Autoregressive Moving Average Models 179
The benefit of ARMA models is a higher-order AR or MA model may
have a parsimonious ARMA representation that is much easier to identify and
estimate. In other words, ARMA models represent observed data with fewer
parameters than suggested by AR or MA models. In the 1970s, George Box
and Gwilym Jenkins popularized the estimation of the ARMA process given
by equation (9.5).^4 Their methodology for estimating ARMA models, referred
to as the Box-Jenkins estimation model, requires the following three steps:
Step 1. Test the series for stationarity.
Step 2. Identify the appropriate order of AR and MA terms.
Step 3. Once an appropriate lag order is identified, determine whether
the model is adequate or not. If the model is adequate, the residuals
(εt) of the model are expected to be white noise.Let’s look at these three steps and use our return time series that we have
studied earlier in this chapter to illustrate them.
The first step is to check for stationarity. There are several methodolo-
gies for doing so and these are discussed in Chapter 10 where we describe the
econometric tool of cointegration. For our discussion here, if fluctuations in the
variable exhibit no pattern over time, then we temporarily consider the vari-
able to be stationary. As an example, Figure 9.1 shows the plot of the weekly
CRSP value-weighted index returns. Although the return series shown in the
figure indicate considerable fluctuations over time, the returns meander around
a constant level close to zero. Thus, the returns exhibit no pattern over time,
and for the purpose of this chapter, we regard this as stationary and proceed to
Step 2 in the methodology for estimating an ARMA process.
The second step involves identifying the order of AR and MA terms.
For illustrative purposes, we start out with AR(1) and MA(1) model which
is referred to as a first-order ARMA(1, 1) model and expressed as
yt = c + ρ 1 yt− 1 + εt + δ 1 εt− 1 (9.6)
Our estimated results of the above model using the weekly index returns
for the period from January 1998 through October 2012 are:
yt^ = 0.13 − 0.79 yt− 1 +^ εt^ + 0.65^ εt− 1
t-statistics (1.40) (−4.17) (3.54)These results clearly show that the first-order autoregressive term for the
index weekly returns is statistically significant and inversely related to the
(^4) George Box and Gwilym Jenkins, Time Series Analysis: Forecasting and Control
(San Francisco: Holden-Day, 1970).