1 of 2 Problem Set 3 MSDA3055 Linear Regression and Time Series Due Date: February 6, 2022 Problem 1 Consider the following models: Model I: Yi = B1 + 2 Xi + wi Model II: Y* = 01 + a2X/ + ui where Y* and X* are standardized variables. Show that &2 = 32 (Sx/Sy) Problem 2 Consider the following models: In Y** = a1 + 02 In Xi + u; In Yi = B1 + 32 In Xi + ui where Y* = wiY; and X* = w2 Xi, the w's being constants. Establish the relationships between the two sets of regression coefficients and their standard errors. Problem 3 A researcher is considering two regression specifications: (1) log Y = B1 + 32 log X + u and: (2) log y = 01 + @2 log X + u where u is a disturbance term. Writing y = log Y, x = log X, and z = log x, and using the same sample of n observations, the researcher fits the two specifications using OLS: (3) y = B1+ B2x and: (4) 2 = 01 + 020 (a) Using the expressions for the OLS regression coefficients, demonstrate that B2 = @2 + 1 (b) Similarly, using the expressions for the OLS regression coefficients, demonstrate that B1 = a1 (c) Hence demonstrate that the relationship between the fitted values of y, the fitted values of z, and the actual values of x, is yi - Ti = Zi (d) Hence show that the residuals for regression (3) are identical to those for (4)