Paper
454-2013
THE BOX-JENKINS
METHODOLOGY FOR TIME SERIES MODELS THERESA HOANG DIEM NGO, WARNER BROS. ENTERTAINMENT
GROUP, BURBANK, CA
ABSTRACT
A time series is a set of values of a particular
variable that occur over a period of time in a certain pattern. The most common
patterns are increasing or decreasing trend, cycle, seasonality, and irregular
fluctuations (Bowerman, O’Connell, and Koehler 2005). To model a time series
event as a function of its past values, analysts identify the pattern with the
assumption that the pattern will persist in the future. Applying the
Box-Jenkins methodology, this paper emphasizes how to identify an appropriate time
series model by matching behaviors of the sample autocorrelation function (ACF)
and partial autocorrelation function (PACF) to the theoretical autocorrelation
functions. In addition to model identification, the paper examines the
significance of the parameter estimates, checks the diagnostics, and validates
the forecasts.
INTRODUCTION
This paper is an introduction to applied time series
modeling for analysts who have minimum experience in model building, but are
not very familiar with time series models. It would help to have a basic
understanding of regression analysis such as simple linear regression or
multiple regressions. The challenge of modeling is to diagnose the problem and
decide on an appropriate model to help answer the real-world questions. It
takes experience to develop an ability to formulate appropriate statistical
models and to interpret the results, but this paper gives a head start on
practicing these techniques.
NON-SEASONAL BOX-JENKINS MODEL IDENTIFICATION
Before identifying the pattern, the time series
values 
must be stationary where its mean and variance
are constant through time. The constant mean and variance can be achieved by
removing the pattern caused by the time dependent autocorrelation. Besides
looking at the plot of the time series values over time to determine stationary
or non-stationary, the sample autocorrelation function (ACF) also gives
visibility to the data. If the ACF of the time series values either cuts off or
dies down fairly quickly (Figure 1(a)), then the time series values should be
considered stationary. On the other hand, if the ACF of the time series values
either cuts off or dies down extremely slowly (Figure 1(b)), then it should be
considered non-stationary. In general, if the original time series values are nonstationary and non-seasonal, perform the first or second differencing
transformation on the original data will usually produce stationary time series
values. 
Second Difference:
