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STAT 804: Lecture 16 Notes

Distribution theory for sample autocovariances

The simplest statistic to consider is

displaymath88

where the sum extends over those t for which the data are available. If the series has mean 0 then the expected value of this statistic is simply

displaymath92

which differs negligibly for T large compared to k from tex2html_wrap_inline98 . To compute the variance we begin with the second moment which is

displaymath100

The expectations in question involve the fourth order product moments of X and depend on the distribution of the X's and not just on tex2html_wrap_inline106 . However, for the interesting case of white noise, we can compute the expected value. For k > 0 you may assume that s < t or s=t since the s > t cases can be figured out by swapping s and t in the s < t case. For s < t the variable tex2html_wrap_inline124 is independent of all 3 of tex2html_wrap_inline126 , tex2html_wrap_inline128 and tex2html_wrap_inline130 . Thus the expectation factors into something containing the factor tex2html_wrap_inline132 . For s=t, we get tex2html_wrap_inline136 . and so the second moment is

displaymath138

This is also the variance since, for k > 0 and for white noise, tex2html_wrap_inline142 .

For k=0 and s < t or s > t the expectation is simply tex2html_wrap_inline150 while for s=t we get tex2html_wrap_inline154 . Thus the variance of the sample variance (when the mean is known to be 0) is

displaymath156

For the normal distribution the fourth moment tex2html_wrap_inline158 is given simply by tex2html_wrap_inline160 .

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 tex2html_wrap_inline164 (in the case of white noise) to prove that

displaymath166

The presence of tex2html_wrap_inline158 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

displaymath174

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

displaymath178

For k > 0 we can apply Slutsky's theorem to conclude that

displaymath182

This justifies drawing lines at tex2html_wrap_inline184 to carry out a 95% test of the hypothesis that the X series is white noise based on the kth sample autocorrelation.

It is possible to verify that subtraction of tex2html_wrap_inline190 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

displaymath194

with tex2html_wrap_inline196 being white noise. Taking

displaymath198

we find that

displaymath200

The expectation is 0 unless either all 4 indices on the tex2html_wrap_inline196 's are the same or the indices come in two pairs of equal values. The first case requires tex2html_wrap_inline204 and tex2html_wrap_inline206 and then tex2html_wrap_inline208 . The second case requires one of three pairs of equalities: tex2html_wrap_inline208 and tex2html_wrap_inline212 or tex2html_wrap_inline214 and tex2html_wrap_inline216 or tex2html_wrap_inline218 and tex2html_wrap_inline220 along with the restriction that the four indices not all be equal. The actual moment is then tex2html_wrap_inline158 when all four indices are equal and tex2html_wrap_inline150 when there are two pairs. It is now possible to do the sum using geometric series identities and compute the variance of tex2html_wrap_inline226 . It is not particularly enlightening to finish the calculation in detail.

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

displaymath230

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 tex2html_wrap_inline232 method or Taylor expansion. In this case for each sample size T you have two estimates, say tex2html_wrap_inline236 and tex2html_wrap_inline238 of two parameters. You want distribution theory for the ratio tex2html_wrap_inline240 . The idea is to write tex2html_wrap_inline242 where f(x,y)=x/y and then make use of the fact that tex2html_wrap_inline236 and tex2html_wrap_inline238 are close to the parameters they are estimates of. In our case tex2html_wrap_inline236 is the sample autocovariance at lag k which is close to the true autocovariance tex2html_wrap_inline98 while the denominator tex2html_wrap_inline238 is the sample autocovariance at lag 0, a consistent estimator of tex2html_wrap_inline258 .

Write

eqnarray63

If we can use a central limit theorem to conclude that

displaymath260

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

displaymath262

has approximately a normal distribution. The notation here is that tex2html_wrap_inline264 denotes differentiation with respect to the jth argument of f. For f(x,y) = x/y we have tex2html_wrap_inline272 and tex2html_wrap_inline274 . When tex2html_wrap_inline142 the term involving tex2html_wrap_inline278 vanishes and we simply get the assertion that

displaymath280

has the same asymptotic normal distribution as tex2html_wrap_inline282 .

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

center Portmanteau tests

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 tex2html_wrap_inline286 .

A typical statistic is

displaymath288

which, for white noise, has approximately a tex2html_wrap_inline290 distribution. (This fact relies on an extension of the previous computations to conclude that

displaymath292

has approximately a standard multivariate distribution. This, in turn, relies on computation of the covariance between tex2html_wrap_inline294 and tex2html_wrap_inline296 .)

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.


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Richard Lockhart
Fri Oct 24 15:33:02 PDT 1997