1 Proofs. Answer the following questions. Show all your work. 1. Show the derivation of the OLS estimator 2. Specify the steps that compose the Two Stage Least Squares estimation. This is the IV estimator 3. Explain in detail the differences between the sharp and the fuzzy regression discontinuity designs. 4. Explain in detail the conditions under which is possible to use the Diff-in-Diff method to perform causal inference. Be as formal as possible in your explanation of how the method produces causal results 2 Application The following questions are based in the information provided in the following paragraphs. You should have in mind the information provided in it and the information that is asked for you to provide as your progress through the different questions. The Mexican government has hired you to evaluate the new flagship social policy program. They provide you with a database on the recipients of a similar preexisting program and also assign you a budget large. enough as to perform and RCT to evaluate the new program and obtain causal results. They are interested in the cognitive development of children, measured through logic tests The new program consists in providing a government voucher to families with children between 3 and 5 years old. The voucher can be used to pay for the services of any early childhood school, either public or private. Meanwhile the preexisting program consists in the free public provision of early childhood education for children between 3 and 5 years old from single mothers. The database that you have has information on the following variables: scores of the children at an end of the year logic test, a variable that identifies if the children attended public or private school (all children go to school), income level of the household, region of the country, educational attainment of the mother, and time spent with the child by the mother. 1. As a first step, the Mexican government wants you to handle it a description of the RCT that you would use to evaluate the new program. The amount of funds allows you to use a sample of 20,000 individuals for the pilot program (that is, the RCT run of the program). They want you to tell them how to select the sample, how to assign the families either to the control or to the treatment group. and which characteristics they should be careful to maintain, on average, equal across groups. Describe the design of the RCT, what conditions need to be met in order to be sure that you can derive causal estimates from it. Also, identify what exactly is the treatment and what is the outcome of interest, and if it is possible that the control group gets contaminated. Discuss how you would prevent that from happening. 2. Since the new program is being promoted by the Social Development Ministry, the Education Ministry wants you to evaluate if their alternative plan is a better option for the government. The alternative plan consists of an expansion of the free public educational centers for small children. To do that evaluation. they suggest you to use the database of the current program, and see if you can identify if the public schools are a better option than the private schools. Write the model that would help you to identify the effect of interest, and what assumptions are necessary to make so you get a causal result. Think about how can you get a causal result without doing an RCT and only using observational data. 3. You estimate a model in which the dependent variable is the test scores and then progressively you include more explanatory variables. The results are shown in the following table. Remember, your interest is to identify if guaranteeing that all children go to public schools is a better policy than just guaranteeing that they go to school (this is important insofar as it implies what is your parameter of interest) Public school Income level Educational attainment of mother Region Time spent with children Intercept R Table 1: Program evaluation Observations (2) Test scores Test scores 12.800 (2.500) 15.02 15.00 (2.300) 30.301 (7.10) 0.500 200,000 (3.250) 5.00 (1.50) 38.300 (5.10) 0.700 200,000 Test scores 12.800 (2.500) 10.420 (2.050) 4.300 (1.54) 13.300 (4.00) 16.300 (6.004) 27.320 (6.12) 0.780 200,000 Perform the necessary hypothesis tests to analyze if the parameter of interest is statistically significant at the 90, 95 and 99 confidence level. Now, compare columns 2 and 3, is there anything that makes you suspect the presence of omitted variable bias? If not, which assumption is holding with respect to the independent variables and the proper identification of the parameter of interest? 4. Based on these results, would you say that expanding the public coverage is a better policy than the vouchers? Provide arguments that use the evidence above and that sustain your answer.