Autoregressive Moving Average Models 177
likelihood function are obtained recursively from the model, starting with
ε 1 = y 1 − μ and ε 2 = y 2 − μ − δ 1 ε 1 and so on.
For example, an estimation of a MA(1) model for CRSP value-weighted
weekly index returns for the period from January 1998 through October
2012 yields
yt = 0.14 + εt − 0.063 εt− 1
t-statistics (1.48) (−1.76)The results show that the first-order moving average term is not statisti-
cally significant. Thus, the MA(1) model in this case is not adequate. Con-
sequently, different lag lengths for the moving average term must be tried.
As with autoregressive models, either the AIC or BIC can be employed
to select the optimal lag length. For the CRSP value-weighted weekly index
returns, Table 9.3 shows that the AIC identifies MA(7) as the optimal model
while the BIC identifies MA(1) as optimal. Since the MA(1) model is not ade-
quate for the return series that we are studying, we tested the residuals of the
MA(7) model for serial correlation. With a computed Q-statistic(12) of 5.97
and with a critical value of 18.54 from the χ^2 distribution, we are unable to
reject the null hypothesis of no autocorrelation. Hence, the MA(7) model
appears to be adequate in modeling the weekly stock index return series.
table 9.3 Moving Average Model: Akaike
Information Criterion (AIC) and Bayesian
Information Criterion (BIC) for the Weekly
Sample Returns of CRSP Value-Weighted Index
(yt) from January 1998 through October 2012Lags AIC BIC
1 2.033 2.045*
2 2.033 2.051
3 2.034 2.058
4 2.034 2.064
5 2.034 2.070
6 2.031 2.073
7 2.026* 2.074
8 2.029 2.083
9 2.030 2.090
10 2.033 2.099
(continued)