Lecture 7

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Questions and Answers

In the context of Fuzzy Regression Discontinuity (FRD) designs, what is the most critical distinction that differentiates it from Sharp Regression Discontinuity (SRD) designs?

FRD allows for a discontinuous jump in the _probability_ of receiving treatment at the threshold, whereas SRD mandates a deterministic switch from no treatment to treatment. (correct)
FRD explicitly models the heterogeneous treatment effects across different subgroups, whereas SRD assumes homogeneous treatment effects.
FRD uses non-parametric methods exclusively, while SRD relies on parametric methods for identification and estimation.
FRD requires the treatment assignment to be perfectly correlated with the assignment variable, while SRD only requires a partial correlation.

Within the framework of Fuzzy Regression Discontinuity (FRD), what fundamental assumption must hold true to ensure the validity of causal inference when employing the threshold as an instrumental variable for treatment status?

The threshold variable must exert a direct influence on the outcome variable, independent of its impact on the treatment.
The threshold variable must be randomly assigned across all observational units to eliminate selection bias.
The threshold variable must only affect the outcome variable through its effect on the treatment status. (correct)
The threshold variable must exhibit a monotonic relationship with the outcome variable across the entire range of the assignment variable.

How does the 'reduced form' version of Fuzzy Regression Discontinuity (FRD) mirror the methodology used in instrumental variables (IV) regressions?

The 'reduced form' FRD estimates the first-stage relationship between the instrument and the treatment variable using a generalized method of moments (GMM) estimator.
In the 'reduced form' FRD, the outcome variable is regressed directly on an interaction term between the assignment variable and treatment status, akin to a two-stage least squares regression in IV.
In the 'reduced form' FRD, the outcome variable is regressed directly on the instrument (the variable indicating whether the threshold is reached), analogous to regressing the outcome directly on the instrument in IV. (correct)
The 'reduced form' FRD models the causal pathway from the instrument to the endogenous variable through a series of structural equations, while IV regressions rely on a single reduced-form equation

In the context of Almond et al.'s (2010) study on the marginal efficiency of health care utilizing a Fuzzy Regression Discontinuity design, which critical assumption allows them to estimate the causal impact of medical expenditures on health outcomes?

They assume that the discontinuity in medical expenditures is as good as randomly assigned, conditional on the assignment variable and other covariates. (A) Signup and view all the answers

Within a Fuzzy Regression Discontinuity framework, considering a scenario where the instrument strength (i.e., the impact of the forcing variable on the probability of treatment) is exceptionally weak, what is the most likely consequence for the estimated treatment effect?

The estimated treatment effect will be heavily influenced by even minor violations of the exclusion restriction, leading to substantial bias. (A) Signup and view all the answers

Suppose a researcher implements a parametric Fuzzy Regression Discontinuity (FRD) design utilizing polynomial functions to approximate the conditional expectation functions. What is the most critical consideration regarding the order (degree) of the polynomial chosen for this approximation?

The optimal order of the polynomial should be determined via cross-validation techniques to balance the trade-off between bias reduction near the threshold and variance inflation in the tails of the distribution. (C) Signup and view all the answers

In a scenario where a researcher aims to implement a non-parametric Fuzzy Regression Discontinuity (FRD) design, employing a Wald estimator, but faces a situation with considerable data sparsity near the threshold, what is the most appropriate methodological adjustment to enhance the robustness and reliability of the estimates?

Increase the bandwidth employed in the kernel smoothing process to encompass a larger neighborhood around the threshold, thereby mitigating the impact of data sparsity. (A) Signup and view all the answers

In the context of fuzzy Regression Discontinuity (RD) designs, the Wald estimator is utilized to estimate the causal effect. Given the formula below, what underlying assumption most critically ensures the validity of interpreting $ρ$ as a Local Average Treatment Effect (LATE)?

$ρ = \frac{\lim_{\delta \to 0} E[y_i | x_0 < x_i < x_0 + \delta] - E[y_i | x_0 - \delta < x_i < x_0]}{\lim_{\delta \to 0} E[D_i | x_0 < x_i < x_0 + \delta] - E[D_i | x_0 - \delta < x_i < x_0]}$

There is a non-zero proportion of 'compliers' near the threshold $x_0$, meaning some individuals' treatment status is influenced by crossing the threshold, and there are no 'defiers'. (A) Signup and view all the answers

When conducting a graphical analysis in a Regression Discontinuity (RD) design, you plot the outcome variable against the forcing variable using binned averages. If the bin sizes are excessively large, what specific threat to the validity of the RD design interpretation is most likely exaggerated, potentially leading to a spurious conclusion?

