# Lasso OLS Computes the two-stage estimator Lasso+OLS (default) or the Lasso estimator (if OLS=FALSE). ```r LassoOLS(x, y, OLS = TRUE, lambda = NULL, fix.lambda = TRUE, cv.method = "cv", nfolds = 10, foldid, cv.OLS = TRUE, tau = 0, parallel = FALSE, standardize = TRUE, intercept = TRUE, ...) ``` ## Arguments - `x`: Input matrix as in glmnet, of dimension nobs x nvars; each row is an observation vector. - `y`: Response variable. - `OLS`: If TRUE, computes Lasso+OLS; otherwise, computes Lasso estimator. The default is TRUE. - `lambda`: A value of lambda - default is NULL. lambda should be given a value when fix.lambda=TRUE. - `fix.lambda`: If TRUE, computes Lasso+OLS (or Lasso) for a fix value of lambda given by the argument "lambda"; otherwise, computes Lasso+OLS (or Lasso) for the value of lambda choosing by cv/cv1se/escv. - `cv.method`: The method used to select lambda -- can be cv, cv1se, and escv; the default is cv. cv.method is useful only when fix.lambda=FALSE. - `nfolds, foldid, cv.OLS, tau, parallel`: Arguments that can be passed to escv.glmnet (useful only when fix.lambda=FALSE). Note that, the default value of cv.OLS is TRUE, which means using Lasso+OLS in the cv fits. - `standardize`: Logical flag for x variable standardization, prior to fitting the model. Default is standardize=TRUE. - `intercept`: Should intercept be fitted (default is TRUE) or set to zero (FALSE). - `...`: Other arguments that can be passed to glmnet. ## Details If OLS=TRUE (default), this function computes the Lasso+OLS estimator for a give value of lambda (if fix.lambda=TRUE) or for the value of lambda choosing by cv/cv1se/escv (if fix.lambda=FALSE). If OLS=FALSE, this function computes the Lasso estimator in the same way as the function "Lasso". Note that, we use the easy-to-understand notation "Lasso+OLS" denoting the "Lasso+mLS" estimator defined in the paper: Liu H, Yu B. Asymptotic Properties of Lasso+mLS and Lasso+Ridge in Sparse High-dimensional Linear Regression. Electronic Journal of Statistics, 2013, 7. ## Returns A list consisting of the following elements is returned. - **beta**: The Lasso+OLS (or Lasso when OLS=FALSE) estimate for the coefficients of variables/predictors. - **beta0**: A value of intercept term. - **lambda**: The value/values of lambda. - **meanx**: The mean vector of variables/predictors if intercept=TRUE, otherwise is a vector of 0's. - **mu**: The mean of the response if intercept=TRUE, otherwise is 0. - **tau**: Tuning parameter in modified Least Squares (mls). ## Examples ```r library("glmnet") library("mvtnorm") ## generate the data set.seed(2015) n <- 200 # number of obs p <- 500 s <- 10 beta <- rep(0, p) beta[1:s] <- runif(s, 1/3, 1) x <- rmvnorm(n = n, mean = rep(0, p), method = "svd") signal <- sqrt(mean((x %*% beta)^2)) sigma <- as.numeric(signal / sqrt(10)) # SNR=10 y <- x %*% beta + rnorm(n) ## Lasso+OLS estimator # for a given value of lambda set.seed(0) obj.escv <- escv.glmnet(x, y) obj <- LassoOLS(x, y, lambda = obj.escv$lambda.cv) # Lasso+OLS estimate of the regression coefficients obj$beta # intercept term obj$beta0 # prediction mypredict(obj, newx = matrix(rnorm(10*p), 10, p)) # for lambda choosing by cross-validation (cv) which uses Lasso+OLS in the cv fit set.seed(0) obj <- LassoOLS(x, y, fix.lambda = FALSE) # for lambda choosing by cross-validation (cv) which uses Lasso in the cv fit set.seed(0) obj <- LassoOLS(x, y, fix.lambda = FALSE, cv.OLS = FALSE) ```

LassoOLS function