Consider the model Yi = β1Xi + εi, i = 1, 2, ..., n with Xi non-random and εi’s satisfying the usual assumptions i.e., E(εi) = 0 and Var(εi) = sigma^2.

a) Show that the least square estimate of β1 for this model is given by:

b1 = ∑XiYi / ∑Xi^2.

=> Need to find the value of b1 that minimizes S = ∑(Yi - βXi)^2. dS/dβ = -2∑Xi(Yi-βXi). We set this equal to zero and get b1 = ∑XiYi / ∑Xi^2.

b) Show that the estimate b1 = ∑XiYi / ∑Xi^2 is unbiased estimator for β1

=> E(b1) = ∑Xi*E(Yi) / ∑Xi^2 = ∑Xi*(βXi)/∑Xi^2 = β.

c) Show that Var(b1) = sigma^2 / ∑Xi^2.

=> Var(b1) = ∑Xi^2*Var(Yi)/(∑Xi^2)^2 = sigma^2/∑Xi^2.

d) Show that Cov(Ybar, b1) = Xbar*sigma^2 / ∑Xi^2.

=> I don't know how to do this one. Help?