Quantile-Regression Inference With Adaptive Control of Size

Quantile-Regression Inference With Adaptive Control of Size

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{\it Keywords:} Regression quantile, asymptotic variance, standard error, conditional density estimation.

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Introduction

Consider an independent and identically distributed (iid) sample $(\bm{X}_{1},Y_{1}),\ldots,(\bm{X}_{n},Y_{n})$, where each $Y_{i}$ is scalar-valued, and where, for some fixed $d$, each $\bm{X}_{i}$ is a $d$-dimensional regressor. We assume that the conditional distribution of the $i$th response variable $Y_{i}$ given $\bm{X}_{i}$ satisfies

almost surely (a.s.) for some fixed quantile $\alpha\in(0,1)$, where $\bm{\beta}(\alpha)\in\mathbb{R}^{d}$ is unknown and $\bm{X}_{i}^{\top}$ denotes the transpose of $\bm{X}_{i}$. The relation ((ref)) specifies a linear $\alpha$-quantile regression model. Models of conditional quantiles, such as the model given above in ((ref)), have taken on an important role in the statistical sciences. They generally offer researchers the possibility of being able to engage in a systematic analysis of the effects of a set of conditioning variables on all aspects of the conditional distribution of a response variable. A notable characteristic of this approach is the ability it gives researchers to model only the quantiles of interest to a given empirical study without the need to construct an explicit model for the other regions of the response density. For example, a researcher may by varying the quantile index $\alpha$ examine the specific effects of regressors on any point of the conditional distribution of the response variable. Thus the differential effects of some medical intervention ($X$) on survival time ($Y$) can be analyzed separately for low-risk and high-risk individuals by constructing estimates of the conditional quantile function of $Y$ given $\bm{X}$ for various quantiles. The monograph of Koenker05 and the volume edited by KoenkerChernozhukovHePeng17 provide comprehensive reviews of quantile-regression methodology, along with illustrative examples of its application in various disciplines.

There are several proposals available for quantile regression inference. Some of these proposals, such as certain methods involving resampling He17, approaches based on the asymptotic behavior of regression rank scores GutenbrunnerJureckova92, direct methods ZhouPortnoy96,FanLiu16 or more recent Bayesian approaches YangHe12,FengChenHe15,YangWangHe16 differ from Wald-type methods by avoiding the need to estimate conditional density functions for the purpose of asymptotic variance estimation of conditional quantile estimators. Wald-type procedures, however, do generally retain the attractive feature of computational simplicity, and perhaps for this reason remain popular in empirical practice.

In this paper we develop a new estimator of the asymptotic covariance matrix of a given regression quantile. The new estimator is explicitly intended to induce the Wald-type tests or confidence regions in which it is embedded to behave as well in large samples as their empirically infeasible counterparts in which the true, as opposed to estimated, conditional densities appear. The asymptotic variance estimator proposed here induces the empirical size distortions of Wald-type tests to vanish at the same rate enjoyed by the corresponding tests incorporating the actual conditional density functions, i.e., the disparity between the actual and nominal sizes of these tests vanishes at the adaptive rate.

There is of course a long history on estimation of the asymptotic variance of quantile regression parameters and the corresponding Wald-type tests. Among existing procedures, two implementations that are particularly popular are those of Powell91 and HendricksKoenker92. We show that the proposals of Powell91 and HendricksKoenker92 both induce Wald-type tests whose empirical size distortions cannot vanish at the adaptive rates that become possible when these tests incorporate the asymptotic variance estimator that we develop below.

The proposed estimator for the conditional density evaluated at the conditional quantile has applications beyond the formulation of Wald-type tests with adaptive control of size. This estimator can be used for counterfactual wage decompositions in a quantile regression setting MachadoMata05. It has been used for developing improved specification tests for linear quantile regression EscancianoGoh14. Semiparametrically efficient inference in linear quantile regression requires, either explicitly or implicitly, an estimator of the so-called efficient score, which involves the conditional density evaluated at the quantile NeweyPowell90,KomunjerVuong10. Finally, estimates of conditional densities are also needed in semiparametric extensions of the basic linear quantile regression model, e.g., MaHe16 and FengZhu16. Further applications of our estimator such as these are of independent interest.

