Polynomial Approximation and Fuzzy RD Application

80 Questions

What are the two main reasons why estimating the relationship between average achievement and class size using OLS may lead to biased results?

Parents from higher socioeconomic backgrounds may put their children in schools with smaller classes, and 2) principals may put weaker students in smaller classes.

In the context of the Maimonides rule, what does the variable 'nsc' play the role of in a fuzzy RD design?

Running variable (Di)

What is the purpose of controlling for the percentage of disadvantaged students in class in the fuzzy RD design?

To account for potential confounding variables

What is the name of the Stata command used to estimate RDD?

No specific command (uses built-in commands)

What is the benefit of using a fuzzy RD design over a sharp RD design?

Allowing for a more flexible relationship between the running variable and the treatment

What is the name of the method used to estimate the causal effect of class size on average achievement in the context of the Maimonides rule?

Local 2SLS

What is the purpose of polynomial approximation in the context of RDD?

To flexibly model the relationship between the running variable and the outcome

What is the role of the variable 'es' in the fuzzy RD design?

Forced variable (Xi)

What is the name of the authors who have made significant contributions to the development of RDD methods?

Melly, Lalive, Cattaneo, and co-authors

What is the benefit of using instrumental variables in the context of RDD?

Allowing for the identification of causal effects in the presence of endogeneity

What is the main assumption required to estimate LATE in a fuzzy RD design, and how does it relate to the standard IV framework?

The same assumptions as in the standard IV framework are required, and we need to exclude local defiers to allow for heterogeneous treatment effects.

In the context of fuzzy RD, what is the doubly local population referred to?

Compliers at x0

What are the two estimation methods used in the sharp RD design, and how do they differ?

The two estimation methods are local linear regression and global polynomial regression. Local linear regression estimates the treatment effect at the threshold, while global polynomial regression estimates the treatment effect over a range of values.

What is the formula for estimating the treatment effect using local linear regression in fuzzy RD?

The estimated treatment effect is (m̂Y+ - m̂Y-) / (m̂D+ - m̂D-)

What is the main difference between local linear regression and local 2SLS in fuzzy RD?

Local 2SLS is numerically equivalent to the 2SLS estimator using only observations close to x0, while local linear regression estimates the treatment effect at the threshold.

What is the purpose of using a kernel function in local linear regression and local 2SLS?

The kernel function is used to weigh the observations closer to the threshold more heavily, to reduce bias and increase precision.

What is the main advantage of using local 2SLS in fuzzy RD, compared to other estimation methods?

Local 2SLS allows for standard standard errors to be used, which can simplify the estimation and inference process.

What is the relationship between the compliers in fuzzy RD and the treatment effect?

The compliers are those whose treatment status changes as we move the value of X from just the left of x0 to just to the right of x0, and the treatment effect is the average effect of the compliers.

What is the main difference between fuzzy RD and sharp RD designs?

In fuzzy RD, the treatment effect can vary at the threshold, while in sharp RD, the treatment effect is assumed to be constant at the threshold.

What is the role of polynomial approximation in fuzzy RD, and how does it relate to local linear regression?

Polynomial approximation can be used to approximate the treatment effect at the threshold, and local linear regression is a special case of polynomial approximation with a linear term.

In a fuzzy regression discontinuity design, what is the relationship between the participation decision and the covariate X?

The participation decision is not completely determined by X, even in a rule-based selection process.

What is the purpose of the exclusion restriction in fuzzy RD?

To ensure continuity of E[Yd|X] at x0.

How does the local binary instrument lead to the local 'Wald' estimator in fuzzy RD?

The discontinuity in Pr(D=1|X) at x0 acts as an instrumental variable for treatment status locally at x0.

What is the role of the first stage relevance condition in fuzzy RD?

To ensure there is a discontinuity in Pr(D=1|X) at x0.

How does the fuzzy RD design generalize the sharp RD design?

Fuzzy RD is a more general case, where Di is no longer deterministically related to crossing a threshold, but there is a jump in the probability of treatment at x0.

What is the assumption required for the local 2SLS estimator to be valid in fuzzy RD?

Homogeneous treatment effect at x0.

How does polynomial approximation relate to fuzzy RD?

Polynomial approximation can be used to model the conditional expectations E[Y0|X] and E[Y1|X] in fuzzy RD.

