Consider the following regression model: Yi = αXi + Ui , i = 1,.., n (2)
The error terms Ui are independently and identically distributedwith E[Ui |X] = 0 and V[Ui |X] = σ^2 .
1. Write down the objective function of the method of leastsquares.
2. Write down the first order condition and derive the OLSestimator αˆ.
Suppose model (2) is estimated, although the (true) populationregression model corresponds to: Yi = β0 + β1Xi + Ui , i = 1, .., nwith β0 different to 0.
3. Derive the expectation of αˆ, E[ˆα], as a function of β0, β1and Xi . Is αˆ an unbiased estimator for β1? [Hint: Derive firstE[ˆα|X].]
4. Derive the conditional variance of αˆ, V[ˆα|X], as a functionof σ^2 and Xi .
