discreteQ: Uniform inference on quantile and quantile effect functions...
In bmelly/discreteQ: Inference on quantile and quantile effect functions for discrete outcomes
Description
Usage
Arguments
Details
Value
References
See Also
Examples
Description
The function discreteQ provides uniform confidence bands for the unconditional
quantile function, the quantile treatment effect function or the
decomposition of the observed difference between the quantile function of an
outcome for two groups. This function implements the algorithms suggested in Chernozhukov et al. (2019).
See also the vignette available with vignette("discreteQ", package="discreteQ").
Usage
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discreteQ(
y,
d = NULL,
x = NULL,
w = NULL,
decomposition = FALSE,
q.range = c(0.05, 0.95),
method = NULL,
bsrep = 200,
alpha = 0.05,
ys = NULL,
cl = NULL,
cluster = NULL,
old.res = NULL,
return.boot = FALSE,
list_of_seeds = NULL,
return.seeds = FALSE,
estim.glm = fastglm::fastglm,
par.estim = NULL
)
Arguments
y
outcome (vector of length n).
d
treatment or group variable (binary vector of length n).
x
matrix of regressors (n x p matrix). x must include a constant if appropriate (i.e. the constant is NOT added automatically). The function model.matrix can be used to create the matrix x if factor variables or interaction terms are present.
w
sampling weights (vector of length n).
decomposition
logical, indicating if the decomposition of the observed
difference between the control and treated quantile functions should be
performed. By default, inference about the quantile treatment effect
function is performed.
q.range
vector of length 2 that provides the lowest and highest
quantile indexes. The uniform bands will cover the whole QF and QE
function in this range. Default is c(0.05,0.95).
method
link function for the distribution regression model. Possible
values: "logit" (the default), "probit", "cloglog", "lpm" (linear
probability model, i.e. OLS estimation of binary regressions), "cauchit", "drp" (incomplete gamma link function suggested in Chernozhukov, Fernandez-Val, Melly and Wütrich (2019). The maximum likelihood estimator of the fully parametric Poisson regression model is used if method = "poisson".
This argument is relevant only if there are regressors in x.
bsrep
number of bootstrap replications. Default: 200.
alpha
confidence level. Default: 0.05.
ys
specifies the thresholds at which the cumulative distribution
function will be estimated. This argument can be specified either as a
scalar that will be interpreted as the number of thresholds or as a vector
that will contain the values of the thresholds. By default, the cdf is
estimated at all distinct observed values of the outcome in the sample if
there are less than 100 unique values and at the 99 empirical percentiles
of the outcomes if there are more than 100 distinct values.
cl
a cluster object as returned by the function makeCluster. Parallel computing is
not used if this argument is not specified.
cluster
vector that specifies to which group each observation belongs.
The cluster weighted bootstrap is used if this argument is specified. Otherwise,
simple random sampling is assumed.
old.res
a discreteQ object (obtained with the argument return.boot set
to TRUE). This argument allows for instance to change the size alpha or the quantile range q.range
without recomputing the estimates.
return.boot
logical scalar. The results of the bootstrap are return in
the matrix F.b when this argument is set to TRUE.
list_of_seeds
list of seeds for L'Ecuyer RNG. The length of this list
must be the same as the value of the argument bsrep.
return.seeds
logical scalar. The list of seeds is returned by the
function if this argument is set to TRUE.
estim.glm
function used to estimate the binary regressions if method is "logit", "probit", "cloglog" or "cauchit". The default is the function fastglm() from the package fastglm. Tested alternatives: glm.fit, glm2::glm.fit2, speedglm::speedglm.wfit.
par.estim
arguments to be passed to the function selected by estim.glm. For instance, the arguments method, tol and maxit of fastglm can be set.
Details
The function discreteQ can be used in three different ways:
-
First, if no treatment variable is specified in the argument d, then the the
command will provide uniform bands for the unconditional quantile and distribution functions of
y.
-
Second, if a treatment variable is specified in the argument d but
decomposition=FALSE, then the command will provide uniform bands that
cover the quantile and distribution functions of the treated and control outcomes and the
quantile treatment effect function (the difference between both quantile functions).
Third, if a treatment variable is
specified in the argument d and decomposition=TRUE, then the command provides
uniform bands that cover both distribution functions, both quantile functions as well as the
decomposition of their difference into an explained (by the regressors
x) and unexplained component.
The output of the function is a long list of step functions (see below for the details). We recommend to use
the plot.discreteQ and summary.discreteQ
functions to analyze the results. For further details see the vignette available with vignette("discreteQ", package="discreteQ").
Value
discreteQ returns an object of class "discreteQ". There are methods available for plotting ("plot", see
plot.discreteQ) and summarizing ("summary", see
summary.discreteQ) "discreteQ" objects. We recommend
using them to analyze the results.
The components contained by an object of class "discreteQ" depend on the way the function has been called. We can distinguish the same three cases as above:
-
If no treatment variable is specified in the argument d, then discreteQ contains the following components:
QThe empirical quantile function of the outcome. This is a function of class stepfun.
lb.QThe lower bound of the uniform confidence band for the unconditional quantile function of the outcome. This is a function of class stepfun.
ub.QThe upper bound of the uniform confidence band for the unconditional quantile function of the outcome. This is a function of class stepfun.
FThe empirical distribution function of the outcome. This is a function of class stepfun.
lb.FThe lower bound of the uniform confidence band for the unconditional distribution function of the outcome. This is a function of class stepfun.
ub.FThe upper bound of the uniform confidence band for the unconditional distribution function of the outcome. This is a function of class stepfun.
q.rangeVector of length 2 that contains the lowest and highest quantile indexes. The uniform bands for the quantile function cover the true quantile function in this quantile range. The uniform bands for the distribution function cover the true function in the range of values of the outcome that are between the quantiles corresponding to these indexes.
ysVector containing the thresholds at which the cumulative distribution has been estimated.
bsrepScalar containing the number of performed bootstrap replications.
modelString scalar that takes the value "univariate" in this case.
methodString scalar that takes the value "empirical" in this case.
F.bMatrix with length(ys) rows and bsrep columns. Each columns contains the estimated distribution function for the corresponding bootstrap replication. This object, which can be voluminous, is returned only if return.boot = TRUE.
seedsList of length bsrep containing the seeds used for L'Ecuyer's RNG in the bootstrap replications. This object is returned only if return.seeds = TRUE.
-
If a treatment variable is specified in the argument d but decomposition=FALSE, then discreteQ contains the components below. Note that the uniform bands jointly cover the true Q0, Q1, F0, F1 and QTE functions with probability 1-alpha.
Q0The estimated unconditional quantile function of the control outcome. This is the quantile function of the outcome that we would observe if all observations had d = 0. If x = NULL, this is simply the empirical quantile function of the outcome for the subsample with d = 0. If regressors have been provided in the argument x, then the estimated conditional distribution of the outcome in the sample with d = 0 is integrated over the distribution of the covariates in the whole sample. This is a function of class stepfun.
lb.Q0The lower bound of the uniform confidence band for the unconditional quantile function of the control outcome. This is a function of class stepfun.
ub.Q0The upper bound of the uniform confidence band for the unconditional quantile function of the control outcome. This is a function of class stepfun.
Q1The estimated unconditional quantile function of the treated outcome. This is the quantile function of the outcome that we would observe if all observations had d = 1. If x = NULL, this is simply the empirical quantile function of the outcome for the subsample with d = 1. If regressors have been provided in the argument x, then the estimated conditional distribution of the outcome in the sample with d = 1 is integrated over the distribution of the covariates in the whole sample. This is a function of class stepfun.
lb.Q1The lower bound of the uniform confidence band for the unconditional quantile function of the treated outcome. This is a function of class stepfun.
ub.Q1The upper bound of the uniform confidence band for the unconditional quantile function of the treated outcome. This is a function of class stepfun.
QTEThe estimated quantile treatment effect function: QTE = Q1 - Q0. This is a function of class stepfun.
lb.QTEThe lower bound of the uniform confidence band for the quantile treatment effect. This is a function of class stepfun.
ub.QTEThe upper bound of the uniform confidence band for the quantile treatment effect. This is a function of class stepfun.
F0The estimated unconditional distribution function of the control outcome. This is the distribution function of the outcome that we would observe if all observations had d = 0. If x = NULL, this is simply the empirical distribution function of the outcome for the subsample with d = 0. If regressors have been provided in the argument x, then the estimated conditional distribution of the outcome in the sample with d = 0 is integrated over the distribution of the covariates in the whole sample. This is a function of class stepfun.
lb.F0The lower bound of the uniform confidence band for the unconditional distribution function of the control outcome. This is a function of class stepfun.
ub.F0The upper bound of the uniform confidence band for the unconditional distribution function of the control outcome. This is a function of class stepfun.
