ECOM30002/90002 Econometrics 计量经济学 assignment 代写

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ECOM30002/90002 Econometrics Semester 2

ASSIGNMENT 2

Instructions and information

 Weight: 7.5%

 Submit online through LMS no later than 2pm Monday 4 September; submission

instructions to follow on LMS.

 Assignments can be completed individually (on your own) or by a group (of up to two

students). Students in a group (pair) do not have to be from the same tute.

 Before submission, pairs must register their group using the group registration link to be

posted on LMS (individuals need not register). Group registration closes (strictly) at

10am Thursday 31 August. Pairs missing the group registration deadline will have to

submit individual assignments.

 Each assignment must include a completed cover page listing every member of the group

with student IDs and tutors’ names.

 Equal marks will be awarded to each member of a group.

 Assignments should be submitted as a fully typed document (pdf or Word). Question

numbers should be clearly indicated.

 All questions requiring explanation can (and should) be answered in no more than three

sentences.

Note

 This assignment has four questions (including an “Appendix Question”) comprising 15

parts and will be marked out of 75 (all question parts are worth 5 marks).

QUESTION 1

Let the true data generating process (DGP) be as follows:

, (1)

where and independent of and .

The coefficient of interest is .

Let the explanatory variable be treated as unobservable. Although will be generated

from equation (1), which includes , calculations of estimators and confidence intervals

will proceed without using . Hence, plays the role of an omitted causal variable; that

is, although it might be involved in the generation of , it is not included in the statistical

analysis of the data.

The variables and have zero means ( and ) and variance–covariance matrix

The correlation between and is (as usual)

Observations on and are sampled from a multivariate normal distribution with

, and .

In addition, .

ECOM30002/90002 Econometrics 计量经济学 assignment 代写

The objective is to simulate an OLS regression of on using 1,000 replications to obtain

the mean and standard deviation (SD) of the OLS estimates of and the width (upper limit

minus lower limit) of the 95% confidence interval (using OLS standard errors) for varying

values of (sample size) and (correlation coefficient).

Carry out your simulations using the following parameter values:

, ,

(a) From your simulations, complete the table below (please present your results in the same

format).

QUESTION 1 Bias (see Note 1) SD CI width (see Note 2)

Notes

1) Bias is the average of 1,000 estimates minus the true value

2) CI width is average 95% confidence interval width (upper limit minus lower limit)

(b) Based on your simulation results, is the OLS estimator of unbiased? Explain your

answer.

settings for the specified DGP.

(d) Based on your simulation results, does the OLS estimator of seem consistent? Explain

your answer.

(e) Use your simulation results to explain the effect of on CI width.

QUESTION 2

(a) Repeat the simulations from Question 1, except now with , and report the same set

of results as in Question 1 (and in the same format).

(b) Explain how and why there are both similarities and differences between the sets of

results in Questions 1 and 2.

mathematically) the differences in bias between the results in Questions 1 and 2.

QUESTION 3

Consider the following causal equation:

(2)

where it is assumed that and are independent of each other and .

Suppose that is not observed but is measured with error so that

(3)

where is observed and represents the measurement error, which is assumed to have a

zero mean ( ) and be independent of both and . Hence, whereas equation (2)

cannot be estimated (because is not observed), the following PRF can be:

(4)

The idea is to examine the relationship between and both analytically and through

simulation. In the context of (4), define the following notation:

(a) Using the definitions above, and having applied the law of iterated expectations (LIE) to

(4) and then in , show that

(5)

(b) Using the same approach as in (a) and the additional notation,

, , show that

(6)

(7)

where is a constant. Hint: use (3) and show that .

(d) Report in a table (as formatted below) the results of a simulation based on 1,000

replications under the following assumptions and settings:

 The true DGP is given by (2)

 (4) is estimated by OLS



 ,

QUESTION 3 MEB SD CI width

Notes:

1MEB is measurement error bias (

2) SD and CI width relate to the OLS estimates of .

(e) Show rigorously the extent to which the results in the MEB column are compatible with

(7).

(f) Use your simulation results to discuss the effects of and

on CI width.

APPENDIX QUESTION

Present your R code as a final-page(s) “Appendix: R code”.

ECOM30002/90002 Econometrics 计量经济学 assignment 代写