Regression Discontinuity World Bank SIEF / APHRC Impact Evaluation Training 2015pubdocs.worldbank.org/en/663221440083705317/08... · Impact Evaluation Training 2015 Owen Ozier Development

Regression Discontinuity

World Bank SIEF / APHRCImpact Evaluation Training

2015

Owen OzierDevelopment Research Group

The World Bank

6 May 2015

Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 1 / 29

Regression discontinuity - basic idea

A precise rule based on a continuous characteristic determines participation in a program.

When do we see such rules? Here are five categories, but surely there are more:

Academic test scores: scholarships or prizes, higher education admission,certificates of merit

Poverty scores: (proxy-)means-tested anti-poverty programs

Land area: fertilizer program or debt relief initiative for owners of plots below acertain area

Date: age cutoffs for pensions; dates of birth for starting school with differentcohorts; date of loan to determine eligibility for debt relief

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















































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 inPractice: Ancillary Material, The World Bank, Washington DC (www.worldbank.org/ieinpractice)


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

IMPACT


Note: Local Average Treatment Effect


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

IMPACT


Note: Local Average Treatment Effect


Regression discontinuity - basic idea (“sharp”)

0.2

.4.6

.81

Pro

babi

lity(

Tre

atm

ent )

-1 -.5 0 .5 1Running variable



5.9

66.

16.

26.

36.

4

-1 -.5 0 .5 1

Y axis: perhaps log earnings; X axis: perhaps qualification for labor market program


Regression discontinuity - basic idea (“fuzzy”)

0.2

.4.6

.81

Pro

babi

lity(

Tre

atm

ent )

-1 -.5 0 .5 1Running variable

Note: “Alwaystakers,” “Nevertakers,” “Compliers,” and the LATE


History - Cook (2008)

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

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

Goldberger (1972; economics) referred to it as deterministic selection on the covariate;

Sacks and Spiegelman (1977,78,80; statistics) studiously avoided naming it;

Rubin (1977; statistics) first wrote about it as part of a larger discussion of treatmentassignment based on the covariate;

Finkelstein et al (1996; biostatistics) called it the risk-allocation design;

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

But starting in the late 1990s, a large amount of research has appeared in economics.Some papers use the technique to find program impacts; others formalize details of themethodology.See Journal of Econometrics, 2008, Volume 142, Number 2 - special issue on RD.


History - Cook (2008)

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

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

Goldberger (1972; economics) referred to it as deterministic selection on the covariate;

Sacks and Spiegelman (1977,78,80; statistics) studiously avoided naming it;

Rubin (1977; statistics) first wrote about it as part of a larger discussion of treatmentassignment based on the covariate;

Finkelstein et al (1996; biostatistics) called it the risk-allocation design;

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

But starting in the late 1990s, a large amount of research has appeared in economics.Some papers use the technique to find program impacts; others formalize details of themethodology.See Journal of Econometrics, 2008, Volume 142, Number 2 - special issue on RD.


Time travel: back to 1960



Observation: academically advanced scholarship winners have “intellectual” attitudes.

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

Outcome: scholarships

Outcome: attitudes



Observation: academically advanced scholarship winners have “intellectual” attitudes.Are their attitudes changed by receiving the scholarship? (Is it a causal link?)


Outcome: attitudes





Outcome: attitudes





Outcome: attitudes


RD, a little more formally

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

Treatment rule in sharp RD:

Di =

{1 if xi ≥ x0

0 if xi < x0(1)

Data generating process:

E [Y0i |xi ] = α + βxi (2)

Y1i = Y0i + ρ (3)

Regression equation:Yi = α + βxi + ρDi + ηi (4)

More general (possibly nonlinear) scenario:

Yi = f (xi ) + ρDi + ηi (5)

What do we do here?





Di =

{1 if xi ≥ x0

0 if xi < x0(1)


E [Y0i |xi ] = α + βxi (2)

Y1i = Y0i + ρ (3)



Yi = f (xi ) + ρDi + ηi (5)

What do we do here?





Di =

{1 if xi ≥ x0

0 if xi < x0(1)


E [Y0i |xi ] = α + βxi (2)

Y1i = Y0i + ρ (3)



Yi = f (xi ) + ρDi + ηi (5)

What do we do here?