Finally, we note that this paper is partly motivated by a recent contribution of Portnoy12 to the effect that the first-order asymptotic normal approximation for regression quantiles is associated with an error bound of order $O_{p}\left( n^{-1/2}(\log n)^{3/2}\right) $. This in turn implies, as we show below, the benchmark $O_{p}\left( n^{-1/2}(\log n)^{3/2}\right)$-rate at which size distortions for Wald-type tests regarding quantile-regression parameters converge when the conditional response densities are assumed to be known. An important point to note is that the error bound of nearly $n^{-1/2}$-order elucidated by Portnoy12 is smaller than the error bound of nearly $n^{-1/4}$-order associated with the classic Bahadur representation for regression quantiles. In particular, the larger error of nearly $n^{-1/4}$-order is in fact larger in magnitude than the estimation error associated with any set of reasonable estimates of the conditional response densities, including those proposed by Powell91 and HendricksKoenker92. This would apparently suggest that the rate-adaptive implementation of Wald-type tests proposed in this paper is at best of second-order importance. The smaller error bound shown by Portnoy12 effectively allows one to consider the question of optimally implementing Wald-type tests in this context as a methodological issue of first-order importance.

The remainder of this paper proceeds as follows. The next section develops the asymptotic properties of our proposed kernel estimator of the conditional response density evaluated at the conditional quantile of interest. Section (ref) analyzes the size distortions of tests of linear restrictions of quantile coefficients based on the asymptotic distribution of regression $\alpha$-quantiles. This section also discusses conditions for our Wald-type tests to exhibit size distortions that decay at the adaptive rate in large samples. Section (ref) presents the results of a series of simulation experiments which illustrate the potential of our methods to deliver accurate and powerful tests, and which are motivated from our empirical application, which in turn is discussed in Section (ref). An online supplement includes precise statements of the assumptions underlying our theoretical results, proofs of those results, additional simulation evidence, details on implementation and further discussion of the empirical example.

The New Estimator

Consider the $\alpha$-quantile regression model given above in ((ref)). For each quantile $\alpha\in(0,1)$, the regression $\alpha$-quantile KoenkerBassett78 is defined as \[ \hat{\bm{\beta}}_{n}(\alpha)\equiv\arg\min_{\bm{b}\in\mathbb{R}^{d}} \sum_{i=1}^{n}\rho_{\alpha}\left( Y_{i}-\bm{X}_{i}^{\top}\bm{b} \right) , \] where $\rho_{\alpha}(u)=u\left( \alpha-1\left\{ u\leq0\right\} \right)$.

For each $i=1,\ldots,n$, let $f_{i}(y)$ and $F_{i}(y)$ denote the conditional density and cumulative distribution function (cdf), respectively, of $Y_{i}$ given $\bm{X}_{i}$, evaluated at $y$. If one assumes that for each $i$, $F_{i}(y)$ is absolutely continuous, and that $f_{i}(y)$ is finite and bounded away from zero at $y=\bm{X} _{i}^{\top}\bm{\beta}(\alpha)$, then under Assumption 1 as given in Appendix A of the supplementary material, the regression $\alpha$-quantile is asymptotically normal with

where $\bm{V}(\alpha)=\alpha(1-\alpha)\bm{G}_{0}^{-1}(\alpha )\bm{HG}_{0}^{-1}(\alpha)$ Koenker05, and where

Standard Wald-type inferential procedures based on ((ref)) naturally require the estimation of the matrix $\bm{G}_{0}(\alpha)$, which in turn requires, at least implicitly, the estimation of the conditional density functions $f_{i}\left( \bm{X}_{i}^{\top}\bm{\beta}(\alpha)\right) $ ($i=1,\ldots,n$).

