help xqcoint_const

Title

xqcoint_const -- Constancy test of the cointegrating vector across
quantiles (Xiao 2009, Section 3.2).

Syntax

xqcoint_const depvar indepvars [if] [in]
[, options]

tau(numlist) quantile grid in (0,1); default 0.05, 0.10, ..., 0.95

ngrid(#) # of grid points if tau() not given; default 19

leads(#) leads of Δx in augmented regression; default 0

lags(#) lags of Δx in augmented regression; default 0

bandwidth(#) long-run variance bandwidth

kernel(name) bartlett (default), parzen, qs

simreps(#) Monte Carlo replications for critical values; default 5000

graph plot V̂_n(τ) process

notable suppress results table

Description

xqcoint_const tests whether the cointegrating vector β is constant across
quantiles:

H0 : β(τ) = β̄ for all τ ∈ T (location-shift cointegration)

H1 : β(τ) varies with τ (quantile-dependent cointegration)

The test process is V̂_n(τ) = (β̂(τ) − β̄)/SE(β̂_OLS), where β̄ is the OLS
cointegrating estimate and SE(β̂_OLS) is the standard OLS standard error.
For one-dimensional β, three functionals are reported:

sup_τ |V̂_n(τ)| — supremum statistic

KS = sup_τ |V̂_n(τ)| (same as sup for one-dim case)

CVM = mean_τ V̂_n(τ)² — Cramer-von Mises analogue

For multi-dimensional β (k > 1), the L∞ norm across coefficients at each
τ is used.

Critical values: under H0, V̂_n(τ) converges to a centered Gaussian
process (Xiao 2009 Theorem 4). The sup/KS/CVM functionals are simulated
via Monte Carlo of standardized Brownian-bridge draws on the τ grid.
The p-values are exact (within Monte Carlo error).

Examples

Test whether the slope is constant across 5 quantiles:

. tsset t

. xqcoint_const y x, tau(0.1 0.25 0.5 0.75 0.9)

With default 19-point grid (τ = 0.05, ..., 0.95):

. xqcoint_const y x, ngrid(19)

With augmented regression and graph:

. xqcoint_const y x, tau(0.1(0.1)0.9) leads(2) lags(2) graph

Stored results

Scalars

e(sup_stat) / e(ks_stat) / e(cvm_stat) test statistics

e(sup_pval) / e(ks_pval) / e(cvm_pval) MC p-values

e(ntau) # of quantiles

e(simreps) # Monte Carlo replications

Matrices

e(Vhat) ntau × k V̂_n process values

e(cv_mat) 3 × 3 matrix: rows sup/KS/CVM × cols cv5/cv1/pval

Reference

Xiao, Z. (2009). Quantile cointegrating regression. Journal of Econometrics
150, 248–260 — Section 3.2 and Theorem 4.

Author

Dr Merwan Roudane

merwanroudane920@gmail.com

February 2026

Also see

xqcoint, xqcoint_robust, qpolycoint, qcointall