Consider the simple linear regression model with no intercept yt = xt β + εt . Assume all the Gauss Markov assumptions hold. Furthermore assume that εt ∼ iidN(0, σε^2 ). Derive the ML estimator of β and discuss its asymptotic properties, constructing the asymptotic covariance matrix.

The Maximum Likelihood (ML) estimator of β in the simple linear regression model yt = xtβ + εt, under the Gauss-Markov assumptions and with εt ~ iidN(0, σε^2), can be derived as follows:

Step 1: Write down the likelihood function The likelihood function for this model is given by the product of the densities of the normal distribution, which is:

L(β, σε^2 | y, X) = Π (1/√(2πσε^2)) exp[-(yt - xtβ)^2 / (2σε^2)]

Step 2: Take the logarithm of the likelihood function The log-likelihood function is easier to maximize. It is given by:

ln L(β, σε^2 | y, X) = -n/2 ln(2π) - n/2 ln(σε^2) - Σ (yt - xtβ)^2 / (2σε^2)

Step 3: Take the derivative of the log-likelihood function with respect to β The derivative of the log-likelihood function with respect to β is:

d ln L(β, σε^2 | y, X) / dβ = Σ xt(yt - xtβ) / σε^2

Step 4: Set the derivative equal to zero and solve for β Setting the derivative equal to zero and solving for β gives the ML estimator of β:

β̂ = (Σ xt yt) / (Σ xt^2)

This is the same as the Ordinary Least Squares (OLS) estimator of β under the Gauss-Markov assumptions.

Asymptotic properties of the ML estimator: Under the Gauss-Markov assumptions, the ML estimator is consistent and asymptotically normal. The asymptotic covariance matrix of the ML estimator is given by:

Asy.Var(β̂) = σε^2 / (Σ xt^2)

This shows that the variance of the ML estimator decreases as the sample size increases, and it depends on the variance of the error term and the variance of the explanatory variable.