We propose an estimator of the conditional response densities $f_{i}\left( \bm{X}_{i}^{\top}\bm{\beta}(\alpha)\right)$, estimates of which in turn are used to specify a new estimator of the matrix $\bm{G}_{0}(\alpha)$ appearing in the asymptotic variance of the regression $\alpha$-quantile. The new estimator of the conditional densities developed here explicitly exploits the behavior of the fitted conditional $U_{j}$-quantiles $\bm{X}_{i} ^{\top}\bm{\hat{\beta}}_{n}\left( U_{j}\right) $ over a range of quantiles $U_{1},\ldots,U_{m}$ that are iid realizations from a uniform distribution on $\mathcal{A}=[a_{1},a_{2}]$. To motivate the new estimator, note the identity $F_{i}(y)=a_{1}+\int_{a_{1}}^{a_{2}}1\left\{ y-F_{i} ^{-1}(\alpha)\geq 0\right\} d\alpha$ for $a_{1}\leq F_{i}(y)\leq a_{2}.$ This suggests using a smooth approximation of the indicator function, which after differentiation leads one to the quantity $\left( a_{2}-a_{1}\right) \cdot h^{-1}E\left[ \left. K\left( h^{-1}\left( y-F_{i}^{-1}(U)\right) \right) \right\vert \bm{X}_{i}\right] $, where $K(\cdot)$ is a smoothing kernel satisfying the conditions of Assumption 2 in the supplementary material and where $\left. U\right\vert \bm{X}_{i}\sim Unif[a_{1},a_{2}]$, where $a_{1}<\alpha<a_{2}$. This quantity should be a good approximation of $f_{i}(y)$ as $h\rightarrow0,$ where $h>0$ is a scalar smoothing parameter. In order to avoid numerical integration, we approximate the integral by a finite sum with $m$ terms. Note that we certainly could take $m=\infty$, but this would require numerical integration. In what follows, we let both $m$ and the scalar smoothing parameter $h$ depend on the sample size $n$, with $m\rightarrow\infty$ and $h\rightarrow 0$ as $n\rightarrow\infty$.

The discussion above leads to the estimator of $f_{i}\left( \bm{X} _{i}^{\top}\bm{\beta}(\alpha)\right) $ given by

for each $i=1,\ldots,n$. The estimators $\hat{f}_{ni}\left( \bm{X} _{i}^{\top}\bm{\hat{\beta}}_{n}(\alpha)\right) $ given in ((ref)) are in turn embedded in the following estimator of the matrix $\bm{G} _{0}(\alpha)$ as given above in ((ref)):

We are now in a position to state the main result of this section. Define for $\alpha\in\mathcal{A}$

$\sigma_{K}^{2}\equiv\int_{-1/2}^{1/2}w^{2}K(w)dw$ and $\Vert K\Vert_{2} \equiv\sqrt{\int_{-1/2}^{1/2}K^{2}(w)dw}$. In addition, we adopt henceforth the notation $g^{(k)}(\bm{X})$ to denote the $k$th-order derivative of any real-valued measurable function $g(\bm{X})$.

The terms $\bm{T}_{1nm}(\alpha)$, $\bm{T}_{2nm}(\alpha)$ and $\bm{T}_{3nm}(\alpha)$ given in the statement of Theorem (ref) are the leading second-order terms in an asymptotic expansion in probability, for a given $\alpha\in\mathcal{A}$, of $\bm{\hat{G}}_{n}(\alpha)$ about the estimand $\bm{G}_{0}(\alpha)$ . Consider

which defines a natural, but empirically infeasible, kernel estimator of $f_{i}\left( \bm{X}_{i}^{\top}\bm{\beta}(\alpha)\right) $ that essentially relies on $\bm{\beta}(\alpha)$ and $\bm{\beta}\left( U_{j}\right) $, where $j\in\{1,\ldots,m\}$, being known. Then the term $\bm{T}_{1nm}(\alpha)$ appearing in the statement of Theorem (ref) reflects the conditional asymptotic biases given $\bm{X}_{i}$ of the estimators $\tilde{f}_{i}\left( \bm{X}_{i}^{\top }\bm{\beta}(\alpha)\right) $, defined above in ((ref)). The magnitude of the term $\bm{T}_{2nm}(\alpha)$, on the other hand, is driven by the conditional variance given $\bm{X}_{i}$ of $\tilde{f}_{i}\left( \bm{X}_{i}^{\top}\bm{\beta}(\alpha)\right) $ about \[ \left( a_{2}-a_{1}\right) \cdot E\left[ \left. h_{m}^{-1}K\left( h_{m}^{-1}\bm{X}_{i}^{\top}\left( \bm{\beta}(U)-\bm{\beta} (\alpha)\right) \right) \right\vert \bm{X}_{i}\right] . \] Lastly, the term $\bm{T}_{3nm}(\alpha)$ corresponds to the error involved in estimating $\bm{\beta}(\alpha)$ with $\bm{\hat{\beta}}_{n}(\alpha)$.