What is the consequence of a weak instrument in fuzzy RD?

A weak instrument leads to small jumps in the probability of treatment at x0, making it difficult to identify the treatment effect.

How does fuzzy RD address the problem of endogeneity in causal inference?

Fuzzy RD uses the discontinuity in the probability of treatment as an instrumental variable to identify the treatment effect.

What is the intuition behind the local 'Wald' estimator in fuzzy RD?

The local 'Wald' estimator exploits the discontinuity in the probability of treatment to identify the treatment effect at x0.

What is the purpose of using a p-order polynomial in the first-stage relationship of a polynomial approximation?

To capture the non-linear relationship between the treatment variable and the outcome variable.

In the context of the Angrist and Lavy (1999) study, what is the causal variable of interest?

Class size

What is the key difference between the traditional RD design and the fuzzy RD design used in Angrist and Lavy (1999)?

The fuzzy RD design exploits discontinuities in average class size, rather than probabilities of a single treatment.

In the Local 2SLS estimator, what is the role of the polynomial in Xi interacted with Ti as control variables?

To control for the non-linear relationship between the treatment variable and the outcome variable.

What is the formula for predicted class size from a strict application of Maimonides' rule in the Angrist and Lavy (1999) study?

msc = int(es/40) + 1

What is the purpose of using multiple discontinuities in the fuzzy RD design?

To increase the precision and robustness of the estimates.

In the context of instrumental variables, what is the role of the first-stage equation?

To identify the causal effect of the treatment variable on the outcome variable.

What is the main advantage of using a polynomial approximation in the context of instrumental variables?

It allows for more flexibility in modeling the relationship between the treatment variable and the outcome variable.

In the context of causal inference, what is the main assumption underlying the 2SLS estimator?

The instrumental variable is exogenous and only affects the outcome variable through its effect on the treatment variable.

What is the main advantage of using the fuzzy RD design in the context of causal inference?

It allows for more robust estimation of the causal effect, particularly when the treatment variable is continuous.

What is the main difference between fuzzy RD and sharp RD designs?

Fuzzy RD design allows for heterogeneous treatment effects, while sharp RD design does not

What is the purpose of using polynomial approximation in the context of fuzzy RD?

To approximate the unknown functional form of the outcome variable

What is the relationship between the compliers in fuzzy RD and the treatment effect?

The compliers are those whose treatment status changes as we move the value of X from just the left of x0 to just to the right of x0

What is the formula for estimating the treatment effect using local linear regression in fuzzy RD?

$\frac{m̂Y+ - m̂Y-}{m̂D+ - m̂D-}$

What is the purpose of using a kernel function in local linear regression and local 2SLS?

To weigh the observations according to their distance from the discontinuity

What is the main advantage of using local 2SLS in fuzzy RD, compared to other estimation methods?

It allows for heterogeneous treatment effects

What is the assumption required for the local 2SLS estimator to be valid in fuzzy RD?

The instrument must be strong

What is the role of polynomial approximation in the context of instrumental variables?

To approximate the unknown functional form of the outcome variable

What is the main difference between global polynomial regression and local linear regression?

Global polynomial regression is a parametric approach, while local linear regression is a non-parametric approach

What is the consequence of a weak instrument in fuzzy RD?

The treatment effect estimate will be biased towards zero

What is the primary purpose of using polynomial approximation in fuzzy RD design?

To flexibly model the relationship between the running variable and the outcome variable

What is the main difference between global polynomial regression and local linear regression in fuzzy RD design?

Global polynomial regression models the entire relationship between the running variable and the outcome variable, while local linear regression models the relationship at the cutoff

What is the role of the p-order polynomial in the first-stage relationship of a polynomial approximation?

To model the relationship between the instrumental variable and the treatment variable

What is the main advantage of using 2SLS estimator in fuzzy RD design?

It is more robust to endogeneity than OLS

What is the purpose of using multiple discontinuities in the fuzzy RD design?

To leverage variation in the treatment effect across different discontinuities

What is the key difference between the traditional RD design and the fuzzy RD design used in Angrist and Lavy (1999) study?

The traditional RD design uses a sharp discontinuity, while the fuzzy RD design uses a fuzzy discontinuity

What is the role of the polynomial in Xi interacted with Ti as control variables in the Local 2SLS estimator?