Artificially attenuating the discontinuity at the cutoff, thus underestimating the true treatment effect by including data points far from the threshold. (A) Signup and view all the answers

In a fuzzy RD design, observing a statistically significant jump in the probability of treatment at the cutoff ($x_0$) is crucial. However, the sole presence of this jump does not guarantee the validity of the RD design. What additional rigorous assessment is most critical to ensure that the observed jump genuinely reflects a causal effect?

Verifying the absence of any statistically significant jump in baseline covariates at the cutoff, thereby supporting the assumption that the forcing variable is as-good-as-random at the threshold. (C) Signup and view all the answers

Consider a scenario where parents strategically enroll their children in specific schools, anticipating smaller class sizes in a particular grade based on predicted enrollment numbers. What key threat to the validity of an RD design is most directly presented by this selective manipulation of the forcing variable (enrollment), and what methodological approach is most appropriate to rigorously address it?

Endogenous manipulation of the forcing variable; conduct a McCrary density test at the threshold to detect discontinuities in the density of the forcing variable. (D) Signup and view all the answers

Suppose you observe an unexpected discontinuity in the relationship between the forcing variable and the outcome variable away from the intended cutoff point ($x_0$) in a Regression Disccontinuity design. Which of the following explanations presents the most challenging threat to the validity of the RD design, and what specific diagnostic test would be most informative in evaluating this threat?

Spurious correlation due to unobserved confounders; conduct a falsification test by examining the effect of the treatment on predetermined covariates. (B) Signup and view all the answers

Within the context of Regression Discontinuity (RD) designs applied to healthcare interventions, what critical assumption must hold true regarding the observed threshold to ensure the validity of causal inferences?

The threshold's cut-off point must have a discontinuous impact on the probability of treatment assignment, while remaining uncorrelated with unobserved determinants of the outcome variable at the threshold. (C) Signup and view all the answers

In a scenario where treatment assignment is determined by a birth weight threshold of 1500 grams for at-risk newborns, and healthcare providers are known to deviate from this guideline based on their clinical judgment, what econometric challenge arises when attempting to estimate treatment effects using a sharp RD design?

The introduction of selection bias due to non-compliance with the birth weight threshold, rendering the assignment of treatment non-random around the cutoff, thus violating the core assumption of RD. (A) Signup and view all the answers

Consider a researcher aiming to implement a Regression Discontinuity (RD) design to assess the impact of Very Low Birth Weight (VLBW) classification on infant mortality, using the indicator function $VLBW_i$ and birth weight $BW$. The researcher estimates the model $y_i = \alpha_0 + \alpha_1 VLBW_i + \alpha_2 BW + \epsilon_i$. What critical threat to the validity of the RD design is not addressed by this model specification?

The inclusion of the linear birth weight term failing to account for complex, non-linear associations with infant mortality, potentially violating the local linearity assumption. (A) Signup and view all the answers

In the context of a fuzzy Regression Discontinuity (RD) design examining the impact of a policy intervention, let $P[D_i = 1 | x_i] = f(x_i)$ represent the probability of receiving the intervention ($D_i$) given the assignment variable ($x_i$). If the function $f(x_i)$ exhibits imperfect compliance around the threshold, meaning that some individuals do not receive the assigned treatment and/or some individuals receive treatment when they should not based on their $x_i$ value, what econometric strategy is most appropriate for consistently estimating the local average treatment effect (LATE)?

Instrumental Variables (IV) estimation, where the assignment based on $x_i$ serves as an instrument for actual treatment receipt, allowing for consistent estimation of the LATE. (A) Signup and view all the answers

When employing a Regression Discontinuity (RD) design to evaluate the effectiveness of a healthcare policy, what methodological consideration is paramount in mitigating potential bias arising from manipulation of the assignment variable near the threshold?

Conducting a McCrary density test to formally assess the continuity of the density function of the assignment variable at the threshold to detect potential manipulation. (D) Signup and view all the answers

In the context of Regression Discontinuity designs, what inherent limitation restricts the extrapolation of estimated treatment effects to populations or contexts beyond the immediate vicinity of the threshold?