Wald-Type Tests With Adaptive Control of Size

We consider the empirical sizes of Wald-type tests of hypotheses of the form

where $\bm{R}$ is a fully specified $(J\times d)$ matrix with rank $J$, $\bm{r}\in\mathbb{R}^{J}$ is fully specified and $\alpha$ is a fixed quantile in $\mathcal{A}=\left[ a_{1},a_{2}\right] $ with $0<a_{1}<a_{2}<1$. Define the following:

where for a generic positive definite matrix $\bm{G}$ we define $\bm{W}_{n}(\bm{G})\equiv(\bm{RG}^{-1}\bm{H}_{n} \bm{G}^{-1}\bm{R}^{\top})^{-1}$ and $\bm{W}(\bm{G} )\equiv(\bm{RG}^{-1}\bm{HG}^{-1}\bm{R}^{\top})^{-1}$ with $\bm{H}_{n}=n^{-1}\sum_{i=1}^{n}\bm{X}_{i}\bm{X}_{i}^{\top}$.

Wald-type tests in this context are based on the asymptotic normality of regression quantiles; as such, attention is naturally directed to the sampling behavior of asymptotically-$\chi_{J}^{2}$ statistics of the form $\{n/[\alpha(1-\alpha)]\}(\bm{R\hat{\beta}}_{n}(\alpha)-\bm{r})^{\top }\bm{W}_{n}(\bm{G}_{n}(\alpha))(\bm{R\hat{\beta}}_{n} (\alpha)-\bm{r})$, where $\bm{G}_{n}(\alpha)$ is a consistent estimator of the matrix $\bm{G}_{0}(\alpha)$. The focus in this section is on the effect estimation of the matrix $\bm{G}_{0}(\alpha)$ exerts on the discrepancy between the empirical and nominal sizes of the associated Wald-type test.

We address the question of whether a Wald-type test of $H_{0}:\,\bm{R\beta }(\alpha)-\bm{r}=0$ admits the possibility of adaptive size control as $n\rightarrow\infty$. In particular, is it possible to implement the estimator $\bm{\hat{G}}_{n}(\alpha)$ given above in ((ref)) in such a way as to make the discrepancy between the actual size and nominal level of a Wald-type test of $H_{0}$ vanish at the same rate as the infeasible test in which the matrix $\bm{G}_{0}(\alpha)$ is actually known? That the answer to this question is positive can be seen by considering the empirical size function of a nominal level-$\tau$ Wald test of $H_{0}$. Let $\chi_{J,\tau }^{2}$ denote the $(1-\tau)$-quantile of a $\chi_{J}^{2}$-distribution, and let $\bm{Z}(\alpha)\sim N(0,\bm{V}(\alpha))$, where the covariance matrix $\bm{V}(\alpha)$ is as given above in ((ref)). Then one can combine the asymptotic normality result in ((ref)) with Theorem (ref) to deduce the following representation of the size function:

where $\Lambda_{1nm}(\alpha,0)$, $\Lambda_{2nm}(\alpha,0)$ and $\Lambda _{3nm}(\alpha,0)$ are $O_{p}(1)$, $\Theta_{n}(0)$ converges to zero at the same rate as the error committed by the first-order asymptotic approximation in ((ref)), and where $\Xi_{nm}(0)=o_{p}\left( h_{m}^{2}+\left[ \log h_{m}^{-1}/\left( mh_{m}\right) \right] ^{1/2}+n^{-1/2}\right) $. Precise expressions for $\Xi_{nm}(0)$, $\Lambda_{knm}(\alpha,0)$ ($k=1,2,3$) and $\Theta_{n}(0)$ are given in (31)--(35) of the supplementary material.