F1The estimated unconditional distribution function of the treated outcome. This is the distribution function of the outcome that we would observe if all observations had d = 1. If x = NULL, this is simply the empirical distribution function of the outcome for the subsample with d = 1. If regressors have been provided in the argument x, then the estimated conditional distribution of the outcome in the sample with d = 1 is integrated over the distribution of the covariates in the whole sample. This is a function of class stepfun.
lb.F1The lower bound of the uniform confidence band for the unconditional distribution function of the treated outcome. This is a function of class stepfun.
ub.F1The upper bound of the uniform confidence band for the unconditional distribution function of the treated outcome. This is a function of class stepfun.
q.rangeVector of length 2 that contains the lowest and highest quantile indexes. The uniform bands for the quantile functions cover the true quantile function in this quantile range. The uniform bands for the distribution functions cover the true function in the range of values of the outcome that are between the quantiles corresponding to thes indexes.
ys0Vector containing the thresholds at which the cumulative distribution of the control outcome has been estimated.
ys1Vector containing the thresholds at which the cumulative distribution of the treated outcome has been estimated.
bsrepScalar containing the number of performed bootstrap replications.
modelString scalar that takes the value "qte" in this case.
methodString scalar. Name of the method used to estimate the conditional distribution functions.
F.bMatrix with length(ys0) + length(ys1) rows and bsrep columns. Each columns contains the estimated distribution functions for the corresponding bootstrap replication. The first length(ys0) rows contains the estimated distribution function for the control outcome F0. The remaining length(ys1) rows contains the estimated distribution function for the treated outcome F1. This object, which can be voluminous, is returned only if return.boot = TRUE.
seedsList of length bsrep containing the seeds used for L'Ecuyer's RNG in the bootstrap replications. This object is returned only if return.seeds = TRUE.
-
If a treatment variable is specified in the argument d and decomposition=TRUE, then discreteQ contains the components below. Note that the uniform bands jointly cover the true functions Q0, Q1, Qc, F0, F1, and Fc as well as the difference between any two of these functions with probability 1-alpha.
Q0The empirical quantile function of the outcome for the group with d = 0. This is a function of class stepfun.
lb.Q0The lower bound of the uniform confidence band for the unconditional quantile function of outcome in the group with d = 0. This is a function of class stepfun.
ub.Q0The upper bound of the uniform confidence band for the unconditional quantile function of outcome in the group with d = 0. This is a function of class stepfun.
Q1The empirical quantile function of the outcome for the group with d = 1. This is a function of class stepfun.
lb.Q1The lower bound of the uniform confidence band for the unconditional quantile function of outcome in the group with d = 1. This is a function of class stepfun.
ub.Q1The upper bound of the uniform confidence band for the unconditional quantile function of outcome in the group with d = 1. This is a function of class stepfun.
QcThe estimated counterfactual quantile function of the outcome that we would observe if the distribution of the covariates was the same as that of the group with d = 0 and the conditional distribution of the outcome given the covariates was the same as that of the group with d = 1. This is a function of class stepfun.
lb.QcThe lower bound of the uniform confidence band for the counterfactual quantile function. This is a function of class stepfun.
ub.QcThe upper bound of the uniform confidence band for the counterfactual quantile function. This is a function of class stepfun.
F0The empirical distribution function of the outcome for the group with d = 0. This is a function of class stepfun.
lb.F0The lower bound of the uniform confidence band for the unconditional distribution function of outcome in the group with d = 0. This is a function of class stepfun.
ub.F0The upper bound of the uniform confidence band for the unconditional distribution function of outcome in the group with d = 0. This is a function of class stepfun.
F1The empirical distribution function of the outcome for the group with d = 1. This is a function of class stepfun.
lb.F1The lower bound of the uniform confidence band for the unconditional distribution function of outcome in the group with d = 1. This is a function of class stepfun.
ub.F1The upper bound of the uniform confidence band for the unconditional distribution function of outcome in the group with d = 1. This is a function of class stepfun.
FcThe estimated counterfactual distribution function of the outcome that we would observe if the distribution of the covariates was the same as that of the group with d = 0 and the conditional distribution of the outcome given the covariates was the same as that of the group with d = 1. This is a function of class stepfun.
lb.FcThe lower bound of the uniform confidence band for the counterfactual distribution function. This is a function of class stepfun.
ub.FcThe upper bound of the uniform confidence band for the counterfactual distribution function. This is a function of class stepfun.
observedThe difference between the observed quantile function for the group d = 1 and and the observed quantile function for the group with d = 0: Q1 - Q0. This is a function of class stepfun.
lb.observedThe lower bound of the uniform confidence band for Q1 - Q0. This is a function of class stepfun.
ub.observedThe upper bound of the uniform confidence band for Q1 - Q0. This is a function of class stepfun.
compositionThe difference between the observed quantile function for the group with d = 1 and the counterfactual quantile function: Q1 - Qc. This is a function of class stepfun.
lb.compositionThe lower bound of the uniform confidence band for Q1 - Qc. This is a function of class stepfun.
ub.compositionThe upper bound of the uniform confidence band for Q1 - Qc. This is a function of class stepfun.
unexplainedThe difference between the counterfactual quantile function and the quantile function for the group with d = 0: Qc - Q0. This is a function of class stepfun.
lb.unexplainedThe lower bound of the uniform confidence band for Qc - Q0. This is a function of class stepfun.
ub.unexplainedThe upper bound of the uniform confidence band for Qc - Q0. This is a function of class stepfun.
q.rangeVector of length 2 that contains the lowest and highest quantile indexes. The uniform bands for the quantile functions cover the true quantile function in this quantile range. The uniform bands for the distribution functions cover the true function in the range of values of the outcome that are between the quantiles corresponding to thes indexes.
ys0Vector containing the thresholds at which the cumulative distribution of the outcome for the group with d = 0 has been estimated.
ys1Vector containing the thresholds at which the cumulative distribution of the outcome for the group with d = 1 and the counterfactual distribution function have been estimated.
bsrepScalar containing the number of performed bootstrap replications.
modelString scalar that takes the value "distribution" in this case.
methodString scalar. Name of the method used to estimate the conditional distribution functions.
F.bMatrix with length(ys0) + 2 * length(ys1) rows and bsrep columns. Each columns contains the estimated distribution functions for the corresponding bootstrap replication. The first length(ys0) rows contains the estimated distribution function for group 0, F0. The next length(ys1) rows contains the estimated distribution function for group 1, F1. The remaining length(ys1) rows contains the estimated counterfactual distribution, Fc. This object, which can be voluminous, is returned only if return.boot = TRUE.
seedsList of length bsrep containing the seeds used for L'Ecuyer's RNG in the bootstrap replications. This object is returned only if return.seeds = TRUE.
References
Chernozhukov, Victor, Iván Fernández-Val, Blaise Melly, and Kaspar Wüthrich. 2019. “Generic Inference on Quantile and Quantile Effect Functions for Discrete Outcomes.” arXiv Preprint, https://arxiv.org/abs/1608.05142.
See Also
For continous outcomes, see also counterfactual from the package Counterfactual.
Examples
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##Example 1: univariate quantile function
#Generate the data
set.seed(1234)
outcome <- rpois(100, 3)
#Estimate the functions and the confidence bands
results1 <- discreteQ(outcome)
#Table containing the estimated quantile function with its confidence band
summary(results1)
#Plot the estimated quantile function with its confidence band
plot(results1)
##Example 2: quantile treatment effect function (QTE)
#Generate the data
set.seed(1234)
treatment <- c(rep(0,100), rep(1,100))
reg <- rbinom(200, 1, 0.4 + treatment*0.2)
outcome <- rpois(200, lambda = 1+reg)
#Estimate the functions and the confidence bands (takes about 1 minute)
results2 <- discreteQ(outcome, treatment, cbind(1, reg))
#Table containing the estimated QTE function with its confidence band
summary(results2)
#Plot the QTE with its confidence band
plot(results2)
#Plot the quantile function of the control outcome
plot(results2, which="Q0")
#Add the quantile function of the treated outcome
plot(results2, which="Q1", add=TRUE, shift=0.2, col.l="dark green", col.b="light green")
##Example 3: decomposition
#Generate the data
set.seed(1234)
group <- c(rep(0,100), rep(1,100))
reg <- rbinom(200, 1, 0.2 + group*0.6)
outcome <- rpois(200, lambda = exp(-3+4*reg))
#Estimate the functions and the confidence bands (takes about 30 seconds)
results3 <- discreteQ(outcome, group, cbind(1, reg), decomposition=TRUE)
#Table containing the unexplained component with its confidence band
summary(results3)
#Table containing the difference between the observed quantile functions
summary(results3, which="observed")
#Plot the observed quantile functions and their decomposition
plot(results3)
#Plot only the composition component
plot(results3, which="composition")
bmelly/discreteQ documentation built on May 22, 2021, 7:55 a.m.
