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11 Consider the equation y-A₂+ B₁x + ₂x² + E(x) = 0, where the explanatory variable x has a standard normal distribution in the population. In particular, E(x)-0, E(x)-Var(x)-1, and E(x)-0. This last condition holds because the standard normal

11 Consider the equation y-A₂+ B₁x + ₂x² + E(x) = 0, where the explanatory variable x has a standard normal distribution in the population. In particular, E(x)-0, E(x)-Var(x)-1, and E(x)-0. This last condition holds because the standard normal

Microeconomic Theory
Microeconomic Theory
12th Edition
ISBN:9781337517942
Author:NICHOLSON
Publisher:NICHOLSON
Chapter19: Externalities And Public Goods
Section: Chapter Questions
Problem 19.12P
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distribution is symmetric about zero. We want to study what we can say about the OLS estimator if we omitx and compute the simple regression estimator of the intercept and slope. (1) Show that we can write y = a₁ + B₁x + v. where E(v) = 0. In particular, find v and the new intercept, (ii) Show that E(v)x) depends on x unless B₂ = 0. (iii) (iv) Show that Cov(x, v) = 0. If, is the slope coefficient from regression y, on x,, is , consistent for B,? Is it unbiased? Explain. (v) Argue that being able to estimate 3, has some value in the following sense: B, is the partial effect of x on E(vix)evaluated at x = 0, the average value of x. (vi) Explain why being able to consistently estimate B, and B, is more valuable than just estimating 3₁.
Transcribed Image Text:distribution is symmetric about zero. We want to study what we can say about the OLS estimator if we omitx and compute the simple regression estimator of the intercept and slope. (1) Show that we can write y = a₁ + B₁x + v. where E(v) = 0. In particular, find v and the new intercept, (ii) Show that E(v)x) depends on x unless B₂ = 0. (iii) (iv) Show that Cov(x, v) = 0. If, is the slope coefficient from regression y, on x,, is , consistent for B,? Is it unbiased? Explain. (v) Argue that being able to estimate 3, has some value in the following sense: B, is the partial effect of x on E(vix)evaluated at x = 0, the average value of x. (vi) Explain why being able to consistently estimate B, and B, is more valuable than just estimating 3₁.
11 Consider the equation y=B₁ + B₁x + B₂x² + u E(x) = 0, where the explanatory variable x has a standard normal distribution in the population. In particular, E(x)=0, E(x¹) - Var(x)=1, and E(x¹) = 0. This last condition holds because the standard normal
Transcribed Image Text:11 Consider the equation y=B₁ + B₁x + B₂x² + u E(x) = 0, where the explanatory variable x has a standard normal distribution in the population. In particular, E(x)=0, E(x¹) - Var(x)=1, and E(x¹) = 0. This last condition holds because the standard normal
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