Autoregressive Moving Average Models 173
The order n for an autoregressive model is unknown and must be deter-
mined. Two approaches are available for determining the value of n:
- Partial autocorrelation function
- The use of some information criterion
partial autocorrelation
The partial autocorrelation (PAC) measures the correlation between yt and
yt−n after controlling for correlations at intermediate lags. In other words,
the PAC at lag n is the regression coefficient on yt−n when yt is regressed on
a constant and yt− 1 ,... ,yt−n.
How does one test for the statistical significance of the PAC for each
lag? This is done by using the Ljung-Box Q-statistic, or simply Q-statistic.
The Q-statistic tests whether the joint statistical significance of accumulated
sample autocorrelations up to any specified lags are all zero. For example,
the Q-statistic for lag 3 is measured as:
Q-statistics(3)=+
−+
−
TT +
y
Ty
Ty
()
()()2
12
12
22
32(()T−
3
where T is the sample size. The statistic is asymptotically distributed as a
chi-square (χ^2 ) with degrees of freedom equal to the number of lags.
If the computed Q-statistic exceeds the critical value from the χ^2 distri-
bution, the null hypothesis of no autocorrelation at the specified lag length
is rejected. Thus, the Q-statistic at lag n is a test statistic for the null hypoth-
esis that there is no autocorrelation up to order n.
The PAC for 24 lags for our earlier illustration using the CRSP value-
weighted weekly index returns are presented in Table 9.1 along with the
results of the Q-statistic. The computed Q-statistic for lags 1 and 2 are 3.67
and 5.85, respectively. The critical values from the χ^2 distribution with 1
and 2 degrees of freedom at the 5% level of significance is 3.84 and 5.99,
respectively. The null hypothesis of no autocorrelation at lags 1 and 2 is
therefore rejected. While the null hypothesis is not rejected at lag 3, it is
again rejected at lags 4 and 5. Given that the results are yielding mixed lag
lengths for an autoregressive model, more formal approaches may provide
us with a better model.
information Criterion
Another approach of selecting an autoregressive model is the use of some
information criterion such as the Akaike information criterion (AIC) or