# Problem Set 4 ECON (Due: Friday, February 28, 2014)

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1 Problem Set 4 ECON 633 (Due: Friday, February 8, 4) Bill Evans Spring 4. Using the psid.dta data set and the areg procedure, run a fixed-effect model of ln(wages) on tenure and union status absorbing the person id. Next, construct within-panel deviations in mean for tenure and union ONLY. Next, run a two-stage least square model of ln(wages) on tenure and union using the within-panel deviations in union and tenure as instruments. Fill in the table below Parameter Estimates and Standard errors Covariates Union Tenure Fixed Effect SLS Kinda neat hah? Why are the standard errors so much lower in the fixed-effect model?. Consider the simple linear model with panel data Y it = X it β + η it where i=,,...n and t=,,...t, and X it is a scaler. Suppose the error term has a two-part structure where η it =u i +ε it and ε it is a zero mean random error with var(ε it )=σ ε and u is an individual-specific error that is potentially correlated with X. a. Show that the variable X ~ it = X it -X i (where X I is the within panel mean of X) is by construction uncorrelated with u i. (Don t overthink this this is not a question about expectations this is a question about finite samples). b. Consider an instrumental variables model where one were to use X ~ it as an instrument for X it to eliminate the potential covariance between X and u. Show that the IV estimate of β where X ~ it is used as an instrument for X is identical to a fixed-effects estimate for β. (This is much easier to write in matrix form use the matrix characterization of the problem). 3. Consider the indirect least squares model where the structural equation of interest is y x w i i i i Where x is a scaler and w i is a g x vector of other exogenous factors. The first-stage is estimated as

2 x z w u i i i i Where z is a scaler. The reduced form is estimated by the equation y z w v i i i i And the SIV estimate of ˆ ˆ ˆ /. Using the delta method and assuming cov( ˆ ˆ, ), what is var( ˆ )? Define t( ˆ ) to be the t-ratio for a parameter. With the results from above, show that t( ˆ ) ˆ t( ) t( ˆ ) 4. In Angrist, Imbens and Rubin, why is the monotonicity assumption important in the definition of the LATE? Consider the results in the Angrist and Evans paper on children and female labor supply. What is the LATE? Suppose that among families with two or more children, there are two types of families those that prefer a mix of children and those that prefer single sex pairings but those that prefer a sex mix outnumber those that prefer same sex child groups which is why there is a first stage. 5. Following the notation in class, consider the RDD model with a structural equation of interest yi xi wi h ( zi ) i Where x is the treatment variable and w is a vector of covariates. The first-stage regression (how does treatment change at the discontinuity) is of the form xi Di wi h ( zi ) vi. Where D i =I(z i z ) and is the coefficient of interest. Assume also that m m j m j i i j j i i j i z in the term zi z we cannot treat m h is defined as h ( z ) [ D ( z z ) ( D ) ( z z ) ]. Show that if we DO NOT subtract off make your life easy, assume linear terms in zi z only. as the impact of crossing the threshold on treatment. To

