Package 'QTE.RD'

QTE.RD: Quantile Treatment Effects under the Regression Discontinuity Design

Description

Provides comprehensive methods for testing, estimating, and conducting uniform inference on quantile treatment effects (QTEs) in sharp regression discontinuity (RD) designs, incorporating covariates and implementing robust bias correction methods of Qu, Yoon, Perron (2024) doi:10.1162/rest_a_01168.

Details

The package QTE.RD includes four main functions:

• rd.qte estimates QTEs and provides uniform confidence bands, with or without covariates, and with or without robust bias correction.

• rdq.test conducts tests for three hypotheses, related to the significance of treatment effects, homogeneous treatment effects, and uniformly positive or negative treatment effects.

• rdq.bandwidth implements two bandwidth selection rules: the cross-validation bandwidth and the MSE optimal bandwidth.

• plot.qte generates figures summarizing the treatment effects along with their confidence bands.

Author(s)

• Zhongjun Qu [email protected]

• Jungmo Yoon [email protected]

References

Zhongjun Qu, Jungmo Yoon, Pierre Perron (2024), "Inference on Conditional Quantile Processes in Partially Linear Models with Applications to the Impact of Unemployment Benefits," The Review of Economics and Statistics; https://doi.org/10.1162/rest_a_01168

Zhongjun Qu and Jungmo Yoon (2019), "Uniform Inference on Quantile Effects under Sharp Regression Discontinuity Designs," Journal of Business and Economic Statistics, 37(4), 625–647; https://doi.org/10.1080/07350015.2017.1407323

---

Cross-validation bandwidth

Description

cv.bandwidth implements the cross-validation bandwidth selection rule. The function rdq.bandwidth calls this function to obtain the CV bandwidth.

Usage

cv.bandwidth(y, x, z, dz, x0, val, xl, order, bdy)

Arguments

y	a numeric vector, the outcome variable.
x	the running variable.
z	additional covariates.
dz	the number of covariates z.
x0	the cutoff point.
val	a set of candidate values for the CV bandwidth.
xl	if xl=0.5, the CV bandwidth is computed using the 50% of observations closest to $x_0$.
order	either 1 or 2. When order=1, a local linear regression is used, and when order=2, a local quadratic regression is used.
bdy	either 0 or 1. When bdy=1, the CV bandwidth is computed by treating x as a boundary point. Otherwise, x is treated as an interior point.

Value

A list with elements:

h.cv

the selected CV bandwidth values at the median.

cand

values of the criterion function evaluated at each of candidate value.

References

Zhongjun Qu, Jungmo Yoon, Pierre Perron (2024), "Inference on Conditional Quantile Processes in Partially Linear Models with Applications to the Impact of Unemployment Benefits," The Review of Economics and Statistics; https://doi.org/10.1162/rest_a_01168

Zhongjun Qu and Jungmo Yoon (2019), "Uniform Inference on Quantile Effects under Sharp Regression Discontinuity Designs," Journal of Business and Economic Statistics, 37(4), 625–647; https://doi.org/10.1080/07350015.2017.1407323

See Also

rdq.bandwidth

Examples

n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
cv.bandwidth(y=y, x=x, z=NULL, dz=0, x0=0, val=c(1,2,3,4), xl=0.5, order=2, bdy=1)
cv.bandwidth(y=y, x=x, z=NULL, dz=0, x0=0, val=c(1,2,3,4), xl=0.5, order=1, bdy=1)

---

Epanechnikov kernel

Description

Epanechnikov kernel

Usage

depa(xx, loc = 0, scale = 1)

Arguments

xx	the evaluation points.
loc	the location parameter (normalized to be 0).
scale	the scale parameter (normalized to be 1).

Value

values.

Examples

depa(0)
depa(seq(-1,1,by=0.05))

---

Uniform confidence bands for QTE

Description

make.band constructs uniform confidence bands using the output of rdq.sim. The function rdq.band calls this function to generates uniform bands.

Usage

make.band(n.sam,Dc.p,Dc.m,Dr.p,Dr.m,dz,cov,taus,hh,Qy.p,Qy.m,
     bias.p,bias.m,alpha,n.sim)

Arguments

n.sam	the sample size.
Dc.p	simulated values from $D_{1,v}(x_{0}^{+},z,\tau)$.
Dc.m	simulated values from $D_{1,v}(x_{0}^{-},z,\tau)$.
Dr.p	simulated values from $D_{1,v}(x_{0}^{+},z,\tau) - D_{2,v}(x_{0}^{+},z,\tau)$.
Dr.m	simulated values from $D_{1,v}(x_{0}^{-},z,\tau) - D_{2,v}(x_{0}^{-},z,\tau)$.
dz	the number of covariates
cov	either 0 or 1. Set cov=1 if covariates are present in the model; otherwise set cov=0.
taus	a vector of quantiles of interest.
hh	the bandwidth values.
Qy.p	estimated conditional quantiles at $(x_{0}^{+},z)$.
Qy.m	estimated conditional quantiles at $(x_{0}^{-},z)$.
bias.p	estimated bias terms at $(x_{0}^{+},z)$.
bias.m	estimated bias terms at $(x_{0}^{-},z)$.
alpha	a number between 0 and 1, the desired significance level.
n.sim	the number of simulation repetitions.

