Nonlinear Econometric Analysis (ECO 722) : Homework 2 Answers. (1 θ) if y i = 0. which can be written in an analytically more convenient way as

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1 Nonlinear Econometric Analysis (ECO 722) : Homework 2 Answers 1. Consider a binary random variable y i that describes a Bernoulli trial in which the probability of observing y i = 1 in any draw is given by θ. Then the probability mass function (pmf) for an observation is given by { θ if y i = 1 f(y i ; θ) = (1 θ) if y i = 0 which can be written in an analytically more convenient way as f(y i ; θ) = θ y i (1 θ) (1 y i). Consider a random sample of n observations drawn from a Bernoulli distribution with unknown θ. a. Write down the log likelihood function. ln L = {y i ln(θ) + (1 y i ) ln(1 θ)} b. What is the first order condition for maximization of the log likelihood with respect to θ? Solve it. So ln L i = = y i θ 1 y i 1 θ = y i y i θ θ + y i θ θ(1 θ) = y i θ θ(1 θ). ln L = 0 { } yi θ = 0 θ(1 θ) = y i nθ = 0 θ = n y i n 1

2 c. Calculate the second derivative of the log likelihood w.r.t. the parameter. Show that the log likelihood function is globally concave. 2 ln L i = ln L { } i yi θ 2 θ(1 θ) = ln L i { (yi θ)θ 1 (1 θ) 1} = y i 2y i θ + θ 2 θ 2 (1 θ) { 2 θ 2 if y θ = 2 (1 θ) 2 i = 0 θ2 2θ+1 = (1 θ)2 if y θ 2 (1 θ) 2 θ 2 (1 θ) 2 i = 1 In both cases, the second derivative is always negative, hence the log likelihood function is globally concave. 2. When independent Bernoulli trials are repeated, each with probability θ of success, the number of trials, y i, it takes to get the first success has a geometric distribution which has a pmf given by f(y i ; θ) = (1 θ) (y i 1) θ for y i = 1, 2, 3,... Consider a random sample of n observations drawn from a geometric distribution with unknown θ. a. Write down the log likelihood function. ln L = {ln(θ) + (y i 1) ln(1 θ)} b. What is the first order condition for maximization of the log likelihood with respect to θ? Solve it. ln L i = 1 θ y i 1 1 θ = 1 θ θ(y i 1) θ(1 θ) = 1 y iθ θ(1 θ). 2

3 So ln L = 0 { } 1 yi θ = = 0 θ(1 θ) = 1 y i θ = 0 = n θ y i = 0 θ = n n y i c. Calculate the second derivative of the log likelihood w.r.t. the parameter. Show that the log likelihood function is globally concave. 2 ln L i = ln L { } i 1 yi θ 2 θ(1 θ) = ln L i { (1 yi θ)θ 1 (1 θ) 1} = 1 2θ + y iθ 2 θ 2 (1 θ) { 2 = (1 θ) 2 if y θ 2 (1 θ) 2 i = 1 < (1 θ)2 if y θ 2 (1 θ) 2 i > 1 In both cases, the second derivative is always negative, hence the log likelihood function is globally concave. 3

4 3. The National Education Longitudinal Study (NELS) is a nationally representative sample of eighth-graders were first surveyed in the spring of A sample of these respondents were then resurveyed through four follow-ups in 1990, 1992, 1994, and 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?. use $datapath/nels.dta.. gen college = (psechoice>1). tab college college Freq. Percent Cum. 0 1, , Total 6, 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?. logit college grades faminc famsiz i.parcoll i.female i.black Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = Iteration 5: log likelihood = Logistic regression Number of obs = 6,649 LR chi2(6) = Log likelihood = Pseudo R2 = college Coef. Std. Err. z P> z [95% Conf. Interval] grades faminc famsiz parcoll female black _cons 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.. sum faminc 4

5 Variable Obs Mean Std. Dev. Min Max faminc 6, local meanfaminc = r(mean). margins, at(black=1 female=1 faminc=`meanfaminc grades=5 famsiz=5 parcoll=1) Adjusted predictions Number of obs = 6,649 Expression : Pr(college), predict() at : grades = 5 faminc = famsiz = 5 parcoll = 1 female = 1 black = 1 Margin Std. Err. z P> z [95% Conf. Interval] _cons margins, at(black=1 female=1 faminc=`meanfaminc grades=10 famsiz=5 parcoll=1) Adjusted predictions Number of obs = 6,649 Expression : Pr(college), predict() at : grades = 10 faminc = famsiz = 5 parcoll = 1 female = 1 black = 1 Margin Std. Err. z P> z [95% Conf. Interval] _cons d. Test the joint hypothesis that black, and female can be omitted from the analysis.. logit college grades faminc famsiz i.parcoll i.female i.black Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = Iteration 5: log likelihood = Logistic regression Number of obs = 6,649 LR chi2(6) = Log likelihood = Pseudo R2 = college Coef. Std. Err. z P> z [95% Conf. Interval] grades faminc famsiz parcoll female black _cons testparm i.parcoll i.female i.black ( 1) [college]1.parcoll = 0 ( 2) [college]1.female = 0 ( 3) [college]1.black = 0 5

