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Dependent Variable Errors (1 pages)

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Dependent Variable Errors

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Lecture number:
22
Pages:
1
Type:
Lecture Note
School:
Cornell University
Course:
Econ 3120 - Applied Econometrics
Edition:
1

Unformatted text preview:

Econ 3120 1st Edition Lecture 22 Outline of Last Lecture I. Specification and Data Issues Outline of Current Lecture II. Dependent Variable Errors Current Lecture 2.1 Dependent Variable Suppose we are interested in running the following regression model: y ∗ = β0 +β1x1 +...+βkxk +u (4) where y ∗ represents a true value of the underlying variable, and y represents what you observe. Measurement is given by e0 = y−y ∗ 2 Where e0 represents mean-zero “noise”, uncorrelated with y ∗ or u. Thus, you run the regression y = β0 +β1x1 +...+βkxk +u+e0 (5) When will estimates using (5) instead of (4) give consistent estimators? This happens as long as e0 is not correlated with the x 0 s that is, if E(e0|x1,..., xk) = 0. Although the estimators under (5) will still be consistent, the standard errors will increase. This occurs because the composite error term is now u+e0. Assuming that u and e0 are uncorrelated, Var(u+e0) = σ 2 u +σ 2 e0 . 2.2 Independent Variable Measurement error in an independent variable is typicaly more difficult to deal with. Consider the bivariate model y = β0 +β1x ∗ 1 +u (6) where, as before, x ∗ 1 is an (unobserved) true value, and x is an observation that includes error: e1 = x1 −x ∗ 1 (7) Assume that e1 has mean zero and is not correlated with u or x ∗ 1 . You run the regression using the observed x1. Substituting (7) into (6) yields y = β0 +β1x1 + (u−β1e1) (8) In this case, however, x1 and e1 are not uncorrelated. In particular: Cov(x1, e1) = E(x1, e1) = E(x ∗ 1 +e1, e1) = E(x ∗ 1 e1) +E(e 2 1 ) = 0+σ 2 e1 = σ 2 e1 3 This implies that the covariance between x1and the composite error term in (8) equals Cov(x1,u−β1e1) = Cov(x1,u)−Cov(x1,β1e1) = Cov(x ∗ 1 +e,u)−Cov(x1,β1e1) = Cov(x ∗ 1 ,u) +Cov(e,u)−Cov(x1,β1e1) = 0+0−Cov(x1,β1e1) = −Cov(x1,β1e1) = −β1Cov(x1, e1) = −β1σ 2 e1 We can derive the plim of ˆβ1 estimated from (8) as follows: ˆβ1 = Covd(x1, y) Vard(x1) = Cov(x1,β1xd1 +u−β1e1) Vard(x1) = Cov(x1,β1x1) +dCov(x1,u−β1e1) Vard(x1) = β1 + Cov(x1d,u−β1e1) Vard(x1) plim( ˆβ1) = β1 + Cov(x1,u−β1e1) Var(x1) = β1 + −β1σ 2 e1 Var(x1) Note that because x1 = x ∗ 1 + e1, and because x ∗ 1 and e1 are uncorrelated, Var(x1) = Var(x ∗ 1 ) + Var(e1). Thus, the plim becomes 4 plim( ˆβ1) = β1 + −β1σ 2 e1 Var(x ∗ 1 ) +Var(e1) = β1 + −β1σ 2 e1 σ 2 x ∗ 1 +σ2 e1 = β1( σ 2 x ∗ 1 +σ 2 e1 σ 2 x ∗ 1 +σ2 e1 − σ 2 e1 σ 2 x ∗ +σ2 e1 ) = β1( σ 2 x ∗ 1 σ 2 x ∗ 1 +σ2 e1 ) Thus, estimates for β1 will be inconsistent, and the plim is closer to zero relative to ...


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