Value

A list with elements:

qte

QTE estimates without bias correction.

qte.r

QTE estimates with bias correction.

uband

uniform confidence band for QTE without bias correction.

uband.r

uniform confidence band for QTE with robust bias correction.

sig

standard errors for the bias-uncorrected QTE estimates.

sig.r

standard errors for the bias-corrected QTE estimates. The values reflect the impact of the bias correction on the estimation precision.

See Also

rdq.band

Examples

n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
tlevel2 = c(0.05,tlevel,0.95)
hh = rep(2,length(tlevel))
hh2 = rep(2,length(tlevel2))
sel = tlevel2 %in% tlevel

ab = rdq(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel2,h.tau=hh2,cov=0)
delta = c(0.05,0.09,0.14,0.17,0.19,0.17,0.14,0.09,0.05)
fp = rdq.condf(x=x,Q=ab$qp.est,bcoe=ab$bcoe.p,taus=tlevel,taul=tlevel2,delta=delta,cov=0)
fm = rdq.condf(x=x,Q=ab$qm.est,bcoe=ab$bcoe.m,taus=tlevel,taul=tlevel2,delta=delta,cov=0)
bp = rdq.bias(y[d==1],x[d==1],dz=0,x0=0,z0=NULL,taus=tlevel,hh,hh,fx=fp$ff[(d==1),],cov=0)
bm = rdq.bias(y[d==0],x[d==0],dz=0,x0=0,z0=NULL,taus=tlevel,hh,hh,fx=fm$ff[(d==0),],cov=0)

sa = rdq.sim(x=x,d=d,x0=0,z0=NULL,dz=0,cov=0,tt=tlevel,hh,hh,fxp=fp$ff,fxm=fm$ff,n.sim=200)
ba = make.band(n,Dc.p=sa$dcp,Dc.m=sa$dcm,Dr.p=sa$drp,Dr.m=sa$drm,dz=0,cov=0,
taus=tlevel,hh,Qy.p=as.matrix(ab$qp.est[sel,]),Qy.m=as.matrix(ab$qm.est[sel,]),
bias.p=bp$bias,bias.m=bm$bias,alpha=0.1,n.sim=200)

---

Uniform confidence bands for conditional quantile processes

Description

make.band.cq constructs uniform confidence bands for conditional quantile processes as functions of tau for each side of the cutoff. See make.band as well. The function rdq.band calls this function to generates uniform bands for conditional quantiles.

Usage

make.band.cq(n.sam,Dc.p,Dc.m,Dr.p,Dr.m,dz,cov,taus,hh,Qy.p,Qy.m,
     bias.p,bias.m,alpha,n.sim)

Arguments

n.sam	the sample size.
Dc.p	simulated values from $D_{1,v}(x_{0}^{+},z,\tau)$.
Dc.m	simulated values from $D_{1,v}(x_{0}^{-},z,\tau)$.
Dr.p	simulated values from $D_{1,v}(x_{0}^{+},z,\tau) - D_{2,v}(x_{0}^{+},z,\tau)$.
Dr.m	simulated values from $D_{1,v}(x_{0}^{-},z,\tau) - D_{2,v}(x_{0}^{-},z,\tau)$.
dz	the number of covariates.
cov	either 0 or 1. Set cov=1 if covariates are present in the model; otherwise set cov=0.
taus	a vector of quantiles of interest.
hh	the bandwidth values.
Qy.p	estimated conditional quantiles at $(x_{0}^{+},z)$.
Qy.m	estimated conditional quantiles at $(x_{0}^{-},z)$.
bias.p	estimated bias terms at $(x_{0}^{+},z)$.
bias.m	estimated bias terms at $(x_{0}^{-},z)$.
alpha	a number between 0 and 1, the desired significance level.
n.sim	the number of simulation repetitions.

Value

A list with elements:

qp

conditional quantile estimates at $x_{0}^{+}$ (i.e., above the cutoff) without bias correction.

qp.r

bias corrected conditional quantile estimates at $x_{0}^{+}$.

qm

conditional quantile estimates at $x_{0}^{-}$ (i.e., below the cutoff) without bias correction.

qm.r

bias corrected conditional quantile estimates at $x_{0}^{-}$.

ubandp

uniform confidence band for conditional quantiles at $x_{0}^{+}$ without bias correction.

ubandp.r

uniform confidence band for conditional quantiles at $x_{0}^{+}$ with robust bias correction.

ubandm

uniform confidence band for conditional quantiles at $x_{0}^{-}$ without bias correction.

ubandm.r

uniform confidence band for conditional quantiles at $x_{0}^{-}$ with robust bias correction.

sp

standard errors of the conditional quantile estimates without bias correction at $x_{0}^{+}$.

sp.r

standard errors of the conditional quantile estimates with robust bias correction at $x_{0}^{+}$.

sm

standard errors of the conditional quantile estimates without bias correction at $x_{0}^{-}$.

sm.r

standard errors of the conditional quantile estimates with robust bias correction at $x_{0}^{-}$.

See Also

make.band

Examples

n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
tlevel2 = c(0.05,tlevel,0.95)
hh = rep(2,length(tlevel))
hh2 = rep(2,length(tlevel2))
sel = tlevel2 %in% tlevel

ab = rdq(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel2,h.tau=hh2,cov=0)
delta = c(0.05,0.09,0.14,0.17,0.19,0.17,0.14,0.09,0.05)
fp = rdq.condf(x=x,Q=ab$qp.est,bcoe=ab$bcoe.p,taus=tlevel,taul=tlevel2,delta,cov=0)
fm = rdq.condf(x=x,Q=ab$qm.est,bcoe=ab$bcoe.m,taus=tlevel,taul=tlevel2,delta,cov=0)
bp = rdq.bias(y[d==1],x[d==1],dz=0,x0=0,z0=NULL,taus=tlevel,hh,hh,fx=fp$ff[(d==1),],cov=0)
bm = rdq.bias(y[d==0],x[d==0],dz=0,x0=0,z0=NULL,taus=tlevel,hh,hh,fx=fm$ff[(d==0),],cov=0)

sa = rdq.sim(x=x,d=d,x0=0,z0=NULL,dz=0,cov=0,tt=tlevel,hh,hh,fxp=fp$ff,fxm=fm$ff,n.sim=200)
ba.cq = make.band.cq(n,Dc.p=sa$dcp,Dc.m=sa$dcm,Dr.p=sa$drp,Dr.m=sa$drm,dz=0,cov=0,
taus=tlevel,hh,Qy.p=as.matrix(ab$qp.est[sel,]),Qy.m=as.matrix(ab$qm.est[sel,]),
bias.p=bp$bias,bias.m=bm$bias,alpha=0.1,n.sim=200)

---

QTE plots

Description

plot.qte generates plots summarizing the QTE estimates and their uniform confidence bands, helping users visualize the results. It also makes plots for conditional quantile processes for each side of the cutoff.

Usage

## S3 method for class 'qte'
plot(x, ptype = 1, ytext = NULL, mtext = NULL, subtext = NULL, ...)

Arguments

x	an object of class "qte" or "summary.qte" produce by rd.qte.
ptype	either 1 or 2. Set ptype=1 for the QTE plots, and ptype=2 for the conditional quantile plots. The default value is 1.
ytext	the y-axis label.
mtext	the title of the plot.
subtext	the subtitles (used for the conditional quantile plots only).
...	optional arguments to plot

Value

plot(s) of the QTE estimates and uniform confidence bands.

Examples

# Without covariate
n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
A <- rd.qte(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel,bdw=2,cov=0,bias=1)
plot(A)

y.text = "test scores"
m.text = "QTE and Uniform band"
plot(A,ytext=y.text,mtext=m.text)

A2 <- summary(A,alpha=0.1)
plot(A2)

z = sample(c(0,1),n,replace=TRUE)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + d*z + rnorm(n)
A <- rd.qte(y=y,x=cbind(x,z),d=d,x0=0,z0=c(0,1),tau=tlevel,bdw=2,cov=1,bias=1)
A2 <- summary(A,alpha=0.1)

y.text = "test scores"
m.text = c("D=0","D=1")
plot(A2,ytext=y.text,mtext=m.text)

# conditional quantile plots
n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
A <- rd.qte(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel,bdw=2,cov=0,bias=1)
A2 <- summary(A,alpha=0.1)

y.text = "test scores"
m.text = "Conditional quantile functions"
sub.text = c("D=0 group","D=1 group")
plot(A2,ptype=2,ytext=y.text,mtext=m.text,subtext=sub.text)

---

Print a band.qte object

Description

Print a band.qte object

Usage

## S3 method for class 'band.qte'
print(x, ...)

Arguments

x	object returned from band.qte.
...	optional arguments.

---

Print a bw.qte object

Description

Print a bw.qte object

Usage

## S3 method for class 'bw.qte'
print(x, ...)

Arguments

x	object returned from bandwidth.qte.
...	optional arguments.

---

Print a qte object.

Description

Print a qte object.

Usage

## S3 method for class 'qte'
print(x, ...)

Arguments

x	object returned from rd.qte.
...	optional arguments.

---

Print a summary.qte object

Description

Print a summary.qte object

Usage

## S3 method for class 'summary.qte'
print(x, ...)

Arguments

x	object returned from summary.qte.
...	optional arguments.

---

Print a test.qte object

Description

Print a test.qte object

Usage

## S3 method for class 'test.qte'
print(x, ...)

Arguments

x	object returned from test.qte.
...	optional arguments.

---

QTE and its uniform confidence band.

Description

rd.qte is the main function of the QTE.RD package. If cov=1, it estimates QTE for each subgroup defined by covariates. If cov=0, it estimate QTE without covariates. If bias=1, it corrects the bias in QTE estimates and obtains the robust confidence band and if bias=0, no bias correction is implemented.

