Box–Jenkins method: Difference between revisions

Revision as of 12:13, 3 December 2016

Revision as of 01:36, 5 December 2016

Line 1:

In [[time series analysis]], the '''Box–Jenkins method,'''<ref>{{cite book |last=Box |first=George |last2=Jenkins |first2=Gwilym |year=1970 |title=Time Series Analysis: Forecasting and Control |___location=San Francisco |publisher=Holden-Day }}</ref> method, named after the [[statistician]]s [[George Box]] and [[Gwilym Jenkins]], applies autoregressive moving average [[Autoregressiveautoregressive moving average|ARMA]] (ARMA) or [[Autoregressiveautoregressive integrated moving average|ARIMA]] (ARIMA) models to find the best fit of a time-series model to past values of a [[time series]].

==Modeling approach==

Line 36:

Specifically, for an [[AR(1)]] process, the sample autocorrelation function should have an exponentially decreasing appearance. However, higher-order AR processes are often a mixture of exponentially decreasing and damped sinusoidal components.

For higher-order autoregressive processes, the sample autocorrelation needs to be supplemented with a partial autocorrelation plot. The partial autocorrelation of an AR(''p'') process becomes zero at lag ''p'' &nbsp;+ &nbsp;1 and greater, so we examine the sample partial autocorrelation function to see if there is evidence of a departure from zero. This is usually determined by placing a 95% [[confidence interval]] on the sample partial autocorrelation plot (most software programs that generate sample autocorrelation plots also plot this confidence interval). If the software program does not generate the confidence band, it is approximately <math>\pm 2/\sqrt{N}</math>, with ''N'' denoting the sample size.

The autocorrelation function of a [[moving average model|MA(''q'')]] process becomes zero at lag ''q'' &nbsp;+ &nbsp;1 and greater, so we examine the sample autocorrelation function to see where it essentially becomes zero. We do this by placing the 95% confidence interval for the sample autocorrelation function on the sample autocorrelation plot. Most software that can generate the autocorrelation plot can also generate this confidence interval.

The sample partial autocorrelation function is generally not helpful for identifying the order of the moving average process.