Help for package rddensity

Type:

Package

Title:

Manipulation Testing Based on Density Discontinuity

Description:

Density discontinuity testing (a.k.a. manipulation testing) is commonly employed in regression discontinuity designs and other program evaluation settings to detect perfect self-selection (manipulation) around a cutoff where treatment/policy assignment changes. This package implements manipulation testing procedures using the local polynomial density estimators: rddensity() to construct test statistics and p-values given a prespecified cutoff, rdbwdensity() to perform data-driven bandwidth selection, and rdplotdensity() to construct density plots.

Version:

2.6

Author:

Matias D. Cattaneo [aut], Michael Jansson [aut], Xinwei Ma [aut, cre]

Maintainer:

Xinwei Ma <x1ma@ucsd.edu>

Imports:

ggplot2, lpdensity (≥ 2.2)

Depends:

R (≥ 3.1.0)

License:

GPL-2

Encoding:

UTF-8

RoxygenNote:

7.3.0

NeedsCompilation:

Packaged:

2024-10-06 05:13:09 UTC; xinweima

Repository:

CRAN

Date/Publication:

2024-10-06 08:30:02 UTC

rddensity: Manipulation Testing Based on Density Discontinuity

Description

This package implements manipulation testing procedures using the local polynomial density estimators proposed in Cattaneo, Jansson and Ma (2020), and implements graphical procedures with valid confidence bands using the results in Cattaneo, Jansson and Ma (2022, 2023). In addition, this package provides complementary manipulation testing based on finite sample exact binomial testing following the esults in Cattaneo, Frandsen and Titiunik (2015) and Cattaneo, Frandsen and Vazquez-Bare (2017).

A companion Stata package is described in Cattaneo, Jansson and Ma (2018).

Commands: rddensity for manipulation (density discontinuity) testing. rdbwdensity for data-driven bandwidth selection, and rdplotdensity for density plots.

Related Stata and R packages useful for inference in regression discontinuity (RD) designs are described in the website: https://rdpackages.github.io/.

Author(s)

Matias D. Cattaneo, Princeton University cattaneo@princeton.edu.

Michael Jansson, University of California Berkeley. mjansson@econ.berkeley.edu.

Xinwei Ma (maintainer), University of California San Diego. x1ma@ucsd.edu.

References

Calonico, S., M. D. Cattaneo, and M. H. Farrell. 2018. On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Inference. Journal of the American Statistical Association 113(522): 767-779. doi:10.1080/01621459.2017.1285776

Calonico, S., M. D. Cattaneo, and M. H. Farrell. 2022. Coverage Error Optimal Confidence Intervals for Local Polynomial Regression. Bernoulli, 28(4): 2998-3022. doi:10.3150/21-BEJ1445

Cattaneo, M. D., B. Frandsen, and R. Titiunik. 2015. Randomization Inference in the Regression Discontinuity Design: An Application to the Study of Party Advantages in the U.S. Senate. Journal of Causal Inference 3(1): 1-24. doi:10.1515/jci-2013-0010

Cattaneo, M. D., M. Jansson, and X. Ma. 2018. Manipulation Testing based on Density Discontinuity. Stata Journal 18(1): 234-261. doi:10.1177/1536867X1801800115

Cattaneo, M. D., M. Jansson, and X. Ma. 2020. Simple Local Polynomial Density Estimators. Journal of the American Statistical Association, 115(531): 1449-1455. doi:10.1080/01621459.2019.1635480

Cattaneo, M. D., M. Jansson, and X. Ma. 2022. lpdensity: Local Polynomial Density Estimation and Inference. Journal of Statistical Software, 101(2): 1–25. doi:10.18637/jss.v101.i02

Cattaneo, M. D., M. Jansson, and X. Ma. 2023. Local Regression Distribution Estimators. Journal of Econometrics, 240(2): 105074. doi:10.1016/j.jeconom.2021.01.006

Cattaneo, M. D., R. Titiunik and G. Vazquez-Bare. 2017. Comparing Inference Approaches for RD Designs: A Reexamination of the Effect of Head Start on Child Mortality. Journal of Policy Analysis and Management 36(3): 643-681. doi:10.1002/pam.21985

McCrary, J. 2008. Manipulation of the Running Variable in the Regression Discontinuity Design: A Density Test. Journal of Econometrics 142(2): 698-714. doi:10.1016/j.jeconom.2007.05.005

Internal function, generate matrices.

