ECON 626: Applied...

ECON 626: Applied Microeconomics

Lecture 5:

Regression Discontinuity

Professors: Pamela Jakiela and Owen Ozier

Regression discontinuity - basic idea

A precise rule based on a continuous characteristic determinesparticipation in a program.

When do we see such rules? Five example categories, but surely more:

• Academic test scores: scholarships or prizes, higher educationadmission, certificates of merit

• Poverty scores: (proxy-)means-tested anti-poverty programs(generally: any program targeting that features rounding or cutoffs)

• Land area: fertilizer program or debt relief initiative for owners ofplots below a certain area

• Date: age cutoffs for pensions; dates of birth for starting school withdifferent cohorts; date of loan to determine eligibility for debt relief

• Elections: fraction that voted for a candidate of a particular party

UMD Economics 626: Applied Microeconomics Lecture 4: Regression Discontinuity, Slide 2

Regression discontinuity - basic idea (“sharp”)Regression Discontinuity Design-Baseline

Not eligible

Eligible

Source: Gertler, P. J.; Martinez, S., Premand, P., Rawlings, L. B. and Christel M. J. Vermeersch,2010, Impact Evaluation in Practice: Ancillary Material, The World Bank, Washington DC

(www.worldbank.org/ieinpractice)

Note: Local Average Treatment Effect

Regression discontinuity - basic idea (“sharp”)Regression Discontinuity Design-Post Intervention

IMPACT

Regression discontinuity - basic idea (“sharp”)Regression Discontinuity Design-Post Intervention

IMPACT

Regression discontinuity - basic idea (“sharp”)

-1 -.5 0 .5 1Running variable

D=program participation = T=treatment assignment

Regression discontinuity - outcome

-1 -.5 0 .5 1

Regression discontinuity - basic idea (“fuzzy”)

D=program participation

Regression discontinuity - basic idea (“fuzzy”)

D=program participation T=treatment assignment

History of the RD design - Cook (2008)

“Several themes stand out in the half century of RDD’s history. One isits repeated independent discovery. ...

• Campbell (1960; psychology / education) first named the designregression-discontinuity;

• Goldberger (1972; economics) referred to it as deterministicselection on the covariate;

• Sacks and Spiegelman (1977,78,80; statistics) studiously avoidednaming it;

• Rubin (1977; statistics) first wrote about it as part of a largerdiscussion of treatment assignment based on the covariate;

• Finkelstein et al (1996; biostatistics) called it the risk-allocationdesign;

• and Trochim (1980; statistics) finished up calling it thecutoff-based design.”

Boom since 1990s in economics: applications and methodology. SeeJournal of Econometrics, 2008 Vol.142 (2) - special issue on RD.

Boom since 1990s in economics: applications and methodology.

SeeJournal of Econometrics, 2008 Vol.142 (2) - special issue on RD.

Thistlethwaite and Campbell (1960)

Observation: scholarship winners have different attitudes.

Are attitudes changed by the scholarship? (Is it a causal link?)

Outcome: scholarshipsOutcome: attitudes

Observation: scholarship winners have different attitudes.Are attitudes changed by the scholarship? (Is it a causal link?)

Outcome: scholarships

Outcome: attitudes

RD, a little more formally

We can (locally) approximate any smooth function:

Yi = f (xi ) + ρDi + ηi (1)

Substitute:f (xi ) ≈ α + β1xi + β2x

2i + ...+ βpx

pi (2)

And thus:

Yi = α + β1xi + β2x2i + ...+ βpx

pi + ρDi + ηi (3)

But because the smooth function may behave differently on either side ofthe cutoff, we will expand on this. First, transform xi notationally (andfor ease of regression). Let

x̃i = xi − x0 (4)

Angrist and Pishke, Chapter 6, pp. 251-267

Yi = f (xi ) + ρDi + ηi (1)

2i + ...+ βpx

pi (2)

And thus:

Yi = α + β1xi + β2x2i + ...+ βpx

pi + ρDi + ηi (3)

x̃i = xi − x0 (4)

Yi = f (xi ) + ρDi + ηi (1)

2i + ...+ βpx

pi (2)

And thus:

Yi = α + β1xi + β2x2i + ...+ βpx

pi + ρDi + ηi (3)

x̃i = xi − x0 (4)

Yi = f (xi ) + ρDi + ηi (1)

2i + ...+ βpx

pi (2)

And thus:

Yi = α + β1xi + β2x2i + ...+ βpx

pi + ρDi + ηi (3)

x̃i = xi − x0 (4)

Yi = f (xi ) + ρDi + ηi (1)

2i + ...+ βpx

pi (2)

And thus:

Yi = α + β1xi + β2x2i + ...+ βpx

pi + ρDi + ηi (3)

x̃i = xi − x0 (4)

RD, a little more formallyAngrist and Pishke, Chapter 6, pp. 251-267

Then, allowing different trends (and indeed, completely differentpolynomials) on either side of the cutoff (with and without the program),we can write the conditional expectation functions:

E [Y0i ] = f0(xi ) = α + β01x̃i + β02x̃2i + ...+ β0p x̃

E [Y1i ] = f1(xi ) = α + ρ+ β11x̃i + β12x̃2i + ...+ β1p x̃

And because Di is a deterministic function of xi (this is important forwriting the conditional expectation):

E [Yi |Xi ] = E [Y0i ] + (E [Y1i ]− E [Y0i ])Di (7)

So, substituting in for the regression equation, we can defineβ∗j = β1j − β0j for any j , and write:

Yi =α + β01x̃i + β02x̃2i + ...+ β0p x̃

pi + (8)

ρDi + β∗1Di x̃i + β∗2Di x̃2i + ...+ β∗p x̃

pi + ηi (9)