3 Empirical Portion of the Problem Set Women with children work less than women without kids. In a model where labor supply is regressed on the number of children in a household, the coefficient on the number of children is negative, large in magnitude, and statistically significant. This does not mean that the drop in work is actually caused by the presence of children in the house. To obtain a consistent estimate of the impact of kids on labor supply, some authors have suggested using whether a mother had twins on their first birth as an instrument for the number of children in the household. Twins are in many respect random and by definition, the realization of a twin increases the number of children in the household by. Using data from the 98 Public Use Micro Sample 5% Census data files, I constructed a sample of women aged - 4 with at least one kid. The 98 PUMS identifies a person s age at the time of then census and their quarter of birth. Because the census is taken on April st, we know a person s year and quarter of birth and we can infer that any two kids in the household with the same age and quarter of birth are twins. There are roughly 6, st births to mothers that are twins. There are over 8, observations in the original data set so to make the problem manageable, I select a random sample of about 6,5 non-twin births for a total of about,5 observations. The STATA data file data is called twinssta.dta. Variable name Description agem Mother's current age in years agefst Mom's age when she first gave birth race =white, =black, 3=other race educm Mother's years of education married Dummy variable for current marital statue, = married, =not kids Number of children ever born to the mother boyst Dummy variable, = if first kid is a boy, = otherwise. twinst Dummy variable, = if the first pregnancy ended in a twin birth weeks Weeks worked in previous year (from -5) worked Dummy variable, = if the Mom worked at all in the previous year lincome Labor income earned in the previous year mysteryz Mystery instrument will be used in part 8. What fraction of women work? What is average weeks worked among women that work? What is median labor earnings for women who worked?. Construct an indicator that equals for women than have a second child. Call this variable SECOND. What fraction of women had a second child? Consider a simple bivariate regression where WEEKS (Y) is regressed on SECOND (X) such as Y = β + β X i + ε i. What is the coefficient for β in this regression? Because of the concern that X and ε are correlated, use twins on st birth (Z) as an instrument for X in an instrumental variables model. What is the first-stage and reduced-form estimates for this model? Interpret these coefficients, that is, what do these coefficients measure? Consider the regression of X on Z. Why is the coefficient on Z not - e..g, don t twins increase the number of kids in the house by? What is the Wald estimate for β and compare the coefficient to the OLS estimate you produced above? Repeat this exercise using whether a mom worked at all as the outcome of interest. What is the R from the st stage regression of SECOND on TWINST? What is the ratio of ˆOLS ( ) / ( ˆ SLS Var Var )? 3 3. A number of authors have used twins as an instrument for fertility in a number of different papers. The argument is that twins are random but the question is whether twins convey information about the mother. Construct three indicators for the mother s race. Run a series of regressions with 6 different outcomes (EDUC, AGEFST, AGEM, and whether the mother is

4 white, black, or some other race) on a single indicator: TWINST. What coefficients are statistically significant? Are these differences economically meaningful, that is, are the coefficients large in magnitude? What do these results suggest about the randomness of twins on first birth? 4. Now that we know twins are correlated with some observed characteristics, run two structural labor supply models with weeks worked and whether a mom worked as outcomes and control for agem, agest, educm, black, other race, and whether the mom had a second child. What is the impact of a second child on labor supply and weeks worked? Now, use twinst as an instrument for SECOND in these models. Compare these estimates to the Wald estimates in b). What has happened to the labor supply impacts of having a second child? 5. The results in 3) suggest that twins might signal something about the mother that is correlated with labor supply, and as a result, the Wald estimates in ) and the estimates SLS estimates in 4) may be more inconsistent than OLS estimates. Calculate the correlation coefficient between Z and X. Given this value, is this a concern? 6. Construct three dummy variables that indicate whether the mother s first birth was before age, between ages and 4, or after age 4. Call these agegrp, agegrp and agegrp3. Next, interact twin first with these three variables to construct three instruments. Call these twinst, twinst and twinst3. Estimate the st stage regression, add agefst and agefst to the model, take out twinst, then add the three instruments agefst agefts agefts3. Using an F test, test two different hypotheses. The first is that the instruments are all the same value and the second being that the instruments are all equal to zero. Can you reject or not reject the null hypotheses in these cases? Is finite sample bias an issue in this case? 7. Using weeks worked and whether the mother worked as outcomes and the same covariates as in (4), the new covariates added in (6) as well as the instruments from (6) in a SLS model where SECOND is considered an endogenous variable. What has happened to the coefficient on SECOND in the WEEKS and WORKED equations? Do tests of over-identifying restrictions for these two models. What is the degrees of freedom on this test statistic? Can you accept or reject the null hypothesis that the model is correctly specified? Provide an intuitive explanation for this result -- you may have to estimate some extra models to answer this question. 8. a. (More difficult) Run an OLS using weeks as the dependent variable with the same covariates as in part 4) (second agem agefst educm black otherrace) and then SLS using mysterz as the instrument. reg weeks second agem agefst educm black otherrace reg weeks second agem agefst educm black otherrace (mystery agem agefst educm black otherrace) b. Next, what you need to do is write a program that draws 5 instruments at random, each from a standard normal distribution, run the first stage (a regression of second on the exogenous variables, mysteryz and the 5 instruments), get the first-stage f-test that mysteryz and the SLS model using mysteryz and the five. Save the first-stage F and the SLS coefficients on second. Do this times. c. Redo part b) drawing random, standard normal instruments. d. Redo part b) drawing 3 random, standard normal instruments. e. Redo part b) drawing random, standard normal instruments but dropping mysterz as the 4

5 instrument. Fill in the table below Instruments Mysteryz replications, mysterz and 5 replications, mysterz and replications, mysterz and 3 replications, random instruments st stage F (actual or average) SLS (actual or average) 5

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