Intro to Econometrics Section 011 HW 4 Due: Wednesday, Mar 9 at the beginning of class Prepare brief and precise answers to the following questions. You are encouraged to discuss the solutions in groups but should write up the solutions independently. 1. Let Ct be consumption and Xt be a predictor of consumption. Suppose you have quarterly data on C and X. Let D1t , D2t , D3t , and D4t be dummy variables such that D1t takes the value 1 in quarter 1 and 0 otherwise, D2t takes the value 1 in quarter 2 and 0 otherwise, etc. Which of the following, if any, suffer from perfect multicollinearity and why? a) Ct = α + βXt + γ1XtD1t + γ2XtD2t + γ3XtD3t + γ4XtD4t + ut b) Ct = α + βXt + γ1XtD1t + γ2XtD2t + γ3XtD3t + γ4Xt(1 − D1t − D2t − D3t) + ut c) Ct = α + δ1D1t + δ2D2t + γ1XtD1t + γ2XtD2t + γ3XtD3t + γ4XtD4t + ut d) Ct = α + δ1D1t + δ2D2t + βXt + γ1XtD1t + γ2XtD2t + γ3XtD3t + ut (There’s nothing special about labeling the parameters with α, β, γ1, γ2, δ1, δ2, etc. We could have labeled them β1, β2, . . . without changing their interpretation.) 2. a) In models (a)–(d) of question 1, what are the slope coefficients of Xt in each of the 4 quarters? b) Suppose you estimate model (c) and wrote down the estimated slope coefficients for Xt in each of the 4 quarters. You then estimate model (d) and write down the estimated slope coefficients for Xt in each of the 4 quarters. Do your estimates change? Why or why not? 3. Determine whether the following are true or false and explain why: a) Adjusted R2 can be negative. b) Adjusted R2 can be larger than 1. 4. For this question you will need to load the data file incomedata.Rda (available from the NYU classes page) using the command load("incomedata.Rda"). Make sure you have the data file stored in your working directory. You will also need to load the package AER using the command require("AER"). The data file contains data on earnings (EARN), education (ED), and gender (GEN) for 1192 individuals in the early 1990s. 1 a) Estimate the regression model: EARNi = β0 + β1GENi + β2EDi + β3GENiEDi + ui and interpret the estimated coefficients and the adjusted R2 . b) Construct 95% confidence intervals for β1 and β3. c) Do a t-test for H0 : β3 = 0 against H1 : β3 > 0 at the 5% level of significance. d) Do a Wald test for H0 : β1 = β3 = 0. e) Now estimate the regression model: LOGINCi = β0 + β1GENi + β2EDi + β3GENiEDi + ui where LOGINC is now the log of the earnings of person i. How does your answer to (c) change? 2

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