Nonlinear Econometric Analysis (ECO 722) : Homework 2
Due Thursday, March 10
You must turn in this assignment before class begins on the due date. For the empirical exercises, you must show all Stata code in addition to output and answers. No photocopies, no loose pages, no paper clips (use a staple).
1 . Consider a model in which a binary variable y i is explained by only one regressor x i
, and no intercept (only one parameter to estimate) and such that the random variable is drawn from
= (0 , 1), P ( y i
= 1) = F ( x i
β ) a logistic (or extreme value) distribution. In other words, y i and exp( x i
β )
F ( x i
β ) =
1 + exp( x i
β ) a . Write down the log likelihood function of the logit model.
b . What is the first order condition for maximization of the log likelihood?
c . Calculate the second derivative of the log likelihood w.r.t. the parameter. Prove that the log likelihood function is globally concave.
2 . The National Education Longitudinal Study (NELS) is a nationally representative sample of eighth-graders were first surveyed in the spring of 1988. A sample of these respondents were then resurveyed through four follow-ups in 1990, 1992, 1994, and 2000. For the three in-school waves of data collection (when most were eighth-graders, sophomores, or seniors), achievement tests in reading, social studies, mathematics and science were administered in addition to student questionnaires. Use NELS.dta
to answer the following questions based on a binary outcome.
a . Define a variable college that equals 1 if a high school graduate chooses either a 2-year or 4-year college, and zero otherwise. What percentage of the high school graduates attend college?
b . Estimate a logit model explaining college using grades, faminc, famsiz, parcoll, female, and black (see the labels in Stata for definitions). Are the signs of the estimated coefficients consistent with your expectations? Explain. Are they statistically significant?
c . Using the estimates in b. predict the probability of attending college for a black female with grades=4, faminc= sample mean, from a family with 5 members and a parent that attended college. Repeat this for grades = 8.
d . Test the joint hypothesis that black and female can be omitted from the analysis.
e . Find the average of marginal effects. What is the effect of at least one parent having a college degree?
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f . What is the marginal effect of another $1000 of income for a black female with grades=4, faminc= sample mean, from a family with 5 members and a parent that attended college.
Repeat this for grades = 8.
3 . Angrist and Lavy (2009) study the effects of an experiment where some Israeli schools with relatively low performing students were randomized to receive incentive payments for improving performance. Graduating high school seniors in Israel take exams to gain eligibility for college admission. This is an example of a group-randomized design. Forty schools were chosen for the study. Twenty schools were randomly assigned to treatment with twenty paired schools assigned to the control group.
One school closed so there are 39 in the dataset. Use angristlavy2009.dta
to answer the following questions.
a . Estimate logit models of zakaibag on treat separately for the samples of boys and girls?
Are the treatment effects statistically significant?
b . Estimate the Average Treatment Effects (ATE) for boys and girls.
c Reestimate the models in a with standard errors adjusted for clustering at the school level.
Are the treatment effects statistically significant?
d . Augment the specification above to include education of the parents ( educem , educav ), whether the school is religious or Arab ( semrel , semarab ), recent immigrant ole5 , whether the child comes from a large family ( sib4 ) and the child’s score on a pretest ( lagscore ). Are the treatment effects statistically significant? Are the signs and significance of the estimated coefficients consistent with your expectations? Explain.
e . Estimate the Average Treatment Effects (ATE) for boys and girls. Are the effects (and significance) different from those estimated with the simpler model?
4 . Write a preliminary proposal for your term project - no more than one page. You must include at least one reference to an article that describes similar work. Similar means that the article should inspire your work along the dimensions of the relationship to be studied, methods, and possibly even data. More specifically, a . Write down, in general form, the relationship you plan to estimate, i.e., the dependent variable(s), key independent variables.
b . What is the economic theory (intuition) behind the hypotheses.
c . Describe the structure of the data you will use - you don’t need to know exactly which data set but you do need to be able to describe whether it will be a survey of individuals, firms, aggregates at the state or country level, etc.
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