Description

Cgenerate generates a matrix.

Usage

Cgenerate(k, p, low = -1, up = 1, kernel = "triangular")

Arguments

k, p

Integer, polynomial order.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns a matrix.

Internal function, generate matrices.

Description

Cminusgenerate generates a matrix.

Usage

Cminusgenerate(k, p, kernel = "triangular")

Arguments

k, p

Integer, polynomial order.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns a matrix.

Internal function, generate matrices.

Description

Cminusgenerate generates a matrix.

Usage

Cplusgenerate(k, p, kernel = "triangular")

Arguments

k, p

Integer, polynomial order.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns a matrix.

Internal function, generate matrices.

Description

Ggenerate generates a matrix.

Usage

Ggenerate(p, low = -1, up = 1, kernel = "triangular")

Arguments

p

Integer, polynomial order.

low, up

Numeric, lower and upper limits of integration, -1 and 1 by default.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns a matrix.

Internal function, generate matrices.

Description

Gminusgenerate generates a matrix.

Usage

Gminusgenerate(p, kernel = "triangular")

Arguments

p

Integer, polynomial order.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns a matrix.

Internal function, generate matrices.

Description

Gplusgenerate generates a matrix.

Usage

Gplusgenerate(p, kernel = "triangular")

Arguments

p

Integer, polynomial order.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns a matrix.

Internal function, generate matrices.

Description

Psigenerate generates a matrix.

Usage

Psigenerate(p)

Arguments

p

Integer, polynomial order.

Details

This is an internal function, and should not be called by users.

Value

Returns a matrix.

Internal function, generate matrices.

Description

Sgenerate generates a matrix.

Usage

Sgenerate(p, low = -1, up = 1, kernel = "triangular")

Arguments

p

Integer, polynomial order.

low, up

Numeric, lower and upper limits of integration, -1 and 1 by default.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns a matrix.

Internal function, generate matrices.

Description

Sminusgenerate generates a matrix.

Usage

Sminusgenerate(p, kernel = "triangular")

Arguments

p

Integer, polynomial order.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns a matrix.

Internal function, generate matrices.

Description

Splusgenerate generates a matrix.

Usage

Splusgenerate(p, kernel = "triangular")

Arguments

p

Integer, polynomial order.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns a matrix.

Internal function, calculates theoretical optimal bandwidth.

Description

h_opt_density calculates theoretical optimal bandwidth.

Usage

h_opt_density(x, p, N, dgp_F1, dgp_Fp1, f_low, f_up, kernel = "triangular")

Arguments

x

Numeric vector or data matrix, the running variable.

p

Integer, polynomial order.

N

Integer, sample size.

dgp_F1, dgp_Fp1

Numeric, theoretical d.g.p.

f_low, f_up

Numeric, lower and upper boundaries.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns MSE optimal bandwidth.

Internal function, calculates theoretical optimal bandwidth.

Description

h_opt_density_res calculates theoretical optimal bandwidth for restricted model.

Usage

h_opt_density_res(
  p,
  N,
  dgp_F1_l,
  dgp_F1_r,
  dgp_Fp1_l,
  dgp_Fp1_r,
  kernel = "triangular"
)

Arguments

p

Integer, polynomial order.

N

Integer, sample size.

dgp_F1_l, dgp_F1_r, dgp_Fp1_l, dgp_Fp1_r

Numeric, theoretical d.g.p.

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

Details

This is an internal function, and should not be called by users.

Value

Returns MSE optimal bandwidth.

Internal function.

Description

Internal function.

Usage

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

Arguments

x

Class CJMrdbwdensity objects.

Internal function.

Description

Internal function.

Usage

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

Arguments

x

Class CJMrddensity objects.

Bandwidth Selection for Manipulation Testing

Description

rdbwdensity implements several data-driven bandwidth selection methods useful to construct manipulation testing procedures using the local polynomial density estimators proposed in Cattaneo, Jansson and Ma (2020).

A companion Stata package is described in Cattaneo, Jansson and Ma (2018).

Companion command: rddensity for manipulation (density discontinuity) testing.

Related Stata and R packages useful for inference in regression discontinuity (RD) designs are described in the website: https://rdpackages.github.io/.

Usage

rdbwdensity(
  X,
  c = 0,
  p = 2,
  fitselect = "",
  kernel = "",
  vce = "",
  massPoints = TRUE,
  regularize = TRUE,
  nLocalMin = NULL,
  nUniqueMin = NULL
)

Arguments

X

Numeric vector or one dimensional matrix/data frame, the running variable.

c

Numeric, specifies the threshold or cutoff value in the support of X, which determines the two samples (e.g., control and treatment units in RD settings). Default is 0.

p

Nonnegative integer, specifies the local polynomial order used to construct the density estimators. Default is 2 (local quadratic approximation).

fitselect

String, specifies the density estimation method. "unrestricted" for density estimation without any restrictions (two-sample, unrestricted inference). This is the default option. "restricted" for density estimation assuming equal distribution function and higher-order derivatives.

kernel

String, specifies the kernel function used to construct the local polynomial estimators. "triangular": K(u)=(1-|u|)*(|u|<=1). This is the default option. "epanechnikov": K(u) = 0.75*(1-u^2)*(|u|<=1). "uniform": K(u) = 0.5 * (|u|<=1).

vce

String, specifies the procedure used to compute the variance-covariance matrix estimator. "plugin" for asymptotic plug-in standard errors. "jackknife" for jackknife standard errors. This is the default option.

massPoints

TRUE (default) or FALSE, specifies whether to adjust for mass points in the data.

regularize

TRUE (default) or FALSE, specifies whether to conduct local sample size checking. When set to TRUE, the bandwidth is chosen such that the local region includes at least nLocalMin observations and at least nUniqueMin unique observations.

nLocalMin

Nonnegative integer, specifies the minimum number of observations in each local neighborhood. This option will be ignored if set to 0, or if regularize=FALSE is used. Default is 20+p+1.

nUniqueMin

Nonnegative integer, specifies the minimum number of unique observations in each local neighborhood. This option will be ignored if set to 0, or if regularize=FALSE is used. Default is 20+p+1.

Value

h

Bandwidths for density discontinuity test, left and right to the cutoff, and asymptotic variance and bias.

N

full: full sample size; left/right: sample size to the left/right of the cutoff.

opt

Options passed to the function.

X_min

Smallest observations to the left and right of the cutoff.

X_max

Largest observations to the left and right of the cutoff.

Author(s)

Matias D. Cattaneo, Princeton University cattaneo@princeton.edu.

Michael Jansson, University of California Berkeley. mjansson@econ.berkeley.edu.

Xinwei Ma (maintainer), University of California San Diego. x1ma@ucsd.edu.

References

Cattaneo, M. D., M. Jansson, and X. Ma. 2018. Manipulation Testing based on Density Discontinuity. Stata Journal 18(1): 234-261. doi:10.1177/1536867X1801800115

Cattaneo, M. D., M. Jansson, and X. Ma. 2020. Simple Local Polynomial Density Estimators. Journal of the American Statistical Association, 115(531): 1449-1455. doi:10.1080/01621459.2019.1635480

Examples

# Generate a random sample
set.seed(42)
x <- rnorm(2000, mean = -0.5)

# Bandwidth selection
summary(rdbwdensity(X = x, vce="jackknife"))

Manipulation Testing Using Local Polynomial Density Estimation

Description

rddensity implements manipulation testing procedures using the local polynomial density estimators proposed in Cattaneo, Jansson and Ma (2020), and implements graphical procedures with valid confidence bands using the results in Cattaneo, Jansson and Ma (2022, 2023). In addition, the command provides complementary manipulation testing based on finite sample exact binomial testing following the esults in Cattaneo, Frandsen and Titiunik (2015) and Cattaneo, Frandsen and Vazquez-Bare (2017). For an introduction to manipulation testing see McCrary (2008).

