EC3304 Tutorial Solutions PDF

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Academic Year 2021/2021 EC3304 Tutorial 1 Solutions...

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Assignment 1 for Weeks 3/4 EC3304: Econometrics II Instructor: Dr. Sorawoot Srisuma E-mail: [email protected]

You will receive full credit for participation if you constructively engage during tutorial discussions when asked regardless whether or not you give the correct answers.

1. Suppose you have a regular six-sided die. First, discuss how would you go about deducing if the die is fair from rolling it many times. Next, suppose the die is fair. Let die roll, i.e.

f1; 2; 3; 4; 5; 6g

takes value from

Y

a high roll so that

X

=1

if

Y

with equal probability, and

takes value from

f4; 5; 6g

and

X

=0

if

Y

Y

X

give the outcome of a indicates whether it is

takes value from

f1; 2; 3g.

Calculate the following objects. (a)

E

(Y );

(b)

E

(Y jX = 1);

(c)

E

(Y jX = 0);

(d)

E

(E (Y jX )).

Ans. The intuitive idea to see if we have a fair die has to be rolling the die many times and look at the proportions of 1, 2, 3, etc and see if they are ALL close to 1/6. We can do this in a more formal setting with hypothesis testing. (a)

E

(Y ) = 61 (1 + 2 + 3 + 4 + 5 + 6);

(b)

E

(Y jX = 1) = 31 (4 + 5 + 6) = 5;

(c)

E

(Y jX = 0) = 31 (1 + 2 + 3) = 2;

(d)

E

(E (Y jX )) =

2. Suppose (a) Do

and

X

(b) Are

X

X

1 2

(2 + 5) = 3:5.

 N (0; 1). Y

and

Y

Let’s

Y

= X .

have identical distribution? independent?

Ans. (a) Both

X

and

Y

are distributed as

N

(0; 1). 1

(b) No. E.g. 3. Let

Y



Pr (X

>

6

0) = 21 = Pr (X ( ),

B ernoulli 

such that

( =

Y

Suppose

f g Yi

n i=1 is an i.i.d. sample from

even observations and n

j

0Y

>

>

1

w.p.

0

w.p.

. Let

Y

0) = 0.





1



: 

be the sample average constructed from just the

b be the sample average constructed from the whole sample. You may assume



is an even number. (a) What is the mean and variance of

?

Y

(b) Compare the mean square errors of



b

and  . Which estimator would you prefer?

(c) Provide one example of a biased estimator of

Ans. (a) V ar

E

 + 0  (1  ) = ) (  ( )) , ( ) =

(Y ) = 1

(Y ) = E (Y 2

(b) We know





E

Y

b

=



E

b

We prefer to  to



2

2

V ar

and

because

f(

Yi ; Xi

V ar



M SE

(c) There are many. One is 4. Suppose

Y



b =   

Also,

(1





E

(Y 2 ) = 12

).



 (1 ) n . Analogously



MSE

b+ 1



.

.

 b.



 

+ 02 =

 (1 

and

V ar

n i=1 is a random sample such that

(

E Y

i

4

)

<

1

and

(

X

Y

X

Y

n

Xi

i=1

X

Y

X



X

Y

X

Yi .

Y

Ans. All you have to do here is send n (a)

E

(Xi + Yi );

(b)

E

(Xi

(c)

E

(Yi

 

E

E

 

4) <

E Xi

Pn ( + ); i i i=1  2 1 Pn (b) ; i i=1 n1   2 1 Pn (c) ; i i=1 n     1 Pn (d) ; i i i=1 n   1 Pn (e) 1

) = =

.

Since

 (1 ) n=2 .



probability limits of: (a) n



.

g

)

E



! 1 and replace sample avg by expectations.

(Xi ))2 ;

(Yi ))2 ; 2

1

. Give the

(d)

E

((Xi  E (Xi )) (Yi  E (Yi )));

(e)

E

((Xi  E (Xi )) (Yi  E (Yi ))).

5. Consider the regression model

Y

where

X

=  0 +  1X +  2D +  3X

is a continuous variable,

interaction of (a)

2

+ 3

(b)

1

+  3D

(c)

1

+  3X

(d)

1

+ 3

(e)

1

X

and

D.

D

D

is a binary variable, and

What is the partial e¤ect of

X

on

Y

+ u: X  D

is the

?

