When Can We Ignore Measurement Error in the Running Variable?

When Can We Ignore Measurement Error in the Running Variable?

Introduction

The key characteristic of \ac{RD} designs is that assignment of units to treatment is determined by whether the value of a particular covariate $X^{*}$, called the running variable, exceeds a fixed threshold $c$. Under weak continuity conditions, comparing units on either side of the threshold identifies the \ac{ATE} for units with $X^{*}=c$. However, in many cases, researchers only observe a noisy version $X$ of the true running variable $X^*$. In our survey of papers published in leading economics journals that featured \ac{RD} designs, 23% of them used a running variable measured with error.\footnote{See (ref) for details.} Most commonly, the noise arose due to rounding or grouping, such as when researchers observe age in years, income reported in income brackets, or job tenure in months, while the administrator assigning treatment uses the exact birthdate, income, or job tenure.

The prevalence of this problem has given rise to a growing literature on measurement error in \ac{RD} designs (see, among others, HuKl10,dong15,BaLiWa16,PeSh17,DaLeBa17,BaBrDi20; or DiBaBr20). As this literature points out, ignoring the measurement error can lead to inconsistent estimates of the usual \ac{RD} estimand, the \ac{ATE} for units with $X^{*}=c$.\footnote{As we discuss in more detail in (ref), an exception is bbrw09, who, in the context of a fuzzy \ac{RD} design, consider measurement error with a point mass at zero, and give conditions under which ignoring the measurement error nonetheless yields a consistent estimate of a fuzzy \ac{RD} analog of this estimand.} This motivated the development of a variety of alternative estimation and inference procedures to recover this \ac{ATE}; the solutions depend on the particular auxiliary assumptions about the form of the measurement error, or the availability of auxiliary datasets.

This paper makes the simple point that under easily interpretable conditions, existing \ac{RD} techniques provide inference for a slightly different estimand, the \ac{ATE} for units with $X=c$. That is, one could interpret the \ac{RD} analysis as if $X$ were the true running variable. For instance, suppose $X^{*}$ is birthdate, but we only observe the year of birth $X$. While standard \ac{RD} analysis does not yield valid inference for the \ac{RD} estimand associated with $X^*$, the \ac{ATE} for individuals born on the cutoff date, it does provide valid inference for the \ac{ATE} for individuals born in the cutoff year.\footnote{Equivalently, this estimand is a weighted average of \acp{ATE} conditional on $X^{*}$, with weights given by the density of the measurement error conditional on $X=c$. In the example, it corresponds to a weighted average of \acp{ATE} for individuals born on each day of the cutoff year, weighted by the relative birthdate frequencies.}

This result relies on two key conditions: (i) using $X$ as a running variable correctly classifies the treatment assignment, and (ii) the conditional means of the potential outcomes are smooth in $X$. The first condition holds automatically for certain types of rounding error; more generally it requires removing observations at or in the immediate vicinity of the threshold, resulting in a “doughnut design”. Our approach is thus most relevant when the support of the measurement error is relatively narrow (which includes most settings with rounding or grouping error); otherwise the doughnut trimming may end up removing too many observations and preclude informative inference.

For the second condition, we give a formal result showing it holds under weaker conditions than those needed for inference on the \ac{ATE} conditional on $X^*=c$, if we were to observe $X^*$: intuitively, the measurement error smooths out kinks or other irregularities in the conditional mean of the outcome given $X^*$. Inference can be conducted using bias-aware inference methods ArKo18optimal,ArKo20,NoRo21, which automatically adapt to the potentially irregular support of $X$. In particular, we can ignore the measurement error in the sense that estimation and inference on the trimmed data can proceed as if $X$ were the running variable used by the administrator to assign treatment.

An appealing feature of focusing on the \ac{ATE} for units with $X=c$ is that valid inference relies only on assumptions about the smoothness of the conditional mean of the potential outcomes given $ X$. Such assumptions are easy to interpret, and are (partially) testable. Furthermore, inference is standard in that we can directly apply existing methods. In contrast, for inference on the average treatment effect for units with $X^{*}=c$, one needs to either make specific assumptions about latent objects, such as the distribution of the measurement error given $X$ or $X^{*}$, or have access to auxiliary data; furthermore, the form of the estimator depends on the exact form of these specific assumptions.

The rest of the paper proceeds as follows. (ref) gives the setup and the main results, and discusses their applicability under different types of measurement error. (ref) illustrates the results in an empirical application. (ref) concludes. Auxiliary results appear in the appendix.

Setup and main results

We are interested in the effect of a treatment $T$ on an outcome $Y$. Let $Y(t)$ denote the potential outcomes, $t\in\{0,1\}$. The observed outcome is given by $Y=Y(0)+T(Y(1)-Y(0))$. An administrator assigns individuals to treatment if their running variable $X^{*}$ crosses a threshold $c$, which we normalize to $c=0$. Let $Z=\1{X^{*}\geq 0}$ denote the indicator for treatment assignment. In a sharp \ac{RD} design, all individuals comply with the treatment assignment, so that $T=Z$. Let $g^{*}_{t}(x):=E[Y(t)\mid X^{*}=x]$ denote the conditional mean of the potential outcomes given $X^{*}$, and let $g^{*}(x):=E[Y\mid X^{*}=x]=\1{x\geq 0}g^{*}_{1}(x)+\1{x <0}g^{*}_{0}(x)$ denote the conditional mean of the observed outcome.

If the conditional means $g^{*}_{t}$ are continuous at $0$, the jump in $g^{*}$ at $0$ identifies an average treatment effect for units at the threshold htv01,

If $X^{*}$ were observed, and we strengthen the continuity assumption by placing nonparametric smoothness assumptions on $g^{*}$, several methods for estimation and inference on $\tau^{*}$ are available ImKa12,CaCaTi14,ArKo18optimal,ArKo20,ImWa19. Under parametric restrictions on $g^{*}$, we can leverage standard parametric regression methods for estimation and inference.

We do not observe $X^{*}$ directly, however; instead we observe $X=X^{*}-e$, where $e$ is measurement error.\footnote{Our setting is distinct from that in eckles2020, where the running variable $X$ observed by the researcher is also the variable used by the administrator, and it can be thought of as a noisy measure of some latent variable $X^{*}$ that affects potential outcomes.} As our leading example, we focus on the case where $e$ represents rounding or grouping error. By rounding, we mean that the observed running variable can be written as $X=r(X^{*})$, where the rounding function $r$ is a monotone step function that is idempotent (rounding a number twice is the same as rounding it once, $r(r(X^{*}))=r(X^{*})$), and the steps are equal-sized. For instance, if we observe a rounded-down version of $X^{*}$, then the rounding function corresponds to the floor function, $r(X^{*})=\lfloor X^{*}\rfloor$, and $e=X^{*}-\lfloor X^{*}\rfloor$. Rounding is a special case of grouping. Grouping allows $r(\cdot)$ to be any idempotent step function, not necessarily with equal-sized steps; it arises when the sample space for $X^{*}$ is partitioned into subsets, and $X=r(X^*)$ is a numeric value representing the subset HeRu91. For instance, under interval reporting, $X$ may correspond to the midpoint or one of the endpoints of the interval that $X^{*}$ belongs to. In spatial \ac{RD}, it is common to observe the centroid of the unit's ZIP code or county instead of its exact location.

Our framework, outlined in (ref) for sharp \ac{RD}, and extended to fuzzy \ac{RD} settings in (ref), only imposes high-level conditions on $e$ that allow the measurement error to take many other forms besides grouping; $X$ may be discrete, continuous, or mixed. (ref) discusses sufficient low-level conditions in particular settings. In addition to grouping, we consider classical measurement error ($e$ is independent of $Y(1), Y(0)$ and $X^{*}$), Berkson measurement error ($e$ is independent of $Y(1), Y(0)$ and $X$), as well as heaping. While under grouping, the grouping function $r(X)$ is the same for each unit (so that the conditional distribution of $X$ given $X^{*}$ is degenerate), under heaping one may round the data according one of several rounding functions $r_{u}(\cdot)$, $u=1,2,\dotsc, K$, with different ranges; we observe $X=r_{U}(X^*)$, where $U$ is a random variable, possibly correlated with potential outcomes or $X^*$, determining which rounding function is used HeRu91. For instance, some parents may report ages of their children rounded to the nearest year or nearest half-year, while others may not round at all and report the exact date of birth HeRu90. The degree of rounding may depend on characteristics of the parents or the age of the child (ages of very young children and children of parents with more education are usually more likely to be precisely reported). The density of the running variable then displays “heaps” at ages that are multiples of $0.5$.

Estimation and inference with measurement error

Our approach is based on the observation that any variable $X$ can serve as a running variable, provided that it correctly classifies the treatment assignment, and provided that the conditional mean functions of the potential outcomes given $X$ are continuous at the cutoff:

We give a detailed discussion of these conditions in our setting, where $X$ is a mismeasured version of $X^{*}$, in (ref) below. Condition (ref) corresponds to the standard \ac{RD} continuity condition from htv01, except it is applied to $X$ rather than $X^{*}$.

To interpret the estimand we consider, let $g^*_{t}(X^*, e):=E[Y(t)\mid X^*, e]$ denote the conditional mean of the potential outcome $Y(t)$, $t=0,1$, given both the true value of the running variable and the measurement error, and let $\tau^*(X^{*}, e)=g^*_{1}(X^{*}, e)-g^*_{0}(X^{*}, e)=E[Y(1)-Y(0) \mid X^{*}, e]$ denote the conditional \ac{ATE}, conditional on both $X^*$ and $e$. We denote the conditional \ac{ATE}, conditional on $X^*$ only, by $\tau^*(x)=g^*_{1}(x)-g^*_{0}(x)=E[Y(1)-Y(0) \mid X^{*}=x]$, so that $\tau^*=\tau^*(0)$. While, as we discuss further below, our approach does allow the measurement error to affect potential outcomes, to clearly link the estimand we consider to the usual \ac{RD} estimand, it is useful to rule this possibility out and assume that the measurement error doesn't affect the conditional \ac{ATE}, at least when $X=0$ (note $E[Y(1)-Y(0) \mid X^{*}=x, X=0]=\tau^{*}(x, x)$):

This condition is slightly weaker than the requirement that the measurement error be non-differential, i.e., independent of $(Y(1), Y(0))$ given $X^{*}$ CaRuSt06; for instance, rounding, grouping, and classical measurement errors are all non-differential.

By (ref), we can effectively “ignore” the measurement error in the observed running variable $X$, in that we can conduct the analysis as if $X$ were the running variable used by the administrator to assign treatment, provided that we align the target of inference accordingly, setting it to $\tau$. Under condition (ref), we can interpret $\tau$ as a weighted average of \acp{ATE} conditional on $X^{*}=x$, $\tau^*(x)$; in contrast $\tau^*$ simply corresponds to the conditional \ac{ATE} at $0$, $\tau^*(0)$. If condition (ref) doesn't hold, then both estimands may be expressed as weighted averages of conditional \acp{ATE}, conditional on both $X^*$ and $e$, but the weighting functions are different, which makes them a little harder to compare. Specifically, by iterated expectations, $\tau =\int \tau^{*}(e, e) dF_{e\mid X}(e\mid 0)$, while $\tau^* =\int \tau^{*}(0, e)dF_{e\mid X^*}(e\mid 0)$, where $F_{e\mid X^*}$ is the conditional distribution of $e$ given $X^*$.

For estimation and inference, we need to strengthen condition (ref) by assuming that $g$ satisfies appropriate parametric or nonparametric smoothness conditions. As a simple parametric approach, one may assume that $ g(x)$ takes the form of a polynomial of degree $q$ on either side of the threshold for values of $x$ within distance $h$ of the threshold. Then one could estimate $\tau$ by a local polynomial regression of $Y$ onto $m_{q}(X)=(\1{X\geq 0}, \1{X\geq 0}X, \dotsc, \1{X\geq 0}X^{q}, 1, \dotsc, X^{q})$ (a polynomial in $X$ interacted with treatment assignment), using \ac{OLS}.\footnote{While this covers a global approach by setting $h=\infty$, as discussed in GeIm19, such an approach may perform poorly relative to local approaches.} Specifically, given a sample $\{Y_{i}, X_{i}\}_{i=1}^{n}$, the estimator is defined as

Under i.i.d.\ sampling, inference can be conducted using \ac{EHW} standard errors, provided that $X$ has at least $q+1$ support points on either side of the threshold.

