Please answer 3 questions a c Consider the random variable

Please answer 3 questions a ~ c

Consider the random variable Y 30+u where is an unknown parameter and u is a random variable with mean zero. a. What is the expected value of Y? b. Suppose you have a random sample Y,Ye Y drawn from the distribution for Y Derive the least squares estimator of 6. (For full credit, you should check the second order condition.) c. Can this estimator 6 also be described as a ments estimator? Explain briefly

Solution

Answer (a).

Expected value of Y would be-

E(Y)= E(+U)

E(Y) = + E(U)

E(Y) = +0, as U has mean zero.

E(Y) =

Answer (b).

For the case in which there is only one IV, the classical OLS regression model can be expressed as follows:

y = b + b x + e

A is constant. Now, in running the regression model, what are trying to do is to minimize the sum of the squared errors of prediction – i.e., of the ei values – across all cases. Mathematically, this quantity can be expressed as:

Specifically, what we want to do is find the values of that minimize the quantity in Equation above.

Formally, for the mathematically inclined, the derivative of y with respect to x – dy/dx – is defined as:

We begin by rearranging the basic OLS equation for the bivariate case so that we can express ei in terms of yi, xi, b0, and b1. This gives us:

where n = the sample size for the data. It is this expression that we actually need to differentiate with respect to a and . Let’s start by taking the partial derivative of SSE with respect to the regression constant, a, i.e.,

In doing this, we can move the summation operator () out front, since the derivative of a sum is equal to the sum of the derivatives:

In order to do this, we treat yi, a, and u as constants. This gives us:

By solving it and putting it equal to zero we will get-