclusion.
4 68 Forecasting Stock Market Series with ARIMA Model 2 Preliminary Unit Root Test The popular way of detecting a unit root is to examine a series, mean and covariance, if the mean is increasing over period of time. Another way would be to perform the t-test and F test to find out whether the autocorrelation function (ACFs) for the variable tend to zero as the length of the lag increases. If the ACFs tend to zero quickly then the variable is stationary; otherwise the variable is non-stationary. Another known method for the testing of non-stationary status is the Dickey-Fuller (DF) test developed by Dickey and Fuller (1979). The test examines the hypothesis that the variable being tested contains a unit root and therefore should be expressed in first difference form. In order to run a DF test, the following equations are estimated. Y = δy + U No drift, no intercept (1) t t 1 t Yt = β0 + δyt 1+ UtIntercept, no drift term (2) Yt = β0 + β1t+ δyt 1+ UtWith intercept and trend (3) However, the Dickey-Fuller test is only valid if the there is no autocorrelation ofu t, the white noise. When U t is auto-correlated then chance of rejecting a correct null hypothesis is high, hence augmenting the test using lags of the dependent variable would be necessary thus the use of the Augmented Dickey-Fuller (ADF) test Said and Dickey (1984). Just like the DF and ADF test can be specified with no drift and no trend; with trend and no drift; lastly with both trend and drift as follows. Yt = δyt 1+ αi Yt 1+ UtNo drift, no intercept (4)
5 F.A. Adebayo, R. Sivasamy and D.K. Shangodoyin 69 Yt = β0 + δyt 1+ αi Yt 1+ UtIntercept, no drift term (5) Yt = β0 + β1t+ δyt 1+ αi Yt 1+ UtWith intercept and trend (6) Furthermore, Phillips-perron unit root tests are used to reinforce the ADF. One advantage that the Phillips-Perron (PP) test has over the ADF test is that it is robust with respect to unspecified autocorrelation and heteroscedasticity in the disturbance process of the test regression, Brooks (2000). Therefore the PP test works well with financial time series. The two tests specify the Null hypothesis ( H 0 ) as that the time series has unit root, thus the time series is non-stationary against the Alternative Hypothesis ( H 1) that the time series has no unit root, thus a stationary time series: H 0 : Time series has a unit root ( δ = 1) H 1: Time series has no unit root ( δ 1) The decision rule of whether to reject or not reject the null hypothesis is as follows; Reject the Null hypothesis if the absolute ADF/PP test statistics < critical value Do not reject the Null hypothesis if the absolute ADF/PP test statistics > critical value. 3 ARIMA Model Selection This procedure is based on fitting an autoregressive integrated moving average (ARIMA) model to the given set of data and then taking conditional expectations. The Box-Jenkins methodology refers to the set of procedures for model identification, estimation, and diagnostic checking ARIMA model with time series data. Forecasts follow directly from the form of the fitted model.
