Econometrics710
Final Exam,Spring2013
1.Take the linear instrumental variables equation
y i=x i 1+z i 2+e i
E(e i j z i)=0
where for simplicity both x i and z i are scalar1 1:
(a)Can the coe¢cients( 1; 2)be estimated by2SLS using z i as an instrument for x i?
Why or why not?
(b)Can the coe¢cients( 1; 2)be estimated by2SLS using z i and z2i as instruments?
(c)For the2SLS estimator suggested in(b),what is the implicit exclusion restriction?
(d)In(b),what is the implicit assumption about instrument relevance?
[Hint:Write down the implied reduced form equation for x i.]
(e)In a generic application,would you be comfortable with the assumptions in(c)and(d)?
2.Take the linear homoskedastic CEF
y i=x0i +e i(1)
E(e i j x i)=0
E(e2i j x i)= 2
and suppose that y i is measured with error.Instead of y i;we observe y i which satis…es
y i=y i+u i
where u i is measurement error.Suppose that e i and u i are independent and
E(u i j x i)=0
E(u2i j x i)= 2u(x i)
(a)Derive an equation for y i as a function of x i.Be explicit to write the error term as a
function of the structural errors e i and u i:What is the e¤ect of this measurement error on the model(1)?
(b)Describe the e¤ect of this measurement error on OLS estimation of in the feasible
regression of the observed Y on X.
(c)Describe the e¤ect(if any)of this measurement error on appropriate standard error
calculation for^ .
3.Take a linear equation with endogeneity and a just-identi…ed linear reduced form
y i=x i +e i(2)
x i= z i+u i(3) where both x i and z i are scalar1 1.Assume that
E(z i e i)=0
E(z i u i)=0
(a)Write down the standard2SLS estimator^ 2SLS for using z i as an instrument for x i:
(b)Find the asymptotic distribution for^ 2SLS:
Write the asymptotic variance as a function of =E(z2i e2i);Q=E(z2i);and
4.In the context of model(2)-(3)from question3:
(a)Derive the reduced form equation
y i=z i +v i:(4) Show that = = if =0;and that
E(z i v i)=0
(b)Let^ denote the OLS estimate from linear regression of Y on Z,and let^ denote the
OLS estimate from linear regression of X on Z.Write =( ; )0and let^ =(^ ;^ )0: De…ne the error vector i= v i u i .Write p n ^ using a single expression as a function of the error i:
(c)Show that E(z i i)=0
(d)Derive the joint asymptotic distribution of p n ^ as n!1:Hint:De…ne = E z2i i 0i
(e)Using the previous result and the Delta Method,…nd the asymptotic distribution of the
Indirect Least Squares estimator^ =^ =^
(f)Bonus:Is the answer in(c)the same as the asymptotic distribution of the2SLS estimator from question3?[Hint:Show that 1 i=e i and 1 1 = ]
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