176 The Basics of financial economeTrics
statistical significance of accumulated sample autocorrelations up to any
specified lags are all zero.
We can test the residual series from the AR(1) and AR(7) models to
see if there is serial correlation in the residuals using the Q-statistic. For
illustrative purposes, the test statistic for 12 lags is presented. The com-
puted Q-statistic of the residuals for the AR(1) and AR(7) models are
Q-statistic(12) = 19.83 and Q-statistic(12) = 3.29, respectively. The criti-
cal value at the 5% significance level from the χ^2 distribution is 18.54.
Based on these values, the null hypothesis of no autocorrelation at the
5% significance level is rejected for the AR(1) model but not for the
AR(7) model. Although the autoregressive model AR(7) is adequate in
explaining the behavior of the weekly index returns series, we do not
know if this model does a good job of explaining the dynamic structure
of the data.
Moving Average Models
The autoregressive models just described for modeling the weekly CRSP
value-weighted index return series may not be the only process that generates
the return series. Suppose we model weekly returns, y, as:
yt = μ + εt + δ 1 εt− 1 (9.3)
where μ = the mean of the series
δ 1 = the parameter of the model
εt, εt− 1 = the white noise error terms
In the model given by equation (9.3), y at time t is equal to a constant
plus a moving average of the current and past error terms. In this case, yt
follows a first-order moving average (denoted by MA(1)) process. The mov-
ing average (MA) models are treated as simple extensions of a white noise
series. In other words, an MA model is a linear regression model of the cur-
rent value of the series against the unobserved white noise error terms or
shocks. A qth order moving average model is represented as
yt = μ + εt + δ 1 εt− 1 +... + δqεt−q (9.4)
Since the error terms in equation (9.4) are not observable, MA models
are usually estimated using the maximum likelihood estimation method
described in Chapter 13. The initial values needed for the shocks in the