Inspection of ((ref)) indicates that should the matrix $\bm{G}_{0}(\alpha)$ be assumed or in fact be known by the researcher, then the magnitude of the term $\Theta_{n}(0)$ indicates the rate of convergence of the size distortion of the infeasible Wald-type test in which $\bm{G} _{0}(\alpha)$ is known, i.e., the adaptive rate of size control as $n\rightarrow\infty$. It follows that the adaptive rate of size control is determined by the accuracy of the first-order asymptotic normal approximation for $\sqrt{n}\left( \bm{\hat{\beta}}_{n}(\alpha )-\bm{\beta}(\alpha)\right)$.

An important question in this connection is whether the adaptive rate of size control is so large as to dominate the estimation error associated with any reasonable estimate of $\bm{G}_{0}(\alpha)$; in this case one might wonder if there is much point in concerning oneself with a size-optimal implementation of a given estimator of $\bm{G}_{0}(\alpha)$. This concern is particularly relevant if the first-order asymptotic normal approximation to $\sqrt{n}\left( \bm{\hat{\beta}}_{n}(\alpha)-\bm{\beta} (\alpha)\right) $ is of nearly $n^{-1/4}$-order, as indicated by traditional analyses of the Bahadur representation for regression quantiles JureckovaSen96. On the other hand, Portnoy12 has recently established that in fact the error associated with the first-order normal approximation is of nearly $n^{-1/2}$-order, which is sufficiently small so as not to dominate strictly the estimation error committed by a typical estimate of $\bm{G}_{0}(\alpha)$ involving local smoothing. It follows that at least under the conditions imposed by Portnoy12, the problem of constructing a size-optimal estimator of $\bm{G}_{0}(\alpha)$ by choice of a smoothing parameter should be of primary concern in empirical practice.

We consider an implementation of the estimator $\bm{\hat{G}}_{n}(\alpha)$ given above in ((ref)) that causes the corresponding Wald-type test of $H_{0}:\,\bm{R\beta}(\alpha)-\bm{r}=0$ to exhibit adaptive size control as $n\rightarrow\infty$. The precise conditions on the bandwidth $h_{m}$ and the grid size $m$ are specified in Assumption 3 in Appendix A of the supplementary material. These conditions suffice to make the size distortion of the Wald-type test of $H_{0}$ vanish at the adaptive rate as $n\rightarrow\infty$:

The same conditions also cause the Wald-type confidence interval for a given linear combination of components of $\bm{\beta}(\alpha)$ to have a level error that vanishes at the rate enjoyed by the corresponding intervals in which $\bm{G}_{0}(\alpha)$ does not need to be estimated.

Practical recommendations on the implementation of bandwidth parameters and grid sizes that satisfy the conditions of Theorem (ref) are given in Section (ref) below and also in Appendix D of the supplementary material. In particular, Wald-type tests embedding our proposed estimator of $\hat{\bm{G}}_n(\alpha)$ implemented with a fixed (i.e., non-random) bandwidth are exhibited in Section (ref) below and in Appendix E of the supplementary material. Appendix D of the supplementary material, on the other hand, derives an empirically feasible data-driven bandwidth that induces corresponding Wald-type tests to exhibit adaptive size control as $n\rightarrow\infty$.

Simulation evidence on the finite-sample performance of Wald-type tests implemented with the data-driven bandwidth are presented in Appendix E of the supplementary material.

The following corollary is immediate from Theorem (ref) and Portnoy12:

Theorem (ref) and Corollary (ref) jointly establish that in this context the adaptive rate of size control of Wald-type tests is of nearly $n^{-1/2}$-order, and that a Wald-type test constructed using the proposed estimator $\hat{\bm{G}}_{n}(\alpha)$ given above in ((ref)) can be implemented to exhibit this rate as $n\rightarrow\infty$.

Finally, Appendix C of the supplementary material shows that the estimators of $\bm{G}_0(\alpha)$ proposed by Powell91 and HendricksKoenker92 cannot induce Wald-type tests that control size adaptively in large samples.