A limitation of the parametric approach is that if $g(x)$ is not exactly polynomial inside the estimation window, the estimator will be biased; consequently, \acp{CI} based on \ac{EHW} standard errors will undercover $\tau$. To address this issue, as the preferred inference method, we propose to use the bias-aware (or “honest”) inference approach developed in ArKo18optimal,ArKo20 and KoRo18. This approach enlarges the \acp{CI} by taking into account the potential finite sample bias of the estimator. In particular, letting $\hat{\sigma}(\hat{\tau}_{h, q}^{Y})$ denote the standard error, a \ac{CI} with level $1-\alpha$ takes the form

Here $\operatorname{cv}_{\alpha}(t)$ is the $1-\alpha$ quantile of a folded normal distribution $\abs{N(t,1)}$, and $B(\hat{\tau}_{h, q}^{Y})$ is a bound on the finite-sample conditional (on $X$) bias of the estimator. As a baseline assumption to bound the potential bias, we replace the parametric assumption that $g$ is polynomial with the weaker non-parametric assumption that $g\in \mathcal{F}_{RD}(M)$, where

Here $\norm{f}_{C^{1}}=\sup_{x\neq x^{\prime}}\abs{f(x)-f(x')}/\abs{x-x'}$ is the Lipschitz constant of $f$ (if $f$ is differentiable, then the constant is the maximum of its derivative, $\norm{f}_{C^{1}}=\sup_{x}\abs{f'(x)}$). The parameter space for $g$ thus corresponds to (the closure of) a family of functions that are twice differentiable on either side of the cutoff, with the second derivative bounded in absolute value by $M$, but are potentially discontinuous at $0$. Under this assumption, it is optimal to run a local linear regression, i.e.\ use the estimator $\hat{\tau}_{h, 1}^{Y}$. The (conditional on $X$) bias of the estimator is maximized at the function $h(x)=Mx^{2}(\1{x< 0}-\1{x\geq 0})/2$, so that $B(\hat{\tau}_{h, 1}^{Y})$ is given by (ref), with $Y_{i}$ replaced by $h(X_{i})$. See ArKo20 and KoRo18 for details. An appealing feature of the bias-aware \ac{CI} is that because it accounts for the exact finite-sample bias of the estimator, it is valid under any bandwidth sequence, including using a fixed bandwidth; for example, the bandwidth $h$ may be selected to minimize the (worst-case over $\mathcal{F}_{RD}(M)$) mean squared error, or the length of the resulting confidence interval.\footnote{We implement this method in the empirical application in (ref), where we also discuss the choice of $M$, the key tuning parameter. See KoRo18, ImWa19, and ArKo20 for a more detailed discussion, including a discussion of implementation issues.}

Conditions for validity of proposed approach

Unlike existing approaches that seek to do inference on $\tau^{*}$ even in presence of measurement error (e.g.\ HuKl10,dong15,DaLeBa17; or PeSh17), we do not impose specific assumptions on the measurement error distribution or require auxiliary data. Instead, our approach is based on the observation that we can use existing parametric or nonparametric methods for inference on $\tau $ provided that condition (ref) holds, and we strengthen condition (ref) by assuming that the conditional mean functions $g$ is smooth (in the sense that it is exactly polynomial inside the estimation window, or else $g\in \mathcal{F}_{RD}(M)$). These assumptions are high-level in that they are exactly the conditions needed to interpret the \ac{RD} design with the observed running variable as a valid \ac{RD} design. In the following remarks, we discuss in detail sufficient conditions for these assumptions in specific measurement errors settings. We also discuss related practical issues.

While conditions (a) and (b) in (ref) are relatively mild, they necessitate subpopulation analysis under heaping. For example, consider using birthweight as a running variable to identify the effects of hospital care on infant health as in adkw10. Suppose that some (but not all) hospitals report rounded rather than precise birthweight---then condition (a) fails since the distribution $F_{e\mid X}(e\mid x)$ changes discontinuously as at round values of $x$. Furthermore, suppose that hospitals with fewer resources are more likely report rounded rather than precise birthweight, as argued in BaLiWa16. Then condition (b) also fails, since $g^{*}_{t}(X^{*}, e)$ is then potentially discontinuous at $e=0$ due to different hospital composition under rounded vs exact reporting. Suppose, however, that the conditional mean functions $g_t$ are smooth in the subpopulation of hospitals that don't round birthweight. Conditions (a) and (b) then hold if we drop the heaping points, only keeping observations with non-round values of $X$, as suggested by BaLiWa16. The estimand $\tau$ corresponds to a conditional \ac{ATE} for units with $X=X^{*}=0$ born in a hospital that reports exact birthweight.

In some instances of heaping, all values of $X$ are rounded, but the rounding precision differs across units. Say some individuals round age to the nearest month, while others round to the nearest year or half-year. To ensure conditions (a) and (b) hold in such cases, we drop individuals with age in months that is a multiple of $6$. The remaining sample then only contains those who report age in months. We can interpret the estimand as the \ac{ATE} for those born in the cutoff month in the subpopulation of individuals who report the running variable with the greatest precision.

