Among the most popular time series forecasting methods that also have been used for financial time series are Box-Jenkins models, also known as ARIMA models.
Many academic papers discuss the use of Box-Jenkins as a method for market time series forecasting, from papers in the early 1990s on the S&P 500 to relatively recent studies, such as the 2008 effort, “Comparing the performance of time series models for forecasting exchange rates,” BRAC University Journal, vol. V no 2, 2008, 55-65, by M.K Newaz. The researcher compares various classic time series models on the India rupee, including the ARIMA models, and finds that ARIMA performs well. Newaz finds that the first difference of the rupee series, not the series itself, is stationary.
ARIMA stands for Autoregressive Integrated-Moving-Average. The “integrated” indicates that the time series is transformed into a stationary series. The “auto” means the transformed series is self-referential. ARIMA represents three different types of models:
- AR, autoregressive
- MA, moving average
- ARMA, both AR and MA in the same model
An AR model is like a simple linear regression model, except that the independent variables are, in practice, time-lagged versions of the dependent variable, time series itself; thus, it is autoregressive. An autoregressive model can have multiple terms and be either linear or nonlinear.
An MA model is a weighted moving average of a fixed number of previously produced forecast errors. Traders expect a moving average to be of the series itself, such as an average of closing prices, but the average in this case is of forecast errors. The term is conceptually used identically across trading and forecasting, but the application differs. The average error is used to correct the error of the regressive model.
Box-Jenkins models are univariate, based on a single time series. The model establishes a relationship between present and past values of the series so the past values can then be used in forecasting. These models require stationary series. Even though Box-Jenkins has been used in many studies to forecast market data, the method is not totally suitable because market data are not stationary and do not have a normal distribution.