[SOLVED] CS 1. (11 Points) Consider the multiplicative model

1. (11 Points) Consider the multiplicative model
Yi =1Xi2ei, i=1,,n
Hint: You can use R for the computations below, but you need to write down your answers in the written part of the exam.
(a) (1 P.) Written answer:
Apply an ln()-transformation to derive the standard simple linear regression model.
(b) (4 P.) Written answers:
Suppose that the following data are observed:
Compute the value of the OLS-estimate = (X 0X )1X 0Y . Additionally, write down the values of the following quantities:
(X 0X ), (X 0X )1 and X 0Y .
(c) (2 P.) Written answers:
Compute the t-test statistic to test (significance level = 0.05)
H0 :2 =0 against H1 :2 6=0 Whats the p-value and whats your test-decision?
(d) (4 P.) Written answers:
Consider the multiplicative model and show that the elasticity of f(x) = E(Y|X = x) with respect to x equals 2. Comment on the role of the independence between i and Xi in your derivations.
Assumptions 1-4 of Chapter 3 of our script are assumed to hold for this model. Additionally, i.i.d. 2
you can assume homoscedastic, spherical errors with i N(0, ), and independence between i and Xi for all i = 1,,n.
2
i
1
2
3
4
5
X i =ln(Xi)
3
1
1
3
5
Y i =ln(Yi)
0
1
4
5
6

2. (11 Points) Consider the multiple linear regression model Yi =Xi0+i
Assumptions 1, 2, 3 4 of Chapter 5 of our script are assumed to hold. Additionally, you can assume homoscedastic, spherical errors i . Unfortunately, we cannot observe the (K 1) dimensional random vector Xi directly, but only
Consider the estimator
n = (X 0X )1X 0Y
0B 1 1C X = B B X i 2 + M i 2 C C ,
i @ . A XiK +MiK
i.i.d.
with measurement errors Mi2,,MiK N(0,1). You can assume that the random vari-
ables X i, Xi, and i are all independent to each other; that is E(i |X i) = E(i |Xi) = E(i) = 0 for all i = 1,,n.
(a) (4 P.) Written answers:
Compute the unconditional mean E(n). Is the estimator n (unconditionally) biased for fixed n?
(b) (3 P.) Written answers:
Lets assume n is inconsistent (i.e., n 6!p as n ! 1). Is sU2B in this case also inconsistent for Var(i )? Explain qualitatively (no mathematical derivations are needed here).
(c) (4 P.) Written answers:
i.i.d
Lets assume Xi is observed directly, but a measurement error, i N(0,1), which
is independent from all other random model components, affects Yi . So you have the
following model
Y i =Xi0+i, where Y i =Yi + i Compute the unconditional mean E( ) of the estimator
= ( X 0 X ) 1 X 0 Y
Is the estimator n (unconditionally) biased for fixed n?
3

3. (15 Points) Consider the following multiple linear regression model Yi =1 +2Xi2 +3Xi3 +i
Assumptions 1, 2, 3 4 of Chapter 5 of our script are assumed to hold. Let the sample size be n = 100 and let
= (1,2,3)0 = (2,3,4)0 Xi2 U[1,4]
Xi3 2Xi2 + Vi
Vi N(0,1)
i N(0,2/3),
where Zi is independent from all other random variables in the model.
Hints: You can use large sample inference. This is how you can draw n = 100 realiza- tions of the regressors and the error term. Dont get confused, the regressor Xi3 has no measurement error.
n <-100X_2 <- runif(n, 1, 4)V <- rnorm(n)X_3 <- 2 * X_2 + Veps <- rnorm(n, sd=sqrt(2/3))(a) (5 P.) R-Coding (no written answers):Write a Monte Carlo (MC) simulation with 500 MC replications to produce 500 re- alizations of the OLS estimator 2 for 2. Compute the empirical mean of the 500 realizations of 2.(b) (5 P.) R-Coding (no written answers):Repeat the MC simulation in (a), but when computing the OLS estimates, omit Xi3 from the estimation formula for all i = 1, … , n. Compute again the empirical mean of the 500 realizations of 2.(c) (5 P.) Written answers:What do the simulation results in (a) and (b) indicate? In (b), we omit Xi3, i = 1, … , n, when computing 2. Does this violate the exogeneity assumption (Assumption 2)? Explain your answer using mathematical derivations.44. (15 Points) Consider the following R code: library(“AER”) data(“Affairs”) ## Estimation lm_obj <- lm(affairs ~ age + yearsmarried + religiousness + rating, data = Affairs)Explanation: The variable affairs contains the number of affairs of a person. The variable rating contains information about how good/bad one rates his/her own marriage.(a) (3 P.) R-Coding (no written answers):Produce a typical regression output table (estimates, standard errors, t-values, and p-values) using HC3 robust standard errors.(b) (4 P.) R-Coding & written answers:Whats the p-value when testing H0: age = 0 versus H1: age < 0? Whats the p-value when testing H0: age = 0 versus H1: age > 0?
(c) (4 P.) R-Coding & written answers:
Test the multiple parameter hypothesis H0: age = yearsmarried = 0 using HC3 robust standard errors. Whats the corresponding R matrix? Can you reject the null hypothe- sis?
(d) (4 P.) R-Coding & written answers:
Test the multiple parameter hypothesis H0: age + yearsmarried = 0 using HC3 robust standard errors. Whats the corresponding R matrix? Can you reject the null hypothe- sis?
5

5. (8 Points)
(a) (2 P.) Written answer:
In case of large sample inference (n ! 1), the t-test statistic is asymptotically normal distributed. Why is it good practice to use nevertheless a t-distribution to compute the critical values? Answer shortly.
(b) (2 P.) Written answer:
Give an example of a conditionally heteroscedastic error term.
(c) (2 P.) Written answer:
Lets say you test the null-hypothesis H0: 2 = 3 = 0 using an F-test and the re- sulting p-value is smaller than your significance level. Whats the interpretation of this significant test result regarding 2 and 3?
(d) (2 P.) Written answer:
Lets assume you do an OLS regression with a small sample sizes of n = 10. All necessary regularity assumptions from Chapter 4 hold. You estimate the parameters of the following simple linear regression model
Yi=1+2Xi+i, i=1,,n (In matrix notation: Y = X + )
where 1 ifiisodd Xi= 0ifiiseven
For this specific example, your professor claims that
2 N(2,2[(X0X)1]22)
Is this correct or is this a typo? Explain your answer shortly.
6