14.32 Final : Spring 2001

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1 14.32 Final : Spring 2001 Please read the entire exam before you begin. You have 3 hours. No books or notes should be used. Calculators are allowed. There are 105 points. Good luck! A. True/False/Sometimes (20 points, 4 each) Indicate whether the following are true, sometimes true or false, with a brief explanation. 1. In a regression model with serially correlated errors, OLS (ordinary least squares) regression will produce biased coefficient estimates. 2. We can test the AR(1) assumption for serially correlated errors. 3. Suppose wages are determined by schooling and ability as follows: = α + βx i + γa i We are interested in estimating the returns to schooling, β, but we don t have information on ability. The OLS regression of wages on schooling alone will produce an estimate of β which is biased downward. 4. Suppose the model of interest is = α + βx 1i + γx 2i, and x 1i happens to be uncorrelated with x 2i. Then β can be estimated consistently by OLS regression of on x 1i alone. 5. The Boston Red Sox are for sale. Mr. Harrington, current owner, is interested in reporting the elasticity of demand for Red Sox tickets to author Stephen King, a potential buyer. Sox general manager Dan the Duke Duqette suggests this elasticity can be computed by simply regressing total tickets sold on ticket prices, using data from every season since Fenway Park opened. Team manager Jimy Williams (who often disagrees with the Duke on this sort of thing) says this won t work, and worries about simultaneous equations bias. Who is right here? B. Short answers (30 points, 15 each) 1. Consider the linear probability model: E[ ] = α + βx i = p i ; i = 1,..., N where = 1 if student i gets an A in and =0 otherwise; and x i = the number of hours spent studying for this course. For the purposes of this study, data was collected on students hours of study and class grades after some students complained about random grading.

2 (a) We want to estimate β by running a regression of the form = α + βx i What is E[ε i ] and V [ε i ] in this model? (b) Will OLS give us unbiased estimates of β here? Will it give us correct standard errors and confidence intervals? (c) Suggest a technique that may provide narrower confidence intervals than OLS. 2. In one of the papers discussed in class, Angrist used a dummy for having a low draft lottery number as an instrument to estimate the causal effect of military service on wages. (a) Why might this be better than simply regressing wages on a dummy for military service (possibly with other controls for age, education etc.)? (b) What are the two conditions which the lottery dummy should satisfn order for this procedure to produce good estimates of the causal effect of military service on wages? (c) Suppose we were able to find a large number of instruments for military service (e.g. instead of a single lottery-number dummy, we use a dummy for each individual lottery number). Is this necessarily a better strategy than using just one instrument? C. Longer questions 1. (15 points) Suppose we want to investigate the relationship between average wages, computer use and college education. We use the following model to describe the conditional expectation of log wages given computer use and college education: = α + βc i + γe i + δc i E i where is log (wages), C i is a dummy for computer use (=1 for computer users) and E i is a dummy for college education (=1 for college educated). (a) What is the average impact of computer use on the wages of college-educated people and non-college educated people in this model? (b) What is the unconditional average impact of computer use on wages? (i.e., without conditioning on college education. Hint: iterate expectations) (c) Suppose you d like to test whether the relationship between education and wages is different for computer-users and non-users. How can this be tested using the model above? 2. (25 points) Consider a simultaneous equations model for supply and demand in the market for breakfast cereal. The sample consists of data for 250 US cities.

3 (demand) q d = α 0 +α 1 p +α 2 I +α 3 FS + ε (supply) q s = β 0 + β 1 p + β 2 D +η where q = log(per-capita consumption of cereal), p = log(average price of cereal in the city), I = average income in the city, FS = average number of children per household in the city, D = distance of the city from Battle Creek, MI. We assume the cereal market is in equilibrium, lest there be cereal riots in the face of shortages! (a) What are the exogenous variables in this system? What are the endogenous variables? (b) Is the demand equation unidentified, exactldentified or over-identified? What about the supply equation? (c) Solve for the reduced form equations. Construct an Indirect Least Squares (ILS) estimate of β 1, if possible. (d) Explain how to construct 2SLS estimates of the supply and demand elasticities. Will these be the same as ILS estimates? (e) What additional exogenous variables might be worth including in this system? 3. (15 points) This question refers to the SAS regression output in the following pages. We have results from two regressions of Science test scores (test_sco) on class size (clsize) as well as the literacy status of the mother (lit_mom=1 if the mother is literate). The listing also shows descriptive statistics and the correlation matrix for these variables as well as the literacy status of the father (lit_dad=1 if the father is literate). The data comes from 2600 students in rural Kenya. (a)the first regression includes no controls, while the second includes a control for whether the student s mother is literate. Which do you think is a better way to capture the impact of class size on test scores? What would be your best guess of the change in average test scores if we were to increase class size by 10 students? (b) Use the two regressions to verify the theoretical relationship between short and long regression coefficients. (You will need to use the covariance matrix and descriptive statistics as well.) (c) Do you think we need to control for father s literacy as well in estimating the impact of class size?

4 Variable Label TEST_SCO Science test score lit_mom Mother literate lit_dad Father literate Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum TEST_SCO CLSIZE lit_mom lit_dad Pearson Correlation Coefficients Prob > r under H0: Rho=0 Number of Observations TEST_SCO CLSIZE lit_mom lit_dad TEST_SCO Science test score CLSIZE Class size lit_mom lit_dad < <.0001 < <.0001

5 The REG Procedure Model: BIVARIATE Dependent Variable: TEST_SCO Science test score Analysis of Variance Source Model Error Corrected Total Sum of Mean Squares Square F Value Pr > F Root MSE Dependent Mean Coeff Var R-Square Adj R-Sq Parameter Estimates Variable Label Parameter Estimate Standard Error t Value Pr > t Intercept Intercept < The REG Procedure Model: MULTIVARIATE Dependent Variable: TEST_SCO Science test score Analysis of Variance Source Sum of Squares Mean Square F Value Pr > F Model Error Corrected Total Root MSE Dependent Mean Coeff Var R-Square Adj R-Sq Variable Label Parameter Estimates Parameter Standard Estimate Error t Value Pr > t Intercept Intercept lit_mom <

SIMULTANEOUS EQUATION MODEL

SIMULTANEOUS EQUATION MODEL ONE Equation Model (revisited) Characteristics: One dependent variable (Y): as a regressand One ore more independent variables (X): as regressors One way causality relationship: