help qpolycoint (also see online)

Title

qpolycoint -- Quantile polynomial cointegration with FM correction
and Wald linearity test (Li, Zheng & Guo 2016).

Syntax

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

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

porder(#) polynomial order p ∈ {2,...,5}; default 3 (Li 2016 recommend 2 or 3)

bandwidth(#) bandwidth M; default = ceil(2*T^(1/3))

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

graph produce combined graph of β̂(τ) and Q(τ)

notable suppress coefficient table

notest suppress Wald test table

savecoef(name) save coefficient matrix

savetest(name) save Wald-statistic vector

Description

qpolycoint implements the polynomial quantile cointegration model of
Li, Zheng & Guo (2016):

Q_y_t(τ|x_t) = α + β(τ)'x_t + Σ_^p γ_j(τ)'(x_t)^j + F⁻¹(τ)

The polynomial regressors (x_t)^j capture nonlinear long-run relationships.
The fully-modified estimator removes endogeneity bias following Phillips & Hansen
(1990). A Wald-type test for linearity is provided:

H_0 : γ_2(τ) = ... = γ_p(τ) = 0 (linear cointegration)

H_1 : at least one γ_j(τ) ≠ 0 (nonlinear)

Under H_0, the Wald statistic Q ⇒ χ²_ where k is the number of I(1)
regressors. Critical values are computed automatically.

Examples

. tsset t

. qpolycoint y x, tau(0.25 0.5 0.75)

. qpolycoint y x, tau(0.1 0.5 0.9) porder(2) graph

Stored results

Scalars

e(N) sample size

e(porder) polynomial order

e(k) number of I(1) regressors

e(nrestr) # Wald restrictions = k·(p-1)

e(bandwidth) bandwidth M

e(cv5_q) / e(cv1_q) χ² critical values

Matrices

e(coef_set) ntau × (1 + kp) FM polynomial coefficients

e(tQ_set) ntau × 1 Wald linearity statistics

e(pval_set) ntau × 1 p-values

e(fm_se) ntau × (1 + kp) FM standard errors

Reference

Li, H., Zheng, C. & Guo, Y. (2016). Estimation and test for quantile nonlinear
cointegrating regression. Economics Letters 148, 27–32.

Author

Dr Merwan Roudane

merwanroudane920@gmail.com

February 2026

Also see

xqcoint, tuqcoint, liqcoint_fc, qcointall, fqardl