This paper aims to evaluate the performance of the ARIMA model to predict the time series of the Bovespa Index, measured by MAPE (mean absolute error percentage) and compare it with other models. Historical data of monthly Bovespa quotations from January 1995 to January 2013 were used. Therefore conclude that the ARIMA(1,1,1) model is the best ARIMA model for the original time series being analyzed ( Naphtha product). The final model is of the following form: Table (6): Estimated model parameters of Naphtha sales model We obtained the model in the form: ˆ 0.6010 1.1713 0 0 (6) Z. t. Z. t 1 t 1 t. Forecasting . After the Multiplicative Seasonal ARIMA The seasonal di erence operator is rs= 1 Bs. Some series show slow decay of ACF only at lags s;2s;:::, which suggests seasonal di erencing. { But note: seasonal means also give slow decay of ACF at those lags. The Multiplicative Seasonal ARIMA model ARIMA(p;d;q) (P;D;Q)s is P(Bs)˚p(B)rDsrdxt= Q(Bs) q(B)wt: 10 APPENDIX 4 The Identification of ARIMA Models As we have established in a previous lecture, there is a one-to-one cor-respondence between the parameters of an ARMA(p,q) model, including thevariance of the disturbance, and the leading p + q + 1 elements of the auto- covariance function. ARIMA Model Parameters Constant Lag 1 ag2 Lag 1 ag2 Seasonal Difference MA, Seasonal Lag 1 Estimate 887 936 1 803 3 423 1 648 No Transformation Mode GapSa es-Mode Number of Predictors Model Statistics Number of Outliers Mode sa s cs Stationary R- squared Ljung-BoxQ( 8) Sa s cs Sasc Stationary R-squared R-squared RMSE MARE Max.APE Max.A With this notation, the AR model reads 1 Xp k=1 kL k! X t = "t: 1.3. MA models. A MA(q) (Moving Average with orders p and q) model is an explicit formula for X t in terms of noise of the form X t = "t + 1" t 1 +:::+ q" t q: The process is given by a (weighted) average of the noise, but not an average from time zero to the B. ARIMA model The main part of the ARIMA model combines AR and MA polynomials into a complex polynomial, as seen in (1) below [9]. The ARIMA (p, d, q) model is applied to all the data points of the TC data. 1 1 1 1 p q t tt t i i y y (1) where the notation is as follows: Before an ARIMA model can be fitted, a good training data set must be selected. This process is crucial because if poor data is selected, the model's forecasts will be naïve. The key to selecting an optimal data set is an awareness of ARIMA's limitations. ARIMA models are known to function best when fitted to trending data, but don't deal ARIMA(0, 1, 0) - known as the random walk model; ARIMA(1, 1, 0) - known as the differenced first-order autoregressive model, and so on. Once the parameters (p, d, q) have been defined, the ARIMA model aims to estimate the coefficients α and θ, which is the result of using previous data points to forecast values. Applications of the ARIMA Using the ARIMA procedure, we can create a forecasting model with predictors, and see if there is a significant difference in predictive ability over the exponential smoothing model with no predictors. With the ARIMA method, you can fine-tune the model by specifying orders of autoregression, differencing, and moving average, as well as seasonal network, hybrid models and ARIMA models for more details refer to (Atsalakis&Kimon, 2009; Mitra, 2009). ARIMA model is a traditional model with widely application in stock market data application (Wang, 2011; Awajan et al., 2017a). This model is generated by two approaches which are statistical an