help xqcoint_robust

Title

xqcoint_robust -- Robust cointegration test in quantile regressions
based on partial-sum process of ψ_τ residuals (Xiao 2009, Section 3.3).

Syntax

xqcoint_robust depvar indepvars [if] [in],
tau(numlist)
[options]

tau(numlist) quantile levels in (0,1); required

leads(#) leads of Δx in augmented regression; default 1

lags(#) lags of Δx in augmented regression; default 1

bandwidth(#) long-run variance bandwidth; default ceil(2*T^(1/3))

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

graph plot KS and CVM functionals vs τ

notable suppress results table

saveks(name) save KS statistic vector

savecvm(name) save CVM statistic vector

Description

xqcoint_robust implements the robust cointegration test of Xiao (2009)
Section 3.3. For each quantile τ, the augmented quantile regression
(with Saikkonen-style leads/lags of Δx) yields residuals ε̂_t(τ). The
partial-sum process is

Y_n(r) = ω̂_ψ⁻¹ · n^(-1/2) · Σ_^{[nr]} ψ_τ(ε̂_jτ)

where ψ_τ(u) = τ − I(u < 0). Two functionals are computed:

KS = sup_r |Y_n(r)|

CVM = ∫_0^1 Y_n(r)² dr

Under H0 of cointegration, Y_n(r) converges to a centered Gaussian process
(closely related to a Brownian bridge). Under H1 of no cointegration, the
statistics diverge. Reference asymptotic critical values from the
Brownian-bridge approximation: KS at 5% = 1.358, 1% = 1.628;
CVM at 5% = 0.461, 1% = 0.743.

Note: Xiao 2009 shows that the EXACT limiting distribution of Y_n(r)
involves stochastic integrals of demeaned Brownian motions and differs
slightly from a Brownian bridge. The CVs reported are conservative.
For exact size, use a Monte Carlo bootstrap. Finite-sample power can be
limited (n ≥ 500 recommended for reliable inference).

This test is DISTINCT from xqcoint's CUSUM (Kuriyama 2016), which
uses FM-residuals rather than augmented residuals.

Examples

. tsset t

. xqcoint_robust y x, tau(0.1 0.5 0.9) leads(2) lags(2) graph

Stored results

Scalars

e(leads) / e(lags) augmentation orders

e(ks_cv5) / e(ks_cv1) KS critical values

e(cvm_cv5) / e(cvm_cv1) CVM critical values

Matrices

e(ks_set) ntau × 1 KS statistics

e(cvm_set) ntau × 1 CVM statistics

Reference

Xiao, Z. (2009). Quantile cointegrating regression. Journal of Econometrics
150, 248–260 — Section 3.3.

Author

Dr Merwan Roudane

merwanroudane920@gmail.com

February 2026

Also see

xqcoint, xqcoint_const, qpolycoint, qcointall