Polynomial Approximation and Fuzzy RD Application

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

Running variable (Di)

To account for potential confounding variables

<p>No specific command (uses built-in commands)</p> Signup and view all the answers

<p>Allowing for a more flexible relationship between the running variable and the treatment</p> Signup and view all the answers

<p>Local 2SLS</p> Signup and view all the answers

<p>To flexibly model the relationship between the running variable and the outcome</p> Signup and view all the answers

<p>Forced variable (Xi)</p> Signup and view all the answers

<p>Melly, Lalive, Cattaneo, and co-authors</p> Signup and view all the answers

<p>Allowing for the identification of causal effects in the presence of endogeneity</p> Signup and view all the answers

<p>The same assumptions as in the standard IV framework are required, and we need to exclude local defiers to allow for heterogeneous treatment effects.</p> Signup and view all the answers

<p>Compliers at x0</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>The estimated treatment effect is (m̂Y+ - m̂Y-) / (m̂D+ - m̂D-)</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>The kernel function is used to weigh the observations closer to the threshold more heavily, to reduce bias and increase precision.</p> Signup and view all the answers

<p>Local 2SLS allows for standard standard errors to be used, which can simplify the estimation and inference process.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>The participation decision is not completely determined by X, even in a rule-based selection process.</p> Signup and view all the answers

<p>To ensure continuity of E[Yd|X] at x0.</p> Signup and view all the answers

<p>The discontinuity in Pr(D=1|X) at x0 acts as an instrumental variable for treatment status locally at x0.</p> Signup and view all the answers

<p>To ensure there is a discontinuity in Pr(D=1|X) at x0.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>Homogeneous treatment effect at x0.</p> Signup and view all the answers

<p>Polynomial approximation can be used to model the conditional expectations E[Y0|X] and E[Y1|X] in fuzzy RD.</p> Signup and view all the answers

<p>A weak instrument leads to small jumps in the probability of treatment at x0, making it difficult to identify the treatment effect.</p> Signup and view all the answers

<p>Fuzzy RD uses the discontinuity in the probability of treatment as an instrumental variable to identify the treatment effect.</p> Signup and view all the answers

<p>The local 'Wald' estimator exploits the discontinuity in the probability of treatment to identify the treatment effect at x0.</p> Signup and view all the answers

<p>To capture the non-linear relationship between the treatment variable and the outcome variable.</p> Signup and view all the answers

<p>Class size</p> Signup and view all the answers

<p>The fuzzy RD design exploits discontinuities in average class size, rather than probabilities of a single treatment.</p> Signup and view all the answers

<p>To control for the non-linear relationship between the treatment variable and the outcome variable.</p> Signup and view all the answers

<p>msc = int(es/40) + 1</p> Signup and view all the answers

<p>To increase the precision and robustness of the estimates.</p> Signup and view all the answers

<p>To identify the causal effect of the treatment variable on the outcome variable.</p> Signup and view all the answers

<p>It allows for more flexibility in modeling the relationship between the treatment variable and the outcome variable.</p> Signup and view all the answers

<p>The instrumental variable is exogenous and only affects the outcome variable through its effect on the treatment variable.</p> Signup and view all the answers

<p>It allows for more robust estimation of the causal effect, particularly when the treatment variable is continuous.</p> Signup and view all the answers

<p>Fuzzy RD design allows for heterogeneous treatment effects, while sharp RD design does not</p> Signup and view all the answers

<p>To approximate the unknown functional form of the outcome variable</p> Signup and view all the answers

<p>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</p> Signup and view all the answers

<p>$\frac{m̂Y+ - m̂Y-}{m̂D+ - m̂D-}$</p> Signup and view all the answers

<p>To weigh the observations according to their distance from the discontinuity</p> Signup and view all the answers

<p>It allows for heterogeneous treatment effects</p> Signup and view all the answers

<p>The instrument must be strong</p> Signup and view all the answers

<p>To approximate the unknown functional form of the outcome variable</p> Signup and view all the answers

<p>Global polynomial regression is a parametric approach, while local linear regression is a non-parametric approach</p> Signup and view all the answers

<p>The treatment effect estimate will be biased towards zero</p> Signup and view all the answers

<p>To flexibly model the relationship between the running variable and the outcome variable</p> Signup and view all the answers

<p>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</p> Signup and view all the answers

<p>To model the relationship between the instrumental variable and the treatment variable</p> Signup and view all the answers

<p>It is more robust to endogeneity than OLS</p> Signup and view all the answers

<p>To leverage variation in the treatment effect across different discontinuities</p> Signup and view all the answers

<p>The traditional RD design uses a sharp discontinuity, while the fuzzy RD design uses a fuzzy discontinuity</p> Signup and view all the answers

<p>To flexibly model the relationship between the covariate and the outcome variable</p> Signup and view all the answers

<p>The instrumental variable is a valid instrument for the treatment variable</p> Signup and view all the answers

<p>It can handle continuous treatment variables</p> Signup and view all the answers

<p>To identify the causal effect of the treatment variable on the outcome variable</p> Signup and view all the answers

<p>Fuzzy RD design exploits discontinuities in the probability of treatment, while sharp RD design does not.</p> Signup and view all the answers

<p>To account for non-linear relationships between the covariate X and the outcome variable Y.</p> Signup and view all the answers

<p>To identify the treatment effect at the threshold x0.</p> Signup and view all the answers

<p>Local linear regression is a special case of local 2SLS.</p> Signup and view all the answers

<p>The treatment effect estimate is biased towards zero.</p> Signup and view all the answers

<p>Fuzzy RD design is more flexible in terms of the functional form of the relationship between X and Y.</p> Signup and view all the answers

<p>To ensure that the instrument is correlated with the treatment status.</p> Signup and view all the answers

<p>It is based on the idea that the instrument is valid only locally at the threshold x0.</p> Signup and view all the answers

<p>The compliers are those who are affected by the treatment, and their response determines the treatment effect.</p> Signup and view all the answers

<p>To model the relationship between the instrument and the treatment status.</p> Signup and view all the answers

<p>It provides a more precise estimate of the treatment effect at the cut-off</p> Signup and view all the answers

<p>To approximate the relationship between the running variable and the outcome variable</p> Signup and view all the answers

<p>Global polynomial regression imposes a global functional form, whereas local linear regression imposes a local functional form</p> Signup and view all the answers

<p>To estimate the local average treatment effect</p> Signup and view all the answers

<p>The instrumental variable is exogenous</p> Signup and view all the answers

<p>It reduces the bias of the instrumental variable</p> Signup and view all the answers

<p>Local linear regression estimates the treatment effect at the cut-off, whereas local 2SLS estimates the local average treatment effect</p> Signup and view all the answers

<p>It weights the observations according to their distance from the cut-off</p> Signup and view all the answers

<p>The estimator is biased towards zero</p> Signup and view all the answers

<p>It provides a more robust estimate of the treatment effect</p> Signup and view all the answers