Question

1. This question is concerned with bias for estimates in an OLS framework.
Consider the model:
yi = Xiβ + i
(a) What is the critical assumption needed for the OLS estimate βˆ to be unbiased?
(b) What is an endogeneity problem?
(c) What is a simulteneity bias problem?
(d) Why does a simulteneity bias problem cause bias? Show analytically.
(e) consider the model above where yi is wage and Xi is marital status.
What is a variable that would endogenously cause upward bias?
(f) Consider a variable that takes on a value of 1 if an individual lives in an urban area and 0 otherwise. Argue why this variable would be positively correlated with marital status. Would this result in bias upward or downward?
(g) argue why such a variable would be negatively correlated with marital status. Would this result in bias upward or downward?
(h) What would be the easiest solution to dealing with the endogeneity problem with the urban variable?
(i) Now consider the model if yi is wage and Xi is tenure at a firm. How would it make sense to specify/record the tenure variable?
(j) Now, argue why there might be upward bias on βˆ
(k) Argue why there might be downward bias on βˆ
(l) Would simulteneity bias be a problem in this model? Why or why not?
(m) Now consider the model where yi is hours worked and Xi is education. Is there an unobservable term that would cause endogeneity? Explain
(n) Discuss a procedure that could handle this endogeneity problem and produce unbiased estimates. Please be specific in what you would specify and use

Answer 1

Answer for a)

A Sample estimator said to be unbiased when Expexted value of sample estimator over the sample is equal to value of population estimator as Sample size tends to infinity.

Answer for b)

Endogenity problem occurs when Independant variable X or explanatory variable X correlated with error term in this case vector Xi is if correlated with i error vector then endogeniety will occur

Answer for c)

SImultaneity bias will occur when dependant variable casues independant variable and independant variable causes depedant variable in other terms it is causality and special case of endogeniety

Answer for d)

Lets assume causality between Y and X

Y= a+bX+e

X=c+dY+u

then Y=a+b(c+dY+u)+e=a+bc+bdY+bu+e

Y-bdY=a+bc+bu+e

Y(1-bd)=a+bc+bu+e

Y=a+bc+v where v=bu+e & g=a+bc

Y=g+v

Now Cov(Y,v)=E(Y)(v)=E(g+v)v=E(gv+v^2)=gE(bu+e)+E(v^2)=gbE(u)+gE(e)+E(b2u2+2bue+e2)=b^2Var(u)+Var(e)

Hence as Variance (Y) will not be equal to lower bound of variance of Y as due to simultenity bias Variance Y will have b^2Var(u)+Var(e) non zero term.