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Description Usage Arguments Details Value References See Also Examples
Description
The function discreteQ provides uniform confidence bands for the unconditional
quantile function, the quantile treatment effect function or the
decomposition of the observed difference between the quantile function of an
outcome for two groups. This function implements the algorithms suggested in Chernozhukov et al. (2019).
See also the vignette available with vignette("discreteQ", package="discreteQ").
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | discreteQ(
y,
d = NULL,
x = NULL,
w = NULL,
decomposition = FALSE,
q.range = c(0.05, 0.95),
method = NULL,
bsrep = 200,
alpha = 0.05,
ys = NULL,
cl = NULL,
cluster = NULL,
old.res = NULL,
return.boot = FALSE,
list_of_seeds = NULL,
return.seeds = FALSE,
estim.glm = fastglm::fastglm,
par.estim = NULL
)
|
Arguments
y |
outcome (vector of length n). |
d |
treatment or group variable (binary vector of length n). |
x |
matrix of regressors (n x p matrix). |
w |
sampling weights (vector of length n). |
decomposition |
logical, indicating if the decomposition of the observed difference between the control and treated quantile functions should be performed. By default, inference about the quantile treatment effect function is performed. |
q.range |
vector of length 2 that provides the lowest and highest quantile indexes. The uniform bands will cover the whole QF and QE function in this range. Default is c(0.05,0.95). |
method |
link function for the distribution regression model. Possible
values: "logit" (the default), "probit", "cloglog", "lpm" (linear
probability model, i.e. OLS estimation of binary regressions), "cauchit", "drp" (incomplete gamma link function suggested in Chernozhukov, Fernandez-Val, Melly and Wütrich (2019). The maximum likelihood estimator of the fully parametric Poisson regression model is used if |
bsrep |
number of bootstrap replications. Default: 200. |
alpha |
confidence level. Default: 0.05. |
ys |
specifies the thresholds at which the cumulative distribution function will be estimated. This argument can be specified either as a scalar that will be interpreted as the number of thresholds or as a vector that will contain the values of the thresholds. By default, the cdf is estimated at all distinct observed values of the outcome in the sample if there are less than 100 unique values and at the 99 empirical percentiles of the outcomes if there are more than 100 distinct values. |
cl |
a cluster object as returned by the function |
cluster |
vector that specifies to which group each observation belongs. The cluster weighted bootstrap is used if this argument is specified. Otherwise, simple random sampling is assumed. |
old.res |
a discreteQ object (obtained with the argument return.boot set
to TRUE). This argument allows for instance to change the size |
return.boot |
logical scalar. The results of the bootstrap are return in the matrix F.b when this argument is set to TRUE. |
list_of_seeds |
list of seeds for L'Ecuyer RNG. The length of this list
must be the same as the value of the argument |
return.seeds |
logical scalar. The list of seeds is returned by the function if this argument is set to TRUE. |
estim.glm |
function used to estimate the binary regressions if |
par.estim |
arguments to be passed to the function selected by |
Details
The function discreteQ can be used in three different ways:
-
First, if no treatment variable is specified in the argument
d, then the the command will provide uniform bands for the unconditional quantile and distribution functions ofy. -
Second, if a treatment variable is specified in the argument
dbutdecomposition=FALSE, then the command will provide uniform bands that cover the quantile and distribution functions of the treated and control outcomes and the quantile treatment effect function (the difference between both quantile functions). Third, if a treatment variable is specified in the argument
danddecomposition=TRUE, then the command provides uniform bands that cover both distribution functions, both quantile functions as well as the decomposition of their difference into an explained (by the regressorsx) and unexplained component.
The output of the function is a long list of step functions (see below for the details). We recommend to use
the plot.discreteQ and summary.discreteQ
functions to analyze the results. For further details see the vignette available with vignette("discreteQ", package="discreteQ").
Value
discreteQ returns an object of class "discreteQ". There are methods available for plotting ("plot", see
plot.discreteQ) and summarizing ("summary", see
summary.discreteQ) "discreteQ" objects. We recommend
using them to analyze the results.
