Q1. Suppose we do not have an intercept parameter in our simple population regression model
y = β1 x + u where E(u) = β0 ≠ 0
^
(a) (14 points) Please use the method of moment to derive the OLS estimator ????. (Note: Do not use the method of least squares method)
(b) Find the expected value of the OLS estimator ??^ under the following four assumptions as usual (except E(u)≠0 in this case):
SLR.1 The model is linear in parameter SLR.2 Random sampling
SLR.3 Sample variations in x SLR.4 E(ulx) = E(u)
Note: we follow the book's approach to take conditional expectation.
Q2. Please prove that sum of squared total (SST) is indeed equal to sum of squared explained (SSE) and sum of squared residuals (SSR) in OLS estimation. Does this relation still hold if we fit a sample regression line that goes through the sample mean of x and y such that the sum of all residuals is equal to zero?
Q3. Consider the following population multiple regression model with two explanatory variables y =- m 0 + 1 x1 + 2 x2 + u
(a) Please use the method of least squares to derive the three first-order conditions for the global minimum point.
(b) A random sample of n = 8 is taken from the population
|
Observation
|
y
|
x1
|
x2
|
|
1
|
12.4
|
28.0
|
18
|
|
2
|
11.7
|
28.0
|
14
|
|
3
|
12.4
|
32.5
|
24
|
|
4
|
10.8
|
39.0
|
22
|
|
5
|
9.4
|
45.9
|
8
|
|
6
|
9.5
|
57.8
|
16
|
|
7
|
8.0
|
58.1
|
1
|
|
8
|
7.5
|
62.5
|
0
|
(b1) Please solve for the OLS estimates from this sample using the equations you derived in part (a).
(b2) Compute the residuals and compute their mean.
(b3) Compute the sample covariance between the residuals and the explanatory variables. (b4) Compute the standard error of the regression.
(b5) Compute the estimated variance of the sampling distribution of the estimator ??^
(c) Suppose the sample turns out to be
|
Observation
|
y
|
x1
|
x2
|
|
1
|
12.4
|
28.0
|
14.0
|
|
2
|
11.7
|
28.0
|
14.0
|
|
3
|
12.4
|
32.5
|
16.25
|
|
4
|
10.8
|
39.0
|
19.50
|
|
5
|
9.4
|
45.9
|
22.95
|
|
6
|
9.5
|
57.8
|
28.90
|
|
7
|
8.0
|
58.1
|
29.05
|
|
8
|
7.5
|
62.5
|
31.25
|
Please solve for the OLS estimates from this sample using the equations you derived in part (a). Discuss the difficulty you may encounter in this case.
Q4. In Lecture 1 spreadsheet, let X= IQ level and Y = Education level, and we are looking at the population.
(a) Compute E(X), E(Y), E(XY), COV (X,Y) in sheet 1
(b) Compute E(X), E(Y), E(XY), COV (X,Y) in sheet 2
Q5.
Starting with the basic concept of covariance: COV (X,Y) = E [(X - X) (Y - - mY)], where X and Y are random variables and X and Y are their expected values, please prove
(a) COV (X, V + W) = COV (X, V) + COV (X, W)
(b) COV (X, aY) = a COV (X, Y)
where X, Y, V, and W are random variables and "a" is a constant