HighDim_rd: High-Dimensional Covariates in RDD Estimation

In akreiss/HighDimRD: RDD Estimation Using High-Dimensional Covariates

High-Dimensional Covariates in RDD Estimation

Description

HighDim_rd function computes the treatment effect estimator in an RD setting when a large list of covariates is supplied by the user.

Usage

HighDim_rd(
  Y,
  X,
  Z,
  c = 0,
  rd = "robust",
  level = 0.95,
  b = NULL,
  bfactor = 1,
  h = NULL,
  tpc,
  kernel = "triangular",
  alpha = 0.05,
  M = NULL,
  L = 100,
  OPC = 50,
  C,
  sclass = "H",
  se.initial = "SilvermanNN"
)

Arguments

Y	Vector of n responses.
X	Vector of n running variables.
Z	n x p - Matrix of covariates (each column corresponds to one covariate).
c	Cutoff point for treatment, individuals with X>=c are treated, default is c=0.
rd	Sets which RD functions should be used to find bandwidths, estimators and confidence sets. If "honest", then the RDHonest package is used. In this case the user has to specify C and should be aware of sclass. If "robust", then the rdrobust package is used.
level	Level of the computed confidence set, between 0 and 1. Default is 0.95.
b	Reference bandwidth used for doing the model selection step. If b=NULL, the default, bandwidth selection from rdrobust without covariates is used to find b. The final bandwidth used for model selection is given by bfactor*b. b can be provided as the output of rdbwselect. It has to be a list with at least the element bws which contains the bandwidth.
bfactor	The bandwidth which is used for model selection is given by bfactor*b (see above), the default is bfactor=1.
h	Bandwidth for doing the actual RD estimation. If h=NULL, the default, bandwidth selection from rdrobust with the selected covariate set is used. See b for the format.
tpc	Method for choosing the tuning the parameter of the Lasso. Possible values are: "CV" for cross-validation. "LV" for the bootstrap-procedure based on Lederer and Vogt (2020), this requires the user specified parameters alpha, M and L. "BCH" for the procedure adapted from Belloni et al. (2013) and "OPC" (observations per covariate) where the number of selected covariates is no larger than n_effective/OPC, where n_effective is the number of observations which receive a positive kernel weight and OPC is specified by the user.
kernel	Specifies which kernel to be used for both model selection and estimation, possible choices are triangular (the default), epanechnikov, uniform.
alpha, M, L	Parameters for "LV" method for tuning parameter choice, ignored if tpc!="LV".
OPC	Parameter for "OPC" method for tuning parameter choice, ignored if tpc!="OPC".
C, sclass, se.initial	Parameters which are used for RDHonest: C is the bound in the smoothness class specified in sclass. The default sclass is H, see RDHonest for details. C must be specified by the user. See RDHonest for a description of se.initial.

Details

For the exact functionality of HighDim_rd, we refer to Kreiss and Rothe (2023). The vectors Y,X contain the responses and the running variable, respectively, that is, an individual is exposed to the treatment if X>=c. The matrix Z contains the covariates (each column contains one covariate). The lengths of Y and X must equal the number of rows in Z which is the number of observations. With b and h the bandwidths which are used for the model selection and the actual estimation can be specified. Note that the bandwidth which is used to do the bias correction in rdrobust will always be the same as the same as the bandwidth which is used for the estimation. The option tpc specifies which specific algorithm shall be used to find the tuning parameter in the Lasso, that is, the model selection, step. Details are provided in the parameter description above. The computationally quickest method is OPC. bfactor allows to deliberately use smaller or larger bandwidths for the model selection.

Value

List with five elements:

rd	Output from rdrobust or RDHonest (depending on the choice of rd) using the selected covariates, see rdrobust or RDHonest for a description of the output, the estimator can be found in rd$Estimate (for rdrobust) or in rd$estimate (for RDHonest).
covs	Vector of selected covariate indices.
b	Bandwidth used for model selection
h	Bandwidth used for the estimation
Z	pars Vector of length p containing the estimated parameters of the covariates.

Authors

Alexander Kreiss, Leipzig University, Germany. [e-mail]

Christoph Rothe, University of Mannheim, Germany

References

Belloni, A., Chernozhukov, V. and Hansen, C. (2013): Inference on treatment effects after selection among high-dimensional controls. The Review of Economic Studies, 81, 608-650

Kreiss, A. and Rothe, C. (2023): Inference in regression discontinuity designs with high-dimensional covariates. The Econometrics Journal, Volume 26, Issue 2, May 2023, Pages 105–123

Lederer, J. and Vogt, M. (2020): Estimating the Lasso's Effective Noise. arxiv 2004.11554

See Also

fourier_basis, interaction_terms, cross_interactions

Examples

## Not run: library(mvtnorm)

set.seed(12345)

n <- 1000
sigma_y <- 0.1295
sigma_z <- 0.1353

## Running Variable
x <- 2*rbeta(n,2,4)-1

## Covariates
Z <- rmvnorm(n,rep(0,5),sigma=diag(sigma_z^2,5))

## Noise
epsilon <- rnorm(n,mean=0,sd=sigma_y)

## Treatment Indicator
T <- as.numeric(x>=0)

## Outcome
Y <- (1-T)*(0.36+0.96*x+5.47*x^2+15.28*x^3+15.87*x^4+5.14*x^5+0.22*Z[,1])+
T *(0.38+0.62*x-2.84*x^2+ 8.42*x^3-10.24*x^4+4.31*x^5+0.28*Z[,1])+epsilon

## High-Dimensional Covariates
Z1 <- fourier_basis(Z,4)
Z_HighDim <- cbind(Z,interaction_terms(Z),Z1,interaction_terms(Z1))

## Compute estimate with model selection using rdrobust
cov_rd_CV_robust  <- HighDim_rd(Y,x,Z_HighDim,tpc="CV" ,rd="robust")
cov_rd_BCH_robust <- HighDim_rd(Y,x,Z_HighDim,tpc="BCH",rd="robust")
cov_rd_LV_robust  <- HighDim_rd(Y,x,Z_HighDim,tpc="LV" ,rd="robust")

## Compute estimate with model selection using RDHonest
cov_rd_CV_honest  <- HighDim_rd(Y,x,Z_HighDim,tpc="CV" ,rd="honest",C=30)
cov_rd_BCH_honest <- HighDim_rd(Y,x,Z_HighDim,tpc="BCH",rd="honest",C=30)
cov_rd_LV_honest  <- HighDim_rd(Y,x,Z_HighDim,tpc="LV" ,rd="honest",C=30)
## End(Not run)

akreiss/HighDimRD documentation built on Nov. 21, 2023, 9:24 a.m.