The components contained by an object of class "discreteQ" depend on the way the function has been called. We can distinguish the same three cases as above:
-
If no treatment variable is specified in the argument
d, thendiscreteQcontains the following components:QThe empirical quantile function of the outcome. This is a function of class
stepfun.lb.QThe lower bound of the uniform confidence band for the unconditional quantile function of the outcome. This is a function of class
stepfun.ub.QThe upper bound of the uniform confidence band for the unconditional quantile function of the outcome. This is a function of class
stepfun.FThe empirical distribution function of the outcome. This is a function of class
stepfun.lb.FThe lower bound of the uniform confidence band for the unconditional distribution function of the outcome. This is a function of class
stepfun.ub.FThe upper bound of the uniform confidence band for the unconditional distribution function of the outcome. This is a function of class
stepfun.q.rangeVector of length 2 that contains the lowest and highest quantile indexes. The uniform bands for the quantile function cover the true quantile function in this quantile range. The uniform bands for the distribution function cover the true function in the range of values of the outcome that are between the quantiles corresponding to these indexes.
ysVector containing the thresholds at which the cumulative distribution has been estimated.
bsrepScalar containing the number of performed bootstrap replications.
modelString scalar that takes the value "univariate" in this case.
methodString scalar that takes the value "empirical" in this case.
F.bMatrix with
length(ys)rows andbsrepcolumns. Each columns contains the estimated distribution function for the corresponding bootstrap replication. This object, which can be voluminous, is returned only ifreturn.boot = TRUE.seedsList of length
bsrepcontaining the seeds used for L'Ecuyer's RNG in the bootstrap replications. This object is returned only ifreturn.seeds = TRUE.
-
If a treatment variable is specified in the argument
dbutdecomposition=FALSE, thendiscreteQcontains the components below. Note that the uniform bands jointly cover the true Q0, Q1, F0, F1 and QTE functions with probability1-alpha.Q0The estimated unconditional quantile function of the control outcome. This is the quantile function of the outcome that we would observe if all observations had
d = 0. Ifx = NULL, this is simply the empirical quantile function of the outcome for the subsample withd = 0. If regressors have been provided in the argumentx, then the estimated conditional distribution of the outcome in the sample withd = 0is integrated over the distribution of the covariates in the whole sample. This is a function of classstepfun.lb.Q0The lower bound of the uniform confidence band for the unconditional quantile function of the control outcome. This is a function of class
stepfun.ub.Q0The upper bound of the uniform confidence band for the unconditional quantile function of the control outcome. This is a function of class
stepfun.Q1The estimated unconditional quantile function of the treated outcome. This is the quantile function of the outcome that we would observe if all observations had
d = 1. Ifx = NULL, this is simply the empirical quantile function of the outcome for the subsample withd = 1. If regressors have been provided in the argumentx, then the estimated conditional distribution of the outcome in the sample withd = 1is integrated over the distribution of the covariates in the whole sample. This is a function of classstepfun.lb.Q1The lower bound of the uniform confidence band for the unconditional quantile function of the treated outcome. This is a function of class
stepfun.ub.Q1The upper bound of the uniform confidence band for the unconditional quantile function of the treated outcome. This is a function of class
stepfun.QTEThe estimated quantile treatment effect function:
QTE = Q1 - Q0. This is a function of classstepfun.lb.QTEThe lower bound of the uniform confidence band for the quantile treatment effect. This is a function of class
stepfun.ub.QTEThe upper bound of the uniform confidence band for the quantile treatment effect. This is a function of class
stepfun.F0The estimated unconditional distribution function of the control outcome. This is the distribution function of the outcome that we would observe if all observations had
d = 0. Ifx = NULL, this is simply the empirical distribution function of the outcome for the subsample withd = 0. If regressors have been provided in the argumentx, then the estimated conditional distribution of the outcome in the sample withd = 0is integrated over the distribution of the covariates in the whole sample. This is a function of classstepfun.lb.F0The lower bound of the uniform confidence band for the unconditional distribution function of the control outcome. This is a function of class
stepfun.ub.F0The upper bound of the uniform confidence band for the unconditional distribution function of the control outcome. This is a function of class
stepfun.F1The estimated unconditional distribution function of the treated outcome. This is the distribution function of the outcome that we would observe if all observations had
d = 1. Ifx = NULL, this is simply the empirical distribution function of the outcome for the subsample withd = 1. If regressors have been provided in the argumentx, then the estimated conditional distribution of the outcome in the sample withd = 1is integrated over the distribution of the covariates in the whole sample. This is a function of classstepfun.lb.F1The lower bound of the uniform confidence band for the unconditional distribution function of the treated outcome. This is a function of class
stepfun.ub.F1The upper bound of the uniform confidence band for the unconditional distribution function of the treated outcome. This is a function of class
stepfun.q.rangeVector of length 2 that contains the lowest and highest quantile indexes. The uniform bands for the quantile functions cover the true quantile function in this quantile range. The uniform bands for the distribution functions cover the true function in the range of values of the outcome that are between the quantiles corresponding to thes indexes.