Since the conditional mean function $g(x)$ is identified over the support of $X$, smoothness assumptions such as $g\in \mathcal{F}_{RD}(M)$ are testable. Problems such as heaping are often apparent from simple plots of undersmoothed binned averages of the outcome against $X$ bglw11, and one can also conduct more formal specification tests KoRo18.

Fuzzy designs

In fuzzy \ac{RD} designs, only a subset of the individuals complies with the treatment assignment, so that $T\neq Z$. In this case, htv01 show that the fuzzy \ac{RD} parameter can be interpreted as a local average treatment effect for individuals who comply with the treatment assignment. Let us reconsider their argument when we use a variable $X$ as the running variable, not necessarily equal to the running variable $X^{*}$ used by the administrator.\footnote{The original argument in htv01 involved defining potential treatments under counterfactual values of the running variable. However, the running variable may not be manipulable (e.g.\ when $X^{*}$ corresponds to a birthdate). We therefore use a slightly different argument, based on manipulation of the treatment assignment. The treatment assignment is typically manipulable, say by moving the cutoff.}

Let $T(1)$ denote the potential treatment status of the individual if they are assigned to treatment, and let $T(0)$ denote their status if they are not assigned to treatment. The observed treatment is given by $T=T(Z)$, and the observed outcome is given by $Y=Y(T(Z))=Y(0)+T(Z)(Y(1)-Y(0))$. Let $\mathfrak{C}$ denote the event that an individual is a complier, that is $T(1)>T(0)$. Finally, in analogy to the conditional means $g$ and $g_{t}$, let $p(x)=E[T\mid X=x]$ and $p_{z}(x)=E[T(z)\mid X=x]$ for $z\in \{0,1\}$.

We replace the sharp \ac{RD} condition that all individuals comply with the treatment assignment ($T=Z$) with the weaker condition that a non-zero fraction of individuals complies with it, and that nobody defies the treatment assignment (in analogy with the monotonicity condition in ImAn94):

Next, we replace the continuity assumption (ref) with a continuity assumption on the first stage and reduced form regression functions:\footnote{Analogous to an instrumental variables regression that uses $Z$ as an instrument, these are (non-parametric) regressions of $T$ and $Y$, respectively, onto $Z$ and $X$.}

Intuitively, if treatment eligibility $Z$ did not change at the cutoff but was instead fixed, this condition implies that the observed outcome $Y=Y(T(Z))$ would be continuous at $0$. As a result, any discontinuity must be due to change in treatment eligibility, which allows for identification of causal effects. Conditions (ref) and (ref) are analogous to the standard fuzzy \ac{RD} assumptions, but applied to $X$ rather than $X^{*}$.

Finally, to link the estimand we consider to the usual \ac{RD} estimand, analogous to condition (ref), it is useful to assume that the measurement error has no effect on the compliance probability or the \ac{ATE} for compliers once we control for $X^{*}$.

This is a slightly weaker requirement that the measurement error $e$ be non-differential, i.e.\ independent of $(Y(1), Y(0), T(1), T(0))$ given $X^{*}$. In analogy to condition (ref) in the sharp case, condition (ref) is helpful for interpreting the estimand, but it is not necessary for validity of our approach.

With this setup, we obtain a fuzzy \ac{RD} analog of (ref).

Under perfect compliance, $T=Z$, (ref) reduces to (ref). In analogy to the sharp case, any variable satisfying conditions (ref), (ref), and (ref) can be used as a running variable. With $X=X^{*}$, we obtain the standard result that

where $p^{\ast}(x)=E[T\mid X^{*}=x]$.

Unless the local average treatment effects $\tau_{F}^{\ast}(x)$ are constant on the support of $e$, $\tau_{F} \neq \tau_{F}^{\ast}$. Since $\tau_{F}^{\ast}(x)$ is given by the ratio of the reduced form effect $E[Y(T(1))-Y(T(0))\mid X^{*}=x]$ to the first stage effect $E[T(1)-T(0)\mid X^{*}=x]$, whether $\tau_{F} ^{\ast}(x)$ is locally constant depends on heterogeneity in both the reduced form and the first stage conditional mean functions. Our empirical results in (ref), for example, are consistent with the reduced form effect being approximately constant, while the first stage effect is approximately linear, i.e., $E[Y(T(1))-Y(T(0))\mid X^{*}=x]\approx a$ and $E[T(1)-T(0)\mid X^{*}=x]\approx b+cx$; further, the measurement error is approximately uniform on $[0,1] $. So $\tau_{F}\approx a/(b+c/2)$, while $\tau_{F}^{\ast}\approx a/b$.

