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Webinar Series in Applied Quantitative Analysis - Updated

https://www.airp3-africa.org/applied-quantitative-analysis-webinar-series

Date

Topic

�February 29�March 7

Session One�Potential Outcomes and Omitted Variable Bias I (Theory) �Potential Outcomes and Omitted Variable Bias II (Application)

�March 21�March 28

Session Two�Difference-in-differences I (Theory)�Difference-in-differences II (Application)

�April 25�May 2

Session Three�Power analysis, clustering and sample size calculations I (Theory)�Power analysis, clustering and sample size calculations II (Application)

 

May 23�May 30

Session Four�Propensity score matching (Theory)�Propensity score matching (Application)

�June 20�June 27

Session Five�Fixed-effects I (Theory)�Fixed-effects II (Application)

�July 25�August 1

Session Six�Instrumental variables I (Theory)�Instrumental Variables II (Application)

 

August 22

August 29

Session Seven�Lagged dependent variables and the Arellano-Bond Estimator I (Theory)�Lagged dependent variables and the Arellano-Bond Estimator II (Application)

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������ �Écoute de l'interprétation d'une langue �Windows | macOS�

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Listening to language interpretation� Windows | macOS

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  1. In your meeting/webinar controls, click Interpretation .
  2. Click the language that you would like to hear. (We will have French) No need to choose English, that is the language in the main Zoom room

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3. (Optional) To hear the interpreted language only, click Mute Original Audio.

Notes:

      • You must join the meeting audio through your computer audio/VoIP. You cannot listen to language interpretation if you use the dial-in or call me phone audio features.
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channel if you unmute your audio and speak.

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Instrumental Variables Theory

Professor Jeremy Moulton

Department of Public Policy

University of North Carolina at Chapel Hill

July 25, 2024

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The Set Up From Last Time

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The Old Solution (Fixed Effects/First Differences)

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Change in the Set Up

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The New Set Up

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The New Solution (Instrumental Variables)

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The New Solution (Instrumental Variables)

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The New Solution (Instrumental Variables)

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The New Solution (Instrumental Variables)

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The New Solution (Instrumental Variables)

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The New Solution (Instrumental Variables)

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The New Solution (Instrumental Variables)

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The New Solution (Instrumental Variables)

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Can There Be Any Problems?

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Small cov(IV, x)

Larger standard errors

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Can There Be Any Problems?

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Can There Be Any Problems?

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Can There Be Any Problems?

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Angrist & Krueger (1991)

  • Question: Returns to Education
  • Method: IV to counter OVB
    • Quarter of Birth
  • Data: U.S. Census
  • Results: IV and OLS similar (no OVB)

 

 

 

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IV = Quarter of Birth

  • Most states have age cutoffs to enter school
    • January 1st during this time
  • States also had cutoffs to exit school
    • At this time most were age 16
  • For example, if you were born at the beginning of the year (QTR 1) you would be the oldest kid in school
    • You can drop out earlier than anyone else in your grade and those born later in the year are forced to get more education.

 

 

 

 

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First Stage

 

 

 

 

 

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First Stage

 

 

 

 

 

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Reduced Form of �2nd Stage

 

 

 

 

 

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Wald Estimator (Simplified IV)

 

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IV Regression

  • E is for Education
  • X is for Controls
  • Y is for Year FE
  • YQ is for Year by Quarter (these are the IV)
  • ln W is for Logged Weekly Wage

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IV Results

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Conclusions

  • Do not see that OLS is biased upward (if anything downward)
  • Do not find effects for Annual Salary or Weeks Worked
  • NOTE that all IV provides a Local Average Treatment Effect (LATE)
    • In this case all variation in X (education) is coming from kids on the margin of dropping out of high school

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Bound, Jaeger, and Baker (the response)

  • Question: Are there issues with instrumental variables analysis?
  • Weak Instruments combined with any (even weak) relationship between Instrument and Y (outside of X) can cause bias
  • In finite samples IV is biased toward OLS
    • Magnitude of bias is related to R2 between IV and X

 

 

 

 

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Back to the Math (OLS)

  • Assume just a single X and no controls for simplicity
  • Y = α + βX + ε
  • Cov(Y,X) = Cov(α + βX + ε,X)
  • Cov(Y,X) = βCov(X,X) + Cov(ε,X)
    • Note: Cov(α,X) = 0 because α does not vary
  • σYX = βσ2X + σεX
  • σYX/σ2X = β + σεX/σ2X
  • βOLS = β + σεX/σ2X
    • Bias if σεX does not = 0 (this could be due to Omitted Variable Bias)

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Same Math, but do IV

  • Y = α + βX + ε
  • Cov(Y,IV) = Cov(α + βX + ε,IV)
  • Cov(Y,IV) = βCov(X,IV) + Cov(ε,IV)
  • σYIV = βσXIV + σεIV
  • σYIV/σXIV = β + σεIV/σXIV
  • βiv = β + σεIV/σXIV
  • In Instrumental Variables we assume that σεIV = 0

 

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Conclusion

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IV

Note that in Bound, Jaeger, Baker’s paper they referred to the IV variable as Z. This is common in the literature.

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What were the F and R2 in Angrist & Krueger?

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Reexamine Angrist & Krueger

    • Attendance Rates
    • Behavioral Difficulties
    • Mental Health
    • Reading, Writing, Math
    • Schizophrenia
    • Mental Retardation
    • Dyslexia
    • Multiple Sclerosis
    • Manic Depression
    • Some show IQ
    • Region of Birth
    • Parental Income
  • Paper hinges on the law being the only reason for the differences in education
  • Problem is that the relationship is weak, so any other reason can be problematic

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Let’s Focus on Just Parental Income

  • Solon (1992): children’s education (X) increases 0.014 for each 1% increase in father’s income
    • Difference in Family Income between QTR 1 and QTR 2 can explain 0.03 education years (this is 1/3 of the IV First Stage, so not really a weak relationship)
  • Intergenerational correlation of long-run income > 0.4
    • Difference in Family Income between QTR 1 and QTR 2 can explain 0.95% lower wages for children
    • A&K: 1980 QTR 1 earn 1.1% less using IV
  • Most of the “effect” can be explained by parental income

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Some Thoughts

  • IV is asymptotically unbiased, but that does not mean unbiased
    • You need a large sample for IV to work correctly
  • IV is an art rather than a science
    • You need to convince the reader that your instrument does not belong in the Second Stage (is not correlated with error – Exclusion Restriction)
    • You will need to appeal to theory, rather than directly test this

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  • Next time
    • More tests – particularly when you have > 1 instrument
    • More Stata – ivregress