Business Forecasting Chapter 10 The Box–Jenkins Method of Forecasting

Chapter Topics The Box–Jenkins Models Forecasting with Autoregressive Models (AR) Forecasting with Moving Average Models (MA) Autoregressive Integrated Moving Average (ARIMA) models Trends and Seasonality in Time Series Trends Seasonal Data Chapter Summary

Univariate Data A majority of the real-world data show certain combinations of the above patterns. Stationary Trend Seasonality Cyclical

Box–Jenkins Method Besides the smoothing techniques, what other methods can we use to forecast univariate data? Using Box–Jenkins Methods Capture the past patternForecast the future

Why Use The Box–Jenkins Method? When facing very complicated data patterns such as a combination of a trend, seasonal, cyclical, and random fluctuations: e.g. Earning data of a corporation e.g. Forecasting stock price e.g. Sales forecasting e.g. Energy forecasting (electricity, gas) e.g. Traffic flow of a city

Why Use the Box–Jenkins Method? When forecasting is the sole purpose of the model. Very reliable especially in short term (0–6 months) prediction; reliable in short-to-mid (6 months–1.5years) -term prediction. Confidence intervals for the estimates are easily constructed.

Pattern I: Stationary Pattern 1: No Trend—Stationary demand seems to cluster around a specific level.

Pattern II: Trend Demand consistently increases or decreases over time. Time

Pattern III: Seasonality

Pattern IV: Cyclical

Box–Jenkins Method Assumption In order to use the B/J method, the time series should be stationary. B/J main idea: Any stationary time series can self-predict its own future from the past data.

Box–Jenkins Method Assumption We know that not all time series are stationary. However, it is easy to convert a trend or a seasonal time series to a stationary time series. Simply use the concept of “Differencing.”

Example of Differencing

Convert a trend time series to stationary time series using the differencing method

Convert a seasonal time series to stationary time series

Differencing Summary To convert trend time series to stationary time series: To convert seasonal time series to stationary time series: Both of the above two methods can be applied/combined to remove the cyclical effects.

How do we decide the model? Use Autocorrelation (AC) and Partial Autocorrelation (PAC) First-order autocorrelation is a measure of how correlated an observation is with an observation one period away, that is: (y t,y t−1 ) Second-order autocorrelation is a measure of how correlated an observation is with an observation two periods away (y t,y t−2 ) etc...

AR Model Fit When the autocorrelation coefficients gradually fall to 0, and the partial correlation has spikes, an AR model is appropriate. The order of the model depends on the number of spikes. An AR(2) model is shown below.

MA Model Fit When the partial correlation coefficients gradually fall to 0, and the autocorrelation has spikes, a MA model is appropriate. The order of the model depends on the number of spikes. An MA(1) model is shown below.

ARIMA Model Fit When both the autocorrelation and the partial correlograms show irregular patterns, then an ARIMA model is appropriate. The order of the model depends on the number of spikes. An ARIMA(1,0,1) model is shown below.

Chapter Summary Box–Jenkins models capture a wide variety of time series patterns. When faced with a complicated time series that includes a combination of trend, seasonal factor, cyclical, as well as random fluctuations, use of the Box–Jenkins is appropriate. This methodology for forecasting is an iterative process that begins by assuming a tentative pattern that is fitted to the data so that error will be minimized. The major assumption of the model is that the data is stationary.

Chapter Summary (continued) “Differencing” could be used to make the data stationary. In using the different models of Box–Jenkins, we depend on the autocorrelation (AC) and partial autocorrelation (PAC) as diagnostic tools. Computer programs such as Minitab, and SPSS provide all the analysis tools for using the Box– Jenkins.