ECON 503 (MSPE)
Spring 2019
Department of Economics
FINAL EXAMINATION
Wednesday, May 8, 2019, 1:30-3:30 pm
A sample of your Palm Regression Lines
1. [15] (Let us start with your palms)
Go back to the cover page. You see 16 random sample (n=16) of your submitted palm pictures. Pay attention to the regression lines in each. Now find (as far as possible) mean, median, and mode of these pictures.
2. [5+5+5+7+8] (Now let us ring the Jingle-Bell (JB))
Consider our standard linear regression model:
Suppose we want to test the normality assumption of εi.
(i) Let us start with the Pearson family of distributions:
where f(εi ) is the pdf of εi. Find the density N(0,σ2 ) as a special case of (1).
(ii) Write down the null hypothesis Ho for testing normality using (1).
(iii) We can use the Rao’s score (RS) test for testing Ho. Write down the RS test principle.
(iv) Suppose l(θ) be the log-likelihood function based on (1), where θ = (β′ ,σ 2 , c1 , c2 )′ . We can show that (you don’t have to derive this) the score function for c1 and c2 are proportional to:
and
where θ is the value of θ evaluated under Ho , i.e.,θ(˜) = (β(ˆ), ˆ(σ)2 , 0, 0)′ , andˆ(µ)j are the estimates of the moments of εi based on the OLS residuals ˆ(ε)i = ei = yi −xi(′)β(ˆ), i = 1,...,n. Interpret the expressions (2) and (3). Based on your answer to part (ii), justify how they can be used to test Ho.
(v) Write down the expression for the JB test with its (asymptotic) distributions under Ho and also under the alternative hypothesis. Interpret each component of JB.
3. [10+5+5+10] (How can we forget some theory?)
In the regression model yt = xt β + εt , let εt follow an autorregressive [AR(1)] process εt = ρεt−1 + ut , where |ρ| < 1 and ut ∼ IID(0,σu(2)), t = 1, 2,...,n.
(i) Derive the variance-covariance matrix Σ of ε = (ε1 ,ε2 ,...,εn )′ .
(ii) From the expression of Σ, identify and interpret Var(εt ), t = 1, 2,...,n.
(iii) Find the Corr(εt ,εt+s) and explain its behavior as “s” increases, s > 0.
(iv) Now let ρ to be random, as ρt = ¯(ρ)+vt , where ¯(ρ) is a fixed parameter and vt ∼ IID(0,σv(2)) and is independent of ut. Find the variance or conditional variance of εt (conditional on past). Compare your result with that in part (ii).
4. [5+5+5+10] (Your Great Expectations)
Recall you Midterm (Spring 2019) expected grades (X) supplied by you and observed grades (Y), both letter grades and actual scores. We converted the letter grades C, C+, B-, B, B+, A-, A and A+ to numbers 1, 2, 3, 4, 5, 6, 7 and 8, respectively. We had two sections: Section 1 (9:30 am) and Section 2 (11:00 am), with class sizes (rather the number of valid respondents) 22 and 34, respectively. Thus the total sample size is 56.
(i) Below are the scatter diagrams and fitted regression lines with observed letter grades and actual scores (Figures 1 and 2, respectively) as the dependent variable. The simple regression results are given in Table 1.
Figure 1: Observed Grades (Letter) vs Expected Grades
Figure 2: Observed Grades (Score) vs Expected Grades
Table 1: Simple Regression: Grade on Expected Grade
Interpret and compare the results using actual scores and letter grades. Which is better (if any)?
(ii) You also provided the overall reliability of the Exam Questions. The results for observed grades vs reliability is given in Figure 3. Compare the results in Figures 2 and 3.
Figure 3: Observed Grades (Score) vs Reliability
(iii) Next, we utilized all the personal information you kindly provided, and have the following big model in Table 2. Interpret the results (very briefly) and suggest what one should do to improve her/his grade.
Table 2: Multiple Regression: Grade on exam reliability and controls
(iv) Finally, describe your project briefly. On the light of what you have done in your project, suggest what further can be done to improve the results in Table 2.