2. (The Efficiency Issue) We can create two correlated Xs using the following procedure. First, write...

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2. (The Efficiency Issue) We can create two correlated Xs using the following procedure. First, write an R code to draw three uniformly distributed random variables v ~ V1 U(-1, 1), v2 ~ U(0, 1) and v3 ~ U(1, 0), with 240 observations each. We then construct X and X2 from them according to the following, X1 = 2v1 v2 and X2 = v + v3. + V2 v1 V3 These two sequences of Xs will not change in the following simulation. Using these two x's, we can generate the Y observations according to the following Data Generating Process (or the true model) where ; ~ N(0, 22). Yi = 1 + 2 1,i + i (1) (a) In each replication, we generate 240 es and compute the corresponding y observations. Using these y and x, we then run the following two regression models: Yi = BA,0 + BA,1 1,i + A,2 2,i+UAi, Model A: Model B: Yi = BB,0 + BB,1 1,i + UB,i You need to record the coefficient estimates and their standard errors. Repeat this process for 1000 times with redrawing of es each time. (Similar to that in my Lecture Note 4) Report the average coefficient estates for each model. What values of coefficients in the above regression models do you expect. Do the average coefficient estimates close to what you have expected (the bias issue)? (b) Compute the standard error o and BA,2 for each of the coefficient estimate in re- BA,O' BA,1 , gression model A, and and BB,1 BB,0 B A, A,0 for each of the coefficient estimate in regression model B using their 1000 coefficient estimates. Comparing 30 to 30, and to to draw conclusion on the efficiency of the two model. (c) Compute the average of the standard error estimates of the corresponding coefficients obtained in each replication. When comparing these standard error estimates to the corresponding ones obtained in (b), are these estimates biased? 2. (The Efficiency Issue) We can create two correlated Xs using the following procedure. First, write an R code to draw three uniformly distributed random variables v ~ V1 U(-1, 1), v2 ~ U(0, 1) and v3 ~ U(1, 0), with 240 observations each. We then construct X and X2 from them according to the following, X1 = 2v1 v2 and X2 = v + v3. + V2 v1 V3 These two sequences of Xs will not change in the following simulation. Using these two x's, we can generate the Y observations according to the following Data Generating Process (or the true model) where ; ~ N(0, 22). Yi = 1 + 2 1,i + i (1) (a) In each replication, we generate 240 es and compute the corresponding y observations. Using these y and x, we then run the following two regression models: Yi = BA,0 + BA,1 1,i + A,2 2,i+UAi, Model A: Model B: Yi = BB,0 + BB,1 1,i + UB,i You need to record the coefficient estimates and their standard errors. Repeat this process for 1000 times with redrawing of es each time. (Similar to that in my Lecture Note 4) Report the average coefficient estates for each model. What values of coefficients in the above regression models do you expect. Do the average coefficient estimates close to what you have expected (the bias issue)? (b) Compute the standard error o and BA,2 for each of the coefficient estimate in re- BA,O' BA,1 , gression model A, and and BB,1 BB,0 B A, A,0 for each of the coefficient estimate in regression model B using their 1000 coefficient estimates. Comparing 30 to 30, and to to draw conclusion on the efficiency of the two model. (c) Compute the average of the standard error estimates of the corresponding coefficients obtained in each replication. When comparing these standard error estimates to the corresponding ones obtained in (b), are these estimates biased?