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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?
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What is the benefit of using a fuzzy RD design over a sharp RD design?
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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?
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What is the purpose of polynomial approximation in the context of RDD?
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What is the role of the variable 'es' in the fuzzy RD design?
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What is the name of the authors who have made significant contributions to the development of RDD methods?
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What is the benefit of using instrumental variables in the context of RDD?
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What is the main assumption required to estimate LATE in a fuzzy RD design, and how does it relate to the standard IV framework?
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In the context of fuzzy RD, what is the doubly local population referred to?
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What are the two estimation methods used in the sharp RD design, and how do they differ?
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What is the formula for estimating the treatment effect using local linear regression in fuzzy RD?
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What is the main difference between local linear regression and local 2SLS in fuzzy RD?
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What is the purpose of using a kernel function in local linear regression and local 2SLS?
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What is the main advantage of using local 2SLS in fuzzy RD, compared to other estimation methods?
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What is the relationship between the compliers in fuzzy RD and the treatment effect?
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What is the main difference between fuzzy RD and sharp RD designs?
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What is the role of polynomial approximation in fuzzy RD, and how does it relate to local linear regression?
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In a fuzzy regression discontinuity design, what is the relationship between the participation decision and the covariate X?
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What is the purpose of the exclusion restriction in fuzzy RD?
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How does the local binary instrument lead to the local 'Wald' estimator in fuzzy RD?
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What is the role of the first stage relevance condition in fuzzy RD?
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How does the fuzzy RD design generalize the sharp RD design?
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What is the assumption required for the local 2SLS estimator to be valid in fuzzy RD?
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How does polynomial approximation relate to fuzzy RD?
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What is the consequence of a weak instrument in fuzzy RD?
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How does fuzzy RD address the problem of endogeneity in causal inference?
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What is the intuition behind the local 'Wald' estimator in fuzzy RD?
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What is the purpose of using a p-order polynomial in the first-stage relationship of a polynomial approximation?
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In the context of the Angrist and Lavy (1999) study, what is the causal variable of interest?
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What is the key difference between the traditional RD design and the fuzzy RD design used in Angrist and Lavy (1999)?
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In the Local 2SLS estimator, what is the role of the polynomial in Xi interacted with Ti as control variables?
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What is the formula for predicted class size from a strict application of Maimonides' rule in the Angrist and Lavy (1999) study?
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What is the purpose of using multiple discontinuities in the fuzzy RD design?
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In the context of instrumental variables, what is the role of the first-stage equation?
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What is the main advantage of using a polynomial approximation in the context of instrumental variables?
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In the context of causal inference, what is the main assumption underlying the 2SLS estimator?
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What is the main advantage of using the fuzzy RD design in the context of causal inference?
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What is the main difference between fuzzy RD and sharp RD designs?
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What is the purpose of using polynomial approximation in the context of fuzzy RD?
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What is the relationship between the compliers in fuzzy RD and the treatment effect?
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What is the formula for estimating the treatment effect using local linear regression in fuzzy RD?
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What is the purpose of using a kernel function in local linear regression and local 2SLS?
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What is the main advantage of using local 2SLS in fuzzy RD, compared to other estimation methods?
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What is the assumption required for the local 2SLS estimator to be valid in fuzzy RD?
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What is the role of polynomial approximation in the context of instrumental variables?
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What is the main difference between global polynomial regression and local linear regression?
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What is the consequence of a weak instrument in fuzzy RD?
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What is the primary purpose of using polynomial approximation in fuzzy RD design?
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What is the main difference between global polynomial regression and local linear regression in fuzzy RD design?
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What is the role of the p-order polynomial in the first-stage relationship of a polynomial approximation?
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What is the main advantage of using 2SLS estimator in fuzzy RD design?
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What is the purpose of using multiple discontinuities in the fuzzy RD design?
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What is the key difference between the traditional RD design and the fuzzy RD design used in Angrist and Lavy (1999) study?
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What is the role of the polynomial in Xi interacted with Ti as control variables in the Local 2SLS estimator?
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What is the main assumption underlying the 2SLS estimator in fuzzy RD design?
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What is the benefit of using a fuzzy RD design over a sharp RD design?
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What is the purpose of using instrumental variables in fuzzy RD design?
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What is the main difference between fuzzy RD and sharp RD designs?
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What is the purpose of polynomial approximation in the context of fuzzy RD design?
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What is the role of the local binary instrument in fuzzy RD design?
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What is the relationship between the local linear regression and local 2SLS estimators in fuzzy RD design?
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What is the consequence of a weak instrument in fuzzy RD design?
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What is the main advantage of using fuzzy RD design over sharp RD design?
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What is the role of the first stage relevance condition in fuzzy RD design?
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What is the intuition behind the local 'Wald' estimator in fuzzy RD design?
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What is the relationship between the compliers in fuzzy RD design and the treatment effect?
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What is the role of polynomial approximation in the context of instrumental variables?
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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?
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What is the purpose of polynomial approximation in the context of fuzzy RD design?
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What is the main difference between global polynomial regression and local linear regression in the context of fuzzy RD design?
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What is the role of the 2SLS estimator in the context of fuzzy RD design?
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What is the main assumption underlying the 2SLS estimator in the context of fuzzy RD design?
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What is the benefit of using a polynomial approximation in the context of instrumental variables?
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What is the main difference between local linear regression and local 2SLS in the context of fuzzy RD design?
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What is the role of the kernel function in local linear regression and local 2SLS in the context of fuzzy RD design?
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What is the consequence of a weak instrument in the context of fuzzy RD design?
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What is the main advantage of using local 2SLS in fuzzy RD design over other estimation methods?
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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 ofDi
,es
plays the role ofXi
, andmsc
plays the role ofTi
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
andcmogram
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.
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Description
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.