Ans. (b) Because, ignoring u (more formally the partial e¤ect calculates @X@ E [Y jX; D ] assuming

@Y 0), @X

=  1 +  3 D.

6. The …le eaef01.dta contains, among others, the following variables, EARNINGS = monthly salary ASVABC = a measure of cognitive ability MALE = 1 if the individual is male, 0 otherwise MARRIED = 1 if individual is married, 0 otherwise ETHBLACK = 1 if individual is black, 0 otherwise ETHHISP = 1 if individual is Hispanic, 0 otherwise S = years of schooling SM = years of schooling of mother SF = years of schooling of father SIBLINGS = number of siblings LIBRARY = 1 if individual has library card, 0 otherwise.

3

[

E ujX; D

]=

(a) Estimate the following model

ln (EARN I N GS ) =  0 +  1 S +  2 MARRI ED +  3 ET H BLAC K + ". (b) Interpret the coe¢cients of S, MARRIED and ETHBLACK. (c) Test whether the coe¢cient of MARRIED is zero at the

5%

(d) Test whether the coe¢cient of ETHBLACK is zero at the

level of signi…cance.

5%

level of signi…cance.

(e) Test whether the coe¢cients of MARRIED and ETHBLACK are both zero at the

5% level of

signi…cance. (f) Estimate the model where you replace the dummies MARRIED and ETHBLACK with dummies that allow wages to di¤er across four groups of people: married and black, married and nonblack, single and black, and single and nonblack. (g) Explain the di¤erence between the equations estimated in parts (a) and (f). (h) Interpret the estimates of the coe¢cient of the dummies in the model in (f). (i) Test whether the coe¢cients of the dummies in the model in (f) are the same.

4

Linear regression

Number of obs F(3, 536) Prob > F R-squared Root MSE

Robust Std. Err.

t

P>|t|

= = = = =

540 37.39 0.0000 0.1941 .53473

lnEARN

Coef.

[95% Conf. Interval]

S MARRIED ETHBLACK

.0983611 .0408474 -.1876029

.0100078 .0472627 .0617053

9.83 0.86 -3.04

0.000 0.388 0.002

.0787019 -.0519954 -.3088168

.1180204 .1336902 -.066389

_cons

1.435829

.1313601

10.93

0.000

1.177785

1.693873

Ans. (a) See a screen shot from STATA in the …gure above. (b)  The coe¢cient of S: 1 year increase in an individual’s schooling is associated with 9.84 percent increase in his/her monthly salary, holding MARRIED and ETHBLACK constant.  The coe¢cient of MARRIED: Compared to that of the individuals who are not married, the monthly salary of individuals who are married is 4.08 percent higher, holding years of schooling (S) and ETHBLACK constant.  The coe¢cient of ETHBLACK: Compared to that of the individuals who are not black, the monthly salary of individuals who are black is 18.76 lower, holding years of schooling (S) and marriage status (MARRIED) constant. (c) The t-statistics of the coe¢cient of MARRIED is 0.86|t|

= = = = =

540 30.31 0.0000 0.1950 .53493

[95% Conf. Interval]

S

.097868

.0100688

9.72

0.000

.0780889

.1176471

MB MNB NMB _cons

-.0777647 .0301799 -.2480591 1.449779

.1015577 .0499714 .0779419 .1337148

-0.77 0.60 -3.18 10.84

0.444 0.546 0.002 0.000

-.2772655 -.0679843 -.4011689 1.187109

.1217361 .1283441 -.0949493 1.71245

(g) While the model in part (a) does allow wages to di¤er across the four groups, the e¤ects of being married or not on wages are actually the same for black and the nonblack. The model in part (f) allows for analogous e¤ects to di¤er between black and nonblack. (h) Using (3) to (6), we have estimates the expected % di¤erence of salary for non-black individuals who are married relative to unmarried.  b  4 estimates the expected % di¤erence of salary for unmarried individuals who are black relative to non-black.  b  2 estimates the expected % di¤erences of salary for black and married individuals relative to non-black and unmarried.  b 3

It is worth noting the positive schooling e¤ect remains prominent. We also see that the socialethnic group that su¤ers the most in terms of income are actually the not married black. In particular, while the estimates on MB is negative it is not statistically signi…cant. (i) The null hypothesis is: 2 = 3 = 4 and the alternative is the negation of the null. The F-statistic, F(2,536) is 7.10 with p-value of 0.0009. Thus we reject the null hypothesis at the 5% level of signi…cance.

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