To flexibly model the relationship between the covariate and the outcome variable

What is the main assumption underlying the 2SLS estimator in fuzzy RD design?

The instrumental variable is a valid instrument for the treatment variable

What is the benefit of using a fuzzy RD design over a sharp RD design?

It can handle continuous treatment variables

What is the purpose of using instrumental variables in fuzzy RD design?

To identify the causal effect of the treatment variable on the outcome variable

What is the main difference between fuzzy RD and sharp RD designs?

Fuzzy RD design exploits discontinuities in the probability of treatment, while sharp RD design does not.

What is the purpose of polynomial approximation in the context of fuzzy RD design?

To account for non-linear relationships between the covariate X and the outcome variable Y.

What is the role of the local binary instrument in fuzzy RD design?

To identify the treatment effect at the threshold x0.

What is the relationship between the local linear regression and local 2SLS estimators in fuzzy RD design?

Local linear regression is a special case of local 2SLS.

What is the consequence of a weak instrument in fuzzy RD design?

The treatment effect estimate is biased towards zero.

What is the main advantage of using fuzzy RD design over sharp RD design?

Fuzzy RD design is more flexible in terms of the functional form of the relationship between X and Y.

What is the role of the first stage relevance condition in fuzzy RD design?

To ensure that the instrument is correlated with the treatment status.

What is the intuition behind the local 'Wald' estimator in fuzzy RD design?

It is based on the idea that the instrument is valid only locally at the threshold x0.

What is the relationship between the compliers in fuzzy RD design and the treatment effect?

The compliers are those who are affected by the treatment, and their response determines the treatment effect.

What is the role of polynomial approximation in the context of instrumental variables?

To model the relationship between the instrument and the treatment status.

What is the main advantage of using a fuzzy RD design over a sharp RD design in the context of estimating the causal effect of class size on average achievement?

It provides a more precise estimate of the treatment effect at the cut-off

What is the purpose of polynomial approximation in the context of fuzzy RD design?

To approximate the relationship between the running variable and the outcome variable

What is the main difference between global polynomial regression and local linear regression in the context of fuzzy RD design?

Global polynomial regression imposes a global functional form, whereas local linear regression imposes a local functional form

What is the role of the 2SLS estimator in the context of fuzzy RD design?

To estimate the local average treatment effect

What is the main assumption underlying the 2SLS estimator in the context of fuzzy RD design?

The instrumental variable is exogenous

What is the benefit of using a polynomial approximation in the context of instrumental variables?

It reduces the bias of the instrumental variable

What is the main difference between local linear regression and local 2SLS in the context of fuzzy RD design?

Local linear regression estimates the treatment effect at the cut-off, whereas local 2SLS estimates the local average treatment effect

What is the role of the kernel function in local linear regression and local 2SLS in the context of fuzzy RD design?

It weights the observations according to their distance from the cut-off

What is the consequence of a weak instrument in the context of fuzzy RD design?

The estimator is biased towards zero

What is the main advantage of using local 2SLS in fuzzy RD design over other estimation methods?

It provides a more robust estimate of the treatment effect

Study Notes

Polynomial Approximation

The first-stage relationship in a polynomial approximation is: Di = γ0 + πTi + ∑ γp Xij + ∑ γxp Ti Xij + ui
The second-stage relationship is: Yi = µ + ρDi + ∑ βp Xij + ∑ βxp Ti Xij + vi

Fuzzy RD Application

Angrist and Lavy (1999) used a fuzzy RD design to analyze the effect of class size on test scores
They exploited discontinuities in average class size instead of probabilities of a single treatment
They used multiple discontinuities, including an old Talmudic rule that classes should be split if they have more than 40 students in Israel

LATE

The Local Average Treatment Effect (LATE) is estimated in RD, assuming the same assumptions as in the standard IV framework
LATE is the average treatment effect of the compliers, who are those whose treatment status changes as we move the value of X from just the left of x0 to just to the right of x0

Local Linear Regression

Two estimation methods can be used in a sharp RD design: local linear regression and global polynomial regression
Local linear regression estimates the treatment effects using four different regression lines

