The Newey-West estimator: uncorrelated and correlated data
I hate long posts but here we by necessity have to go through all ideas and calculations, to understand what is going on. One page of formulas in A. Patton's guide to FN3142 Quantitative Finance in my rendition becomes three posts.
Preliminaries and autocovariance function
Let be random variables. We need to recall that the variance of the vector is
With the help of this matrix we derived two expressions for variance of a linear combination:
for uncorrelated variables and
when there is autocorrelation.
In a time series context are observations along time. stand for moments in time and the sequence is called a time series. We need to recall the definition of a stationary process. Of that definition, we will use only the part about covariances: depends only on the distance between the time moments For example, in the top right corner of (1) we have which depends only on
Preamble. Let be a stationary times series. Firstly, depends only on Secondly, for all integer denoting we have
Definition. The autocovariance function is defined by
(5) for all integer
(6) for all
The preamble shows that definition (5) is correct (the right side in (5) depends only on and not on ). Because of (4) we have symmetry so negative can be excluded from consideration.
With (5) and (6) for a stationary series (1) becomes
Estimating variance of a sample mean
Uncorrelated observations. Suppose are uncorrelated observations from the same population with variance From (2)
This is a theoretical relationship. To actually obtain an estimator of the sample variance, we need to replace by some estimator. It is known that
consistently estimates Plugging it in (8) we see that variance of the sample mean is consistently estimated by
This is the estimator derived on p.151 of Patton's guide.
Correlated observations. In this case we use (3):
Here visualization comes in handy. The sums in the square brackets include all terms on the main diagonal of (7) and above it. That is, we have copies of copies of ,..., 2 copies of and 1 copy of The sum in the brackets is
Thus we obtain the first equation on p.152 of Patton's guide (it's up to you to match the notation):
As above, this is just a theoretical relationship. is estimated by (9). Ideally, the estimator of is obtained by replacing all population means by sample means:
There are two problems with this estimator, though. The first problem is that when runs from to runs from to To exclude out-of-sample values, the summation in (11) is reduced:
The second problem is that the sum in (12) becomes too small when is close to For example, for (12) contains just one term (there is no averaging). Therefore the upper limit of summation in (10) is replaced by some function that tends to infinity slower than The result is the estimator
where is given by (9) and is given by (12). This is almost the Newey-West estimator from p.152. The only difference is that instead of they use , and I have no idea why. One explanation is that for low , can be zero, so they just wanted to avoid division by zero.