6. Runs in the Stock Market 77
Lo and MacKinlay slightly change the second variance estimator
above, by dividing by two. So let us define
ˆ22
σ =
1
n
n/2−1
i=0
(
X2i+2 X2i μ
)2
.
Under the assumption of independent increments, the theoretical
value of σ2
2
(for which
ˆ22
σ is an estimator) equals the value of
σ2.
Thus, still under this assumption, the ratio of the estimators should
be close to 1. Of course, one can, for any integer q 1, define the
estimator
ˆq2
σ in the same way. Once again, under the assumption of
independent increments, σq
2
=
σ2.
Lo and MacKinlay modify the above set-up in one additional way.
Instead of using non-overlapping time increments in the definition of
ˆq2,
σ they use all of the time increments of length q, obtaining the
following definition (note that we no longer need to assume that q
divides n):
ˆq2
σ =
1
q(n q + 1)
(n−q+1)
i=0
(
Xi+q−1 Xi
qˆ)2
μ .
The test statistic for the variance ratio is denoted by Mr(q) (the
subscript refers to the fact that we are dealing with a ratio), and is
defined by
Mr(q) =
ˆq2
σ
ˆ2 σ
1 .
We have seen that under the assumption of independent increments,
this statistic should typically be close to 0. In order for it to be a useful
statistic in testing this assumption, we need to know something about
how this statistic is distributed. Lo and MacKinlay show that for large
n, the statistic

nqMr(q) is approximately normally distributed with
mean 0 and variance
2(2q 1)(q 1)
3q
.
(In fact, to be strictly accurate, they make one further modification
to ensure that the individual variance estimators are unbiased. We
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