Numerical Evidence

We present in this section the results of a series of Monte Carlo simulations that are motivated by the empirical question examined in Section (ref). These simulations evaluate the performance of Wald-type tests for testing the heterogeneity of quantile treatment effects Doksum74 in covariates. We naturally focus attention on the relative performance of Wald-type tests incorporating our proposed estimator of $\bm{G}_{0}(\alpha )$. We compare the empirical size and size-corrected power performance of our tests to those of ten alternative testing procedures available in version 5.35 of the \url{quantreg} package Koenker18 for the R statistical computing environment RCoreTeam16. The simulations presented here are all implemented in R; in particular, we make use of the \url{quantreg} package to generate simulations for each of the competing testing procedures that we considered. R code to implement the simulations presented here is included in the supplementary material.

We consider the data-generating process $Y=1+\sum_{j=1}^{4}X_{j}+D+\delta _{a}(U)DX_{1}+F^{-1}(U)$, where $\{X_{j}\}_{j=1}^{4}$ are iid standard normal and independent of a treatment indicator $D,$ which follows a Bernoulli distribution with probability $1/2,$ where $U$ is an independent U[0,1] and where $a\in\mathbb{R}$ denotes the parameter indexing the family of functions $\left\{\delta_a(\cdot):\,a\in\mathbb{R}\right\}$. In this model the QTE for a given setting of $a$, expressed as a function of a quantile of interest $\alpha$, is given by $QTE(\alpha )=1+\delta _{a}(\alpha )X_{1}$.

It follows that for a given quantile $\alpha$, a test of the hypothesis $H_{0}:\delta _{a}(\alpha )=0$ against $H_{1}:\delta_{a}(\alpha )\neq 0$ corresponds to a test of the homogeneity of the $\alpha$-QTE in $X_{1}$ against the alternative of heterogeneity.

We set $F$ in the simulations presented here to a standard normal distribution; results in which $F$ denotes a Student-$t$ distribution with three degrees of freedom are given in Appendix E.3 of the supplement. We consider the following specifications of the heterogeneity parameter $\delta _{a}(\alpha )$:

Each of these models satisfies the null hypothesis of treatment homogeneity when $a=0$. Under the null, all models but Models 3 and 6 are pure location models. The alternative hypothesis corresponds to $a\neq 0.$ Size-corrected power performance is considered against alternatives corresponding to the settings $a=0.50,$ $1.00$ and $1.50$. The corresponding heterogeneity parameters for Models 1--6 under $\alpha=.50$ are plotted in Figure (ref) for the case where $a=1.50$. It is clear that our specifications of Models 1--6 imply QTEs with very different functional forms.

The simulations presented below consider the size and power performance over 1000 Monte Carlo replications of nominal 5%-level tests for $\alpha$-quantile regression parameters, where $\alpha\in\{.25,.50,.75\}$. Average CPU times over 1000 replications required to implement each of the tests examined here are also reported. We considered simulated samples of size $n\in\{100,300\}$. The techniques used to compute the tests considered are as follows:

The Wald-type tests computed using the \url{wiid}, \url{wnid} and \url{wker} methods were all implemented using the default bandwidth setting in the \url{quantreg} package Koenker18, namely the HallSheather88 rule-of-thumb-bandwidth appropriate for inference regarding a population quantile. In addition, the bootstrap tests were all implemented with the default setting of $200$ bootstrap resamples.

Each of \url{wiid}, \url{wnid}, \url{wker}, \url{bxy}, \url{bpwy}, \url{bmcmb}, \url{bwxy} and \url{bwild} was implemented by direct computation of the corresponding test statistic using the corresponding standard error returned by the \url{summary.rq} feature of \url{quantreg}. On the other hand, the rank-based procedures \url{riid} and \url{rnid} both involved direct inversion of the corresponding confidence interval obtained from the \url{summary.rq} feature.

The corresponding simulation results are displayed in Tables (ref)--(ref). These results include average CPU times in seconds over 1000 replications taken to compute each test statistic. These average timings correspond to simulations under the null (i.e., the setting $a=0$) when the quantile of interest is given by $\alpha=0.5$. Average timings for simulations in which $a\neq 0$ or $\alpha\neq 0.5$ are virtually identical.