Usage

rd.qte(y, x, d, x0, z0=NULL, tau, bdw, cov, bias)

Arguments

y	a numeric vector, the outcome variable.
x	a vector (or a matrix) of covariates, the first column is the running variable.
d	a numeric vector, the treatment status.
x0	the cutoff point.
z0	the value of the covariates at which to evaluate the effects. For example, if a female dummy z is included, z0 = 1 may indicate the female subgroup.
tau	a vector of quantiles of interest.
bdw	the bandwidth value(s). If 'bdw' is a scalar, it is interpreted as the bandwidth for the median. See the function rdq.bandwidth for how to select this bandwidth. The bandwidths for the rest of the quantiles are computed automatically using the formula of Yu and Jones (1998). If it is a vector with the same dimension as 'tau', the function will use these values for the respective quantiles accordingly.
cov	either 0 or 1. Set cov=1 when covariates are present in the model; otherwise set cov=0.
bias	either 0 or 1. If bias=1, the QTE estimate is bias corrected and the robust confidence band in Qu, Yoon, and Perron (2024) is produced. If bias=0, no bias correction is implemented.

Value

A list with elements:

qte

QTE estimates.

uband

uniform confidence band for QTE. If bias=1, the band is robust capturing the effect of the bias correction. If bias=0, no bias correction is implemented.

sigma

standard errors for each quantile level. If bias=1, its value captures the effect of the bias correction. If bias=0, no bias correction is implemented.

qp.est

conditional quantile estimates on the right side of $x_{0}$ (or for the $D=1$ group).

qm.est

conditional quantile estimates on the left side of $x_{0}$ (or for the $D=0$ group).

References

Zhongjun Qu, Jungmo Yoon, Pierre Perron (2024), "Inference on Conditional Quantile Processes in Partially Linear Models with Applications to the Impact of Unemployment Benefits," The Review of Economics and Statistics; https://doi.org/10.1162/rest_a_01168

Zhongjun Qu and Jungmo Yoon (2019), "Uniform Inference on Quantile Effects under Sharp Regression Discontinuity Designs," Journal of Business and Economic Statistics, 37(4), 625–647; https://doi.org/10.1080/07350015.2017.1407323

Keming Yu and M. C. Jones (1998), “Local Linear Quantile Regression,” Journal of the American Statistical Association, 93(441), 228–237; https://doi.org/10.2307/2669619

Examples

# Without covariate
n <- 500
x <- runif(n,min=-4,max=4)
d <- (x > 0)
y <- x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel <- seq(0.1,0.9,by=0.1)
A <- rd.qte(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel,bdw=2,cov=0,bias=1)

# (continued) With covariates
z <- sample(c(0,1),n,replace=TRUE)
y <- x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + d*z + rnorm(n)
A <- rd.qte(y=y,x=cbind(x,z),d=d,x0=0,z0=c(0,1),tau=tlevel,bdw=2,cov=1,bias=1)

---

Estimate the QTE under the RDD

Description

rdq estimates QTE under the RDD with or without covariates. This function is used by rd.qte to generate QTE estimates.

Usage

rdq(y, x, d, x0, z0 = NULL, tau, h.tau, cov)

Arguments

y	a numeric vector, the outcome variable.
x	a vector (or a matrix) of covariates, the first column is the running variable.
d	a numeric vector, the treatment status.
x0	the cutoff point.
z0	the value of the covariates at which to evaluate the effects. For example, if a female dummy is included, z0 = 1 may indicate the female subgroup.
tau	a vector of quantiles of interest.
h.tau	the bandwidth values (specified for each quantile level).
cov	either 0 or 1. Set cov=1 if covariates are present in the model; otherwise set cov=0.

Value

A list with elements:

qte

QTE estimates.

qp.est

conditional quantile estimates on the right side of $x_{0}$ (or for the D=1 group).

qm.est

conditional quantile estimates on the left side of $x_{0}$ (or for the D=0 group).

bcoe.p

quantile regression coefficients on the right side of $x_{0}$.

bcoe.m

quantile regression coefficients on the left side of $x_{0}$.

Examples

# Without covariate
n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
hh = rep(2,length(tlevel))
rdq(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel,h.tau=hh,cov=0)

# (continued) With covariates
z = sample(c(0,1),n,replace=TRUE)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + d*z + rnorm(n)
rdq(y=y,x=cbind(x,z),d=d,x0=0,z0=c(0,1),tau=tlevel,h.tau=hh,cov=1)

---

Uniform confidence bands for QTE

Description

rdq.band produces uniform confidence bands for QTEs with and without bias correction. This function is used by rd.qte to generate uniform bands.

Usage

rdq.band(y, x, d, x0, z0 = NULL, tau, bdw, cov, alpha)

Arguments

y	a numeric vector, the outcome variable.
x	a vector (or a matrix) of covariates, the first column is the running variable.
d	a numeric vector, the treatment status.
x0	the cutoff point.
z0	the value of the covariates at which to evaluate the effects. For example, if a female dummy is included, z0 = 1 may indicate the female subgroup.
tau	a vector of quantiles of interest.
bdw	the bandwidth value(s). If 'bdw' is a scalar, it is interpreted as the bandwidth for the median. The bandwidths for the rest of the quantiles are computed automatically using the formula in Yu and Jones (1998). If it is a vector with the same dimension as 'tau', the function will use these values for the respective quantiles accordingly.
cov	either 0 or 1. Set cov=1 when covariates are present in the model; otherwise set cov=0.
alpha	a number between 0 and 1, the desired significance level.

