In economics we are often faced with causal problems where the endogeneity arises because of simultaneity. A classic example is that if you are interested in estimating a demand curve, the issue is what you observe in the data are equli- birum and prices and quantity which is not only a function of demand but also a function of supply. In this excercise we will generalize the problem of simul- taneity bias. Consider a situation where we are interested in the effect of T; on Y₁, i.e. obtain- ing a consistent estimate of a. But Y; also effects T;₁. Let X; be some exogenous
In economics we are often faced with causal problems where the endogeneity arises because of simultaneity. A classic example is that if you are interested in estimating a demand curve, the issue is what you observe in the data are equli- birum and prices and quantity which is not only a function of demand but also a function of supply. In this excercise we will generalize the problem of simul- taneity bias. Consider a situation where we are interested in the effect of T; on Y₁, i.e. obtain- ing a consistent estimate of a. But Y; also effects T;₁. Let X; be some exogenous
Chapter1: The Role And Method Of Economics
Section: Chapter Questions
Problem 5P
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Transcribed Image Text:Problem 2 In economics we are often faced with causal problems where the endogeneity
arises because of simultaneity. A classic example is that if you are interested in
estimating a demand curve, the issue is what you observe in the data are equli-
birum and prices and quantity which is not only a function of demand but also
a function of supply. In this excercise we will generalize the problem of simul-
taneity bias.
Consider a situation where we are interested in the effect of T; on Y₁, i.e. obtain-
ing a consistent estimate of a. But Y; also effects T₁. Let X; be some exogenous
1
covariates affecting both Y; and T₁. Let us imagine you have an exogenous vari-
able Z which only effects T; but does not directly effect Y₁.
In particular, consider the following structural equations:
=
Y₁
Ti
where E (u; | X₁, Z;) = 0 and E (v; | Xį, Zi) = 0
aTi + Xiß + ui
pY; + X₁Y+Z₁8+ Vi
(1)
(2)
![(a) Show why you cannot you use OLS to estimate a consistently in model 1?
(b) Solve for the reduced form equation for T¡ i.e. use (1) to plug in for Y; in
(2). Simplify and denote the coefficients in front of X₁, Z¡ as ß½ and 82 and
denote the composite error term as v. Show why can you get a consistent
estimate of Band 82 by regressing T on X and Z. [Hint: Use the assumptions
on the conditional expectations above to show that exogeneity will hold in this case
with v.1
(c) Solve for the reduced form equation fo Y; i.e. use the reduced for equation
for T; derived above and plug in (1). Simplify and denote the the coefficients
in front of X¡, Z¡ as ߆ and 8† and denote the composite error term as už.
Show why can you get a consistent estimate of ß₁ and 8 by regressing Y
on X and Z. [Hint: Use the assumptions on the conditional expectations above to
show that exogeneity will hold in this case with už.]
(d) Now with consistent estimates obtained in steps (b) and (c), show that you
can get a consistent estimate of a.
(e) Would this have been possible if you did not have Z;? Explain in words
what is the role of Z; in solving the simultaneity bias problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffce179e1-f5f1-448d-a1d1-81cf5103f29a%2F8218797d-051d-4144-ae7a-ed9e00c88a81%2F7wxhg8h_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Show why you cannot you use OLS to estimate a consistently in model 1?
(b) Solve for the reduced form equation for T¡ i.e. use (1) to plug in for Y; in
(2). Simplify and denote the coefficients in front of X₁, Z¡ as ß½ and 82 and
denote the composite error term as v. Show why can you get a consistent
estimate of Band 82 by regressing T on X and Z. [Hint: Use the assumptions
on the conditional expectations above to show that exogeneity will hold in this case
with v.1
(c) Solve for the reduced form equation fo Y; i.e. use the reduced for equation
for T; derived above and plug in (1). Simplify and denote the the coefficients
in front of X¡, Z¡ as ߆ and 8† and denote the composite error term as už.
Show why can you get a consistent estimate of ß₁ and 8 by regressing Y
on X and Z. [Hint: Use the assumptions on the conditional expectations above to
show that exogeneity will hold in this case with už.]
(d) Now with consistent estimates obtained in steps (b) and (c), show that you
can get a consistent estimate of a.
(e) Would this have been possible if you did not have Z;? Explain in words
what is the role of Z; in solving the simultaneity bias problem.
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