# ECON338 - Assignment 1

## Problem 3 - Estimating returns to schooling using IV

Use the data in WAGE2.dta for this exercise.

$$lwage = \beta_0 + \beta_1 educ + \beta_2 exper + u$$

### 1. Question 1

. reg lwage sibs exper

. outreg, ctitles("Variables","lwage")

-------------------------
Variables     lwage
-------------------------
sibs          -0.028
(4.77)**
exper         0.003
(0.94)
_cons         6.828
(165.09)**
R2             0.02
N              935
-------------------------
* p<0.05; ** p<0.01



This is reduced form coefficients and not the return of schooling (see the lecture: the IV estimate is equal to the ratio of the reduced form coefficient on the instrument to the first stage coefficient).

### Question 2

The variable brthord is birth order (brthord is one for a first-born child, two for a second- born child, and so on). Explain why educ and brthord might be negatively correlated. Regress educ on brthord to determine whether there is a statistically significant negative correlation.

. reg educ brthord

Source |       SS           df       MS      Number of obs   =       852
-------------+----------------------------------   F(1, 850)       =     37.29
Model |  173.087012         1  173.087012   Prob > F        =    0.0000
Residual |  3945.88364       850  4.64221605   R-squared       =    0.0420
Total |  4118.97066       851  4.84015353   Root MSE        =    2.1546

------------------------------------------------------------------------------
educ |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
brthord |  -.2826441   .0462882    -6.11   0.000    -.3734967   -.1917915
_cons |   14.14945   .1286754   109.96   0.000     13.89689    14.40201
------------------------------------------------------------------------------



Negative effect of brthord on education. Give the reason.

### Question 3

Use brthord as an IV for educ in equation (1). Report and interpret the results.

. ivregress 2sls lwage exper    (educ = brthord), vce(robust)  noheader
------------------------------------------------------------------------------
|               Robust
lwage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
educ |   .1630714   .0386999     4.21   0.000      .087221    .2389218
exper |   .0378465   .0098738     3.83   0.000     .0184943    .0571988
_cons |   4.152993   .6325998     6.56   0.000      2.91312    5.392866
------------------------------------------------------------------------------
Instrumented:  educ
Instruments:   exper brthord



### Question 4

Now, suppose that we include number of siblings as an explanatory variable in the wage equation; this controls for family background, to some extent:

$$lwage = \beta_0 + \beta_1 educ + \beta_2 sibs + u$$

Suppose that we want to use brthord as an IV for educ, assuming that sibs is exogenous. The reduced form for educ is

$$educ = \pi_0 + \pi_1 sibs + \pi_2 brthord + \varepsilon$$

• Run the first stage and perform any postestimation to verify if it is weak intruments
. reg educ sibs brthord

. predict educ_hat

. outreg, ctitles("Variables","First stage")

--------------------------
Variables   First stage
--------------------------
sibs          -0.153
(3.83)**
brthord       -0.153
(2.67)**
_cons         14.296
(107.26)**
R2             0.06
N               852
--------------------------
* p<0.05; ** p<0.01

. ivregress 2sls lwage sibs   (educ = brthord), vce(robust)  noheader

.  estat firststage

First-stage regression summary statistics
--------------------------------------------------------------------------
Variable |   R-sq.       R-sq.        R-sq.      F(1,849)   Prob > F
-------------+------------------------------------------------------------
educ |  0.0583      0.0561       0.0084       7.58507    0.0060
--------------------------------------------------------------------------



### Question 5

. ivregress 2sls lwage sibs   (educ = brthord), vce(robust)  noheader
------------------------------------------------------------------------------
|               Robust
lwage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
educ |   .1369941   .0766359     1.79   0.074    -.0132095    .2871978
sibs |   .0021108   .0179424     0.12   0.906    -.0330557    .0372772
_cons |   4.938527   1.084886     4.55   0.000      2.81219    7.064865
------------------------------------------------------------------------------
Instrumented:  educ
Instruments:   sibs brthord



## Question 6

. corr educ_hat sibs
(obs=852)

| educ_hat     sibs
-------------+------------------
educ_hat |   1.0000
sibs |  -0.9295   1.0000