Value

qte

QTE estimates without bias correction.

qte.cor

bias corrected QTE estimates.

uband

uniform confidence band for QTE without bias correction.

uband.robust

uniform confidence band for QTE with robust bias correction.

sig

standard errors for each quantile level for estimates without bias correction.

sig.r

standard errors for each quantile level for estimates with robust bias correction.

uband.p

uniform confidence band for the conditional quantile estimates on the right side of the cutoff, without bias correction.

uband.robust.p

uniform confidence band for the conditional quantile estimates on the right side of the cutoff, robust to the bias correction.

uband.m

uniform confidence band for the conditional quantile estimates on the left side of the cutoff, without bias correction.

uband.robust.m

uniform confidence band for the conditional quantile estimates on the left side of the cutoff, robust to the bias correction.

References

Zhongjun Qu, Jungmo Yoon, Pierre Perron (2024), "Inference on Conditional Quantile Processes in Partially Linear Models with Applications to the Impact of Unemployment Benefits," The Review of Economics and Statistics; https://doi.org/10.1162/rest_a_01168

Zhongjun Qu and Jungmo Yoon (2019), "Uniform Inference on Quantile Effects under Sharp Regression Discontinuity Designs," Journal of Business and Economic Statistics, 37(4), 625–647; https://doi.org/10.1080/07350015.2017.1407323

Keming Yu and M. C. Jones (1998), “Local Linear Quantile Regression,” Journal of the American Statistical Association, 93(441), 228–237; https://doi.org/10.2307/2669619

See Also

rd.qte

Examples

# Without covariate
n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
D = rdq.band(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel,bdw=2,cov=0,alpha=0.1)

# (continued) With covariates
z = sample(c(0,1),n,replace=TRUE)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + d*z + rnorm(n)
D = rdq.band(y=y,x=cbind(x,z),d=d,x0=0,z0=c(0,1),tau=tlevel,bdw=2,cov=1,alpha=0.1)

---

Bandwidth estimation

Description

rdq.bandwidth implements two bandwidth selection rules and obtains the cross-validation (CV) bandwidth and the MSE optimal bandwidth.

Usage

rdq.bandwidth(y, x, d, x0, z0=NULL, cov, cv, val,hp=NULL,pm.each=1,
     bdy=1,p.order=1,xl=0.5)

Arguments

y	a numeric vector, the outcome variable.
x	a vector (or a matrix) of covariates, the first column is the running variable.
d	a numeric vector, the treatment status.
x0	the cutoff point.
z0	the value of the covariates at which to evaluate the effects.
cov	either 0 or 1. Set cov=1 when covariates are present in the model; otherwise set cov=0.
cv	either 0 or 1. When cv=1, both the CV and MSE optimal bandwidths are produced. When cv=0, the MSE optimal bandwidth is produced.
val	a set of candidate values for the CV bandwidth.
hp	a pilot bandwidth to estimate nuisance parameters for the MSE optimal bandwidth. It will be used only if cv=0. If cv=1, the CV bandwidth will be used as the pilot bandwidth to compute the MSE optimal bandwidth.
pm.each	either 0 or 1. When pm.each=1, the CV bandwidths for each side of the cutoff will be obtained separately.
bdy	either 0 or 1. When bdy=1, the CV bandwidth uses the boundary point procdure.
p.order	either 1 or 2. When p.order=1, a local linear regression is used, and when p.order=2, a local quadratic regression is used.
xl	if xl=0.5, the CV bandwidth use the 50% of observations closest to $x_0$.

Value

A list with elements:

cv

the selected CV bandwidth at the median.

opt.p

the MSE optimal bandwidth at the median from the right side of $x_0$.

opt.m

the MSE optimal bandwidth at the median from the left side of $x_0$.

References

Zhongjun Qu, Jungmo Yoon, Pierre Perron (2024), "Inference on Conditional Quantile Processes in Partially Linear Models with Applications to the Impact of Unemployment Benefits," The Review of Economics and Statistics; https://doi.org/10.1162/rest_a_01168

Zhongjun Qu and Jungmo Yoon (2019), "Uniform Inference on Quantile Effects under Sharp Regression Discontinuity Designs," Journal of Business and Economic Statistics, 37(4), 625–647; https://doi.org/10.1080/07350015.2017.1407323

Examples

# Without covariate
n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
rdq.bandwidth(y=y,x=x,d=d,x0=0,z0=NULL,cov=0,cv=1,val=(1:4))
rdq.bandwidth(y=y,x=x,d=d,x0=0,z0=NULL,cov=0,cv=0,val=(1:4),hp=2)

# (continued) With covariates
z = sample(c(0,1),n,replace=TRUE)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + d*z + rnorm(n)
rdq.bandwidth(y=y,x=cbind(x,z),d=d,x0=0,z0=c(0,1),cov=1,cv=1,val=(1:4),bdy=1,p.order=1)

---

Bias estimation

Description

rdq.bias estimates the bias terms using the local quadratic quantile regression.

