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Econ 281 Instrumental Variables

# Econ 281 Instrumental Variables - Assessing Regression...

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1 Assessing Regression Studies Introduction to Econometrics Econ 281

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2 Assessing Studies There are usually two areas of interest Internal Validity External Validity Internal Validity statistical inferences about the causal effects are valid for the population being studied External Validity inferences and conclusions can be generalized from the population studied to other populations and settings
3 Threats to External Validity Mainly two causes Differences in population i.e. Studies on mice might not be good indicator for effect on men Differences in Settings Different legal institutions, time period, countries, cultures etc. Let’s look at the Test Score Example

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4 Threats to External Validity How far can we generalize class size results from California school districts? Differences in populations California in 2005? Massachusetts in 2005? Mexico in 2005? Differences in settings different legal requirements concerning special education different treatment of bilingual education differences in teacher characteristics
5 Threats to Internal Validity Omitted Variables Bias We already discussed this Measurement Error in independent variables Simultaneous Equation Bias

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6 Measurement Error in Independent Variables Assume that instead of the true variable X you observe X plus an error term True Model You estimate 2 with ~ (0, ) XX i i d ε εσ =+ ± 01 YX v ββ = +⋅ + ± u β = +
7 Measurement Error in Independent Variables Rewrite the model you estimate in terms of the true model Can we estimate without bias? 01 YX v ββ = +⋅ + ± ( ) 1 () v X u ε β =+ + + + ± ²³´³µ ²³´³ µ 1

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8 Calculate ( ) , Cov v X ± ( ) ( ) 1 ,, Cov X v Cov X u εβ ε =+ + ± ( ) ( ) () 1 2 1 , Cov X Cov X u Cov u βε βσ =− ⋅ + −⋅ + ( ) ( ) ( ) Assume: , 0 , , 0 Cov X u Cov X Cov u εε = == 2 1 0
9 Bias Estimator will be biased because How much bias will result asymptotically? Let’s use our formula for omitted variable bias ( ) ,0 Cov v X ± 1 ˆ β ˆ p v Xv X σ ββ ρ ⎯⎯ →+ ± ±

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10 11 ˆ p v Xv X σ ββ ρ ⎯⎯ →+ ± ± ( ) 22 2 , vv v XX X X X X Cov X v ε σβ σσ ⋅− ⋅= ⋅⋅ ± ±± ± ± ± ± ± 2 1 1 2 ˆ p X βσ ⎯⎯ →− ± Let’s plug our result into the term on the far right This simplifies even further…….
11 22 2 X X ε σ σσ = + ± Note that (independence) Finally ……. 1 11 ˆ 1 p XX εε βσ ββ β ⎛⎞ ⎯⎯ →− = ⎜⎟ ⎝⎠ ±± 1 2 ˆ 1 p X X = + ±

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12 Note that the denominator is larger than the numerator With measurement error the OLS estimator is biased towards zero 2 1 22 1 ˆ p X X ε σ ββ < ⎯⎯ →⋅ + ± ²³ ²´
13 Solutions to Measurement Error Get accurate measure Usually not possible Develop a model of measurement error Complicated Might be wrong Instrumental Variables Chapter 12

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14 Simultaneous Causality
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Econ 281 Instrumental Variables - Assessing Regression...

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