conquer: Convolution-Type Smoothed Quantile Regression

Description

Estimation and inference for conditional linear quantile regression models using a convolution smoothed approach. Efficient gradient-based methods are employed for fitting both a single model and a regression process over a quantile range. Normal-based and (multiplier) bootstrap confidence intervals for all slope coefficients are constructed.

Usage

conquer(
  X,
  Y,
  tau = 0.5,
  kernel = c("Gaussian", "logistic", "uniform", "parabolic", "triangular"),
  h = 0,
  checkSing = FALSE,
  tol = 1e-04,
  iteMax = 5000,
  stepBounded = TRUE,
  stepMax = 100,
  ci = c("none", "bootstrap", "asymptotic", "both"),
  alpha = 0.05,
  B = 1000
)

Value

An object containing the following items will be returned:

coeff: A \((p + 1)\)-vector of estimated quantile regression coefficients, including the intercept.
ite: Number of iterations until convergence.
residual: An \(n\)-vector of fitted residuals.
bandwidth: Bandwidth value.
tau: Quantile level.
kernel: Kernel function.
n: Sample size.
p: Number of covariates.
perCI: The percentile confidence intervals for regression coefficients. Only available if ci = "bootstrap" or ci = "both".
pivCI: The pivotal confidence intervals for regression coefficients. Only available if ci = "bootstrap" or ci = "both".
normCI: The normal-based confidence intervals for regression coefficients. Only available if ci = "bootstrap" or ci = "both".
asyCI: The asymptotic confidence intervals for regression coefficients. Only available if ci = "asymptotic" or ci = "both".

Arguments

X: An \(n\) by \(p\) design matrix. Each row is a vector of observations with \(p\) covariates. Number of observations \(n\) must be greater than number of covariates \(p\).
Y: An \(n\)-dimensional response vector.
tau: (optional) The desired quantile level. Default is 0.5. Value must be between 0 and 1.
kernel: (optional) A character string specifying the choice of kernel function. Default is "Gaussian". Choices are "Gaussian", "logistic", "uniform", "parabolic" and "triangular".
h: (optional) Bandwidth/smoothing parameter. Default is \(\max\{((log(n) + p) / n)^{0.4}, 0.05\}\). The default will be used if the input value is less than or equal to 0.
checkSing: (optional) A logical flag. Default is FALSE. If checkSing = TRUE, then it will check if the design matrix is singular before running conquer.
tol: (optional) Tolerance level of the gradient descent algorithm. The iteration will stop when the maximum magnitude of all the elements of the gradient is less than tol. Default is 1e-04.
iteMax: (optional) Maximum number of iterations. Default is 5000.
stepBounded: (optional) A logical flag. Default is TRUE. If stepBounded = TRUE, then the step size of gradient descent is upper bounded by stepMax. If stepBounded = FALSE, then the step size is unbounded.
stepMax: (optional) Maximum bound for the gradient descent step size. Default is 100.
ci: (optional) A character string specifying methods to construct confidence intervals. Choices are "none" (default), "bootstrap", "asymptotic" and "both". If ci = "none", then confidence intervals will not be constructed. If ci = "bootstrap", then three types of confidence intervals (percentile, pivotal and normal) will be constructed via multiplier bootstrap. If ci = "asymptotic", then confidence intervals will be constructed based on estimated asymptotic covariance matrix. If ci = "both", then confidence intervals from both bootstrap and asymptotic covariance will be returned.
alpha: (optional) Miscoverage level for each confidence interval. Default is 0.05.
B: (optional) The size of bootstrap samples. Default is 1000.

References

Barzilai, J. and Borwein, J. M. (1988). Two-point step size gradient methods. IMA J. Numer. Anal., 8, 141–148.

Fernandes, M., Guerre, E. and Horta, E. (2021). Smoothing quantile regressions. J. Bus. Econ. Statist., 39, 338-357.

He, X., Pan, X., Tan, K. M., and Zhou, W.-X. (2022+). Smoothed quantile regression for large-scale inference. J. Econometrics, in press.

Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33-50.

Examples

Run this code

n = 500; p = 10
beta = rep(1, p)
X = matrix(rnorm(n * p), n, p)
Y = X %*% beta + rt(n, 2)

## Smoothed quantile regression with Gaussian kernel
fit.Gauss = conquer(X, Y, tau = 0.5, kernel = "Gaussian")
beta.hat.Gauss = fit.Gauss$coeff

## Smoothe quantile regression with uniform kernel
fit.unif = conquer(X, Y, tau = 0.5, kernel = "uniform")
beta.hat.unif = fit.unif$coeff

## Construct three types of confidence intervals via multiplier bootstrap
fit = conquer(X, Y, tau = 0.5, kernel = "Gaussian", ci = "bootstrap")
ci.per = fit$perCI
ci.piv = fit$pivCI
ci.norm = fit$normCI