rlassoEffects: rigorous Lasso for Linear Models: Inference

In hdm: High-Dimensional Metrics

Description

Estimation and inference of (low-dimensional) target coefficients in a high-dimensional linear model.

Usage

rlassoEffects(x, ...)

## Default S3 method:
rlassoEffects(x, y, index = c(1:ncol(x)),
  method = "partialling out", I3 = NULL, post = TRUE, ...)

## S3 method for class 'formula'
rlassoEffects(formula, data, I, method = "partialling out",
  included = NULL, post = TRUE, ...)

rlassoEffect(x, y, d, method = "double selection", I3 = NULL, post = TRUE,
  ...)

Arguments

x	matrix of regressor variables serving as controls and potential treatments. For rlassoEffect it contains only controls, for rlassoEffects both controls and potential treatments. For rlassoEffects it must have at least two columns.
...	parameters passed to the rlasso function.
y	outcome variable (vector or matrix)
index	vector of integers, logicals or variables names indicating the position (column) of variables (integer case), logical vector of length of the variables (TRUE or FALSE) or the variable names of x which should be used for inference / as treatment variables.
method	method for inference, either 'partialling out' (default) or 'double selection'.
I3	For the 'double selection'-method the logical vector I3 has same length as the number of variables in x; indicates if variables (TRUE) should be included in any case to the model and they are exempt from selection. These variables should not be included in the index; hence the intersection with index must be the empty set. In the case of partialling out it is ignored.
post	logical, if post Lasso is conducted with default TRUE.
formula	An element of class formula specifying the linear model.
data	an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which the function is called.
I	An one-sided formula specifying the variables for which inference is conducted.
included	One-sided formula of variables which should be included in any case (only for method="double selection").
d	variable for which inference is conducted (treatment variable)

Details

The functions estimates (low-dimensional) target coefficients in a high-dimensional linear model. An application is e.g. estimation of a treatment effect α_0 in a setting of high-dimensional controls. The user can choose between the so-called post-double-selection method and partialling-out. The idea of the double selection method is to select variables by Lasso regression of the outcome variable on the control variables and the treatment variable on the control variables. The final estimation is done by a regression of the outcome on the treatment effect and the union of the selected variables in the first two steps. In partialling-out first the effect of the regressors on the outcome and the treatment variable is taken out by Lasso and then a regression of the residuals is conducted. The resulting estimator for α_0 is normal distributed which allows inference on the treatment effect. It presents a wrap function for rlassoEffect which does inference for a single variable.

Value

The function returns an object of class rlassoEffects with the following entries:

coefficients	vector with estimated values of the coefficients for each selected variable
se	standard error (vector)
t	t-statistic
pval	p-value
samplesize	sample size of the data set
index	index of the variables for which inference is performed

References

A. Belloni, V. Chernozhukov, C. Hansen (2014). Inference on treatment effects after selection among high-dimensional controls. The Review of Economic Studies 81(2), 608-650.

Examples

library(hdm); library(ggplot2)
set.seed(1)
n = 100 #sample size
p = 100 # number of variables
s = 3 # nubmer of non-zero variables
X = matrix(rnorm(n*p), ncol=p)
colnames(X) <- paste("X", 1:p, sep="")
beta = c(rep(3,s), rep(0,p-s))
y = 1 + X%*%beta + rnorm(n)
data = data.frame(cbind(y,X))
colnames(data)[1] <- "y"
fm = paste("y ~", paste(colnames(X), collapse="+"))
fm = as.formula(fm)                 
lasso.effect = rlassoEffects(X, y, index=c(1,2,3,50))
lasso.effect = rlassoEffects(fm, I = ~ X1 + X2 + X3 + X50, data=data)
print(lasso.effect)
summary(lasso.effect)
confint(lasso.effect)
plot(lasso.effect)

Example output

Warning messages:
1: In grep(cn[I.c], colnames(X)) :
  argument 'pattern' has length > 1 and only the first element will be used
2: In grep(cn[I.c], colnames(X)) :
  argument 'pattern' has length > 1 and only the first element will be used

Call:
rlassoEffects.formula(formula = fm, data = data, I = ~X1 + X2 + 
    X3 + X50)

Coefficients:
      X1       X10       X11       X12       X13       X14       X15       X16  
 2.94448  -0.03038   0.10862  -0.01007   0.02001   0.09558  -0.01426  -0.10897  
     X17       X18       X19      X100  
-0.04645  -0.06300  -0.10824   0.03385  

[1] "Estimates and significance testing of the effect of target variables"
     Estimate. Std. Error t value Pr(>|t|)    
X1     2.94448    0.08815  33.404   <2e-16 ***
X10   -0.03038    0.07485  -0.406    0.685    
X11    0.10862    0.07595   1.430    0.153    
X12   -0.01007    0.08012  -0.126    0.900    
X13    0.02001    0.07325   0.273    0.785    
X14    0.09558    0.09106   1.050    0.294    
X15   -0.01426    0.07608  -0.187    0.851    
X16   -0.10897    0.07383  -1.476    0.140    
X17   -0.04645    0.07942  -0.585    0.559    
X18   -0.06300    0.06676  -0.944    0.345    
X19   -0.10824    0.07372  -1.468    0.142    
X100   0.03385    0.08973   0.377    0.706    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


           2.5 %     97.5 %
X1    2.77171308 3.11724213
X10  -0.17709146 0.11632283
X11  -0.04024408 0.25748432
X12  -0.16709410 0.14695616
X13  -0.12356150 0.16358294
X14  -0.08288835 0.27404185
X15  -0.16338572 0.13485904
X16  -0.25367299 0.03574284
X17  -0.20209814 0.10920353
X18  -0.19384187 0.06784675
X19  -0.25273372 0.03624496
X100 -0.14201026 0.20971450
Warning: Ignoring unknown aesthetics: width, h

hdm documentation built on May 1, 2019, 7:56 p.m.