Yi =α + β01x̃i + β02x̃2i + ...+ β0p x̃

pi + (8)

pi + ηi (9)

Yi =α + β01x̃i + β02x̃2i + ...+ β0p x̃

pi + (8)

pi + ηi (9)

But this can all really be simplified in many practical cases. For smallvalues of ∆:

E [Yi |x0 −∆ < xi < x0] ≈ E [Y0i |xi = x0] (10)

E [Yi |x0 ≤ xi < x0 + ∆] ≈ E [Y1i |xi = x0] (11)

and then, in the most extreme case, we can take the limit:

lim∆→0

E [Yi |x0 ≤ xi < x0 + ∆]− E [Yi |x0 −∆ < xi < x0] = E [Y1i − Y0i |xi = x0]

So the difference in means in an extremely (vanishingly!) narrow band oneach side of the cutoff might be enough to estimate the effect of theprogram, ρ.

In practice, usually include linear terms and use a narrow region aroundthe cutoff.

E [Yi |x0 −∆ < xi < x0] ≈ E [Y0i |xi = x0] (10)

E [Yi |x0 ≤ xi < x0 + ∆] ≈ E [Y1i |xi = x0] (11)

lim∆→0

E [Yi |x0 −∆ < xi < x0] ≈ E [Y0i |xi = x0] (10)

E [Yi |x0 ≤ xi < x0 + ∆] ≈ E [Y1i |xi = x0] (11)

lim∆→0

E [Yi |x0 −∆ < xi < x0] ≈ E [Y0i |xi = x0] (10)

E [Yi |x0 ≤ xi < x0 + ∆] ≈ E [Y1i |xi = x0] (11)

lim∆→0

What if the assignment rule is discontinuous, but does not completelydetermine treatment status?

Prob(Di = 1|xi ) =

{g1(xi ) if xi ≥ x0

g0(xi ) if xi < x0,where g1(x0) 6= g0(x0) (13)

We need a different notation for being on the left or the right of thecutoff, now that Di doesn’t jump from zero to one. Let Ti = I(xi ≥ x0).Now, following the equations in the text, we arrive at two (piecewise)polynomial approximations:

Yi = µ+ κ1xi + κ2x2i + ...+ κpx

pi + πρTi + ζ2i (14)

Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (15)

So to estimate ρ, we use instrumental variables, and in essence divide thecoefficient estimate on Ti in the “first stage” regression (variations onEquation 15) by the coefficient estimage on Ti in the “reduced form”regression (variations on Equation 14). Again, as in IV: Exclusionrestriction, standard errors

Prob(Di = 1|xi ) =

We need a different notation for being on the left or the right of thecutoff, now that Di doesn’t jump from zero to one. Let Ti = I(xi ≥ x0).

Now, following the equations in the text, we arrive at two (piecewise)polynomial approximations:

Yi = µ+ κ1xi + κ2x2i + ...+ κpx

Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (15)

Prob(Di = 1|xi ) =

Yi = µ+ κ1xi + κ2x2i + ...+ κpx

Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (15)

Prob(Di = 1|xi ) =

Yi = µ+ κ1xi + κ2x2i + ...+ κpx

Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (15)

Prob(Di = 1|xi ) =

Yi = µ+ κ1xi + κ2x2i + ...+ κpx

Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (15)

So to estimate ρ, we use instrumental variables, and in essence divide thecoefficient estimate on Ti in the “first stage” regression (variations onEquation 15) by the coefficient estimage on Ti in the “reduced form”regression (variations on Equation 14).

Again, as in IV: Exclusionrestriction, standard errors

Prob(Di = 1|xi ) =

Yi = µ+ κ1xi + κ2x2i + ...+ κpx

Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (15)

Prob(Di = 1|xi ) =

Yi = µ+ κ1xi + κ2x2i + ...+ κpx

Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (15)

Practical considerations

Five basic issues are highlighted by Guido Imbens and Thomas Lemieuxin their paper, Regression discontinuity designs: A guide to practice:

• Specification tests: density, covariates, other jumps

• Density: analogy to attrition. This is conceptually important.

• Visualization

• Specification: polynomial order (linear in many cases), “kernel”

• Bandwidth

• Standard errors (confidence interval)

Methodological updates and extensions:

• Cattaneo, Calonico, and Titiunik series (SE’s, visualization)

• Card, Lee, Pei, and Weber (Kink design)

• Visualization

• Bandwidth

• Visualization

• Bandwidth

• Visualization

• Bandwidth

• Visualization

• Bandwidth

• Visualization

• Bandwidth

• Visualization

• Bandwidth

• Visualization

• Bandwidth

• Visualization

• Bandwidth

Visualization: Dube, Giuliano, Leonard example

Manipulation of the running variable

What if the population of potential program participantsis able to precisely influence the running variable,and knows the program assignment rule?

Example from Camacho and Conover (2011) in Colombia:program rule became known in 1997;watch what happens.

Manipulation of the running variable

What if the population of potential program participantsis able to precisely influence the running variable,and knows the program assignment rule?

Example from Camacho and Conover (2011) in Colombia:program rule became known in 1997;watch what happens.

Poverty score distribution - Camacho and Conover(2011) in Colombia

VOL. 3 NO. 2 43CAMACHO AND CONOVER: MANIPULATION OF SOCIAL PROGRAM ELIGIBILITY

Figure 1. Poverty Index Score Distribution 1994–2003, Algorithm Disclosed in 1997

Notes: Each !gure corresponds to the interviews conducted in a given year, restricting the sample to urban house-holds living in strata levels below four. The vertical line indicates the eligibility threshold of 47 for many social programs.

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

0 7 14 21 28 35 42 49 56 63 70 77 84 91 98

Poverty index score

An example.

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