The aim this study is discussed on the detection and correction of data containing the additive outlier (AO) on the model ARIMA (p, d, q). The process of detection and correction of data using an iterative procedure popularized by Box, Jenkins, and Reinsel (1994). By using this method we obtained an ARIMA models were fit to the data containing AO, this model is added to the original model of ARIMA coefficients obtained from the iteration process using regression methods. In the simulation data is obtained that the data contained AO initial models are ARIMA (2,0,0) with MSE = 36,780, after the detection and correction of data obtained by the iteration of the model ARIMA (2,0,0) with the coefficients obtained from the regression       1 2 1 2 3 0,106 0, 204 0, 401 329 115 35,9 t t t Z Z Z X t X t X t         and MSE = 19,365. This shows that there is an improvement of forecasting error rate data.

This study is to expose the sharia concept in the Islamic market, especially on the practice of the equilibrium model or the Capital Asset Pricing Model (CAPM). Islamic index and sharia market are introduced to answer the Islamic investment. However, we cannot apart from the interest rate, which is related to ‘riba’ and prohibited in Islam religion. Many references proposed the Islamic theory into the CAPM, so the model has been modified and adjusted to deliver the new solution on sharia investment. We provide a general illustration to explain how the sharia concept in CAPM as an equilibrium model and its implementation in Jakarta Islamic Index (JII). The result shows that the range of return is various, while the risk both beta and standard deviation have remained steady. The result shows that the Sharia version with the Sukuk rate performs better than the others from the expected return.

Forecasting is a part of statistical modelling that is widely used in various fields because of its benefits in decision-making. The purpose of forecasting is to predict the future values of certain variables that vary with time using its previous values. Forecasting is related to the formation of models and methods that can be used to produce a good forecast. This research is a survey paper research that used a systematic mapping study and systematic literature review. Generally, time series research uses linear time series models, specifically the autoregressive integrated moving average model that has long been used because it has good forecasting accuracy. The successfulness of the Box–Jenkins methodology is based on the reality that various models can imitate the behaviour of various types of series, usually without requiring many parameters to be estimated in the final choice of the model. However, the assumption of stationarity that must be met makes this method inflexible to use. With the advances in computers, forecasting methods from stochastic models to soft computing continue to develop and extend. Soft computing for forecasting can provide more accurate results than traditional methods. Moreover, soft computing has many advantages in terms of the amount of data that can be analysed and the time- and cost-effectiveness of the process.

NARNN is a type of ANN model consisting of a limited number of parameters and widely used for various applications. This study aims to determine the appropriate NARNN model, for the selection of input variables of nonlinear autoregressive neural network model for time series data forecasting, using the stepwise method. Furthermore, the study determines the optimal number of neurons in the hidden layer, using a trial and error method for some architecture. The NARNN model is combined in three parts, namely the learning method, the activation function, and the ensemble operator, to get the best single model. Its application in this study was conducted on real data, such as the interest rate of Bank Indonesia. The comparison results of MASE, RMSE, and MAPE values with ARIMA and Exponential Smoothing models shows that the NARNN is the best model used to effectively improve forecasting accuracy.

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