HOME EXAM in GRA6039 ECONOMETRICS WITH PROGRAMMING Please read this section carefully before you proceed

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HOME EXAM in GRA6039

ECONOMETRICS WITH PROGRAMMING

Please read this section carefully before you proceed. Your report is to be formatted according to BI’s guidelines. Failure to use this formatting can result in a substantial subtraction on your total score.  Summarized, these requirements are:

 

• Format: A4
• Margins: 5cm left margin, 2cm right margin, top and bottom
• Line spacing: 1 1/2
• Font type and size: Times New Roman 12 pt.

 

Page-limits, which are specified in detail in each assignment, assume that you use the above mentioned format. The easiest way to follow these guidelines is to use the template, found on the site linked to above. 

 

All R-code is to be attached twice: Firstly, as an appendix of your report (copy-paste the code into your report as text), formatted nicely so that we easily see where the code for each problem starts and ends. Secondly, as a single R “.R”-file with all code from Part A. The reason that we want two copies is that the report is delivered to us in PDF-format, and sometimes it is problematic to copy-paste the code from a PDF. We therefore also want the code in text-format so that it can be easily evaluated. 

You may place comments in your R code. Do not include unnecessary R code that you do not use. Please only include R code for the “final product”. Do not include code for first tries etc.

 

Page limits are absolute. We will not read what is written beyond the page limit, which means you may be heavily disadvantaged by not keeping the page limit. Please also remember that some of the best answers can be short and concise. Reaching the limit does not necessarily mean you have provided a better answer.

 

               

PART A (20 Marks) Page limit: ½ pages, plus R code in the Appendix of no more than 2 pages Consider the following model:

???? = ???? + ???? ???? + ???? ;  ???? = 1, … , ????,

where ???? ~????. ????. ????. ????(0,???? ), and ???? ~????. ????. ????. ????(1,2). Set ????            = −1, ???? = 1, ???? = 1, ???? = 20.  Define the OLS estimator as

                                                                       ???? = (????′????)     ???? ????,

where ???? = ???? , ???? ′, ???? = (???? , … , ???? )′ with ???? = (1, ???? )′, and ???? = (???? , … , ???? )′.  The variance-covariance matrix of the OLS estimator is given by ???????????? ????????? = ???? (????′????) .

Simulate the model above ???? times, where ???? = 10000. To ensure you can replicate the results, set the seed equal to the last three digits of your student ID number (if in a group, choose one ID number at random among the group members).

Each time you generate a dataset, estimate ???? = (???? , ???? )′ using OLS and store the values in a matrix of dimension ???? × 2.  

Moreover, each time you generate a dataset, estimate ???????????? ???? ????? and conduct a t-test for the null hypothesis ???? : ???? = 1 using the 5% level of significance. Create a new variable that equals 1 if the null hypothesis is rejected and 0 otherwise.

Hint: To estimate ???? use the formula ???? = ∑ ???? /(???? − ????), where ???? denotes the number of parameters estimated.

Repeat the same exercise with ???? = 100 and ???? = 400.

1. Using the results you have obtained, fill in the following table:

	Simulation results for ????
	Average	Standard Deviation	Empirical Size
???? = 20
???? = 100
???? = 400

where ‘Average’ denotes the average value of ???? over ???? replications, ‘Standard Deviation’ denotes the standard deviation of ???? over ???? replications, and ‘Empirical Size’ denotes the empirical probability of rejecting the null hypothesis when the null is correct, i.e. it reports the number of times the null has been rejected over ???? replications, divided by the total number of replications.

2. Comment on the performance of the OLS estimator in terms of mean, standard deviation and size across different sample sizes. Does the performance improve as the sample size increases?

 

               

PART B

Question 1 (10 marks) Page limit: 1 page

State with brief reasoning whether the following statements are true or false. (Note: failure to provide a reasoning, or providing invalid reasoning, yields 0 marks).

 

1. The normality assumption in the errors of a regression model is necessary for the OLS estimator to be BLUE among all other linear unbiased estimators. (2 marks)
2. Let ???? = ???????? + ???? , where ????????????(???? , ???? ) ≠ 0. The OLS estimator is biased but consistent. (2 marks)
3. For the same model as in 2., let ???? be a random variable such that conditional on ???? , ????????????(???? , ???? ) ≠

0. The IV estimator that uses ???? as an instrument is inconsistent. (2 marks)

4. The power of a test is the probability of rejecting the null hypothesis when the null is correct. (2 marks)
5. A weak instrument is a random variable that is weakly correlated with the error of the regression. (2 marks)

 

 Question 2 (10 marks) Page limit: 1 page

1. Let ???? = ???? ???? + ???? , where Assumptions 1-4 of the linear regression model are satisfied. Define

???? =

.

 

Show that ???? is consistent. (2 marks)

2. Let ???? = ???? + ???? ???? + ???? , where ????????????(???? , ???? ) ≠ 0 and let ???? be an instrument that yields the following IV estimator 

                                                                          ∑       (???? − ?????)(???? − ????)

                                                             ???? _,      = .

                                                                          ∑       (???? − ?????)(???? − ?????)

Discuss a situation where the variance of the ????        estimator equals the variance of the OLS estimator. What implications does this have on the properties of the instrument, ???? ? (5 marks) 3. Consider the following model: 

???????????????????? = ???? + ???? ???????????????????????? + ???? ???????????????? + ???? ???????????? + ???? ???????????????????? + ???? ,

where ???????????????????? denotes the sales price of property ????, ???????????????????????? denotes the evaluation price of the same property prior to being sold, and ???????????????? , ???????????? and ???????????????????? denote lot size, size of the interior of the property and the number of bedrooms, respectively. Explain how you would go about testing whether house price assessments are rational. List null and alternative hypotheses, report the restricted and unrestricted models, and explain the steps required to implement the test. (3 marks)