If the measurement error is differential, and condition (ref) doesn't hold, then both $\tau^{*}_{F}$ and $\tau_{F}$ may be expressed as weighted averages of conditional \acp{ATE} for compliers, conditional on both $X^*$ and $e$, $\tau_{F} ^{\ast}(X^{*}, e) :=E[Y(1)-Y(0)\mid \mathfrak{C}, X^{*}, e]$. Specifically, by iterated expectations, $\tau_{F}= \frac{\int \tau_{F} ^{\ast}(e, e) p_{\mathfrak{C}}(e, e) dF_{e\mid X}(e\mid 0)}{\int p_{\mathfrak{C}}(e, e) dF_{e\mid X}(e\mid 0)}$, while $\tau_{F}^{*}=\frac{\int \tau_{F}^{*}(0, e)p_{\mathfrak{C}}(0,e) dF_{e\mid X^{*}}(e\mid 0)}{\int p_{\mathfrak{C}}(0,e) dF_{e\mid X^{*}}(e\mid 0)}$, where $p_{\mathfrak{C}}(X^{*}, e):=P(\mathfrak{C}\mid X^{*}, e)$. (ref) is related to the result in bbrw09 who show that if we replace condition (ref) with the assumption that the measurement error is non-differential, and has a point mass at zero but is otherwise smooth, $\tau_{F}=\tau_{F}^{*}$. If the measurement error is differential, but affects the potential outcomes and potential treatments smoothly, the arguments in bbrw09 imply that $\tau_{F}=\tau_{F}^{*}(0,0)$; this was shown in clpw15 in the context of fuzzy regression kink designs.

Similarly to the sharp case, if we assume that the conditional mean functions $g(x)$ and $p(x)$ are polynomial inside a window $h$ of the threshold, then we can estimate $\tau_{F}$ as a ratio of local polynomial estimators

with $\hat{\tau}_{h, q}^{Y}$ defined in (ref), and $\hat{\tau}_{h, q}^{T}$ defined analogously.\footnote{Equivalently, as noted in htv01, the estimator can be computed as a two-stage least squares estimator in a regression of $Y$ onto $T$ using $\1{X\geq 0}$ and instrument, and the remaining elements of $m_{q}(X)$ as exogenous covariates, using observations inside the estimation window.} If there are at least $q+1$ support points for $X$ on either side of the threshold and inside the estimation window, then under i.i.d.\ sampling, standard errors for $\hat{\tau}_{h, q}$ can be constructed based on the \ac{EHW} covariance matrix for $(\hat{\tau}_{h, q}^{Y}, \hat{\tau}_{h, q}^{T})$ using the delta method.

Our preferred approach weakens the polynomial assumptions on $g(x)$ and $ p(x) $ by instead assuming that $g\in\mathcal{F}_{RD}(M_{y})$, and $p\in \mathcal{F}_{RD}(M_{t})$. While this assumption only delivers set identification if the support of $X$ is irregular (see (ref)), we can use the bias-aware inference approach for constructing \acp{CI} that are asymptotically valid whether $\tau_{F}$ is point identified, set identified, or unidentified.\footnote{The parameter $\tau_{F}$ is unidentified if the instrument $\1{X\geq 0}$ is irrelevant in the sense that $P(T(1)>T(0)\mid X=0)=0$. Since the expression for $\tau_{F}$ in (ref) not well-defined in this case, one can define $\tau_{F}$ in an arbitrary way.} In particular, following NoRo21, we can test the hypothesis $H_{0}\colon\tau_{F}=\tau_{F, 0}$ by checking whether $0$ is in the bias-aware confidence interval based on $\hat{\tau}_{h,1}^{Y-\tau_{F, 0}T}$, and noting that the smoothness assumptions on $g$ and $p$ imply $ E[Y-\tau_{F, 0}T\mid X=x]\in\mathcal{F}_{RD}(M_{y}+\abs{\tau_{F, 0}}M_{t})$. The confidence set for $\tau_{F}$ is constructed by collecting all values of $\tau_{F, 0}$ that are not rejected, similar to the construction of AnRu49 confidence set in standard linear instrumental variables model.\footnote{We implement this method in our empirical application in (ref), where we also discuss the choice of the smoothness constants $M_{t}$ and $M_{y}$. See NoRo21 for a detailed discussion of implementation issues.}

