Question

Suppose that the regression model is yi=μ+εi, where E(εi∣xi)=0,cov(εi,εj∣xi,xj)=0 for i≠j, but Var(εi∣xi)=σ²xi², with xi>0.

Write down Var(ε∣X)=E(εε′∣X). Given a sample of observations on yi and xi for i=1,2,…,N, what is the most efficient estimator of μ ? What is its variance?

Answer 1

Var(ε | X) = diag(σ²x_1², σ²x_2², ..., σ²x_N²)

The most efficient estimator of μ is μ = 1/N * Σ y_i

Var(μ) = 1/Σ (1/x_i²)

Variance of the error term and Estimator of μ

1. Var(ε|X) = E(εε' | X):

Since the errors are conditionally uncorrelated given X, the conditional variance-covariance matrix of the errors is diagonal:

Var(ε | X) = E(εε' | X) = diag(Var(ε_1 | X), Var(ε_2 | X), ..., Var(ε_N | X))

where Var(ε_i | X) = σ²x_i².

2. Most efficient estimator of μ:

Under the given assumptions, the ordinary least squares (OLS) estimator for μ is unbiased and has the minimum variance among all linear unbiased estimators. Therefore, the most efficient estimator of μ is:

μ= 1/N * Σ y_i

3. Variance of the estimator of μ:

The variance of the OLS estimator for μ can be derived using the formula:

Var(μ) = σ²/N² * Σ (1/Var(ε_i | X)) = σ²/N² * Σ (1/σ²x_i²) = 1/Σ (1/x_i²)

In summary:

Var(ε | X) = diag(σ²x_1², σ²x_2², ..., σ²x_N²)

The most efficient estimator of μ is μ = 1/N * Σ y_i

Var(μ) = 1/Σ (1/x_i²)

These results show that the variance of the error term depends on the corresponding X values, while the most efficient estimator for μ is simply the average of the y values and its variance depends on the reciprocal of the squared X values.