The following two equations describe the interactions between fertility rates an

Question

The following two equations describe the interactions between fertility rates and average income of women in a cross-section of countries: Fertilityi = a0 + a1Incomei + a2Educationi + a3Rurali + u1

Incomei = b0 + b1Fertilityi + b2Educationi + u2

where the fertility rate (measured by average number of births per woman in country i) is a function of average female income in that country, the average years of female education, and the fraction of country i’s population that lives in rural areas, and female income is a function of the fertility rate and female education.

a. If you were to estimate these two equations separately with OLS, would you get consistent parameter estimates? Explain why or why not.

b. Which of the two equations is fully identified? Explain why.

c. Describe how you would consistently estimate the identified equation. d. Come up with another explanatory variable, Zi, that you could add to this system so that both equations can be estimated. What two conditions must Zi satisfy? Explain why the variable you selected meets these two criteria.

(All questions should be answered)

Explanation / Answer

a. If you were to estimate these two equations separately with OLS, would you get consistent parameter estimates? Explain why or why not.
Fertilityi = a0 + a1Incomei + a2Educationi + a3Rurali + u1
Incomei = b0 + b1Fertilityi + b2Educationi + u2
Substituting Fertilityi from the first equation.
Incomei = b0 + b1(a0 + a1Incomei + a2Educationi + a3Rurali + u1) + b2Educationi + u2
We can see that Incomei is a linear function of u1 and hence will be correalted with u1. This violates the model assumptions and the OLS estimator a1 will be biased.

Also, one of the assumptions of the OLS regression is that the independent variables are not too strongly collinear (correlated)
Incomei and Educationi are the independent variables of the first equation and they are correlated as per the second equation.
Fertilityi and Educationi are the independent variables of the second equation and they are correlated as per the first equation.

So, it will not be a good model and parameter estimates would not be consistent.

b. Which of the two equations is fully identified? Explain why.
Number of predetermined variables in the system are Educationi and Rurali.
So,total number of predetermined variables in the system = 2
Number of slope coefficients in the equation 1 is 3
Number of slope coefficients in the equation 2 is 2
As number of slope coefficients in the equation 1 is greater than total number of predetermined variables in the system, it is not exactly identified
As number of slope coefficients in the equation 2 equals total number of predetermined variables in the system, it exactly identified.

c. Describe how you would consistently estimate the identified equation.

Incomei = b0 + b1(a0 + a1Incomei + a2Educationi + a3Rurali + u1) + b2Educationi + u2
Incomei = b0 + a0*b1 + a1*b1Incomei + a2*b1Educationi + a3*b1Rurali + u1*b1 + b2Educationi + u2
(1-a1*b1)Incomei = b0 + a0*b1 + a2*b1Educationi + a3*b1Rurali + u1*b1 + b2Educationi + u2
Incomei = b0/(1-a1*b1) + a0*b1/(1-a1*b1) + a2*b1/(1-a1*b1) Educationi + a3*b1/(1-a1*b1) Rurali + u1*b1/(1-a1*b1) + b2/(1-a1*b1) Educationi + u2/(1-a1*b1)
Incomei = b0/(1-a1*b1) + a0*b1/(1-a1*b1) + (a2*b1+b2)/(1-a1*b1) Educationi + a3*b1/(1-a1*b1) Rurali + u1*b1/(1-a1*b1) + u2/(1-a1*b1)
Incomei = B0 + B1Educationi + B2Rurali + U2

where B0 = (b0+a0*b1)/(1-a1*b1)
B1 = (a2*b1+b2)/(1-a1*b1)
B2 = a3*b1/(1-a1*b1)
U2 = u1*b1+u1/(1-a1*b1)
Similarly, for the equation 1, the reduced form is Fertilityi = A0 + A1Educationi + A2Rurali + U1
We can solve for b0, b1 and b2 by these set of equations, as below.
b0 = B0 - (A0/A1)B1
b1 = B1/A1
b2 = B2 - (A2/A1)B1

d. Come up with another explanatory variable, Zi, that you could add to this system so that both equations can be estimated.
What two conditions must Zi satisfy? Explain why the variable you selected meets these two criteria.

We need to find a variable Zi that determines Incomei but not influences Fertilityi.
For e.g. let Zi be GDPi which denotes GDP of the country i (GDP shows the economic status of a country)
Fertilityi = a0 + a1Incomei + a2Educationi + a3Rurali + u1
Incomei = b0 + b1Fertilityi + b2Educationi + u2
If we find a variable Zi (GDPi), then the above equation can be written as
Incomei = c0 + c1Zi + c2Educationi + u3 => Incomei = c0 + c1GDPi + c2Educationi + u3
Then we would estimate the above equation using OLS. All the variables GDPi and Educationi are exogenous variables, and the estimated Incomei will be given as
Incomei' = c0' + c1' GDPi + c2' Educationi + u3'
And Incomei = Incomei' + v'
where Incomei' is the estimated Incomei from the equation Incomei' = c0' + c1' GDPi + c2' Educationi + u3'

Now estimated Incomei' can be substituted for Incomei in the first equation.
Fertilityi = a0 + a1(Incomei' + v') + a2Educationi + a3Rurali + u1
Fertilityi = a0 + a1Incomei' + a2Educationi + a3Rurali + u1 + a1v'

The new equation can be estimated for Fertilityi using OLS. This will produce consistent estimates of all the parameters, including a1. The estimates will not be biased.
The two conditions Zi must satisfy:
Corr(GDPi,Fertilityi) = 0
Corr(GDPi,Incomei) <> 0
We assume that the correlation between GDPi and Fertilityi is 0 and the correlation between GDPi and Incomei is not 0.

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