 # You are encouraged to work and submit solutions in groups (no more ...

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You are encouraged to work and submit solutions in groups (no more than people). Hint: bricf, clear and precise ansurers are the leey your score. 1. Let and be negatively correlated random variables. Determine whether the following is true false and explain why: a) and Ly are positively correlated b) -x and are positively correlated c) the covariance of -x and d) the covariance of e) the covariance between f) the slope coefficient obtained by positive g) the slope coefficient obtained by regressing positive h) increasing will produce increase i) increasing will produce reduction 2. Suppose you have three random variables, Cor(Y, Z) <0. and a) Compute the sign of Cor(X,Z) b) Compute the c) 3. Let be random 5.X and Z=-2+5X. a) Compute E(Y). b) Compute V(Y c) Show 4. Suppose you are interested estimating the population mean of variable interest x You are given random sample of size N from the population {X, and two estimators: a) Describe words how each estimator works b) Show that unbinsed c) Show that consistent d) Suppose that the sample 250 Which estimator would you prefer? Why? 5. A large class of students just sat an exam of 25 true-false questions The TA recorded the number answers (X) and the number of incorrect answers (Y for each student. Suppose we a) Write the population b) What will RP7 Explain c) What will be the standard regression? Explain d) general, what the difference between of the regression? Answer more than sentences. 6. Consider the regression model 30 a) Suppose you know for that formula for the least- squares esti mator that makes Hint: write the function, then minimize obtain a b) Show your estimator 7. Consider the Tennessee school experiment from class. There, students were randomly assigned small class 1) ra class (X -0). Explain why assumption I holds that setting. a) Does assumption lof the OLS hold in this setting? b) Can learn the causal eflect of small class test scores from the Tennessee school experiment? 8. Consider the regression of class size on test score weran using the Tennessee data in class. Let large denote binary variable equal to l the student assigned to large class and equalte zero otherwise. Suppose you regressed argacore on large using the same data. a) Write the sample regression function you would obtain. Hint: you should be able work out what the estimated intercept and slope coefficient would be using the results from class, without actually having run the regression. b) Would the R² you obtained be different from the R² you obtained regressing argacore on amal[? Explain No need to shom this formally using algebra just give the intuition R.1 Does increasing expenditure or education improve outcomes? Using the Califor nian schools data, we want estimate model: teataer, when expustu, the expenditure per student (in school district In answering list all the R commands you use (where appropriate). a) Write the population regression function b) Interpret Br. c) Does have causal interpretation? Why/why not? d) Estimate the above regression model and write the sample regression line. e) Is the sign of what you would expect? Explain. f) What the estimated expected change test score for \$1000 increase per-student expenditure? g) Interpret the n° standard error of the R.2 Does greater access to information technology outcomes? Using the Cal- iforninn schools data, estimate the where compatu the number computers per student school district In answering list all commands you use (where appropriate). a) Write the population regression function. b) Interpret 32. c) Does have causal interpretation? Why/why not? d) Estimate the above regression model and write the sample regression line. e) Is the sign what you would expeet? f) Interpret the n° and standard error of the g) Comparing with the previous question, does or computing re- sources seem to explain more of the variation R.3 In this exerciso we are going analyze the sample variance estimator through simulations We are going consider two estimators: a) Generate fictitious population of size 10000 from normal distribution with mean and variance a² 30 Plot histogram the population distribution b) Fix the sample size to N = 10 and plot the sampling distribution of both estimators with histogram (use enough samples that the distribution looks smooth enough) Which estimator closer the true population value? this related consistency or unbiasedness the estimatori c) Now fix the sample size 9500 and plot the sampling distribution both estima- tors with histogram (use enoughsamples that the distribution looks smooth enough) Which estimator closer the true population value? this related consistency or inbiasedness the estimator?

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