Autoregressive Moving Average Models 183
of 1998 through December 2012 (783 observations) used in our illustration
are obtained from DataStream.^7
The first step is to check for stationarity in the S&P 500 index returns.
The plotted weekly S&P 500 index returns are presented in Figure 9.2. The
returns fluctuate considerably but always around a constant level close to
zero and exhibit no pattern over time. As stated earlier, for the purpose of
this chapter, we regard this as stationary and proceed to Step 2 in the meth-
odology of estimating an ARMA process.
The next step is to identify a possible model that best fits the data. We
tried a combination of 12 AR and MA models and used the AIC to select the
model that describes the data optimally. The results are presented in Table 9.6.
For the weekly S&P 500 return series, the AIC is at its minimum when the
model has three autoregressive terms and two moving average terms.
The final step involves estimating the model identified in Step 2 and
testing the residuals for the presence of autocorrelation. Table 9.7 shows
the results when we fit an ARMA(3,2) to the weekly S&P 500 index return
series. As can be seen, the first and second AR terms and both MA terms
are statistically significant. To ensure that the model is describing the data
adequately, we checked to see if the residuals of the model are white noise.
With a computed Q-statistic(12) of 3.75 and with a critical value of 18.54
table 9.5 Autoregressive Moving Average Model, Weekly Sample Returns of
CRSP Value-Weighted Index from January 1998 through October 2012
Variable Coefficient t-Statistic p-Value
c 0.12 1.23 0.22
ρ 1 0.94 3.60 0.00
ρ 2 −0.72 −2.30 0.02
ρ 3 0.34 1.51 0.13
ρ 4 0.19 1.03 0.30
ρ 5 0.02 0.34 0.73
δ 1 −1.00 −3.81 0.00
δ 2 0.84 2.52 0.01
δ 3 −0.51 −1.97 0.05
δ 4 −0.10 −0.48 0.63(^7) DataStream is a comprehensive time series database available from Thomson
Reuters.