The inherent focus on the Local Average Treatment Effect (LATE), which specifically quantifies the treatment effect for individuals near the threshold, lacking external validity for broader populations. (C) Signup and view all the answers

Given the model: $P[D_i = 1 | x_i] = f(x_i)$, and assuming that $f(x_i)$ is a well-behaved, continuous function in the neighborhood around the threshold, what condition must hold for a valid Regression Discontinuity design?

$\lim_{x_i \to x_0^-} f(x_i) \neq \lim_{x_i \to x_0^+} f(x_i)$, indicating a significant jump in the probability of treatment ($D_i$) at the threshold ($x_0$). (C) Signup and view all the answers

Assuming a sharp Regression Discontinuity design, how can researchers best address the possibility of functional form misspecification when modeling the relationship between the assignment variable and the outcome?

By exploring non-parametric estimation techniques, such as local linear regression, to allow the data to determine the functional form without imposing strong parametric assumptions. (B) Signup and view all the answers

How should healthcare researchers address the challenge when they only observe treatment guidelines that suggest treatment based on a birth weight threshold, but without direct observation of actual treatment?

Implement a fuzzy Regression Discontinuity (RD) design, recognizing the treatment guideline as an instrument for actual treatment, to address the imperfect compliance. (A) Signup and view all the answers

In the context of Angrist and Lavy's (1999) fuzzy regression discontinuity (RD) design, what is the most critical assumption required for the validity of using class size thresholds as an instrument for actual class size ($C_{isc}$)?

Conditional on enrollment ($e_s$), the class size thresholds ($T_s$) should not directly influence test scores ($y_{isc}$) except through their effect on actual class size ($C_{isc}$), satisfying the exclusion restriction. (B) Signup and view all the answers

Within Angrist and Lavy's instrumental variable (IV) framework, what econometric issue would arise if the 'Maimonides' rule' regarding class size caps were perfectly enforced, leading to a sharp, deterministic relationship between enrollment and predicted class size?

The instrumental variable would become perfectly predictive, collapsing the 2SLS estimator to ordinary least squares (OLS). (A) Signup and view all the answers

In the context of the provided equations from Angrist and Lavy (1999), what is the most significant threat to the validity of using Maimonides’ rule as an instrument for class size, potentially violating the exclusion restriction?

Parents strategically sort their children into schools based on perceived class size benefits, which are correlated with both enrollment and test scores. (A) Signup and view all the answers

Assuming the exclusion restriction holds in Angrist and Lavy's fuzzy RD design, but there exists heterogeneity in the treatment effect of class size on student achievement, what is the most accurate interpretation of the Local Average Treatment Effect (LATE) identified by their 2SLS estimator?

The treatment effect of reduced class size for students induced to be in smaller classes due to crossing the class size thresholds dictated by Maimonides' rule. (D) Signup and view all the answers

In the context of fuzzy Regression Discontinuity (RD) designs used as Instrumental Variables (IV), what fundamental assumption is most critical for valid causal inference, distinguishing it from the sharp RD design?

The exclusion restriction, adjusted for fuzziness, mandating that the instrument affects the outcome only through the intended treatment, with any direct effects meticulously accounted for. (A) Signup and view all the answers

In a fuzzy RD design where financial aid eligibility for university applicants is determined by a numerical score ($x_i$) with a cutoff $c$, how does the discontinuity at $c$ enable causal inference, considering that financial aid receipt is not a deterministic function of $x_i$?

The discontinuity provides an instrumental variable (IV) which identifies a Local Average Treatment Effect (LATE) for those whose treatment status is changed by crossing the threshold $c$. (B) Signup and view all the answers

If Angrist and Lavy (1999) had employed a sharp RD design instead of a fuzzy RD, how would this have altered the interpretation and estimation of the effect of class size on student outcomes, assuming perfect compliance with Maimonides’ rule?

The estimated effect would directly measure the causal impact of class size on student outcomes, without the need for instrumental variables or two-stage least squares. (B) Signup and view all the answers

When implementing a fuzzy RD design using Two-Stage Least Squares (2SLS), what is the most precise interpretation of the first-stage equation in the context of estimating the impact of university financial aid on college enrollment?

The first-stage estimates the effect of crossing the eligibility threshold ($x_i > c$) on the probability of receiving university financial aid. Conditional on $x_i$. (A) Signup and view all the answers

Suppose that in Israeli schools, wealthier communities consistently lobby to ensure their schools receive additional resources that allow for smaller class sizes, irrespective of Maimonides’ rule. How would this endogenous policy response affect the validity of Angrist and Lavy's fuzzy RD design?

