Postscript version of these notes

Distribution theory for sample autocovariances

The simplest statistic to consider is

where the sum extends over those

which differs negligibly for

The expectations in question involve the fourth order product moments of

This is also the variance since, for

For *k*=0 and *s* <*t* or *s*> *t* the expectation is simply
while for *s*=*t* we get
.
Thus the variance of the sample variance (when the mean is known
to be 0) is

For the normal distribution the fourth moment is given simply by .

Having computed the variance it is usual to look at the large
sample distribution theory. For *k*=0 the usual central limit theorem
applies to
(in the case of white noise) to prove that

The presence of in the formula shows that the approximation is quite sensitive to the assumption of normality.

For *k*> 0 the theorem needed is called the *m*-dependent central
limit theorem; it shows that

In each of these cases the assertion is simply that the statistic in question divided by its standard deviation has an approximate normal distribution.

The sample autocorrelation at lag *k* is

For

This justifies drawing lines at to carry out a 95% test of the hypothesis that the

It is possible to verify that subtraction of from the observations before computing the sample covariances does not change the large sample approximations, although it does affect the exact formulas for moments.

When the *X* series is actually not white noise the situation is
more complicated. Consider as an example the model

with being white noise. Taking

we find that

The expectation is 0 unless either all 4 indices on the 's are the same or the indices come in two pairs of equal values. The first case requires

There are versions of the central limit theorem called
mixing central limit theorems which can be used for ARMA(*p*,*q*) processes
in order to conclude that

has asymptotically a standard normal distribution and that the same is true when the standard deviation in the denominator is replaced by an estimate. To get from this to distribution theory for the sample autocorrelation is easiest when the true autocorrelation is 0.

The general tactic is the
method or Taylor expansion. In this
case for each sample size *T* you have two estimates, say *N*_{T} and *D*_{T}of two parameters. You want distribution theory for the ratio
*R*_{T} = *N*_{T}/*D*_{T}. The idea is to write
*R*_{T}=*f*(*N*_{T},*D*_{T}) where
*f*(*x*,*y*)=*x*/*y* and then make use of the fact that *N*_{T} and *D*_{T} are
close to the parameters they are estimates of. In our case *N*_{T}is the sample autocovariance at lag *k* which is close to the
true autocovariance *C*_{X}(*k*) while the denominator *D*_{T} is the
sample autocovariance at lag 0, a consistent estimator of *C*_{X}(0).

Write

If we can use a central limit theorem to conclude that

has an approximately bivariate normal distribution and if we can neglect the remainder term then

has approximately a normal distribution. The notation here is that

has the same asymptotic normal distribution as .

Similar ideas can be used for the estimated sample partial ACF.

In order to test the hypothesis that a series is white noise using the
distribution theory just given, you have to produce a single statistic
to base youre test on. Rather than pick a single value of *k* the
suggestion has been made to consider a sum of squares or a weighted
sum of squares of the
.

A typical statistic is

which, for white noise, has approximately a distribution. (This fact relies on an extension of the previous computations to conclude that

has approximately a standard multivariate distribution. This, in turn, relies on computation of the covariance between and .)

When the parameters in an ARMA(*p*,*q*) have been estimated by maximum likelihood
the degrees of freedom must be adjusted to *K*-*p*-*q*. The resulting
test is the Box-Pierce test; a refined version which takes better account
of finite sample properties is the Box-Pierce-Ljung test. S-Plus plots the
*P*-values from these tests for 1 through 10 degrees of freedom as
part of the output of *arima.diag*.

1999-10-13