DDL {DDL}

Point estimation and inference for a single regression coefficient in the high-dimensional linear model with hidden confounders.

Description

Computes the Doubly Debiased Lasso estimator of a single regression coefficient in the high-dimensional linear model with hidden confounders. It also constructs the confidence interval for the target regression coefficient.

Usage

DDL(X, Y, index, rho = 0.5, rhop = 0.5)

Arguments

X

the covariates matrix, of dimension n\times p

Y

the outcome vector, of length n

index

the vector of indexes for the regression coefficient of interest

rho

the trim level for X, default is 0.5

rhop

the trim level for X_{-j}, default is 0.5

Value

index

the vector of indexes for the regression coefficient of interest

est_ddl

The vector of the Doubly Debiased Lasso estimator of the target regression coefficient

se

The vector of the standard error of the Doubly Debiased Lasso estimator

est_init

The vector of the spectral deconfounding estimator of the whole regression vector

Examples

index = c(1,2,10)
n=100
p=200
s=5
q=3
sigmaE=2
sigma=2
pert=1

H = pert*matrix(rnorm(n*q,mean=0,sd=1),n,q,byrow = TRUE)
Gamma = matrix(rnorm(q*p,mean=0,sd=1),q,p,byrow = TRUE)
#value of X independent from H
E = matrix(rnorm(n*p,mean=0,sd=sigmaE),n,p,byrow = TRUE)

#defined in eq. (2), high-dimensional measured covariates
X = E + H %*% Gamma

delta = matrix(rnorm(q*1,mean=0,sd=1),q,1,byrow = TRUE)

#px1 matrix, creates beta with 1s in the first s entries and the remaining p-s as 0s
beta = matrix(rep(c(1,0),times = c(s,p-s)),p,1,byrow = TRUE)

#nx1 matrix with values of mean 0 and SD of sigma, error in Y independent of X
nu = matrix(rnorm(n*1,mean=0,sd=sigma),n,1,byrow = TRUE)

#eq. (1), the response of the Structural Equation Model
Y = X %*% beta + H %*% delta + nu

result = DDL(X, Y, index)
summary(result)