Question

Consider regression though the origin (i.e., straight line regression with population

intercept known to be zero) with Var(ei | xi) =xi2 (sigma)2

The corresponding regression model is Yi = b xi + ei (i=1,2,...n) .

Find an explicit expression for the weighted least squares estimate of b .

Answer 1

The weighted least squares estimate of b is found by summing the products of xi and Yi, weighted by 1 over xi squared, and dividing by the sum of xi squared.

Step-by-step explanation:

The question asks for the explicit expression of the weighted least squares estimate of b in a linear regression model without an intercept, where the variance of the error terms is proportional to x2 and a constant σ2.

The model is represented by Yi = b xi + ei, and weighted least squares (WLS) is used to address the heteroscedasticity of errors.

To find the WLS estimate of b, each term in the ordinary least squares (OLS) equation is weighted by 1/xi2. The formula for the WLS estimator of b, ^b, is given by:

^b = Σ(xiYi)/(xi2)

Thus, the estimate is the sum of the products of xi and Yi divided by the sum of the squares of xi.