Di =

{1 if xi ≥ x0

0 if xi < x0(1)


E [Y0i |xi ] = α + βxi (2)

Y1i = Y0i + ρ (3)



Yi = f (xi ) + ρDi + ηi (5)

What do we do here?





Di =

{1 if xi ≥ x0

0 if xi < x0(1)


E [Y0i |xi ] = α + βxi (2)

Y1i = Y0i + ρ (3)



Yi = f (xi ) + ρDi + ηi (5)

What do we do here?





Di =

{1 if xi ≥ x0

0 if xi < x0(1)


E [Y0i |xi ] = α + βxi (2)

Y1i = Y0i + ρ (3)



Yi = f (xi ) + ρDi + ηi (5)

What do we do here?



We can (locally) approximate any smooth function:

Yi = f (xi ) + ρDi + ηi (6)

Substitute:f (xi ) ≈ α + β1xi + β2x

2i + ...+ βpx

pi (7)

And thus:Yi = α + β1xi + β2x

2i + ...+ βpx

pi + ρDi + ηi (8)

But because the smooth function may behave differently on either side of the cutoff, wewill expand on this. First, transform xi notationally (and for ease of regression). Let

x̃i = xi − x0 (9)





Yi = f (xi ) + ρDi + ηi (6)


2i + ...+ βpx

pi (7)


2i + ...+ βpx

pi + ρDi + ηi (8)


x̃i = xi − x0 (9)





Yi = f (xi ) + ρDi + ηi (6)


2i + ...+ βpx

pi (7)


2i + ...+ βpx

pi + ρDi + ηi (8)


x̃i = xi − x0 (9)





Yi = f (xi ) + ρDi + ηi (6)


2i + ...+ βpx

pi (7)


2i + ...+ βpx

pi + ρDi + ηi (8)


x̃i = xi − x0 (9)





Yi = f (xi ) + ρDi + ηi (6)


2i + ...+ βpx

pi (7)


2i + ...+ βpx

pi + ρDi + ηi (8)


x̃i = xi − x0 (9)



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

Then, allowing different trends (and indeed, completely different polynomials) on eitherside of the cutoff (with and without the program), we can write the conditionalexpectation functions:

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

pi (10)

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

pi (11)

And because Di is a deterministic function of xi (this is important for writing theconditional expectation):

E [Yi |Xi ] = E [Y0i ] + (E [Y1i ]− E [Y0i ])Di (12)

So, substituting in for the regression equation, we can define β∗ = β1 − β0, and write:

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

pi + (13)

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

pi + ηi (14)






pi (10)


pi (11)




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

pi + (13)


pi + ηi (14)






pi (10)


pi (11)




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

pi + (13)


pi + ηi (14)






pi (10)


pi (11)




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

pi + (13)


pi + ηi (14)




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

E [Yi |x0 −∆ < xi < x0] ≈ E [Y0i |xi = x0] (15)

E [Yi |x0 ≤ xi < x0 + ∆] ≈ E [Y1i |xi = x0] (16)

and then, in the most extreme case:

lim∆→0

E [Yi |x0 ≤ xi < x0 + ∆]− E [Yi |x0 −∆ < xi < x0] = E [Y1i − Y0i |xi = x0] (17)

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

In practice, usually include linear terms and use a narrow region around the cutoff.Angrist and Pishke, Chapter 6, pp. 251-267




E [Yi |x0 −∆ < xi < x0] ≈ E [Y0i |xi = x0] (15)

E [Yi |x0 ≤ xi < x0 + ∆] ≈ E [Y1i |xi = x0] (16)


lim∆→0







E [Yi |x0 −∆ < xi < x0] ≈ E [Y0i |xi = x0] (15)

E [Yi |x0 ≤ xi < x0 + ∆] ≈ E [Y1i |xi = x0] (16)


lim∆→0







E [Yi |x0 −∆ < xi < x0] ≈ E [Y0i |xi = x0] (15)

E [Yi |x0 ≤ xi < x0 + ∆] ≈ E [Y1i |xi = x0] (16)


lim∆→0



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





E [Yi |x0 −∆ < xi < x0] ≈ E [Y0i |xi = x0] (15)

E [Yi |x0 ≤ xi < x0 + ∆] ≈ E [Y1i |xi = x0] (16)


lim∆→0






What if the assignment rule is discontinuous, but does not completely determinetreatment status?

Prob(Di = 1|xi ) =

{g1(xi ) if xi ≥ x0

g0(xi ) if xi < x0,whereg1(x0) 6= g0(x0) (18)

We need a different notation for being on the left or the right of the cutoff, now that Di

doesn’t jump from zero to one. Let Ti = 1(xi ≥ x0).Now, following the equations in the text, we arrive at two (piecewise) polynomialapproximations:

Yi = µ+ κ1xi + κ2x2i + ...+ κpx

pi + πρTi + ζ2i (19)

Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (20)

So to estimate ρ, we use instrumental variables, and in essence divide the coefficientestimate on Ti in the “first stage” regression (variations on Equation 20) by thecoefficient estimage on Ti in the “reduced form” regression (variations on Equation 19).Again, as in IV: Exclusion restriction, standard errorsAngrist and Pishke, Chapter 6, pp. 251-267




Prob(Di = 1|xi ) =




doesn’t jump from zero to one. Let Ti = 1(xi ≥ x0).

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

Yi = µ+ κ1xi + κ2x2i + ...+ κpx


Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (20)





Prob(Di = 1|xi ) =





Yi = µ+ κ1xi + κ2x2i + ...+ κpx


Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (20)





Prob(Di = 1|xi ) =





Yi = µ+ κ1xi + κ2x2i + ...+ κpx


Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (20)





Prob(Di = 1|xi ) =





Yi = µ+ κ1xi + κ2x2i + ...+ κpx


Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (20)

So to estimate ρ, we use instrumental variables, and in essence divide the coefficientestimate on Ti in the “first stage” regression (variations on Equation 20) by thecoefficient estimage on Ti in the “reduced form” regression (variations on Equation 19).

Again, as in IV: Exclusion restriction, standard errorsAngrist and Pishke, Chapter 6, pp. 251-267




Prob(Di = 1|xi ) =





Yi = µ+ κ1xi + κ2x2i + ...+ κpx


Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (20)

So to estimate ρ, we use instrumental variables, and in essence divide the coefficientestimate on Ti in the “first stage” regression (variations on Equation 20) by thecoefficient estimage on Ti in the “reduced form” regression (variations on Equation 19).Again, as in IV: Exclusion restriction, standard errors





Prob(Di = 1|xi ) =





Yi = µ+ κ1xi + κ2x2i + ...+ κpx


Di = γ0 + γ1xi + γ2x2i + ...+ γpx

pi + πTi + ζ1i (20)



Practical considerations

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

Visualization

Specification: polynomial order, “kernel”

Bandwidth

Standard errors (confidence interval)

Specification tests: density, covariates, other jumps




Visualization


Bandwidth






Visualization


Bandwidth






Visualization


Bandwidth






Visualization


Bandwidth






Visualization


Bandwidth




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.

Perc

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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






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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






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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






Perc

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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






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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






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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






Perc

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1996 1997

1998 1999

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2002 2003

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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






Perc

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1994 1995

1996 1997

1998 1999

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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






Perc

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1994 1995

1996 1997

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Perc

ent

6

5

4

3

2

1

0

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






Perc

ent

1994 1995

1996 1997

1998 1999

2000 2001

2002 2003

6

5

4

3

2

1

0

Perc

ent

6

5

4

3

2

1

0

Perc

ent

6

5

4

3

2

1

0

Perc

ent

6

5

4

3

2

1

0

Perc

ent

6

5

4

3

2

1

0

Perc

ent

6

5

4

3

2

1

0

Perc

ent

6

5

4

3

2

1

0

Perc

ent

6

5

4

3

2

1

0

Perc

ent

6

5

4

3

2

1

0

Perc

ent

6

5

4

3

2

1

0

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


Documents

Regression Discontinuity World Bank SIEF / APHRC Impact Evaluation Training 2015pubdocs.worldbank.org/en/663221440083705317/08... · Impact Evaluation Training 2015 Owen Ozier Development