Autoregressive Moving Average Models 181
current weekly return. The moving average is positively related to this week’s
returns and the coefficient is statistically significant. Hence, both AR(1) and
MA(1) are statistically significant.
Now the issue is the identification of optimal lags. To identify the opti-
mal lag, a combination of ARMA models up to a specified number of lags is
calculated and then one of the information criteria is calculated for each of
these models. For our weekly return series, we tried up to 12 lags and then
used the AIC. The results are presented in Table 9.4. For the weekly return
series, the AIC is at its minimum when the model has five autoregressive
terms and four moving average terms.
The third and final step in ARMA modeling involves using the AR and
MA terms found in Step 2 and then testing the residuals. If the model is
adequate, the residuals should not exhibit serial correlation. This is done
by testing whether the residuals can be characterized as being white noise.
In our illustration, in Step 2 we have identified five AR terms and four MA
terms as optimal lags. Table 9.5 shows the results when we fit an ARMA(5,4)
to the time series of weekly returns. As can be seen, the first and the second
AR terms are statistically significant and the first three MA terms are statis-
tical 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 Q-statistic(12)
of 8.93, we are unable to reject the null hypothesis of no autocorrelation.
Hence, the ARMA(5,4) model appears to be adequate in modeling the
weekly stock index return series.
arMa ModeliNg to ForeCast
s&p 500 WeeKlY iNdeX returNs
As long as asset returns exhibit trends,^5 ARMA modeling can be employed
to predict these trends. There are investors who believe that stock returns,
commodity returns, and currency returns exhibit trends and these trends can
be forecasted and then used to design highly profitable trading strategies.
Those investors who seek to capitalize on trends are said to be technical
traders and follow an investment approach known as technical analysis.
For illustrative purposes, we will use an ARMA^6 model to forecast weekly
S&P 500 stock index returns. The weekly S&P 500 returns from January
(^5) See Chapter 5 for the definition of trends.
(^6) ARMA modeling is only one of the ways trends can be predicted and technical trad-
ers may or may not use ARMA to extract trends.