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422, ECON Spring 2007 Department of Economics, University of Maryland Jessica Hennessey Problem Set #2 Question 1 Let kids denote the number of children ever born to a woman, and let educ denote years of education for the woman. A simple model relating to fertility to years of education is kids = where u is the unobserved error. (a) What kinds of factors are contained in u? Are these likely to be correlated with level of education? (b) Will a simple regression analysis uncover the ceteris paribus e ect of education on fertility? Explain. 0 + 1 educ +u Question 2 Write down the formula for the least squares estimators of all i. (a) What happens to the denominator of 1 0 and 1. Suppose that xi = c, a constant, for in this case? (b) Why do we assume that xi 6= c for all i? Question 3 Wooldridge utilizes a dataset called BWGHT that contains data on births to women in the US. Two variables of interest are the dependent variable, infant birth weight in ounces (bwght), and an explanatory variable, average number of cigarettes the mother smoked per day during pregnancy (cigs). The following simple regression was estimated using data on n = 1388 births: d bwght = 119:77 0:514 cigs (a) What is the predicted birth weight when cigs = 0? What about when cigs = 20 (one pack per day)? Comment on the di erence. (b) Does this simple regression necessarily capture a causal relationship between the child birth weight s and the mother smoking habits? Explain. s (c) To predict a birth weight of 125 ounces, what would cigs have to be? Comment. (d) The proportion of women in the sample who do not smoke while pregnant is about .85. Does this help reconcile your nding from part (c)? 1 Question 4 Suppose you are interested in estimating the e ect of hours spent in an SAT preparation course (hours) on total SAT score (sat). The population is all college-bound high seniors school for a particular year. (a) Suppose you are given a grant to run a controlled experiment. Explain how you would structure the experiment in order to estimate the causal e ect of hours on sat: (b) Consider the more realistic case where students choose how much time to spend in a prep course, and you can only randomly sample sat and hours from the population. Write the population model as sat = 0 + 1 hours + u where we can assume E(u) = 0. List at least two factors contained in u. Are these likely to have positive or negative correlation with hours? (c) In the equation above, what should be the sign of 1 if the prep course is e ective? 0? (d) In the population model, what is the interpretation of Question 5 State the ve assumptions of the simple linear regression model. (a) What does the assumption concerning E(ujx) imply about the correlation between x and u? (b) Which of these assumptions are necessary to show that the least squares estimators are unbiased? (c) Which of these assumptions are necessary to derive the variance of 1? Question 6 Write the expression for the variance of c . How in practice would you estimate the variance of c ? 1 1 Question 7 Using data from 1988 for houses sold in Andover, Massachusetts, from Kiel and McClain (1995), the following equation relates housing price (price) to the distance from a recently built garbage incinerator (dist): d log(price) = 9:40 + 0:312 log(dist) n = 135; R2 = 0:162 (a) Interpret the coe cient on log(dist). Is the sign of this estimate what you expect it to be? (b) Do you think simple regression provides an unbiased estimator of the ceteris paribus elasticity of price with respect to dist? (Think about the city decision on where to put the incinerator.) s (c) What other factors about a house a ect its price? Might these be correlated with distance from the incinerator? 2

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