A companion Stata package is described in Cattaneo, Jansson and Ma (2018).

Companion commands: rdbwdensity for data-driven bandwidth selection, and rdplotdensity for density plots.

Related Stata and R packages useful for inference in regression discontinuity (RD) designs are described in the website: https://rdpackages.github.io/.

Usage

rddensity(
  X,
  c = 0,
  p = 2,
  q = 0,
  fitselect = "",
  kernel = "",
  vce = "",
  massPoints = TRUE,
  h = c(),
  bwselect = "",
  all = FALSE,
  regularize = TRUE,
  nLocalMin = NULL,
  nUniqueMin = NULL,
  bino = TRUE,
  binoW = NULL,
  binoN = NULL,
  binoWStep = NULL,
  binoNStep = NULL,
  binoNW = 10,
  binoP = 0.5
)

Arguments

X

Numeric vector or one dimensional matrix/data frame, the running variable.

c

Numeric, specifies the threshold or cutoff value in the support of X, which determines the two samples (e.g., control and treatment units in RD settings). Default is 0.

p

Nonnegative integer, specifies the local polynomial order used to construct the density estimators. Default is 2 (local quadratic approximation).

q

Nonnegative integer, specifies the local polynomial order used to construct the bias-corrected density estimators. Default is p+1 (local cubic approximation for default p=2).

fitselect

kernel

vce

massPoints

TRUE (default) or FALSE, specifies whether to adjust for mass points in the data.

h

Numeric, specifies the bandwidth used to construct the density estimators on the two sides of the cutoff. If not specified, the bandwidth h is computed by the companion command rdbwdensity If two bandwidths are specified, the first bandwidth is used for the data below the cutoff and the second bandwidth is used for the data above the cutoff.

bwselect

String, specifies the bandwidth selection procedure to be used. "each" based on MSE of each density estimator separately (two distinct bandwidths, hl and hr). "diff" based on MSE of difference of two density estimators (one common bandwidth, hl=hr). "sum" based on MSE of sum of two density estimators (one common bandwidth, hl=hr). "comb" bandwidth is selected as a combination of the alternatives above. This is the default option. For fitselect="unrestricted", it selects median(each,diff,sum). For fitselect = "restricted", it selects min(diff,sum).

all

TRUE or FALSE (default), if specified, will report two testing procedures: conventional test statistic (not valid when using MSE-optimal bandwidth choice) and robust bias-corrected statistic.

regularize

nLocalMin

Nonnegative integer, specifies the minimum number of observations in each local neighborhood. This option will be ignored if set to 0, or if regularize=FALSE is used. Default is 20+p+1.

nUniqueMin

Nonnegative integer, specifies the minimum number of unique observations in each local neighborhood. This option will be ignored if set to 0, or if regularize=FALSE is used. Default is 20+p+1.

bino

TRUE (default) or FALSE, specifies whether to conduct binomial tests. By default, the initial (smallest) window contains at least 20 observations on each side, and its length is also used as the increment for subsequent windows. This feature is based on the binom.test function.

binoW

Numeric, specifies the half length(s) of the initial window. If two values are provided, they will be used for the data below and above the cutoff separately.

binoN

Nonnegative integer, specifies the minimum number of observations on each side of the cutoff used for the binomial test. This option will be ignored if binoW is provided.

binoWStep

Numeric, specifies the increment in half length(s).

binoNStep

Nonnegative integer, specifies the minimum increment in sample size (on each side of the cutoff). This option will be ignored if binoWStep is provided.

binoNW

Nonnegative integer, specifies the total number of windows. Default is 10.

binoP

Numeric, specifies the null hypothesis of the binomial test. Default is 0.5.

