# qrs.fast.bt Algorithm 4: bootstrap algorithm with preprocessing and quantile grid reduction for Quantile Regression with Selection (QRS). ```r qrs.fast.bt( y, x, d, z, w0 = NULL, Q1, Q2, P = 10, link, family, gridtheta, m, b0, reps, alpha ) ``` ## Arguments - `y`: = Dependent variable (N x 1) - `x`: = Regressors matrix (N x K) - `d`: = Participation variable (N x 1) - `z`: = Regressors and instruments matrix for the propensity score (N x Kz) - `w0`: = Sample weights (N x 1) - `Q1`: = Number of quantiles in reduced grid - `Q2`: = Number of quantiles in large grid - `P`: = Number of evaluated values of parameter with large quantile grid - `link`: = Link function to compute the propensity score - `family`: = Parametric copula family - `gridtheta`: = Grid of values for copula parameter (T x 1) - `m`: = Parameter to select interval of observations in top and bottom groups - `b0`: = Initial values of the beta coefficients for all quantiles in the reduced quantile grid (K x Q1) - `reps`: = Number of bootstrap repetitions - `alpha`: = Significance level ## Returns gammase = Bootstrapped standard error of gamma coefficients (Kz x 1) gammaub = Bootstrapped upper bound of confidence interval of gamma coefficients (Kz x 1) gammalb = Bootstrapped lower bound of confidence interval of gamma coefficients (Kz x 1) betase = Bootstrapped standard error of beta coefficients (K x Q) betaub = Bootstrapped upper bound of confidence interval of beta coefficients (K x Q) betalb = Bootstrapped lower bound of confidence interval of beta coefficients (K x Q) thetase = Bootstrapped standard error of theta coefficients (1 x 1) thetaub = Bootstrapped upper bound of confidence interval of theta coefficients (1 x 1) thetalb = Bootstrapped lower bound of confidence interval of theta coefficients (1 x 1) gamma = Bootstrapped estimated theta coefficients (Kz x reps) beta = Bootstrapped estimated beta coefficients (K x Q2 x reps) theta = Bootstrapped estimated copula parameter (1 x reps) objf = Bootstrapped value of objective function at the optimum (1 x reps) ## Examples ```r set.seed(1) N <- 100 x <- cbind(1, 2 + runif(N)) z <- cbind(x, runif(N)) cop <- copula::normalCopula(param = -0.5, dim = 2) copu <- copula::rCopula(N, cop) v <- copu[,1] u <- copu[,2] gamma <- c(-1.5, 0.05, 2) beta <- cbind(qnorm(u), u^0.5) prop <- exp(z %*% gamma) / (1 + exp(z %*% gamma)) d <- as.numeric(v <= prop) y <- d * rowSums(x * beta) w <- matrix(1, nrow = N, ncol = 1) Q1 <- 9 Q2 <- 19 P <- 2 m <- 1 gridtheta <- seq(-1, 0, by = 0.1) link <- "probit" family <- "Gaussian" reps <- 10 alpha <- 0.05 est <- qrs.fast(y, x[,-1], d, z[,-1], w, Q1, Q2, P, link, family, gridtheta, m) bt <- qrs.fast.bt(y, x[,-1], d, z[,-1], w, Q1, Q2, P, link, family, gridtheta, m, est$b1, reps, alpha) summary(bt) ```

qrs.fast.bt function