HOME EXAM in GRA6039 ECONOMETRICS WITH PROGRAMMING FALL 2022 This assessment is to be undertaken by groups of 3 to 4 students

Subject:EconomicsPrice:38.99 Bought3

Share With

HOME EXAM in GRA6039 ECONOMETRICS WITH PROGRAMMING FALL 2022

This assessment is to be undertaken by groups of 3 to 4 students. Please read this section carefully before you proceed. Your report should be formatted according to the following guidelines. Failure to use this formatting can result in a substantial subtraction on your total score. Summarized, these requirements are: • Format: A4, pdf file • Margins: 2.5 cm left and right margins, 2.5 cm top and bottom margins • Line spacing: 1.2 • Font type and size: Calibri (Body) 11 pt. Page-limits, which are specified in detail in each assignment, assume that you use the abovementioned format. Answers need to be typed using a text editor, like MS Word. Hand-written answers (or photos of hand-written notes) will not be accepted. Your report should be converted in pdf before you submit it. 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 in a clear manner. Secondly, as a single R “.R”-file with all code from Part A. The reason two copies are required 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. Please do NOT use readily available functions in R, such as the lm() function used to fit linear models or the ivreg() function used to estimate models using IV and 2SLS. You must rely on the matrix expressions provided below. 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 (30 Marks) Page limit: 1/2 pages, plus the table of simulations results. The R code should enter in an Appendix of no more than 2 pages. Consider the following model: ???????????????? = ????????0 + ????????1???????????????? + ????????????????; ???????? = 1, … , ????????, and define ???????????????? = (????????????????, ????????????????, ????????????????)′, such that ????????????????~????????(????????, ????????). Set ???????? = (0,1,1)′ and ???????? = ? ???????????????? 2 ????????????????,???????? ????????????????,???????? ????????????????,???????? ???????????????? 2 ????????????????,???????? ????????????????,???????? ????????????????,???????? ???????????????? 2 ? = ? 1 0.5 0 0.5 1 ????????????????,???????? 0 ????????????????,???????? 1 ?, where ???????????????? 2 denotes the variance of ????????????????, ????????????????,???????? denotes the covariance between ???????????????? and ????????????????, and so on. Set ????????0 = −2, and ????????1 = 1. When it comes to the correlation (covariance) between ???????? and ????????, distinguish between the following two cases: Case 1 where ????????????????,???????? = 0.8 (instrument is strong) and Case 2 where ???????????????? 2 = 0.2 (instrument is weak). Define the OLS estimator as ????????? = (????????′????????)−1????????′ ????????, where ????????? = ?????????? 0, ????????? 1?′, ???????? = (????????1, … , ????????????????)′ with ???????????????? = (1, ????????????????)′, and ???????? = (????????1, … , ????????????????)′, and the IV estimator as ????????????????????????? = (????????′????????)−1????????′ ????????, where ????????????????????????? = ?????????? 0,????????????????, ????????? 1,?????????????????′, ???????? = (????????1, … , ????????????????)′ with ???????????????? = (1, ????????????????)′, and ???????? = (????????1, … , ????????????????)′. Therefore, both ???????? and ???????? are ???????? × 2 matrices. Set ???????? = 20 and simulate the stochastic process 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 ???????? = (????????0, ????????1)′ using OLS and IV, and store the values in two matrices, each of dimension ???????? × 2. Repeat the same exercise with ???????? = 100 and ???????? = 400. 1. Using the results you have obtained, fill in the following table: Simulation Results Instrument is strong Instrument is weak ????????? 1 ????????? 1,???????????????? ????????? 1,???????????????? Average Standard Deviation Average Standard Deviation Average Standard Deviation ???????? = 20 ???????? = 100 ???????? = 400 where ‘Average’ denotes the average value of ????????? 1 or ????????? 1,???????????????? over ???????? replications, and ‘Standard Deviation’ denotes the standard deviation of ????????? 1 and ????????? 1,???????????????? over ???????? replications. 2. Discussthe performance of the OLS ad IV estimatorsin terms of the average and standard deviation, across different sample sizes. Your analysis should distinguish between Case 1 and Case 2 (instrument is strong and weak, respectively). Also, the discussion should draw a connection between the results and the properties of the explanatory variable and/or instrument. 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 reasoning, or providing invalid reasoning, yields 0 marks even if you chose the right answer). 1. Let ???????????????? = ???????????????????????? + ????????????????, ???????? = 1, … , ????????, where ???????????????? = ???????????????? + ????????????????, such that ????????(????????????????) = ????????(????????????????) = 0 and ????????(????????????????????????????????) = 0. The OLS estimator for ???????? is biased but consistent. (2 marks) 2. For the same model as in 1., let ???????????????? be a random variable such that ????????????????????????????????(????????????????, ????????????????) = 1, where ????????????????????????????????(. , . ) is the correlation coefficient between two random variables. The IV estimator that uses ???????????????? as an instrument is consistent. (2 marks) 3. The normality assumption in the errors is violated in models where the dependent variable is binary. (2 marks) 4. Consider the following linear model ????????????????????????????????????????????????? = ????????0 + ????????1???????????????????????????????????????????????????????? + ????????2????????????????????????????????????????????????????????? + ????????????????, where ????????????????????????????????????????????????? denotes output per hour for worker ????????, ???????????????????????????????????????????????????????? denotes weekly salary and ????????????????????????????????????????????????????????? denotes weekly working hours. Adding hourly pay as a variable would violate assumption MLR.4. (2 marks) 5. Let ???????????????? = ????????0 + ????????1????????1???????? + ????????2????????2???????? + ????????????????. The OLS estimate for ????????1 can be obtained using the following twostage procedure: in the first stage, ????????1???????? is regressed on ????????2???????? and an intercept. In the second stage, ???????????????? is regressed on the fitted value obtained from the first-stage, omitting ????????2????????. (2 marks) Question 2 (10 marks) Page limit: 1 page 1. Let ???????????????? = ????????1???????????????? + ????????????????, ???????? = 1, … , ????????, where Assumptions 1-5 of the simple linear regression model are satisfied. Define ????????? 1 = ????????1+???????????????? ????????1+???????????????? . Explain whether ????????? 1 is consistent or not in this setup. (2 marks) 2. For the same model as above, show that the variance of ????????? 1 is larger than the variance of ????????? 1, where ????????? 1 denotes the OLS estimator. Please refrain from referring to the Gauss-Markov theorem as part of your answer. (4 marks) 3. Consider the following estimated regression model: ?????????????????????????????????????????????????? = 431.25 − 9.264?????????????????????????????????????????????????????????, where ????????????????????????????????????????????????? denotes output per hour for worker ????????, ???????? = 1, … , ????????, and ????????????????????????????????????????????????????????? denotes weekly working hours. Let ?????????(?????????????????????????????????????????????????) = 82.92, ?????????????????????????(?????????????????????????????????????????????????) = 307.45, ?????????(?????????????????????????????????????????????????????????) = 37.6 and ?????????????????????????(?????????????????????????????????????????????????????????) = 3.217, where ?????????(. ) denotes the sample average and ?????????????????????????(. ) denotes the sample variance. Find the expected weekly output of a typical worker, i.e. estimate ????????(????????????????????????????????????????????????? × ?????????????????????????????????????????????????????????). (4 marks)