Value

hat

left/right: density estimate to the left/right of cutoff; diff: difference in estimated densities on the two sides of cutoff.

sd_asy

left/right: standard error for the estimated density to the left/right of the cutoff; diff: standard error for the difference in estimated densities. (Based on asymptotic formula.)

sd_jk

left/right: standard error for the estimated density to the left/right of the cutoff; diff: standard error for the difference in estimated densities. (Based on the jackknife method.)

test

t_asy/t_jk: t-statistic for the density discontinuity test, with standard error based on asymptotic formula or the jackknife; p_asy/p_jk: p-value for the density discontinuity test, with standard error based on asymptotic formula or the jackknife.

hat_p

Same as hat, without bias correction (only available when all=TRUE).

sd_asy_p

Same as sd_asy, without bias correction (only available when all=TRUE).

sd_jk_p

Same as sd_jk, without bias correction (only available when all=TRUE).

test_p

Same as test, without bias correction (only available when all=TRUE).

N

full: full sample size; left/right: sample size to the left/right of the cutoff; eff_left/eff_right: effective sample size to the left/right of the cutoff (this depends on the bandwidth).

h

left/right: bandwidth used to the left/right of the cutoff.

opt

Options passed to the function.

bino

Binomial test results. leftWindow/rightWindow: window lengths. leftN/rightN: number of observations. pval: p-values.

X_min

left/right: the samllest observation to the left/right of the cutoff.

X_max

left/right: the largest observation to the left/right of the cutoff.

Author(s)

Matias D. Cattaneo, Princeton University cattaneo@princeton.edu.

Michael Jansson, University of California Berkeley. mjansson@econ.berkeley.edu.

Xinwei Ma (maintainer), University of California San Diego. x1ma@ucsd.edu.

References

Cattaneo, M. D., M. Jansson, and X. Ma. 2018. Manipulation Testing based on Density Discontinuity. Stata Journal 18(1): 234-261. doi:10.1177/1536867X1801800115

Cattaneo, M. D., M. Jansson, and X. Ma. 2020. Simple Local Polynomial Density Estimators. Journal of the American Statistical Association, 115(531): 1449-1455. doi:10.1080/01621459.2019.1635480

Cattaneo, M. D., M. Jansson, and X. Ma. 2022. lpdensity: Local Polynomial Density Estimation and Inference. Journal of Statistical Software, 101(2): 1–25. doi:10.18637/jss.v101.i02

Cattaneo, M. D., M. Jansson, and X. Ma. 2023. Local Regression Distribution Estimators. Journal of Econometrics, 240(2): 105074. doi:10.1016/j.jeconom.2021.01.006

McCrary, J. 2008. Manipulation of the Running Variable in the Regression Discontinuity Design: A Density Test. Journal of Econometrics 142(2): 698-714. doi:10.1016/j.jeconom.2007.05.005

Examples

### Continuous Density
set.seed(42)
x <- rnorm(2000, mean = -0.5)
rdd <- rddensity(X = x, vce = "jackknife")
summary(rdd)

### Bandwidth selection using rdbwdensity()
rddbw <- rdbwdensity(X = x, vce = "jackknife")
summary(rddbw)

### Plotting using rdplotdensity()
# 1. From -2 to 2 with 25 evaluation points at each side
plot1 <- rdplotdensity(rdd, x, plotRange = c(-2, 2), plotN = 25)

# 2. Plotting a uniform confidence band
set.seed(42) # fix the seed for simulating critical values
plot2 <- rdplotdensity(rdd, x, plotRange = c(-2, 2), plotN = 25, CIuniform = TRUE)

### Density discontinuity at 0
x[x > 0] <- x[x > 0] * 2
rdd2 <- rddensity(X = x, vce = "jackknife")
summary(rdd2)
plot3 <- rdplotdensity(rdd2, x, plotRange = c(-2, 2), plotN = 25)

Internal function.