Usage

rdq.bias(y, x, dz, x0, z0, taus, h.tau, h.tau2, fx, cov)

Arguments

y	a numeric vector, the outcome variable.
x	a vector (or a matrix) of covariates, the first column is the running variable.
dz	the number of covariates.
x0	the cutoff point.
z0	the value of the covariates at which to evaluate the effects.
taus	a vector of quantiles of interest.
h.tau	the bandwidth values (specified for each quantile level), for estimating conditional quantiles.
h.tau2	the bandwidth values for the local quadratic quantile regression, for estimating the bias terms.
fx	conditional density estimates.
cov	either 0 or 1. Set cov=1 if covariates are present in the model; otherwise set cov=0.

Value

A list with elements:

bias

the bias estimates.

b.hat

the estimate of the $B_{v}(x,z,\tau)$ term. See Qu, Yoon, and Perron (2024).

References

Zhongjun Qu, Jungmo Yoon, Pierre Perron (2024), "Inference on Conditional Quantile Processes in Partially Linear Models with Applications to the Impact of Unemployment Benefits," The Review of Economics and Statistics; https://doi.org/10.1162/rest_a_01168

Examples

n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
tlevel2 = c(0.05,tlevel,0.95)
hh = rep(2,length(tlevel))
hh2 = rep(2,length(tlevel2))

ab = rdq(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel2,h.tau=hh2,cov=0)
delta = c(0.05,0.09,0.14,0.17,0.19,0.17,0.14,0.09,0.05)
hh = rep(2,length(tlevel))
fe = rdq.condf(x,Q=ab$qp.est,bcoe=ab$bcoe.p,taus=tlevel,taul=tlevel2,delta=delta,cov=0)
be = rdq.bias(y[d==1],x[d==1],dz=0,x0=0,z0=NULL,taus=tlevel,hh,hh,fx=fe$ff[(d==1),],cov=0)

---

Conditional density estimation

Description

rdq.condf estimates conditional density functions by using the differencing method.

Usage

rdq.condf(x, Q, bcoe, taus, taul, delta, cov)

Arguments

x	a vector (or a matrix) of covariates.
Q	a vector of estimated conditional quantiles.
bcoe	quantile regression coefficient estimates.
taus	a vector of quantiles of interest.
taul	a vector of quantiles used for the conditional density estimation. It is needed to estimate the tail parts of conditional density functions more precisely.
delta	bandwidths for estimating the conditional density.
cov	either 0 or 1. Set cov=1 if covariates are present in the model; otherwise set cov=0.

Value

conditional density function estimates

Examples

n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
hh = rep(2,length(tlevel))

ab = rdq(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel,h.tau=hh,cov=0)
delta = 0.186
fe = rdq.condf(x=x,Q=ab$qp.est,bcoe=ab$bcoe.p,taus=0.5,taul=tlevel,delta=delta,cov=0)

---

Simulation the asymptotic distributions

Description

rdq.sim produces iid draws from the asymptotic distribution of the conditional quantile process estimate.

Usage

rdq.sim(x, d, x0, z0, dz, cov, tt, hh, hh2, fxp, fxm, n.sim)

Arguments

x	a vector (or a matrix) of covariates.
d	a numeric vector, the treatment status.
x0	the cutoff point.
z0	the value of the covariates at which to evaluate the effects.
dz	the number of covariates.
cov	either 0 or 1. Set cov=1 if covariates are present in the model; otherwise set cov=0.
tt	a vector of quantiles.
hh	the bandwidth values (specified for each quantile level).
hh2	the bandwidth values for the local quadratic quantile regression.
fxp	conditional density estimates on the right side of $x_0$.
fxm	conditional density estimates on the left side of $x_0$.
n.sim	the number of simulation repetitions.

Value

A list with elements:

dcp

realizations from the asymptotic distribution of the conditional quantile process, from the right side of $x_0$.

dcm

realizations from the asymptotic distribution of the conditional quantile process, from the left side of $x_0$.

drp

realizations from the asymptotic distribution of the bias corrected conditional quantile process, from the right side of $x_0$.

drm

realizations from the asymptotic distribution of the bias corrected conditional quantile process, from the left side of $x_0$.

Examples

n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
tlevel2 = c(0.05,tlevel,0.95)
hh = rep(2,length(tlevel))
hh2 = rep(2,length(tlevel2))

ab = rdq(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel2,h.tau=hh2,cov=0)
delta = c(0.05,0.09,0.14,0.17,0.19,0.17,0.14,0.09,0.05)
fp = rdq.condf(x=x,Q=ab$qp.est,bcoe=ab$bcoe.p,taus=tlevel,taul=tlevel2,delta,cov=0)
fm = rdq.condf(x=x,Q=ab$qm.est,bcoe=ab$bcoe.m,taus=tlevel,taul=tlevel2,delta,cov=0)
sa = rdq.sim(x=x,d=d,x0=0,z0=NULL,dz=0,cov=0,tt=tlevel,hh,hh,fxp=fp$ff,fxm=fm$ff,n.sim=200)

---

tests for QTE

Description

rdq.test provides testing results for hypotheses on the treatment effects concerning (i) treatment significance, (ii) homogeneity of effects over quantiles, and (iii) positive or negative dominance hypothesis.

