Consider the population regression model without an intercept Yi = αXi + Ui.
1. Obtain the least squares estimator of the parameter α in this model. You will first need to form the least squares criterion that is the sum of squares of the differences between Yi and aXi . You will then need to minimize it by differentiating the criterion function with respect to the parameter a. You will then equate the derivative to zero, and solve the resulting equation to obtain the estimator. If you do everything correctly you will obtain the estimator
αˆ = Pn P i=1 XiYi n i=1 X^2 i.
2. Assume that the Least Squares Assumptions 1-3 hold. Show that the estimator is an unbiased estimator of the population parameter. You will want to first obtain the expression for a and then use the law of iterated expectations.) 3. Assume that the Least Squares Assumptions 1-3. Show that the estimator à is a cousistent estimator of the population parameter. (You will want to first obtain the expression for à - a and of large numbers to prove the mumerator converges to 0 and denominator converges to some positive mumber. Then apply continuous mapping theorem.)