Description

Find unique elements and their frequencies in a numeric vector. This function has a similar performance as the built-in R function unique.

Usage

rddensityUnique(x)

Arguments

x

Numeric vector, already sorted in ascending order.

Value

unique

A vector containing unique elements in x.

freq

The frequency of each element in x.

index

The last occurrence of each element in x.

Internal function, normal distribution related quantities.

Description

rddensity_H calculates normal densities and derivatives.

Usage

rddensity_H(x, p)

Arguments

x

Numeric, point of evaluation.

p

Integer, polynomial order

Details

This is an internal function, and should not be called by users.

Value

Returns density or derivatives.

Internal function, implements density test.

Description

rddensity_fV generates primitive results for the density test.

Usage

rddensity_fV(
  Y,
  X,
  Nl,
  Nr,
  Nlh,
  Nrh,
  hl,
  hr,
  p,
  s,
  kernel,
  fitselect,
  vce,
  massPoints
)

Arguments

Y

Numeric vector or data matrix, the estimated c.d.f.

X

Numeric vector or data matrix, the running variable

Nl, Nr

Integers, sample sizes to the left and right of cutoff.

Nlh, Nrh

Integers, sample sizes to the left and right of cutoff, within bandwidth

hl, hr

Numeric, bandwidth to the left and right of cutoff.

p

Integer, polynomial order

s

Integer, higher order derivative estimate

kernel

String, the kernel function, can be triangular (default), uniform or epanechnikov.

fitselect

String, the model, can be restricted or unrestricted

vce

String, specifies the procedure used to compute the variance-covariance matrix estimator. Options are: "plugin" for asymptotic plug-in standard errors. "jackknife" for jackknife standard errors.

massPoints

Boolean, whether whether point estimates and standard errors

Details

This is an internal function, and should not be called by users. NOTE: data is assumed to be on ascending order.

Value

Returns a data frame for further use.

RD Senate Data

Description

Extract of the dataset constructed by Cattaneo, Frandsen, and Titiunik (2015), which include measures of incumbency advantage in the U.S. Senate for the period 1914-2010.

Format

Numeric vector containing 1390 observations:

margin: Numeric vector. See Cattaneo, Frandsen and Titiunik (2015) regarding details about this dataset.

Source

Density Plotting for Manipulation Testing

Description

rdplotdensity constructs density plots. It is based on the local polynomial density estimator proposed in Cattaneo, Jansson and Ma (2020, 2023). A companion Stata package is described in Cattaneo, Jansson and Ma (2018).

Companion command: rddensity for manipulation (density discontinuity) testing.

Related Stata and R packages useful for inference in regression discontinuity (RD) designs are described in the website: https://rdpackages.github.io/.

Usage

rdplotdensity(
  rdd,
  X,
  plotRange = NULL,
  plotN = 10,
  plotGrid = c("es", "qs"),
  alpha = 0.05,
  type = NULL,
  lty = NULL,
  lwd = NULL,
  lcol = NULL,
  pty = NULL,
  pwd = NULL,
  pcol = NULL,
  CItype = NULL,
  CIuniform = FALSE,
  CIsimul = 2000,
  CIshade = NULL,
  CIcol = NULL,
  bwselect = NULL,
  hist = TRUE,
  histBreaks = NULL,
  histFillCol = 3,
  histFillShade = 0.2,
  histLineCol = "white",
  title = "",
  xlabel = "",
  ylabel = "",
  legendTitle = NULL,
  legendGroups = NULL,
  noPlot = FALSE
)

Arguments

rdd

Object returned by rddensity

X

Numeric vector or one dimensional matrix/data frame, the running variable.

plotRange

Numeric, specifies the lower and upper bound of the plotting region. Default is [c-3*hl,c+3*hr] (three bandwidths around the cutoff).

