Problem 2 In economics we are often faced with causal problems where the endogeneityarises because of simultaneity. A classic example is that if you are interested inestimating 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 alsoa 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 exogenous1covariates 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:=Y₁Tiwhere E (u; | X₁, Z;) = 0 and E (v; | Xį, Zi) = 0aTi + Xiß + uipY; + 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 anddenote the composite error term as v. Show why can you get a consistentestimate of Band 82 by regressing T on X and Z. [Hint: Use the assumptionson the conditional expectations above to show that exogeneity will hold in this casewith v.1(c) Solve for the reduced form equation fo Y; i.e. use the reduced for equationfor T; derived above and plug in (1). Simplify and denote the the coefficientsin front of X¡, Z¡ as ß† and 8† and denote the composite error term as už.Show why can you get a consistent estimate of ß₁ and 8 by regressing Yon X and Z. [Hint: Use the assumptions on the conditional expectations above toshow that exogeneity will hold in this case with už.](d) Now with consistent estimates obtained in steps (b) and (c), show that youcan get a consistent estimate of a.(e) Would this have been possible if you did not have Z;? Explain in wordswhat is the role of Z; in solving the simultaneity bias problem.

Given information Yi=αTi+βXi'+uiTi=ρYi+γXi'+δZi+ViHere Yi is affected by Ti and Xi'On the other hand…

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

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Chapter1: Making Economics Decisions

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Question

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 ß† 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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