6 70 Forecasting Stock Market Series with ARIMA Model The first step in the Box-Jenkins procedure is to difference the data until it is stationary. A pth-order autoregressive model: AR (p), which has general form where Y φ φy φ Y φ Y ε t= t 1+ 2 t p t p+ t Y t = Response (dependent) variable at time t. Y, Y,... Y t 1 t 2 t p = Response variable at time lags t-1, t-2,..., t-p, respectively. φ0, φ1, φ2,..., φ p = Coefficients to be estimated ε t = Error term at time t. A qth-order moving average model: MA (q), which has the general form Y µ ε θε θε θ ε t= + t 1 t 1 2 t 2... q t q where Y t = Response (dependent) variable at time t µ = constant mean of the process θ1, θ2,... θ q = coefficient to be estimated ε t = Error term at time t ε, ε,..., ε t 1 t 2 t q = Errors in previous time periods that are incorporated in the response form Y t Autoregressive Moving Average Model: ARMA (p,q), which has the general Y = φ + φy + φy φ Y + ε θε θε... θ ε t 0 1 t 1 2 t 2 p t p t 1 t 1 2 t 2 q t q We can use the graph of the sample autocorrelation function (ACF) and the sample partial autocorrelation function (PACF) to determine the model which processes can be summarized as follows:
7 F.A. Adebayo, R. Sivasamy and D.K. Shangodoyin 71 Model ACF PACF AR(p) Dies down Cut off after lag q MA (q) Cut off after lag p Dies down ARMA (p,q) Dies down Dies down 4 Main Results For empirical illustration we use the quarterly data on stock market data that were collected from Botswana and Nigeria during 2002 to We carried out unit root test to determine the order of integration of series. We present the empirical result of our data analysis in Table 1 and 2 below. Botswana Stock Market Series Result. ARIMA Model (p,d,q) Table 1a show Statistical Result for ARIMA model φ t-test P-value p θ t-test P-value q (1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (2,1,1) (2,1,2) (2,1,3) s(2,1,4) (2,1,5)
8 72 Forecasting Stock Market Series with ARIMA Model (3,1,1) (3,1,2) (3,1,3) (3,1,4) (3,1,5) The summary of the various statistics obtained from fitting the model are tabulated in Table 2. Table 1b summary of Portmanteau Test of stock market in Botswana Variable (1,1,2) (1,1,3) (1,1,5) (2,1,3) (2,1,4) (2,1,5) (3,1,1) (3,1,3) (3,1,4) (3,1,5) AIC SIC HQC RMSE MAE In selecting the best ARIMA model of stock market series in Botswana we subject all the ARIMA model to Akaike Information Criterion (AIC) and Schwarz Bayesian Information Criterion (SBIC). The above result of table 1b show that ARIMA (3,1,1) is preferred to others since it has the lowest value of AIC as well, as compared with (p, d, q) (3,1,1) FORECAST EVALUATION FOR ARIMA (3,1,1) MODEL In the next step, the forecast of Botswana stock market series using ARIMA (3,1,1) model was conducted. The duration of forecasts is from 2008Q1 to 2013Q3. Then those forecasted value are plotted in Figure 1.
9 F.A. Adebayo, R. Sivasamy and D.K. Shangodoyin Forecast: DIVIDEND_YF Actual: DIVIDEND_YIELD Forecast sample: 2008Q1 2013Q3 Adjusted sample: 2009Q1 2013Q3 Included observations: 19 Root Mean Squared Error Mean Absolute Error Mean Abs. Percent Error Theil Inequality Coefficient Bias Proportion Variance Proportion Covariance Proportion I II III IV I II III IV I II III IV I II III IV I II III DIVIDEND_YF ± 2 S.E. Figure 1: Forecast of Stock Market series by ARIMA (3.1.1) model. In Figure 1 the middle line represents the forecast value of stock market. Meanwhile, the line which are above or below the forecast quarterly stock market series show the forecast with ± 2 of standard errors. Some forecast evaluation measures such as root mean square error (RMSE), mean absolute error (MAE) and Theil Inequality coefficient value were used to select ARIMA (3,1,1) model. Thus, smaller RMSE, MAE and Theil Inequality coefficient are preferred.
10 74 Forecasting Stock Market Series with ARIMA Model Nigeria stock market Result ARIMA Model (p,d,q) Table 2a show Statistical Result for ARIMA Models φ t-test p-value p θ q t-test p-value (1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (2,1,1) (2,1,2) (2,1,3) (2,1,4) (2,1,5) (3,1,1) (3,1,2) (3,1,3) (3,1,4) (3,1,5) In selecting the best ARIMA model of stock market series we subject all the ARIMA model to Akaike Information Criterion (AIC) and Schwarz Bayesian Information Criterion (BIC). The result above show that ARIMA (1,1,4) is preferred to others since it has the lowest values of AIC and BIC.
