Problem Set

1. Problem 1

Import the data from the excel spreadsheet "Fertility-small.xls" (on NYU classes/Data) into your statistical software.

The data is a random sample with n=30,000 women between the age 21 and 35 drawn from the data analyzed in Angrist and Evans (American Economic Review, 1998). The main objective of the following exercise is to estimate the link between fertility and labor supply. Labor supply is measured with the number of weeks that a mother has worked in 1979. The coefficient of interest is the effect of having more than two children; that is, the coefficient γ for the dummy variable "morekids", which takes the value 1 if a mother has more than 2 children. Please take a look at the documentation file (also NYU classes/Data) for a description of the other covariates.

Part 1 - Consider the following model for the dependent variable:

weeksm1_i = β₀ + γ morekids_i + β₁ agem1_i + β₂ black_i + β₃ hispan_i + β₄ othrace_i + β₅ boy1st_i + β₆ boy2nd_i + U_i            (1)

1. Estimate regression model (1) using the OLS command of your statistical software. Write down the coefficient estimates.

2. Interpret the coefficient β₂. What is the race of the reference category?

3. Interpret the coefficient γ.

4. Provide an intuitive explanation for why the variable "morekids" might be endogenous.

5. What are the implications of an endogenous regressor?

Part 2 - Consider now the two-stage least squares (2SLS) estimation:

weeksm1_i = β₀ + γ morekids^ˆ_i + β₁ agem1_i + β₂ black_i + β₃ hispan_i + β₄ othrace_i + β₅ boy1st_i + β₆ boy2nd_i + U_i        (2)

morekids_i = α₀ + α₁ agem1_i + α₂ black_i + α₃ hispan_i + α₄ othrace_i + α₅ boy1st_i + α₆ boy2nd_i + α₇ samesex_i + ν_i         (3)

1. Is the 2SLS model under-, exactly or over-identified? Justify your answer.

2. Write down formally the two assumptions that a good instrument must satisfy.

3. Estimate the first stage of the 2SLS model (Equation 3) using the OLS command of your statistical software. Write down the coefficient estimates.

4. Is the variable "samesex" a relevant instrument for the variable "morekids"? Justify your answer with the appropriate p-value.

5. Estimate the second stage of the 2SLS model (Equation 2) using the OLS command of your statistical software. Write down the coefficient estimates.

Hint: You wil l ?rst have to create the new variable "morekids^ˆ_i".

6. Compare the coefficient estimates γ^ˆ from the OLS and the 2SLS estimation.

7. What are the consequences of using weak instruments? Compute the Hausman test to test for the exogeneity of the variable "morekids".

8. Write down the estimated regression model.

9. Write down the hypotheses.

10. Write down the p-value. What do you conclude?

Part 3 - Create two new dummy variables, "twoboys" and "twogirls" using the following formulae:

twoboys = boy1st × boy2nd

twogirls = (1 - boy1st) × (1 - boy2nd)

The variable "twoboys" ("twogirls") takes the value 1 if the first two children of a mother are 2 boys (girls) and 0 otherwise.

Consider now the two-stage least squares estimation:

weeksm1_i = β₀ + γ morekids^ˆ_i + β₁ agem1_i + β₂ black_i + β₃ hispan_i + β₄ othrace_i + β₅ boy1st_i + U_i     (4)

morekids_i = α₀ + α₁ agem1_i + α₂ black_i + α₃ hispan_i + α₄ othrace_i + α₅ boy1st_i + α₆ twoboys_i + α₇ twogirls_i + ν_i         (5)

1. Explain why the variable "boy2nd" must be dropped from the first stage regression.

2. Is the 2SLS model under-, exactly or over-identified? Justify your answer.

3. Estimate the first stage of the 2SLS model (Equation 5) using the OLS command of your statistical software. Write down the coefficient estimates.

4. Estimate the second stage of the 2SLS model (Equation 4) using the OLS command of your statistical software. Write down the coefficient estimates.

Compute the test for over-identifying restrictions.

5. Write down the estimated regression model and the null hypothesis.

(Bonus question) Write down the p-value. What do you conclude?

Attachment:- Assignment Files.rar