plotN

Numeric, specifies the number of grid points used for plotting on the two sides of the cutoff. Default is c(10,10) (i.e., 10 points are used on each side).

plotGrid

String, specifies how the grid points are positioned. Options are es (evenly spaced) and qs (quantile spaced).

alpha

Numeric scalar between 0 and 1, the significance level for plotting confidence regions. If more than one is provided, they will be applied to the two sides accordingly.

type

String, one of "line" (default), "points" or "both", how the point estimates are plotted. If more than one is provided, they will be applied to the two sides accordingly.

lty

Line type for point estimates, only effective if type is "line" or "both". 1 for solid line, 2 for dashed line, 3 for dotted line. For other options, see the instructions for ggplot2 or par. If more than one is provided, they will be applied to the two sides accordingly.

lwd

Line width for point estimates, only effective if type is "line" or "both". Should be strictly positive. For other options, see the instructions for ggplot2 or par. If more than one is provided, they will be applied to the two sides accordingly.

lcol

Line color for point estimates, only effective if type is "line" or "both". 1 for black, 2 for red, 3 for green, 4 for blue. For other options, see the instructions for ggplot2 or par. If more than one is provided, they will be applied to the two sides accordingly.

pty

Scatter plot type for point estimates, only effective if type is "points" or "both". For options, see the instructions for ggplot2 or par. If more than one is provided, they will be applied to the two sides accordingly.

pwd

Scatter plot size for point estimates, only effective if type is "points" or "both". Should be strictly positive. If more than one is provided, they will be applied to the two sides accordingly.

pcol

Scatter plot color for point estimates, only effective if type is "points" or "both". 1 for black, 2 for red, 3 for green, 4 for blue. For other options, see the instructions for ggplot2 or par. If more than one is provided, they will be applied to the two sides accordingly.

CItype

String, one of "region" (shaded region, default), "line" (dashed lines), "ebar" (error bars), "all" (all of the previous) or "none" (no confidence region), how the confidence region should be plotted. If more than one is provided, they will be applied to the two sides accordingly.

CIuniform

TRUE or FALSE (default), plotting either pointwise confidence intervals (FALSE) or uniform confidence bands (TRUE).

CIsimul

Positive integer, the number of simulations used to construct critical values (default is 2000). This option is ignored if CIuniform=FALSE.

CIshade

Numeric, opaqueness of the confidence region, should be between 0 (transparent) and 1. Default is 0.2. If more than one is provided, they will be applied to the two sides accordingly.

CIcol

Color of the confidence region. 1 for black, 2 for red, 3 for green, 4 for blue. For other options, see the instructions for ggplot2 or par. If more than one is provided, they will be applied to the two sides accordingly.

bwselect

String, the method for data-driven bandwidth selection. Available options are (1) "mse-dpi" (mean squared error-optimal bandwidth selected for each grid point); (2) "imse-dpi" (integrated MSE-optimal bandwidth, common for all grid points); (3) "mse-rot" (rule-of-thumb bandwidth with Gaussian reference model); and (4) "imse-rot" (integrated rule-of-thumb bandwidth with Gaussian reference model). If omitted, bandwidths returned by rddensity will be used.

hist

TRUE (default) or FALSE, whether adding a histogram to the background.

histBreaks

Numeric vector, giving the breakpoints between histogram cells.

histFillCol

Color of the histogram cells.

histFillShade

Opaqueness of the histogram cells, should be between 0 (transparent) and 1. Default is 0.2.

histLineCol

Color of the histogram lines.

title, xlabel, ylabel

Strings, title of the plot and labels for x- and y-axis.

legendTitle

String, title of legend.

legendGroups

String Vector, group names used in legend.

noPlot

No density plot will be generated if set to TRUE.

