rlasso: rlasso: Function for Lasso estimation under homoscedastic and...

In hdm: High-Dimensional Metrics

rlasso: Function for Lasso estimation under homoscedastic and heteroscedastic non-Gaussian disturbances

Description

The function estimates the coefficients of a Lasso regression with data-driven penalty under homoscedasticity and heteroscedasticity with non-Gaussian noise and X-dependent or X-independent design. The method of the data-driven penalty can be chosen. The object which is returned is of the S3 class rlasso.

Usage

rlasso(x, ...)

## S3 method for class 'formula'
rlasso(
  formula,
  data = NULL,
  post = TRUE,
  intercept = TRUE,
  model = TRUE,
  penalty = list(homoscedastic = FALSE, X.dependent.lambda = FALSE, lambda.start = NULL,
    c = 1.1, gamma = 0.1/log(n)),
  control = list(numIter = 15, tol = 10^-5, threshold = NULL),
  ...
)

## S3 method for class 'character'
rlasso(
  x,
  data = NULL,
  post = TRUE,
  intercept = TRUE,
  model = TRUE,
  penalty = list(homoscedastic = FALSE, X.dependent.lambda = FALSE, lambda.start = NULL,
    c = 1.1, gamma = 0.1/log(n)),
  control = list(numIter = 15, tol = 10^-5, threshold = NULL),
  ...
)

## Default S3 method:
rlasso(
  x,
  y,
  post = TRUE,
  intercept = TRUE,
  model = TRUE,
  penalty = list(homoscedastic = FALSE, X.dependent.lambda = FALSE, lambda.start = NULL,
    c = 1.1, gamma = 0.1/log(n)),
  control = list(numIter = 15, tol = 10^-5, threshold = NULL),
  ...
)

Arguments

x	regressors (vector, matrix or object can be coerced to matrix)
...	further arguments (only for consistent defintion of methods)
formula	an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted in the form y~x
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 rlasso is called.
post	logical. If TRUE, post-Lasso estimation is conducted.
intercept	logical. If TRUE, intercept is included which is not penalized.
model	logical. If TRUE (default), model matrix is returned.
penalty	list with options for the calculation of the penalty. c and gamma constants for the penalty with default c=1.1 and gamma=0.1 homoscedastic logical, if homoscedastic errors are considered (default FALSE). Option none is described below. X.dependent.lambda logical, TRUE, if the penalization parameter depends on the the design of the matrix x. FALSE, if independent of the design matrix (default). numSim number of simulations for the dependent methods, default=5000 lambda.start initial penalization value, compulsory for method "none"
control	list with control values. numIter number of iterations for the algorithm for the estimation of the variance and data-driven penalty, ie. loadings, tol tolerance for improvement of the estimated variances. threshold is applied to the final estimated lasso coefficients. Absolute values below the threshold are set to zero.
y	dependent variable (vector, matrix or object can be coerced to matrix)

Details

The function estimates the coefficients of a Lasso regression with data-driven penalty under homoscedasticity / heteroscedasticity and non-Gaussian noise. The options homoscedastic is a logical with FALSE by default. Moreover, for the calculation of the penalty parameter it can be chosen, if the penalization parameter depends on the design matrix (X.dependent.lambda=TRUE) or independent (default, X.dependent.lambda=FALSE). The default value of the constant c is 1.1 in the post-Lasso case and 0.5 in the Lasso case. A special option is to set homoscedastic to none and to supply a values lambda.start. Then this value is used as penalty parameter with independent design and heteroscedastic errors to weight the regressors. For details of the implementation of the Algorithm for estimation of the data-driven penalty, in particular the regressor-independent loadings, we refer to Appendix A in Belloni et al. (2012). When the option "none" is chosen for homoscedastic (together with lambda.start), lambda is set to lambda.start and the regressor-independent loadings und heteroscedasticity are used. The options "X-dependent" and "X-independent" under homoscedasticity are described in Belloni et al. (2013).

The option post=TRUE conducts post-lasso estimation, i.e. a refit of the model with the selected variables.

Value

rlasso returns an object of class rlasso. An object of class "rlasso" is a list containing at least the following components:

• coefficients parameter estimates

• beta parameter estimates (named vector of coefficients without intercept)

• intercept value of the intercept

• index index of selected variables (logical vector)

• lambda data-driven penalty term for each variable, product of lambda0 (the penalization parameter) and the loadings

• lambda0 penalty term

• loadings loading for each regressor

• residuals residuals, response minus fitted values

• sigma root of the variance of the residuals

• iter number of iterations

• call function call

• options options

• model model matrix (if model = TRUE in function call)

References

A. Belloni, D. Chen, V. Chernozhukov and C. Hansen (2012). Sparse models and methods for optimal instruments with an application to eminent domain. Econometrica 80 (6), 2369-2429.

A. Belloni, V. Chernozhukov and C. Hansen (2013). Inference for high-dimensional sparse econometric models. In Advances in Economics and Econometrics: 10th World Congress, Vol. 3: Econometrics, Cambirdge University Press: Cambridge, 245-295.

Examples

set.seed(1)
n = 100 #sample size
p = 100 # number of variables
s = 3 # nubmer of variables with non-zero coefficients
X = Xnames = matrix(rnorm(n*p), ncol=p)
colnames(Xnames) <- paste("V", 1:p, sep="")
beta = c(rep(5,s), rep(0,p-s))
Y = X%*%beta + rnorm(n)
reg.lasso <- rlasso(Y~Xnames)
Xnew = matrix(rnorm(n*p), ncol=p)  # new X
colnames(Xnew) <- paste("V", 1:p, sep="")
Ynew =  Xnew%*%beta + rnorm(n)  #new Y
yhat = predict(reg.lasso, newdata = Xnew)

hdm documentation built on May 29, 2024, 3:35 a.m.