Specify the reduced form for educ in terms of unknown

Question

Statistics & Econometrics-

Consider  the  following  wage  equation  where  tog(rage)  is  the  logarithm  of  monthly  earnings,  educ  denotes the years of education, age the individual's age, married is a binary variable taking the value of one if the individual is married and zero otherwise, and black is a binary variable taking the value of one if the individual is black and zero otherwise:

log(wage) = β₀ + β₁educ + β₂age + β₃married + β₄black + u            (1)

The  individual's  innate  ability  is  an  unobserved  determinant  of  log(wage). We  are  concerned  that  educ  is potentially endogenous as the years of education are expected to be associated with the individual's innate ability that forms part of the error term, u. Apart from educ, all explanatory variables appearing in equation (1) are assumed to be exogenous.

We have two instrumental variable candidates for educ namely, urban, a binary variable taking the value of one if the individual lives in a city with a population that exceeds £00,000 inhabitants and zero otherwise, and ƒeduc, the years of education of the individual's father. The following estimations were obtained using Ordinary Least Squares (OLS) and Two Stage Least Squares (2SLS).

Model 1: OLS, using observations 1-663 Dependent variable:  log(wage)

	Coefficient	Std. Error	t-ratio	p-value
constant	5.14054	0.183951	27.9451	0
Educ	0.0513631	0.006617	8.6691	0
Age	0.0218306	0.004806	4.5421	0
married	0.202588	0.049123	4.1241	0
Black	-0.165176	0.053972	-3.0604	0.0023

R²                0.171908          Adjusted R²        0.166874

F (4, 658)      34.14932           P-value(F )          6.43e-26

 Model 2: OLS, using observations 1-663 Dependent variable:  educ

	Coefficient	Std. Error	t-ratio	p-value
constant	9.19633	0.930052	9.888	0
age	0.054128	0.0257582	2.1014	0.036
married	-0.387216	0.262459	-1.4753	0.1406
black	-0.340275	0.293375	-1.1599	0.2465
urban	0.2592	0.176233	1.4708	0.1418
feduc	0.281173	0.0244978	11.4775	0

R²                   0.192530       Adjusted R²        0.186385

F (5, 663)         31.33049        P-value(F )          1.20e-28

Model 3: OLS, using observations 1-663 Dependent variable:  educ

	Coefficient	Std. Error	t-ratio	p-value
constant	9.33107	0.926347	10.073	0
age	0.0544051	0.0257802	2.1103	0.0352
married	-0.399517	0.262557	-1.5216	0.1286
black	-0.297948	0.292218	-1.0196	0.3083
feduc	0.286062	0.0242927	11.7757	0

R²              0.189871       Adjusted R²        0.184947

F (4, 658)    38.55416        P-value(F )          5.21e-9

Model 4: OLS, using observations 1-663 Dependent variable:  log(wage)

	Coefficient	Std. Error	t-ratio	p-value
constant	4.54067	0.262276	17.3126	0
educ	0.102993	0.0157511	6.5388	0
age	0.0203746	0.004795	4.2492	0
married	0.224984	0.0492879	4.5647	0
black	-0.122178	0.0552719	-2.2105	0.0274
Model 3 Residuals	-0.552454	0.0173315	-3.1876	0.0015

R²                0.184519      Adjusted R²        0.178313

F (5, 657)      29.73192      P-value(F )          2.89e-27

Model 5: 2SLS, using observations 1-663 Dependent variable: log(wage) Instrumented: educ,

Instruments: constant, age, married, black, feduc,

	Coefficient	Std. Error	t-ratio	p-value
constant	4.54067	0.273471	16.6038	0
educ	0.102993	0.0164235	6.2711	0
age	0.0203746	0.0069997	4.0752	0
married	0.224984	0.0513919	4.3778	0
black	-0.122178	0.0576313	-2.12	0.034

R²              0.159800       Adjusted R²        0.154693

F (4, 658)    24.15740        P-value(F )          1.14e-18

(a) Specify the reduced form for educ in terms of unknown parameters and explain why reduced form parameters can be estimated by OLS.

(b) Test for instrumental variable relevance. Are urban and ƒeduc suitable instruments for educ?

(c) Using the relevant output carry out Hausman's endogeneity test to determine whether educ is endogenous.

(d) Based on your conclusion regarding the endogeneity test outcome specify which is your preferred estimate of the impact of an additional year of education on wage. Explain your choice.

(e) How would you implement Sargan's test for over identifying restrictions? Is it applicable here?