It would invalidate the design by violating the exclusion restriction, as community wealth directly affects student outcomes independently of class size. (B) Signup and view all the answers

In the van der Klaauw (2002) study, the assignment of applicants into groups $G_i$ based on discretized numerical scores ($x_i$) introduces a specific methodological challenge. What is the most salient econometric issue arising from this grouping?

It may introduce bias due to the selection of the bandwidth around the cutoff points, potentially distorting the LATE estimates. (B) Signup and view all the answers

In the context of Angrist and Lavy's (1999) study, if unobserved teacher quality is systematically correlated with both class size and student test scores, how might this bias the 2SLS estimates, and what specific strategies could be employed to mitigate this bias?

This would bias the estimates; including school fixed effects or controlling for observable teacher characteristics in both stages could help mitigate the bias. (D) Signup and view all the answers

Assuming that the effect of class size on student achievement varies significantly across different subjects (e.g., math vs. reading), how could Angrist and Lavy refine their model to account for this heterogeneity and obtain more nuanced estimates of the treatment effect?

Run separate 2SLS regressions for each subject, using the same instrumental variable but different outcome variables. (C) Signup and view all the answers

What key modification to the standard 2SLS framework is essential when implementing fuzzy RD in the presence of spatial correlation, such as in Angrist and Lavy's (1999) study of classroom size effects, to ensure the validity of statistical inference?

Using clustered standard errors that account for the spatial dependence structure within schools or districts. (D) Signup and view all the answers

What potential bias could arise in Angrist and Lavy’s estimates if school administrators strategically manipulate enrollment figures around the threshold of 40 students to receive additional resources or optimize teacher assignments, and what econometric technique might be employed to detect such manipulation?

This manipulation could invalidate the RD design by creating a spurious discontinuity; employing a McCrary density test on the enrollment variable would help detect such manipulation. (A) Signup and view all the answers

When compared to sharp RD designs, what represents the foremost econometric challenge introduced by fuzzy RD designs in terms of identification and estimation?

The necessity of estimating a Local Average Treatment Effect (LATE) rather than the Average Treatment Effect (ATE), limiting the generalizability of the findings. (D) Signup and view all the answers

In a fuzzy RD context, suppose the first stage F-statistic in a 2SLS regression is exceptionally low (e.g., less than 4). What is the most critical consequence of this situation for the reliability of the RD estimates?

It indicates weak instrument bias, leading to inconsistent and potentially severely biased estimates of the treatment effect. (C) Signup and view all the answers

Suppose that in later years, Israeli schools implemented a policy that provided additional resources to schools just below the enrollment thresholds (e.g., 39 students) to prevent them from exceeding the 40-student limit. How would this policy change affect the interpretation of the fuzzy RD estimates from Angrist and Lavy's original (1999) study?

The original estimates would now capture the combined effect of class size reduction and the additional resources provided to schools just below the threshold. (B) Signup and view all the answers

How does the interpretation of the LATE differ in fuzzy RD designs compared to standard IV settings, considering the running variable and cutoff point?

In fuzzy RD, the LATE represents the treatment effect for those induced to cross the cutoff, whereas in standard IV it represents the effect for those induced to take the treatment by any instrument. (B) Signup and view all the answers

In the presence of multiple cutoffs, as alluded to in the grouping structure $G_i$ in van der Klaauw (2002), what advanced econometric technique could be employed to more efficiently estimate treatment effects across the various discontinuity points, acknowledging potential heterogeneity?

A Bayesian hierarchical model, allowing for borrowing of strength across cutoffs while accommodating heterogeneity in treatment effects. (B) Signup and view all the answers

Beyond the standard threats to IV validity, what specific concern is most pertinent in fuzzy RD designs when the running variable is constructed from multiple components (e.g., SAT scores, grades, etc.), potentially introducing measurement error?

Weak instrument bias amplified by the measurement error, exacerbated by the fuzzy nature of the treatment assignment. (B) Signup and view all the answers

Flashcards

Fuzzy Regression Discontinuity (FRD)

The probability of receiving treatment changes at the threshold, but not necessarily from zero to one.

Fuzzy RD: Incentive Shift

Incentives to participate in a program change at a threshold, but are not strong enough to move everyone to treatment.