ys0Vector containing the thresholds at which the cumulative distribution of the control outcome has been estimated.
ys1Vector containing the thresholds at which the cumulative distribution of the treated outcome has been estimated.
bsrepScalar containing the number of performed bootstrap replications.
modelString scalar that takes the value "qte" in this case.
methodString scalar. Name of the method used to estimate the conditional distribution functions.
F.bMatrix with
length(ys0) + length(ys1)rows andbsrepcolumns. Each columns contains the estimated distribution functions for the corresponding bootstrap replication. The firstlength(ys0)rows contains the estimated distribution function for the control outcome F0. The remaininglength(ys1)rows contains the estimated distribution function for the treated outcome F1. This object, which can be voluminous, is returned only ifreturn.boot = TRUE.seedsList of length
bsrepcontaining the seeds used for L'Ecuyer's RNG in the bootstrap replications. This object is returned only ifreturn.seeds = TRUE.
-
If a treatment variable is specified in the argument
danddecomposition=TRUE, thendiscreteQcontains the components below. Note that the uniform bands jointly cover the true functions Q0, Q1, Qc, F0, F1, and Fc as well as the difference between any two of these functions with probability1-alpha.Q0The empirical quantile function of the outcome for the group with
d = 0. This is a function of classstepfun.lb.Q0The lower bound of the uniform confidence band for the unconditional quantile function of outcome in the group with
d = 0. This is a function of classstepfun.ub.Q0The upper bound of the uniform confidence band for the unconditional quantile function of outcome in the group with
d = 0. This is a function of classstepfun.Q1The empirical quantile function of the outcome for the group with
d = 1. This is a function of classstepfun.lb.Q1The lower bound of the uniform confidence band for the unconditional quantile function of outcome in the group with
d = 1. This is a function of classstepfun.ub.Q1The upper bound of the uniform confidence band for the unconditional quantile function of outcome in the group with
d = 1. This is a function of classstepfun.QcThe estimated counterfactual quantile function of the outcome that we would observe if the distribution of the covariates was the same as that of the group with
d = 0and the conditional distribution of the outcome given the covariates was the same as that of the group withd = 1. This is a function of classstepfun.lb.QcThe lower bound of the uniform confidence band for the counterfactual quantile function. This is a function of class
stepfun.ub.QcThe upper bound of the uniform confidence band for the counterfactual quantile function. This is a function of class
stepfun.F0The empirical distribution function of the outcome for the group with
d = 0. This is a function of classstepfun.lb.F0The lower bound of the uniform confidence band for the unconditional distribution function of outcome in the group with
d = 0. This is a function of classstepfun.ub.F0The upper bound of the uniform confidence band for the unconditional distribution function of outcome in the group with
d = 0. This is a function of classstepfun.F1The empirical distribution function of the outcome for the group with
d = 1. This is a function of classstepfun.lb.F1The lower bound of the uniform confidence band for the unconditional distribution function of outcome in the group with
d = 1. This is a function of classstepfun.ub.F1The upper bound of the uniform confidence band for the unconditional distribution function of outcome in the group with
d = 1. This is a function of classstepfun.FcThe estimated counterfactual distribution function of the outcome that we would observe if the distribution of the covariates was the same as that of the group with
d = 0and the conditional distribution of the outcome given the covariates was the same as that of the group withd = 1. This is a function of classstepfun.lb.FcThe lower bound of the uniform confidence band for the counterfactual distribution function. This is a function of class
stepfun.ub.FcThe upper bound of the uniform confidence band for the counterfactual distribution function. This is a function of class
stepfun.observedThe difference between the observed quantile function for the group
d = 1and and the observed quantile function for the group withd = 0: Q1 - Q0. This is a function of classstepfun.lb.observedThe lower bound of the uniform confidence band for Q1 - Q0. This is a function of class
stepfun.ub.observedThe upper bound of the uniform confidence band for Q1 - Q0. This is a function of class
stepfun.compositionThe difference between the observed quantile function for the group with
d = 1and the counterfactual quantile function: Q1 - Qc. This is a function of classstepfun.lb.compositionThe lower bound of the uniform confidence band for Q1 - Qc. This is a function of class
stepfun.ub.compositionThe upper bound of the uniform confidence band for Q1 - Qc. This is a function of class
stepfun.unexplainedThe difference between the counterfactual quantile function and the quantile function for the group with
d = 0: Qc - Q0. This is a function of classstepfun.lb.unexplainedThe lower bound of the uniform confidence band for Qc - Q0. This is a function of class
stepfun.ub.unexplainedThe upper bound of the uniform confidence band for Qc - Q0. This is a function of class
stepfun.q.rangeVector of length 2 that contains the lowest and highest quantile indexes. The uniform bands for the quantile functions cover the true quantile function in this quantile range. The uniform bands for the distribution functions cover the true function in the range of values of the outcome that are between the quantiles corresponding to thes indexes.