We also examined in unreported work implementations of \url{wiid}, \url{wnid}, \url{wker} and \url{riid} available from the \url{anova.rq} feature of \url{quantreg}, but the resulting tests were found to exhibit empirical rejection probabilities that were virtually identical to those of the corresponding implementations of these tests using \url{summary.rq}. We also noticed that \url{anova.rq} has a noticeable tendency to run more slowly than \url{summary.rq} for \url{wiid}, \url{wnid} and \url{wker}, and more quickly than \url{summary.rq} for \url{riid}.

We see that the empirical size of the proposed method is accurate even with samples of sizes as small as $n=100$, and is often more accurate than alternative methods, including resampling methods. We also see that the proposed Wald test has good size-corrected power across all six models, three quantiles and two sample sizes for relatively small deviations from the null, i.e. when the constant $a$ is small. It seems clear that an analytical comparison of the asymptotic local relative efficiencies of the different tests considered here with that of the asymptotically uniformly most powerful test ChoiHallSchick96 would be interesting, although such an analysis seems beyond the scope of this paper. We note in passing that the conditional density estimator embedded in our method of inference can also be instrumental in estimating the efficient score NeweyPowell90 and thus in developing asymptotically optimal inference for quantile regression.

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The simulations presented here, along with further simulations reported in the supplementary material, indicate the potential of Wald-type tests based on our proposed method to deliver good size accuracy and reasonable power across a range of quantiles and data-generating processes. These simulations also support the theoretical results presented earlier in Section (ref) inasmuch as the size accuracy of the test tends to outperform those of the other Wald-type tests considered over the three different quantiles and six data-generating processes considered in our simulations.

Empirical Example

We consider the reemployment bonus experiments conducted in Pennsylvania by the United States Department of Labor between July 1988 and October 1989 CorsonDeckerDunstanKerachsky92. This experiment involved the randomized assignment of new claimants for unemployment insurance (UI) benefits into one of several treatment groups or a control group. Claimants assigned to the control group were handled according to the usual procedures of the unemployment insurance system, while claimants assigned to treatment were awarded cash bonuses if they were able to demonstrate full-time reemployment within a specified qualifying period.

The corresponding data were previously analyzed using quantile-regression methods by KoenkerBilias01 and KoenkerXiao02; KoenkerBilias01 also discuss older literature evaluating similar experiments. We follow KoenkerXiao02 by focusing solely on a single treatment group, which combined with the control group yields a sample of size $n=6384$. The corresponding dataset is publicly available and can be downloaded from \url{http://www.econ.uiuc.edu/ roger/research/inference/Penn46.ascii}. Claimants for unemployment benefits that were assigned to this treatment were offered a bonus equal to six times the usual weekly benefit if they secured full-time employment within 12 weeks. Because approximately 20% of the subjects were reemployed within one week and another 20% were not reemployed within a 26-week follow-up window, KoenkerXiao02 assume a quantile-regression specification of the form $F_{\log T|\bm{X}}^{-1}(\alpha)=\bm{X}^{\top}\bm{\beta}(\alpha)$, where $\alpha\in[.20,.80]$, where $T$ denotes the duration of unemployment in weeks and where the regressors contained in $\bm{X}$ include a constant term, an indicator for assignment to treatment and the fourteen demographic or socioeconomic control variables listed in KoenkerXiao02.

We depart from the specification of KoenkerXiao02 by including interactions of the treatment indicator with each of the control variables used by these authors. We also include interactions of the indicator for gender with indicators for race, Hispanic ethnicity and number of dependents. We consider, for a given quantile in the interval $[.20,.80]$, the hypothesis that the treatment interaction terms in $\bm{X}$ are jointly insignificant, i.e., that the effect of treatment at a given quantile in $[.20,.80]$ does not vary with any of the control variables included in $\bm{X}$. Appendix F of the supplementary material presents some additional evidence specific to the question of whether the effect of treatment in this context varies by age or by participants' stated expectation of being recalled to a previously held job.