Usage

rdq.test(y,x,d,x0,z0=NULL,tau,bdw,cov,bias,alpha,type,std.opt=1)

Arguments

y	a numeric vector, the outcome variable.
x	a vector (or a matrix) of covariates, the first column is the running variable.
d	a numeric vector, the treatment status.
x0	the cutoff point.
z0	the value of the covariates at which to evaluate the effects. For example, if a female dummy is included, z0 = 1 indicates the female subgroup.
tau	a vector of quantiles of interest.
bdw	the bandwidth value(s). If 'bdw' is a scalar, it is interpreted as the bandwidth for the median. The bandwidths for the rest of the quantiles are computed automatically using the formula in Yu and Jones (1998). If it is a vector with the same dimension as 'tau', the function will use these values for the respective quantiles accordingly.
cov	either 0 or 1. Set cov=1 when covariates are present in the model; otherwise set cov=0.
bias	either 0 or 1. If bias=1, the QTE estimate is bias corrected and the robust confidence band in Qu, Yoon, and Perron (2024) is produced. If bias=0, no bias correction is implemented.
alpha	a number between 0 and 1, the desired significance level. When alpha=0.1, one will get a 90% uniform band.
type	a value in 1–4. Set type to 1 to test the null hypothesis of a zero treatment effect against the alternative hypothesis of significant treatment effects; set type to 2 to test the null hypothesis of homogeneous treatment against heterogeneous treatment effects; set type to 3 to test the null hypothesis of uniformly non-negative treatment effects against the presence of negative effects; and set type to 4 to test the null hypothesis of uniformly non-positive treatment effects against the presence of positive effects at some quantiles.
std.opt	either 0 or 1. If std.opt=1, the test statistic is standardized so that the variance is equalized across quantiles; if std.opt=0, the test is not standardized.

Value

A list with elements:

statistic

test statistics.

cr.va

critical values.

p.value

p values.

References

Zhongjun Qu, Jungmo Yoon, Pierre Perron (2024), "Inference on Conditional Quantile Processes in Partially Linear Models with Applications to the Impact of Unemployment Benefits," The Review of Economics and Statistics; https://doi.org/10.1162/rest_a_01168

Zhongjun Qu and Jungmo Yoon (2019), "Uniform Inference on Quantile Effects under Sharp Regression Discontinuity Designs," Journal of Business and Economic Statistics, 37(4), 625–647; https://doi.org/10.1080/07350015.2017.1407323

Examples

# Without covariate
n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
B = rdq.test(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel,bdw=2,cov=0,bias=1,alpha=c(0.1,0.05),type=c(1,2,3))

# (continued) With covariates
z = sample(c(0,1),n,replace=TRUE)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + d*z + rnorm(n)
B = rdq.test(y=y,x=cbind(x,z),d=d,x0=0,z0=c(0,1),tau=tlevel,bdw=2,cov=1,bias=1,
alpha=c(0.1,0.05),type=c(3,4))

---

Run tests

Description

run.test performs hypothesis testing. The function rdq.test calls this function to run tests.

Usage

run.test(n.sam,dz,taus,hh,Dc.p,Dc.m,Dr.p,Dr.m,Qy.p,Qy.m,bias.p,bias.m,
     cov,bias,alpha,n.sim,test.type,std.opt)

Arguments

n.sam	the sample size.
dz	the number of covariates.
taus	a vector of quantiles of interest.
hh	the bandwidth values.
Dc.p	simulated values from $D_{1,v}(x_{0}^{+},z,\tau)$.
Dc.m	simulated values from $D_{1,v}(x_{0}^{-},z,\tau)$.
Dr.p	simulated values from $D_{1,v}(x_{0}^{+},z,\tau) - D_{2,v}(x_{0}^{+},z,\tau)$.
Dr.m	simulated values from $D_{1,v}(x_{0}^{-},z,\tau) - D_{2,v}(x_{0}^{-},z,\tau)$.
Qy.p	estimated conditional quantiles at $(x_{0}^{+},z)$.
Qy.m	estimated conditional quantiles at $(x_{0}^{-},z)$.
bias.p	estimated bias terms at $(x_{0}^{+},z)$.
bias.m	estimated bias terms at $(x_{0}^{-},z)$.
cov	either 0 or 1. Set cov=1 if covariates are present in the model; otherwise set cov=0.
bias	either 0 or 1. If bias=1, the QTE estimate is bias corrected and the robust confidence band in Qu, Yoon, and Perron (2024) is produced. If bias=0, no bias correction is implemented.
alpha	a number between 0 and 1, the desired significance level.
n.sim	the number of simulation repetitions.
test.type	a value in 1–4. Set type to 1 to test the null hypothesis of a zero treatment effect against the alternative hypothesis of significant treatment effects; set type to 2 to test the null hypothesis of homogeneous treatment against heterogeneous treatment effects; set type to 3 to test the null hypothesis of uniformly non-negative treatment effects against the presence of negative effects; and set type to 4 to test the null hypothesis of uniformly non-positive treatment effects against the presence of positive effects at some quantiles.
std.opt	either 0 or 1. If std.opt=1, the test statistic is standardized so that the variance is equalized across quantiles; if std.opt=0, the test is not standardized.

