AIC, BIC, etc. To determine the order of a non-seasonal ARIMA model, a useful criterion is the Akaike information criterion (AIC). It is written as where L is the likelihood of the data, p is the order of the autoregressive part and q is the order of the moving average part. The k represents the intercept of the ARIMA model. For AIC, if k = 1 then there is an intercept in the ARIMA model (c ≠ 0) and if k = 0 then there is no intercept in the ARIMA model (c = 0). The corrected AIC for ARIMA models can be written as The Bayesian Information Criterion (BIC) can be written as The objective is to minimize the AIC, AICc or BIC values for a good model. The lower the value of one of these criteria for a range of models being investigated, the better the model will suit the data. The AIC and the BIC are used for two completely different purposes. While the AIC tries to approximate models towards the reality of the situation, the BIC attempts to find the perfect fit. The BIC approach is often criticized as there never is a perfect fit to real-life complex data; however, it is still a useful method for selection as it penalizes models more heavily for having more parameters than the AIC would. AICc can only be used to compare ARIMA models with the same orders of differencing. For ARIMAs with different orders of differencing, RMSE can be used for model comparison. Estimation of coefficients Forecasts using ARIMA models The ARIMA model can be viewed as a "cascade" of two models. The first is non-stationary: while the second is wide-sense stationary: Now forecasts can be made for the process , using a generalization of the method of autoregressive forecasting. Forecast intervals The forecast intervals (confidence intervals for forecasts) for ARIMA models are based on assumptions that the residuals are uncorrelated and normally distributed. If either of these assumptions does not hold, then the forecast intervals may be incorrect. For this reason, researchers plot the ACF and histogram of the residuals to check the assumptions before producing forecast intervals. 95% forecast interval: , where is the variance of . For , for all ARIMA models regardless of parameters and orders. For ARIMA(0,0,q), In general, forecast intervals from ARIMA models will increase as the forecast horizon increases. Variations and