ys0Vector containing the thresholds at which the cumulative distribution of the outcome for the group with
d = 0has been estimated.ys1Vector containing the thresholds at which the cumulative distribution of the outcome for the group with
d = 1and the counterfactual distribution function have been estimated.bsrepScalar containing the number of performed bootstrap replications.
modelString scalar that takes the value "distribution" in this case.
methodString scalar. Name of the method used to estimate the conditional distribution functions.
F.bMatrix with
length(ys0) + 2 * length(ys1)rows andbsrepcolumns. Each columns contains the estimated distribution functions for the corresponding bootstrap replication. The firstlength(ys0)rows contains the estimated distribution function for group 0, F0. The nextlength(ys1)rows contains the estimated distribution function for group 1, F1. The remaininglength(ys1)rows contains the estimated counterfactual distribution, Fc. This object, which can be voluminous, is returned only ifreturn.boot = TRUE.seedsList of length
bsrepcontaining the seeds used for L'Ecuyer's RNG in the bootstrap replications. This object is returned only ifreturn.seeds = TRUE.
References
Chernozhukov, Victor, Iván Fernández-Val, Blaise Melly, and Kaspar Wüthrich. 2019. “Generic Inference on Quantile and Quantile Effect Functions for Discrete Outcomes.” arXiv Preprint, https://arxiv.org/abs/1608.05142.
See Also
For continous outcomes, see also counterfactual from the package Counterfactual.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | ##Example 1: univariate quantile function
#Generate the data
set.seed(1234)
outcome <- rpois(100, 3)
#Estimate the functions and the confidence bands
results1 <- discreteQ(outcome)
#Table containing the estimated quantile function with its confidence band
summary(results1)
#Plot the estimated quantile function with its confidence band
plot(results1)
##Example 2: quantile treatment effect function (QTE)
#Generate the data
set.seed(1234)
treatment <- c(rep(0,100), rep(1,100))
reg <- rbinom(200, 1, 0.4 + treatment*0.2)
outcome <- rpois(200, lambda = 1+reg)
#Estimate the functions and the confidence bands (takes about 1 minute)
results2 <- discreteQ(outcome, treatment, cbind(1, reg))
#Table containing the estimated QTE function with its confidence band
summary(results2)
#Plot the QTE with its confidence band
plot(results2)
#Plot the quantile function of the control outcome
plot(results2, which="Q0")
#Add the quantile function of the treated outcome
plot(results2, which="Q1", add=TRUE, shift=0.2, col.l="dark green", col.b="light green")
##Example 3: decomposition
#Generate the data
set.seed(1234)
group <- c(rep(0,100), rep(1,100))
reg <- rbinom(200, 1, 0.2 + group*0.6)
outcome <- rpois(200, lambda = exp(-3+4*reg))
#Estimate the functions and the confidence bands (takes about 30 seconds)
results3 <- discreteQ(outcome, group, cbind(1, reg), decomposition=TRUE)
#Table containing the unexplained component with its confidence band
summary(results3)
#Table containing the difference between the observed quantile functions
summary(results3, which="observed")
#Plot the observed quantile functions and their decomposition
plot(results3)
#Plot only the composition component
plot(results3, which="composition")
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