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Page 1: Section 3. Simple Regression - Omitted Variable Biaseconweb.ucsd.edu/~elib/120b/less8_simplereg_ovb.pdf · Section 3. Simple Regression - Omitted Variable Bias 1. Sampling Properties

Section 3. Simple Regression - Omitted Variable Bias

1. Sampling Properties of OLS estimators2. What’s a Covariance?3. More sampling properties of OLS

estimators4. Standard Errors of OLS estimators5. Confidence intervals6. CLT demonstration7. Omitted Variables and Omitted Variable

Bias (prelude to Section 4)

Page 2: Section 3. Simple Regression - Omitted Variable Biaseconweb.ucsd.edu/~elib/120b/less8_simplereg_ovb.pdf · Section 3. Simple Regression - Omitted Variable Bias 1. Sampling Properties

Copyright © 2003 by Pearson Education, Inc. 4-2

1. Sampling Properties of OLS estimators

Review: • Unbiased: E(b0) = β0, E(b1) = β1

• Consistent: plim(b0) = β0, plim(b1) = β1

New: • Asymptotically Normal by the CLT

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Derivation

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Derivation

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2. What’s a Covariance?

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3. Sampling Properties of OLS estimators (cont.)

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Why asymptotically N(.)?

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Copyright © 2003 by Pearson Education, Inc. 4-8

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4. Standard Errors

• CLT: b1 approx ~ N(β1, V(b1))b0 approx ~ N(β0, V(b0))

• “Standard Errors” are consistent estimators of standard deviations of boand b1 . (S&W p. 133, 151,180).

• So “t-statistics” have a standard normal distribution.

Page 10: Section 3. Simple Regression - Omitted Variable Biaseconweb.ucsd.edu/~elib/120b/less8_simplereg_ovb.pdf · Section 3. Simple Regression - Omitted Variable Bias 1. Sampling Properties

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5. Confidence Intervals

..and the same for β0

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Confidence Intervals

Confidence intervals for β0,β1

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6. CLT demonstrations using Stata

..and the same for b0

• Stata doesn’t mind running a regression a few thousand times, - which allows us to observe a sampling distribution for b1e.g., bootregh00.do

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CLT in action: sampling distributions for b1

Frac

tion

n=20, 2000 repsb

-.1 -.05 0 .05

0

.128

Frac

tion

n=30, 2000 repsb

-.05 0 .05

0

.1075

Frac

tion

n=40, 2000 repsb

-.05 0 .05

0

.126

Note: In developing countriesthis slope is about -0.2 childrenper year of education. Vg

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7. Omitted Variables and Omitted Variable Bias

• What if you left out an important variable?

• Many interesting relationships have more than 2 dimensions

• Multivariate regression:

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7a. OLS Multivariate regression

Look familiar? Same criterion with more variables.

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7b. Properties of OLS estimators in Multivariate Regression

• Consistent• Unbiased• Approximately N(.) in large samples• Claim:

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7c. Omitted Variable “Bias”

• Short regressiony = b0

s + b1s x1 + eS (SR)

• Long regressiony = b0

L + b1L x1 + b2

L x2 + eL (LR) • Claim:

b1s = b1

L + b2L b21 ,

b21 is slope of a regression of x2 on x1


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