6 chi2( 3) = e. Find the average of marginal effects. What is the effect of at least one parent having a college degree?. margins, dydx(*) Average marginal effects Number of obs = 6,649 Expression : Pr(college), predict() dy/dx w.r.t. : grades faminc famsiz 1.parcoll 1.female 1.black grades faminc famsiz parcoll female black Note: dy/dx for factor levels is the discrete change from the base level. Having at least one parent with a college degree increases the probability of attending college by 10 percentage points. Note that I have used the factor notation (i.) to define all the indicator variables in the regressions. This does not matter for the interpretation of the coefficients, nor for testing, but it does matter for the way in which Stata calculates marginal effects. If the variable is declared as an indicator (discrete), Stata does not use derivatives, instead it uses the discrete difference operator to calculate marginal effects. 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.. margins, dydx(faminc) noatlegend /// > at(black=1 female=1 grades=5 faminc=`meanfaminc famsiz=5 parcoll=1) Conditional marginal effects Number of obs = 6,649 Expression : Pr(college), predict() dy/dx w.r.t. : faminc faminc margins, dydx(faminc) noatlegend /// > at(black=1 female=1 grades=10 faminc=`meanfaminc famsiz=5 parcoll=1) Conditional marginal effects Number of obs = 6,649 Expression : Pr(college), predict() dy/dx w.r.t. : faminc 6

7 faminc 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?. use $datapath/angristlavy2009.dta, clear.. logit zakaibag i.treat if boy==1 Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Logistic regression Number of obs = 1,960 LR chi2(1) = 0.33 Prob > chi2 = Log likelihood = Pseudo R2 = treated _cons logit zakaibag i.treat if boy==0 Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Logistic regression Number of obs = 1,861 LR chi2(1) = Log likelihood = Pseudo R2 = treated _cons The effect is statistically significant and positive for girls. It is not significantly different from zero for boys. b. Estimate the Average Treatment Effects (ATE) for boys and girls.. quietly logit zakaibag i.treat if boy==1 7

8 . margins, dydx(treat) Conditional marginal effects Number of obs = 1,960 Expression : Pr(zakaibag), predict() dy/dx w.r.t. : 1.treated 1.treated Note: dy/dx for factor levels is the discrete change from the base level.. quietly logit zakaibag i.treat if boy==0. margins, dydx(treat) Conditional marginal effects Number of obs = 1,861 Expression : Pr(zakaibag), predict() dy/dx w.r.t. : 1.treated 1.treated Note: dy/dx for factor levels is the discrete change from the base level. The financial incentive increases the probability of girls passing the test by almost 10.9 percentage points. c. Reestimate the models in a with standard errors adjusted for clustering at the school level. Are the treatment effects statistically significant?. logit zakaibag i.treat if boy==1, cluster(school_id) Iteration 0: log pseudolikelihood = Iteration 1: log pseudolikelihood = Iteration 2: log pseudolikelihood = Logistic regression Number of obs = 1,960 Wald chi2(1) = 0.04 Prob > chi2 = Log pseudolikelihood = Pseudo R2 = (Std. Err. adjusted for 34 clusters in school_id) Robust 1.treated _cons logit zakaibag i.treat if boy==0, cluster(school_id) Iteration 0: log pseudolikelihood = Iteration 1: log pseudolikelihood = Iteration 2: log pseudolikelihood = Iteration 3: log pseudolikelihood = Logistic regression Number of obs = 1,861 Wald chi2(1) = 3.23 Prob > chi2 = Log pseudolikelihood = Pseudo R2 = (Std. Err. adjusted for 34 clusters in school_id) Robust 8

9 1.treated _cons The clustering adjustment makes a difference. The effect for boys is still not significant. The effect for girls is now only significant at the 10% level. d. Augment the specification above to include education of the parents (educem, educav), whether the school is religious or Arab (semrel, semarab), recent immigrantole5, 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.. gen sib4 = m_ahim>=4. logit zakaibag i.treat educem educav semrel semarab ole5 sib4 lagscore if boy==1, cluster(school > _id) Iteration 0: log pseudolikelihood = Iteration 1: log pseudolikelihood = Iteration 2: log pseudolikelihood = Iteration 3: log pseudolikelihood = Iteration 4: log pseudolikelihood = Iteration 5: log pseudolikelihood = Logistic regression Number of obs = 1,960 Wald chi2(8) = Log pseudolikelihood = Pseudo R2 = (Std. Err. adjusted for 34 clusters in school_id) Robust 1.treated educem educav semrel semarab ole sib lagscore _cons logit zakaibag i.treat educem educav semrel semarab ole5 sib4 lagscore if boy==0, cluster(school > _id) Iteration 0: log pseudolikelihood = Iteration 1: log pseudolikelihood = Iteration 2: log pseudolikelihood = Iteration 3: log pseudolikelihood = Iteration 4: log pseudolikelihood = Iteration 5: log pseudolikelihood = Logistic regression Number of obs = 1,861 Wald chi2(8) = Log pseudolikelihood = Pseudo R2 = (Std. Err. adjusted for 34 clusters in school_id) Robust 1.treated educem

10 educav semrel semarab ole sib lagscore _cons In this analysis, after controlling for a number of covariates, the effect is statistically significant for girls at 5%. e. Estimate the Average Treatment Effects (ATE) for boys and girls. Are the effects (and significance) different from those estimated with the simpler model?. quietly logit zakaibag i.treat educem educav semrel semarab ole5 sib4 lagscore if boy==1, cluste > r(school_id). margins, dydx(treat) Average marginal effects Number of obs = 1,960 Model VCE : Robust Expression : Pr(zakaibag), predict() dy/dx w.r.t. : 1.treated 1.treated Note: dy/dx for factor levels is the discrete change from the base level.. quietly logit zakaibag i.treat educem educav semrel semarab ole5 sib4 lagscore if boy==0, cluste > r(school_id). margins, dydx(treat) Average marginal effects Number of obs = 1,861 Model VCE : Robust Expression : Pr(zakaibag), predict() dy/dx w.r.t. : 1.treated 1.treated Note: dy/dx for factor levels is the discrete change from the base level. The estimates of the treatment effects are not so different from those obtained from the simple model, but for girls, inference can be made with greater confidence (the effect is significant at 5% rather than only at 10%. 10