Problem 3 A researcher is considering two regression specifications: (1) log Y = B1 + 32 log X + u and: (2) Y log x = a1 + 02 log X + u where u is a disturbance term. Writing y = log Y, x = log X, and z = log , and using the same sample of n observations, the researcher fits the two specifications using OLS: (3) y = B1+ B2x and: (4) 2 = a1+ @2x (a) Using the expressions for the OLS regression coefficients, demonstrate that B2 = a2 + 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 - Xi = Zi (d) Hence show that the residuals for regression (3) are identical to those for (4). 2