• More

### Identification

• $Cov(brthord,educ)\neq 0$
• $Cov(brthord,u)= 0$ $$lwage = \beta_0 + \beta_1 \left(\pi_0 + \pi_1 sibs + \pi_2 brthord + \varepsilon \right) + \beta_2 sibs + u$$ $$lwage = \underbrace{ \beta_0 + \beta_1\pi_0 }_{\alpha_0}+ \left( \beta_1\pi_1 +\beta_2\right) sibs + \beta_1\pi_2 brthord + \beta_1\varepsilon + u$$

## Problem 3 - Fertility

The data in FERTIL2.dta include, for women in Botswana during 1988, information on number of children, years of education, age, and religious and economic status variables.

### Question 1

Estimate the model

$$children = \beta_0 + \beta_1 educ + \beta_2 age + \beta_3 age^2+ u$$ by OLS, and interpret the estimates. In particular, holding age fixed, what is the estimated effect of another year of education on fertility? If 100 women receive another year of education, how many fewer children are they expected to have?

. reg children educ age agesq,noheader
------------------------------------------------------------------------------
children |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
educ |  -.0905755   .0059207   -15.30   0.000     -.102183   -.0789679
age |   .3324486   .0165495    20.09   0.000     .3000032     .364894
agesq |  -.0026308   .0002726    -9.65   0.000    -.0031652   -.0020964
_cons |  -4.138307   .2405942   -17.20   0.000    -4.609994    -3.66662
------------------------------------------------------------------------------

. estimates store OLS



### Question 2

The variable frsthalf is a dummy variable equal to one if the woman was born during the first six months of the year. Assuming that frsthalf is uncorrelated with the error term from part (1), show that frsthalf is a reasonable IV candidate for educ. (Hint: You need to do a regression.)

. reg educ frsthalf ,noheader
------------------------------------------------------------------------------
educ |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
frsthalf |  -.9376578   .1184912    -7.91   0.000    -1.169961   -.7053548
_cons |   6.362774    .087111    73.04   0.000     6.191993    6.533556
------------------------------------------------------------------------------



### Question 3

Estimate the model from part (1) by using frsthalf as an IV for educ. Compare the estimated effect of education with the OLS estimate from part (1).

. ivregress 2sls  children  age agesq (educ = frsthalf),noheader

. estimates store IV

. estimates table OLS IV , b(%7.4f) star(.05 .01 .001) stats(N r2_a)

----------------------------------------
Variable |    OLS           IV
-------------+--------------------------
educ | -0.0906***   -0.1715**
age |  0.3324***    0.3236***
agesq | -0.0026***   -0.0027***
_cons | -4.1383***   -3.3878***
-------------+--------------------------
N |    4361         4361
r2_a |  0.5684       0.5499
----------------------------------------
legend: * p<.05; ** p<.01; *** p<.001



### Question 4

Add the binary variables electric, tv, and bicycle to the model and assume these are exogenous. Estimate the equation by OLS and 2SLS and compare the esti- mated coefficients on educ. Interpret the coefficient on tv and explain why television ownership has a negative effect on fertility.

. reg children educ age agesq tv electric bicycle,noheader

. outreg,ctitles("Variables", "OLS")

. ivregress 2sls  children  age agesq  tv electric bicycle (educ = frsthalf),noheader

. outreg,ctitles("Variables", "IV") merge

-----------------------------------
Variables      OLS        IV
-----------------------------------
educ         -0.077     -0.164
(12.08)**   (2.50)*
age           0.340      0.328
(20.69)**  (17.23)**
agesq        -0.003     -0.003
(10.01)**  (9.85)**
tv           -0.253     -0.003
(2.77)**    (0.01)
electric     -0.303     -0.107
(3.97)**    (0.64)
bicycle       0.318      0.332
(6.44)**   (6.45)**
_cons        -4.390     -3.591
(18.27)**  (5.57)**
R2            0.58       0.56
N             4,356      4,356
-----------------------------------
* p<0.05; ** p<0.01