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Homework answers / question archive / Your report should be formatted according to the following guidelines

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 above-mentioned 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 Im() 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:

Yi= f3o /31.xi + = 1, , n, and define Iv, = xi, ziY, such that wi—N(p,E). Set p = (0,1,1)' and uti (Tux uu,z 1 0.5 0 E = [crux crx,z = 0.5 1 Ox,z1, crzz 4 0 O„,z 1 where Qu denotes the variance of ui, denotes the covariance between ui and xi, and so on. Set = —2, and f31 = 1. When it comes to the correlation (covariance) between x and z, distinguish between the following two cases: Case 1 where ax,z = 0.8 (instrument is strong) and Case 2 where QZ = 0.2 (instrument is weak).

Define the OLS estimator as

= (X'X)-1X'y, where = (/'0,l(1)', X = (x1, , xtt)' with xi = (1, xi)' and Y = and the IV estimator as

fiiv = (Z'X)-1Z'y, where Ply = Cgo,w,fitivY, Z = (z1, ...,zn)' with zi = (1, zi)', and Y = (Y1, --A)'. Therefore, both X and Z are n x 2 matrices.

Set n = 20 and simulate the stochastic process above r times, where r = 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 (I = 660, iglY using OLS and IV, and store the values in two matrices, each of dimension r x 2.

Repeat the same exercise with n = 100 and n = 400. 1. Using the results you have obtained, fill in the following table:

Simulation Results Instrument is strong Instrument is weak #1 &iv ili, Average Standard Deviation Average Standard Deviation Average Standard Deviation n = 20 n = 100 n = 400

where 'Average' denotes the average value of 4', or /3, over r replications, 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.

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