Empirical Application

In this section, we use data from HoHi16 to estimate the impact of preregistration on youth turnout in an election. HoHi16 leverage the fact that in Florida, individuals who were ineligible to vote in the 2008 election (those born after November 4, 1990) were nonetheless eligible to preregister to be added to the voter rolls for the next election. Those born before November 4, 1990 were already eligible to register regularly and vote in 2008. This motivates a fuzzy \ac{RD} design, where the treatment $T$ is an indicator for preregistering, the outcome $Y$ is an indicator for voting in the 2012 election, and the running variable $X^{*}$ is the proximity to the eligibility cutoff in days.

To illustrate the effects of measurement error in the running variable, we compare this design to a fuzzy \ac{RD} design in which we (pretend to) only observe individuals' month of birth, and hence use proximity to November 1990 in months, $X$, as a running variable. We discard individuals born in November 1990, since their eligibility cannot be determined by month of birth alone (see (ref)). We show that, consistent with the discussion in (ref), (i) using proximity in days vs months yields different estimates, reflecting the impact of the rounding error on the estimand, and (ii) using month of birth generally leads to wider \acp{CI}.

We first visualize both versions of the \ac{RD} design. In each case, the sample size is 186,575, consisting of individuals born within 6 months of the eligibility cutoff. (ref) presents the first stage, plotting preregistration rate against proximity in days (panel (a)) or in months (panel (b)). For ineligible individuals, the preregistration rate is essentially 0, while for eligible individuals, the preregistration rate is downward slopping: those born further away from the cutoff preregister with lower probability. There is a clear jump in the registration rate at the eligibility threshold in either panel. (ref) shows the reduced form, plotting the proportion who voted in the 2012 election against proximity to eligibility. In both panels, there is a small jump in the voting probability at the cutoff.

We use five specifications to compute the fuzzy \ac{RD} estimator in (ref), the sharp \ac{RD} estimators of the first stage and reduced form effects, and the associated confidence intervals. For ease of comparison across specifications, all specifications use a uniform kernel and local linear regression ($q=1$). The first specification follows HoHi16, and uses bandwidth set to $h=60$ days (or $h=2$ months), and the confidence intervals to not account for the potential bias of the estimator. The second specification differs only in that it uses a slightly larger bandwidth, $h=90$ days (or $h=3$ months).\footnote{These specifications can be interpreted as imposing a parametric linear functional form inside the estimation window. Alternatively, one can justify them by an “undersmoothing” argument: the specifications implicitly assume that the constants $M_{y}$ and $M_{t}$ are small enough so that the bias is negligible at these bandwidth choices.} The third specification uses the \ac{RBC} method of CaCaTi14. For proximity in days, we use the default “MSE optimal” bandwidth provided by their software package; for proximity in months we use $h=3$.\footnote{The formal arguments justifying the \ac{RBC} method and the default bandwidth selector require the running variable to be continuous, which is not the case in either design. When proximity is measured in months, the discreteness causes implementation issues with the default “MSE optimal” bandwidth calculations.}

The last two methods implement the bias-aware approach. We use confidence intervals given in (ref) for the first stage and reduced form effects; for inference on the fuzzy \ac{RD} estimand, we use the NoRo21 construction. Implementing these methods requires a choice of smoothness bounds for the first stage ($M_{t}$) and the reduced form ($M_{y}$). The results of low97 and ArKo18optimal imply that picking $M_t$ and $M_y$ in a data-driven way without violating coverage requires further non-convex restrictions on the parameter spaces $\mathcal{F}_{RD}(M_{y})$ and $\mathcal{F}_{RD}(M_{t})$ for $g$ and $p$.\footnote{The problem of choosing the smoothness constants is essentially a non-parametric model selection problem. Echoing the difficulties with conducting valid post-model selection inference in parametric contexts LePo05, ArKo18optimal show that bias-aware confidence intervals that assume the worst-case smoothness are in fact highly efficient at smooth functions. Thus, there is little scope for improvement by using data-driven choices of the smoothness constants.} A natural way of doing this is to relate the global smoothness of $g$ and $p$ to the local smoothness constants $M_y$ and $M_t$. We consider two ways of formalizing how the global and local smoothness relate. In particular, the fourth specification assumes, following the proposal in ArKo20, that $M_y$ is bounded by the smoothness of a global quartic approximation to $g$ on either side of the cutoff, as measured by the largest (in absolute value) second derivative of the fitted line; we impose an analogous assumption on $M_t$ and $p$. The fifth specification follows the suggestion in ImWa19 to use a global quadratic regression instead, and, additionally, multiply the largest second derivative of the fitted line by some moderate factor, taken here to be $2$.

