WEIGHTED LEAST SQUARES

To apply the same decay factor logic to linear regression analysis, we simply need to multiply all of the sample data, both the regressors and regressands, by the appropriate decay factors. Recall from Chapter 8 that, for a multivariate regression, the ordinary least squares (OLS) estimator is defined as:

(10.14)

where X is a t × n matrix for our regressor, and Y is a t × 1 matrix for our regressand. To integrate our decay factor into this analysis, we start by defining λ as the square root of our decay factor, δ. Next, we construct a diagonal weight matrix, W, whose diagonal elements are a geometric progression of λ:

(10.15)

We can then form a new estimator for our regression parameters:

(10.16)

This estimator is known as the weighted least squares estimator.