11 F.A. Adebayo, R. Sivasamy and D.K. Shangodoyin 75 The summary of the various statistics obtained from fitting the model are tabulated in Table 2b. ARIMA (1,1,1) (1,1,4) (2,1,4) (3,1,2) (3,1,4) Model AIC SBIC HQC RME MAE Forecast Evaluation for ARIMA (1,1,4) Model The forecast of Nigeria stock market series using ARIMA (1,1,4) model was conducted. The duration of forecasts is from 2008Q1 to 2012Q4. The forecast are plotted in Figure Forecast: DIVIDENDF Actual: DIVIDEND Forecast sample: 2008Q1 2012Q4 Adjusted sample: 2008Q3 2012Q4 Included observations: 18 Root Mean Squared Error Mean Absolute Error Mean Abs. Percent Error Theil Inequality Coefficient Bias Proportion Variance Proportion Covariance Proportion III IV I II III IV I II III IV I II III IV I II III IV DIV IDENDF ± 2 S.E. Figure 2: Forecast of Stock Market series by ARIMA (1.1,4) model.
12 76 Forecasting Stock Market Series with ARIMA Model In Figure 1 the middle line represents the forecast value of stock market. Meanwhile, the line which are above or below the forecasted quarterly stock market series show the forecast with ± 2 of standard errors. Some forecast evaluation measure such as root mean square error (RMSE), mean absolute error (MAE) and Theil Inequality coefficient value were used to select ARIMA (1,1,4) model. Thus, smaller RMSE, MAE and Theil Inequality coefficient are preferred. 5 Conclusion This study investigates the choice of the best ARIMA model in forecasting the stock market series in Botswana and Nigeria. Having a reliable statistical forecast model for predicting stock market is a very crucial issue among investors, portfolio managers and developing economies such as Botswana and Nigeria as it helps the investors as well as other stakeholders that are connected to the business of stock market to make informed decision about investment in stock market. The empirical analysis indicated that the ARIMA (3,1,1) and (1,1,4) models are the best models for forecasting stock market series in Botswana and Nigeria. References  H.A. Al-Zeaud, Modeling & Forecasting Volatility using ARIMA model, European Journal of Economics, Finance and Administrative Science, 35, (2011),  A.K. Azad and M. Mahsin, Forecasting Exchange Rates of Bangladesh using ANN and ARIMA models: A comparative study, International Journal of Advanced Engineering Science & Technologies, 10(1), (2011),
13 F.A. Adebayo, R. Sivasamy and D.K. Shangodoyin 77  J. Contreras, R. Espinola, F.J. Nogales and A.J. Conejo, ARIMA models to predict Next Day Electricity Prices, IFEE Transactions on power system, 18(3), (2003),  K. Datta, ARIMA Forecasting of Inflation in the Bangladesh Economy, The IUP Journal of Bank Management, X(4), (2011),  Hsien-Lun Wong, Yi-HsienTu and Chi-Chen Wang, Application of fuzzy time series models for forecasting the amount of Taiwan export, Experts Systems with Applications, (2010),  K. Kumar, A.K. Yadav, M.P. Singh, H. Hassan and V. K. Jain, Forecasting Daily Maximum Surface Ozone, (2004).  Q. Liv, X. Liu, B. Jiang and W. Yang, Forecasting incidence of hemorrhagic fever with renal syndrome in China using ARIMA model, Biomed Central, (2011), 1-7.  N. Merh, V. P. Saxena and K. R. Pardasani, Next Day Stock market Forecasting: An Application of ANN & ARIMA, The IUP Journal of Applied Science, 17(1), (2011),  E.V. Tsitsika, C. D. Maravelias and J. Haralatous, Modeling & forecasting pelagic fish production using univariate and multivariate ARIMA models, Fisheries Science, 73. (2007),  A.K. Uko and E. Nkoro, Inflation Forecasts with ARIMA, Vector Autoregressive and Error Correction Models in Nigeria, European Journal of Economics, Finance & Administrative Science, 50, (2012),
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