pqrfe (version 1.3)

pqr: Penalized quantile regression with fixed effects

Description

Estimate regression parameters and tuning parameters for quantile, expectile, or M-quantile regression.

Remarks:

1. If the first column of `x` is entirely equal to 1, then the first element of `beta` represents the common intercept. Otherwise, there is no default common intercept (unlike the default behavior in `lm`).

2. If there is a common intercept and `effect` is `"fixed"` or `"lasso"`, a 'sum-to-zero constraint' is applied on the `alpha` parameters: $$\sum_{i=1}^{n} \alpha_i = 0$$ This follows the approach in Danilevicz (2025).

Usage

pqr(x, y, subj, tau = 0.5, effect = "fixed", c = 0)

Value

alpha Numeric vector, intercepts' coefficients.

beta Numeric vector, exploratory variables' coefficients.

lambda Numeric, estimated lambda.

res Numeric vector, percentile residuals.

tau Numeric scalar, the percentile.

penalty Numeric scalar, indicate the chosen effect.

c Numeric scalar, indicate the chosen c.

sig2_alpha Numeric vector, intercepts' standard errors.

sig2_beta Numeric vector, exploratory variables' standard errors.

Tab_alpha Data.frame, intercepts' summary.

Tab_beta Data.frame, exploratory variables' summary.

Mat_alpha Numeric matrix, intercepts' summary.

Mat_beta Numeric matrix, exploratory variables' summary.

Arguments

x

Numeric matrix, covariates

y

Numeric vector, outcome.

subj

Numeric vector, identifies the unit to which the observation belongs.

tau

Numeric scalar between zero and one, identifies the percentile.

effect

Factor, "simple" simple regression, "fixed" regression with fixed effects, "lasso" penalized regression with fixed effects.

c

Numeric, 0 is quantile, Inf is expectile, any number between zero and infinite is M-quantile.

References

Danilevicz, I.M., Bondon, P., Reisen, V.A. (2025) "Adaptive LASSO Quantile Regression with Fixed Effects", Appl. Math. Model., xx (xx), <doi:10.1016/j.apm.2025.116600>

Danilevicz, I.M., Reisen, V.A., Bondon, P. (2024) "Expectile and M-quantile regression for panel data", Stat. Comput., 34 (97), <doi:10.1007/s11222-024-10396-7>

Koenker, R. (2004) "Quantile regression for longitudinal data", J. Multivar. Anal., 91(1): 74-89, <doi:10.1016/j.jmva.2004.05.006>

Examples

n = 10
m = 5
d = 4
N = n*m
x = matrix(rnorm(d*N), ncol=d, nrow=N)
subj = rep(1:n, each=m)
alpha = rnorm(n)
beta = rnorm(d)
eps = rnorm(N)
y = as.vector(x %*% beta + rep(alpha, each=m) + eps)
m1 = pqr(x=x, y=y, subj=subj, tau=0.75, effect="lasso", c = 0)
m1$Tab_beta