Figure (ref) reports $p$-values for the hypothesis of covariate homogeneity in treatment over each quantile in a grid of 300 points in $[.20,.80]$. Our test is implemented using our proposed method with the data-driven bandwidth with $k=5$ discussed in detail in Appendix D of the supplement. We also compare the $p$-values from tests implemented using our method with the corresponding $p$-values from the alternative testing methods considered in the simulations reported above. In particular, the \url{wiid}, \url{wnid}, \url{wker}, \url{bxy}, \url{bpwy}, \url{bmcmb}, \url{bwxy} and \url{bwild} methods were implemented by direct computation of the corresponding Wald-type statistic using the estimated asymptotic covariance matrix generated by the \url{summary.rq} feature of version 5.35 of the \url{quantreg} package Koenker18 for the R statistical computing environment RCoreTeam16. The \url{riid} method, on the other hand, was implemented by direct invocation of the \url{anova.rq} feature of \url{quantreg}.

One can see from Figure (ref) that our proposed procedure implies significant covariate-heterogeneity in quantile treatment effects at the .10-level over nearly all quantiles between .43 and .74. Unreported results indicate that the joint significance observed at these quantiles is driven largely by the significance of two covariates, namely the interaction between treatment and an indicator variable for being younger than 35 years of age, and the interaction between treatment and an indicator for whether a given participant expected to be recalled to previous employment. Additional results reported in Appendix F of the supplement reveal significant differences in quantile treatment effects between participants younger than 35 and those aged 35 and older for nearly all quantiles between .50 and .80. In particular, the corresponding participants aged 35 and older are shown to exit unemployment significantly more slowly than those younger than 35.

Significant differences in quantile treatment effects between participants expecting recall to a previous job and those not expecting recall are also shown in Appendix F to exist for nearly all quantiles between .43 and .74. This last result is potentially important in evaluating the cost-effectiveness of the program given the experiment's exclusion of all claimants for unemployment insurance for whom inclusion in the treatment group was deemed not to provide a sufficient encouragement “to search for work more diligently and to accept suitable employment more rapidly than would be the case otherwise” CorsonDeckerDunstanKerachsky92. The experimenters specifically excluded from the study all claimants who indicated a definite expectation of being recalled to a previous employer on a specific date within 60 days of filing their applications for UI benefits. These claimants were deemed to be so secure in their expectation of future full-time employment that any bonus paid to them upon resuming full-time employment would be interpreted as a windfall. Included in the experiment, however, were those claimants who indicated some expectation of being recalled to a previous job, although with no definite date of recall. The experimenters deemed claimants in this category to be similar to claimants with no stated expectation of returning to a previous job in terms of their assumed response to a promised bonus payment upon resuming full-time employment within the qualifying period. The results presented in Appendix F of the supplement indicate that UI claimants who indicated some expectation of being recalled, although not to the extent of having a specific date of recall, in fact differ in their responses to treatment than those claimants who indicated no expectation of recall whatsoever.

Figure (ref) also shows that the other testing methods considered varied in the extent to which the hypothesis of covariate-homogeneity in the treatment effect was rejected over quantiles in the interval $[.20,.80]$. In particular, none of the additional inference methods considered was seen to imply the same range of quantiles corresponding to covariate heterogeneity in the corresponding quantile treatment effects that was revealed by our method. For example, \url{wiid} yielded significance at all quantiles greater than .53. We note in addition that some $p$-values for tests implemented using \url{wker} in fact exceed .98 for most quantiles above .78, which suggests that the corresponding regression-quantile covariance matrices were not well estimated by \url{wker}.

In view of the rejection, reported by KoenkerXiao02, of the null of a linear location-shift model for quantiles on the interval $[.25,.75]$, we interpret the \url{wiid} method's conclusion of significance at all quantiles greater than .53 as misleading, and likely driven by misspecification of the assumed location-shift model. As such, inferences resulting from other methods that assume a linear location-shift model (i.e., \url{riid} and \url{bmcmb}) are similarly likely to be misleading.

In summary, we have used our proposed method of inference to show that the effect of treatment on the duration of employment tends to vary with individual characteristics of the experimental subjects only over a relatively narrow range of quantiles between .43 and .74. These ranges of quantiles corresponding to covariate heterogeneity in the effect of treatment is not matched by any of the other testing methods considered. It follows that our proposed method permits an understanding of the effectiveness of a particular unemployment relief policy distinct from that produced by other methods of inference.