Value

A list with elements:

test.stat

test statistics.

cr.value

critical values.

p.val

p values.

References

Zhongjun Qu, Jungmo Yoon, Pierre Perron (2024), "Inference on Conditional Quantile Processes in Partially Linear Models with Applications to the Impact of Unemployment Benefits," The Review of Economics and Statistics; https://doi.org/10.1162/rest_a_01168

See Also

rdq.test

Examples

n = 500
x = runif(n,min=-4,max=4)
d = (x > 0)
y = x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
tlevel2 = c(0.05,tlevel,0.95)
hh = rep(2,length(tlevel))
hh2 = rep(2,length(tlevel2))
sel = tlevel2 %in% tlevel

ab = rdq(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel2,h.tau=hh2,cov=0)
delta = c(0.05,0.09,0.14,0.17,0.19,0.17,0.14,0.09,0.05)
fp = rdq.condf(x=x,Q=ab$qp.est,bcoe=ab$bcoe.p,taus=tlevel,taul=tlevel2,delta,cov=0)
fm = rdq.condf(x=x,Q=ab$qm.est,bcoe=ab$bcoe.m,taus=tlevel,taul=tlevel2,delta,cov=0)
bp = rdq.bias(y[d==1],x[d==1],dz=0,x0=0,z0=NULL,taus=tlevel,hh,hh,fx=fp$ff[(d==1),],cov=0)
bm = rdq.bias(y[d==0],x[d==0],dz=0,x0=0,z0=NULL,taus=tlevel,hh,hh,fx=fm$ff[(d==0),],cov=0)

sa = rdq.sim(x=x,d=d,x0=0,z0=NULL,dz=0,cov=0,tt=tlevel,hh,hh,fxp=fp$ff,fxm=fm$ff,n.sim=200)
bt <- run.test(n,dz=0,taus=tlevel,hh,Dc.p=sa$dcp,Dc.m=sa$dcm,Dr.p=sa$drp,Dr.m=sa$drm,
Qy.p=as.matrix(ab$qp.est[sel,]),Qy.m=as.matrix(ab$qm.est[sel,]),bias.p=bp$bias,bias.m=bm$bias,
cov=0,bias=1,alpha=0.1,n.sim=200,test.type=1,std.opt=1)

---

Uniform confidence band for QTE.

Description

summary.qte returns uniform confidence bands and standard errors for QTE estimates.

Usage

## S3 method for class 'qte'
summary(object, alpha, ...)

Arguments

object	It is an object of class "qte" produced by rd.qte.
alpha	a number between 0 and 1, the desired significance level. For example, when alpha=0.1, one will get a 90% uniform band.
...	optional arguments.

Value

A list with elements:

qte

QTE estimates.

uband

uniform confidence band for QTE. If bias=1, the band is robust capturing the effect of the bias correction. If bias=0, no bias correction is implemented.

sigma

standard errors for each quantile level. If bias=1, its value captures the effect of the bias correction. If bias=0, no bias correction is implemented.

qp.est

conditional quantile estimates on the right side of $x_{0}$ (or for the $D=1$ group).

qm.est

conditional quantile estimates on the left side of $x_{0}$ (or for the $D=0$ group).

uband.p

uniform confidence band for conditional quantiles on the right side of $x_{0}$.

uband.m

uniform confidence band for conditional quantiles on the left side of $x_{0}$.

References

Zhongjun Qu, Jungmo Yoon, Pierre Perron (2024), "Inference on Conditional Quantile Processes in Partially Linear Models with Applications to the Impact of Unemployment Benefits," The Review of Economics and Statistics; https://doi.org/10.1162/rest_a_01168

Zhongjun Qu and Jungmo Yoon (2019), "Uniform Inference on Quantile Effects under Sharp Regression Discontinuity Designs," Journal of Business and Economic Statistics, 37(4), 625–647; https://doi.org/10.1080/07350015.2017.1407323

Examples

# Without covariate
n <- 500
x <- runif(n,min=-4,max=4)
d <- (x > 0)
y <- x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + rnorm(n)
tlevel = seq(0.1,0.9,by=0.1)
A <- rd.qte(y=y,x=x,d=d,x0=0,z0=NULL,tau=tlevel,bdw=2,cov=0,bias=1)
A2 <- summary(A,alpha=0.1)

# (continued) With covariates
z <- sample(c(0,1),n,replace=TRUE)
y <- x + 0.3*(x^2) - 0.1*(x^3) + 1.5*d + d*z + rnorm(n)
A <- rd.qte(y=y,x=cbind(x,z),d=d,x0=0,z0=c(0,1),tau=tlevel,bdw=2,cov=1,bias=1)
A2 <- summary(A,alpha=0.1)

---