Newey-West Covariance Matrix (Part 2)

In the previous part we went over how Newey and West's insight can be used to describe a single random variable across time. More precisely, we produced a unbiased and asymptotically consistent estimate of variance for a single heteroskedastic and autocorrelated variable.

This process was quite involved and involved approximating covariance terms of the variable with itself across time.

In this part, we'll cover how to generalize the intuition for our variance estimation procedure to a group of random variables to produce a covariance matrix that has the properties of being both unbiased and asymptotically consistent.

The Multivariate Case

Instead of having random variable $X$ suppose we have a vector of random variables $\pmb x$ with corresponding sample means $\bar{\pmb x}$ and population means $\pmb \mu$. The CLT implies that:

We assume:

Which allows us to cross out terms in our covariance expression:

(Note: The given formula for term counts holds when $q > 0$ but when $q = 0$ all term counts are $1$).

Which we collect like so:

To make this more collection of covariance terms more manageable, we collect them like so:

Where

In order to remove the undesirable properties of the estimator, discussed in the previous part , while preserving the desirable properties of the estimator, we add a weighting term

Recalling that $\sigma^2_{\bar x} = \sigma^2/n \implies \sigma^2 = n \sigma^2_{\bar x}$ we can write out the variance for our un-aggregated observations:

We observe each variable at each slice of time exactly once, therefore our strategy for computing each (co)variance term is: