Post on 21-Aug-2020
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 2 / 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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 2 / 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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 2 / 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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 2 / 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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 2 / 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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 2 / 29
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)
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 3 / 29
Regression discontinuity - basic idea (“sharp”)Regression Discontinuity Design-Post Intervention
IMPACT
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)
Note: Local Average Treatment Effect
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 4 / 29
Regression discontinuity - basic idea (“sharp”)Regression Discontinuity Design-Post Intervention
IMPACT
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)
Note: Local Average Treatment Effect
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 4 / 29
Regression discontinuity - basic idea (“sharp”)
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Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 5 / 29
Regression discontinuity - basic idea
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Y axis: perhaps log earnings; X axis: perhaps qualification for labor market program
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 6 / 29
Regression discontinuity - basic idea (“fuzzy”)
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Note: “Alwaystakers,” “Nevertakers,” “Compliers,” and the LATE
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 7 / 29
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.
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 8 / 29
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.
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 8 / 29
Time travel: back to 1960
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 9 / 29
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 10 / 29
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 10 / 29
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 10 / 29
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 10 / 29
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?
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 11 / 29
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?
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 11 / 29
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?
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 11 / 29
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?
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 11 / 29
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?
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 11 / 29
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?
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 11 / 29
RD, a little more formally
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)
Angrist and Pishke, Chapter 6, pp. 251-267
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 12 / 29
RD, a little more formally
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)
Angrist and Pishke, Chapter 6, pp. 251-267
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 12 / 29
RD, a little more formally
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)
Angrist and Pishke, Chapter 6, pp. 251-267
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 12 / 29
RD, a little more formally
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)
Angrist and Pishke, Chapter 6, pp. 251-267
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 12 / 29
RD, a little more formally
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)
Angrist and Pishke, Chapter 6, pp. 251-267
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 12 / 29
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)
Angrist and Pishke, Chapter 6, pp. 251-267
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 13 / 29
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)
Angrist and Pishke, Chapter 6, pp. 251-267
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 13 / 29
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)
Angrist and Pishke, Chapter 6, pp. 251-267
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 13 / 29
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)
Angrist and Pishke, Chapter 6, pp. 251-267
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 13 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 14 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 14 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 14 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 14 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 14 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 15 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 15 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 15 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 15 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 15 / 29
RD, a little more formally
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 errors
Angrist and Pishke, Chapter 6, pp. 251-267
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 15 / 29
RD, a little more formally
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 15 / 29
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 16 / 29
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 16 / 29
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 16 / 29
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 16 / 29
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 16 / 29
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
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 16 / 29
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.
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 17 / 29
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.
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 17 / 29
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
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 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 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 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 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 score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Poverty index score
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 18 / 29
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
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 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 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 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 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 score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Poverty index score
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 19 / 29
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
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 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 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 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 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 score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Poverty index score
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 20 / 29
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
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 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 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 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 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 score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Poverty index score
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 21 / 29
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
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 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 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 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 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 score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Poverty index score
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 22 / 29
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
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 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 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 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 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 score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Poverty index score
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 23 / 29
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
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 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 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 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 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 score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Poverty index score
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 24 / 29
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
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 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 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 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 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 score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Poverty index score
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 25 / 29
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
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 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 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 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 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 score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Poverty index score
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 26 / 29
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
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 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 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 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 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 score0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
Poverty index score
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 27 / 29
An example
Owen Ozier (The World Bank) Regression Discontinuity 6 May 2015 28 / 29