Consider the regression model y = a + ßx+u, where x is an endogenous variable. a) Write x = x₁ + ê; and show that x =x. b) Suppose there are k valid instruments: Z1, Z2, ..., Zk. Let x₁= πo+1zil + Â2zi2+...+îkZzik. Show that the 2SLS estimator is equivalent to an IV estimator using as an instrument: Σ₁ (₁-P)(vi- V) _ Σ₁₁(Xi—Ñ)(vi— V) Σ₁ (₁-x)² 21i-ễ)(x-1) [Hint: OLS first order conditions imply Σ₁ ê¡Â¡ = 0 and Σ¼₁ ê¡= 0.] c) Suppose there is only one instrument z and ✰¡ = âto+ftizi. Show that the 2SLS estimator is numerically identical to the IV estimator: n Σ²₁(Ρ¯ Ã)(Yi-Y) _ Σ²²₁(²₁−7)(Yi-Y) Σ1 (xi-x)² Ei=1(²₁-7)(xi-x)* [Hint: Use the result in (b) and plug in ¡ and ☎ = înto+ Â₁Z.] n
Consider the regression model y = a + ßx+u, where x is an endogenous variable. a) Write x = x₁ + ê; and show that x =x. b) Suppose there are k valid instruments: Z1, Z2, ..., Zk. Let x₁= πo+1zil + Â2zi2+...+îkZzik. Show that the 2SLS estimator is equivalent to an IV estimator using as an instrument: Σ₁ (₁-P)(vi- V) _ Σ₁₁(Xi—Ñ)(vi— V) Σ₁ (₁-x)² 21i-ễ)(x-1) [Hint: OLS first order conditions imply Σ₁ ê¡Â¡ = 0 and Σ¼₁ ê¡= 0.] c) Suppose there is only one instrument z and ✰¡ = âto+ftizi. Show that the 2SLS estimator is numerically identical to the IV estimator: n Σ²₁(Ρ¯ Ã)(Yi-Y) _ Σ²²₁(²₁−7)(Yi-Y) Σ1 (xi-x)² Ei=1(²₁-7)(xi-x)* [Hint: Use the result in (b) and plug in ¡ and ☎ = înto+ Â₁Z.] n
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Related questions
Question
![Consider the regression model
y = a + ßx+u,
where x is an endogenous variable.
a) Write x₁ = x + ê; and show that = x.
b) Suppose there are k valid instruments: Z1, Z2, ..., Zk.
Let x₁= πo+f1zil + π2Zi2+...+îkzik. Show that the 2SLS estimator is equivalent
to an IV estimator using as an instrument:
Σ₁₁(&i- Ñ)(vi- V) _ Σ₁(Ri—Ñ)(Yi-Y)
Ei=1(xi-x)² 21(®i- Đ)(xi-x)
=
[Hint: OLS first order conditions imply №₁ ê¡Â¡ = 0 and Σ¼_₁ ê¡= 0.]
c) Suppose there is only one instrument z and ✰; = fo+ fizi. Show that the 2SLS
estimator is numerically identical to the IV estimator:
=
Σ₁ (Îi—Ñ)(vi-V) – Σ₁(²₁-7)(vi-V)
Σ=1(ât− x)2 Σ₁ (Z₁-7)(xi-x)*
n
[Hint: Use the result in (b) and plug in â; and ☎ = ñîo+â¦Z.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5a2af565-536b-41b6-a437-1b0d1b7c6de2%2Ff3298933-9260-4d47-9456-2f6a63589a48%2F6w1gx2l_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the regression model
y = a + ßx+u,
where x is an endogenous variable.
a) Write x₁ = x + ê; and show that = x.
b) Suppose there are k valid instruments: Z1, Z2, ..., Zk.
Let x₁= πo+f1zil + π2Zi2+...+îkzik. Show that the 2SLS estimator is equivalent
to an IV estimator using as an instrument:
Σ₁₁(&i- Ñ)(vi- V) _ Σ₁(Ri—Ñ)(Yi-Y)
Ei=1(xi-x)² 21(®i- Đ)(xi-x)
=
[Hint: OLS first order conditions imply №₁ ê¡Â¡ = 0 and Σ¼_₁ ê¡= 0.]
c) Suppose there is only one instrument z and ✰; = fo+ fizi. Show that the 2SLS
estimator is numerically identical to the IV estimator:
=
Σ₁ (Îi—Ñ)(vi-V) – Σ₁(²₁-7)(vi-V)
Σ=1(ât− x)2 Σ₁ (Z₁-7)(xi-x)*
n
[Hint: Use the result in (b) and plug in â; and ☎ = ñîo+â¦Z.]
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