Since there are many other reasonable ways of formalizing the idea that the local and global smoothness are related, we view these methods as merely \acp{ROT} for selecting the smoothness constants. To assess these rules, we use the visualization approach proposed in NoRo21, described and implemented in (ref). These visualizations suggest that the ArKo20 \ac{ROT} is quite conservative, and allows for $g$ and $p$ to be quite non-smooth. The second \ac{ROT} delivers more optimistic smoothness bounds that generate reasonably smooth conditional mean functions. To make the smoothness constants comparable across the specifications, we report the implied smoothness constants after rescaling the running variable to have support $[-1,1]$ (which amounts to multiplying the original smoothness constants by $\max_{i}X_{i}^{2}$ and $\max_{i}(X_{i}^{*})^{2}$, respectively). Given a choice of the smoothness constants, the bandwidth is selected so that the point estimate defined in (ref) minimizes the worst-case (over the chosen smoothness class) finite-sample MSE of the estimator.

Results

(ref) presents the first stage estimates, the effect of the preregistration eligibility on preregistration. The estimates are stable across the specifications, in the range of 38--40% when using proximity in days; the estimates using proximity in months are slightly lower, in the range 36--37%, but still indicating a clear jump in the preregistration rate at the eligibility threshold. This is consistent with our theory, discussed in (ref), and reflects the difference between the parameters $E[T(1)-T(0)\mid X^{*}=0]$ (the effect for those born on the cutoff date, November 4) and $E[T(1)-T(0)\mid X=0]$ (the effect for those born in November, the cutoff month). In particular, the latter estimand averages over individuals born further away from the cutoff date, and (ref) suggests that the preregistration probability is decreasing with the distance to the cutoff. Since $X$ is discrete, the parameter $E[T(1)-T(0)\mid X=0]$ is not point identified. The confidence intervals for the bias-aware specifications, which account for this, are correspondingly wider than those in panel A, albeit they still remain quite tight.

(ref) presents the reduced form estimates, the effect of preregistration eligibility on voting. The point estimates are about 3% for both designs and stable across specifications. In line with the discussion in (ref), the \acp{CI} are produced by the bias-aware specifications are wider when using proximity in months, reflecting the loss of point identification.

(ref) presents the fuzzy \ac{RD} estimates of the effect of preregistration on voting. When eligibility is measured in months, the smaller first stage estimates in panel B of (ref) translate to larger estimates of the effect of preregistration on voting, around 10%, compared to 7--8% when eligibility is measured in days. When eligibility is measured in months, the fuzzy \ac{RD} estimand, $\tau_{F}$, is the \ac{ATE} for compliers born in November 1990, and thus averages over individuals born further away from the cutoff than the estimand $\tau_{F}^{*}$ when eligibility is measured in days, which corresponds to the \ac{ATE} for compliers born on November 4, 1990. If the treatment effect for compliers born $x$ days from the eligibility threshold, $\tau_{F} ^{\ast}(x)=E[Y(1)-Y(0)\mid \mathfrak{C}, X^{*}=x]$, is increasing in $x$, then $\tau_{F}$ will be larger than $\tau_{F}^{*}$, which is consistent with the results in (ref). However, the bias-aware confidence intervals are fairly wide, and also consistent with $\tau_{F} ^{\ast}(x)$ being constant.

Summary and conclusions

Measurement error is a common feature of \ac{RD} applications. We show that its presence does not have deleterious effects on the validity of existing inference methods, provided that one employs doughnut trimming to ensure that the observed running variable $X$ correctly classifies the treatment assignment. Care needs to be taken when interpreting the estimand: it corresponds to the \ac{ATE} for units with the observed running variable $X$ equal to the cutoff, rather than the usual parameter, the \ac{ATE} for units with the latent running variable $X^{*}$ equal to the cutoff. We illustrate this point in an empirical application.

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