Local 2SLS

Local 2SLS is a method that uses a uniform kernel and the same bandwidth for all four estimators
The estimator is numerically equivalent to the 2SLS estimator using only observations close to x0 with Ti ≡ 1 (Xi ≥ x0) as an instrument for Di

Maimonides Rule and Actual Class Size

Maimonides rule is used to predict class size from a strict application of the rule
Predicted class size is: msc = int(es/40) + 1

Econometric Specification

The relationship between average achievement and class size is estimated using: Yisc = α0 + ρnsc + ηisc
OLS may lead to biased results due to correlation between class size and the error term
Two main reasons for this correlation are:
- Parents from higher socioeconomic backgrounds may put their children in schools with smaller classes
- Principals may put weaker students in smaller classes

Fuzzy RD Design

Angrist and Lavy used the Maimonides rule in a fuzzy RD design
The variables relate to the previous description as follows: nsc plays the role of Di, es plays the role of Xi, and msc plays the role of Ti

2SLS Results

The 2SLS results are used to estimate the relationship between average achievement and class size

RDD in Stata

RDD can be estimated in Stata using standard built-in commands
Additional commands are available, including rd and cmogram

Instrumental Variable Framework

In the instrumental variable (IV) framework, one needs the same assumptions as in the standard IV framework to allow for heterogeneous treatment effects at the discontinuity.
Local defiers need to be excluded to estimate LATE (Local Average Treatment Effect): the average treatment effect of the compliers.

Regression Discontinuity (RD) Design

In RD, the compliers are those whose treatment status changes as we move the value of X from just to the left of x0 to just to the right of x0.
The doubly local population consists of compliers at x0.

Estimation Methods

Two estimation methods can be used:
- Local linear regression
- Global polynomial regression
Local linear regression can be used to estimate the 4 elements in the LATE formula.

Local Linear Regression

The estimated treatment effects is calculated as: m̂Y+ - m̂Y- / (m̂D+ - m̂D-)
In the simplest case, a uniform (rectangular) kernel can be used with the same bandwidth for m̂Y+, m̂Y-, m̂D+, and m̂D-.
The estimator is numerically equivalent to the 2SLS estimator using only observations close to x0 with Ti ≡ 1 (Xi ≥ x0) as an instrument for Di while controlling for Xi and Ti · Xi.

Polynomial Approximation

A p-order polynomial can be used to write the first-stage relationship: Di = γ0 + πTi + ∑ γp Xij + ∑ γxp Ti Xij + ui
The second stage is: Yi = µ + ρDi + ∑ βp Xij + ∑ βxp Ti Xij + vi
The 2SLS estimator uses polynomial in Xi interacted with Ti as control variables.

Application of Fuzzy RD on Class Sizes

Angrist and Lavy (1999) use a fuzzy RD design to analyze the effect of class size on test scores.
They extend RD in two ways:
- The causal variable of interest (class size) takes on many values.
- The first stage exploits discontinuities in average class size instead of probabilities of a single treatment.
They use multiple discontinuities and exploit an old Talmudic rule that classes should be split if they have more than 40 students in Israel.

Maimonides Rule and Actual Class Size

The predicted class size from a strict application of Maimonides rule is: msc = int(es / 40) + 1
Actual class size may differ from predicted class size due to various factors.

Econometric Specification

The relationship between average achievement and class size is estimated as: Yisc = α0 + ρnsc + ηisc
Estimating this relationship with OLS may lead to biased results because class size is likely to be correlated with the error term.
Two main reasons for this correlation are:
- Parents from higher socioeconomic backgrounds may put their children in schools with smaller classes.
- Principals may put weaker students in smaller classes.

Fuzzy RD Design

Angrist & Lavy use the Maimonides rule in a fuzzy RD design.
The variables relate to the previous description as follows:
- nsc plays the role of Di
- es plays the role of Xi
- msc plays the role of Ti
They also control for the percentage of disadvantage students in class.

2SLS Results

2SLS results from the fuzzy RD design are presented.

RDD in Stata

RDD can be estimated in Stata with standard built-in commands.
Additional commands are available, including ssc install rd and ssc install cmogram.
Users can also read more about recent developments in the RDD literature.

This quiz covers polynomial approximation and its application in fuzzy regression discontinuity design, including its use in analyzing the effect of class size on test scores.

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