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 − 2ˆ 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