Details

Bias correction is only used for the construction of confidence intervals/bands, but not for point estimation. The point estimates, denoted by f_p, are constructed using local polynomial estimates of order p, while the centering of the confidence intervals/bands, denoted by f_q, are constructed using local polynomial estimates of order q. The confidence intervals/bands take the form: [f_q - cv * SE(f_q) , f_q + cv * SE(f_q)], where cv denotes the appropriate critical value and SE(f_q) denotes a standard error estimate for the centering of the confidence interval/band. As a result, the confidence intervals/bands may not be centered at the point estimates because they have been bias-corrected. Setting q and p to be equal results on centered at the point estimate confidence intervals/bands, but requires undersmoothing for valid inference (i.e., (I)MSE-optimal bandwdith for the density point estimator cannot be used). Hence the bandwidth would need to be specified manually when q=p, and the point estimates will not be (I)MSE optimal. See Cattaneo, Jansson and Ma (2022, 2023) for details, and also Calonico, Cattaneo, and Farrell (2018, 2022) for robust bias correction methods.

Sometimes the density point estimates may lie outside of the confidence intervals/bands, which can happen if the underlying distribution exhibits high curvature at some evaluation point(s). One possible solution in this case is to increase the polynomial order p or to employ a smaller bandwidth.

Value

Estl, Estr

Matrices containing estimation results: (1) grid (grid points), (2) bw (bandwidths), (3) nh (number of observations in each local neighborhood), (4) nhu (number of unique observations in each local neighborhood), (5) f_p (point estimates with p-th order local polynomial), (6) f_q (point estimates with q-th order local polynomial, only if option q is nonzero), (7) se_p (standard error corresponding to f_p), and (8) se_q (standard error corresponding to f_q). Variance-covariance matrix corresponding to f_p. Variance-covariance matrix corresponding to f_q. A list containing options passed to the function.

Estplot

A stadnard ggplot object is returned, hence can be used for further customization.

Author(s)

Matias D. Cattaneo, Princeton University cattaneo@princeton.edu.

Michael Jansson, University of California Berkeley. mjansson@econ.berkeley.edu.

Xinwei Ma (maintainer), University of California San Diego. x1ma@ucsd.edu.

References

Calonico, S., M. D. Cattaneo, and M. H. Farrell. 2022. Coverage Error Optimal Confidence Intervals for Local Polynomial Regression. Bernoulli, 28(4): 2998-3022. doi:10.3150/21-BEJ1445

Cattaneo, M. D., M. Jansson, and X. Ma. 2018. Manipulation Testing based on Density Discontinuity. Stata Journal 18(1): 234-261. doi:10.1177/1536867X1801800115

Cattaneo, M. D., M. Jansson, and X. Ma. 2020. Simple Local Polynomial Density Estimators. Journal of the American Statistical Association, 115(531): 1449-1455. doi:10.1080/01621459.2019.1635480

Cattaneo, M. D., M. Jansson, and X. Ma. 2022. lpdensity: Local Polynomial Density Estimation and Inference. Journal of Statistical Software, 101(2): 1–25. doi:10.18637/jss.v101.i02

Cattaneo, M. D., M. Jansson, and X. Ma. 2023. Local Regression Distribution Estimators. Journal of Econometrics, 240(2): 105074. doi:10.1016/j.jeconom.2021.01.006

Examples

# Generate a random sample with a density discontinuity at 0
set.seed(42)
x <- rnorm(2000, mean = -0.5)
x[x > 0] <- x[x > 0] * 2

# Estimation
rdd <- rddensity(X = x)
summary(rdd)

# Density plot (from -2 to 2 with 25 evaluation points at each side)
plot1 <- rdplotdensity(rdd, x, plotRange = c(-2, 2), plotN = 25)

# Plotting a uniform confidence band
set.seed(42) # fix the seed for simulating critical values
plot3 <- rdplotdensity(rdd, x, plotRange = c(-2, 2), plotN = 25, CIuniform = TRUE)

Internal function.

Description

Internal function.

Usage

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

Arguments

object

Class CJMrdbwdensity objects.

Internal function.

Description

Internal function.

Usage

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

Arguments

object

Class CJMrddensity objects.