The Box-Jenkins methodology  is a five-step process for identifying, selecting, and assessing conditional mean models (for discrete, univariate time series data).
Establish the stationarity of your time series. If your series is not stationary, successively difference your series to attain stationarity. The sample autocorrelation function (ACF) and partial autocorrelation function (PACF) of a stationary series decay exponentially (or cut off completely after a few lags).
Identify a (stationary) conditional mean model for your data. The sample ACF and PACF functions can help with this selection. For an autoregressive (AR) process, the sample ACF decays gradually, but the sample PACF cuts off after a few lags. Conversely, for a moving average (MA) process, the sample ACF cuts off after a few lags, but the sample PACF decays gradually. If both the ACF and PACF decay gradually, consider an ARMA model.
Specify the model, and estimate the model parameters. When fitting nonstationary models in Econometrics Toolbox™, it is not necessary to manually difference your data and fit a stationary model. Instead, use your data on the original scale, and create an
arimamodel object with the desired degree of nonseasonal and seasonal differencing. Fitting an ARIMA model directly is advantageous for forecasting: forecasts are returned on the original scale (not differenced).
Conduct goodness-of-fit checks to ensure the model describes your data adequately. Residuals should be uncorrelated, homoscedastic, and normally distributed with constant mean and variance. If the residuals are not normally distributed, you can change your innovation distribution to a Student’s t.
After choosing a model—and checking its fit and forecasting ability—you can use the model to forecast or generate Monte Carlo simulations over a future time horizon.
 Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
- Implement Box-Jenkins Model Selection and Estimation Using Econometric Modeler App
- Detect Serial Correlation Using Econometric Modeler App
- Select ARIMA Model for Time Series Using Box-Jenkins Methodology
- Assess Predictive Performance
- Check Fit of Multiplicative ARIMA Model
- Box-Jenkins Differencing vs. ARIMA Estimation