help pnardl version 1.1.0

Title

pnardl -- Panel Nonlinear ARDL (Panel NARDL) Estimation

Version

Version 1.1.0, 12 February 2026

Author: Dr Merwan Roudane (merwanroudane920@gmail.com)

Based on:

Shin, Y., Yu, B., and Greenwood-Nimmo, M. (2014). Modelling Asymmetric
Cointegration and Dynamic Multipliers in a Nonlinear ARDL Framework.
In: Sickles, R., Horrace, W. (eds) Festschrift in Honor of Peter Schmidt.
Springer, New York. pp. 281-314.

Salisu, A.A. and Isah, K.O. (2017). Revisiting the oil price and
stock market nexus: A nonlinear Panel ARDL approach.
Economic Modelling 66: 258-271.
doi:10.1016/j.econmod.2017.07.010

Requires: xtpmg version 2.0.1 or later

What's new in version 1.1.0:

- Dynamic Multipliers: multip() traces cumulative m+(h) and m-(h) over h periods

- Asymmetry Table: asytable side-by-side beta+/beta- and gamma+/gamma- with Wald tests

- IRF for +/- Shocks: irfshock() separate impulse responses for positive and negative shocks

- Per-Panel Coefficients: panelcoef ECT, half-life, convergence status per panel

- Graph Visualizations: graph publication-quality graphs (ECT bars, LR asymmetry, dynamic multipliers)

Syntax

pnardl depvar [indepvars] [if] [in],
lr(varlist) asymmetric(varlist) [options]

Required
lr(varlist) variables for the long-run cointegrating vector

asymmetric(varlist) variables to decompose into positive and negative partial sums

Model
pmg Pooled Mean Group estimator (default)

mg Mean Group estimator

dfe Dynamic Fixed Effects estimator

ec(name) name for the error-correction term; default is ECT

replace overwrite existing decomposed variables and EC term

noconstant suppress constant term

cluster(varname) clustered standard errors

constraints(numlist) apply constraints

full display all panel regressions

Testing
hausman perform Hausman test (MG vs PMG)

noasymtest suppress asymmetry Wald tests

ML Options (PMG only)
technique(algorithm) ML maximization technique

difficult alternative stepping algorithm

Reporting
level(#) set confidence level; default is level(95)

Diagnostics (New in 1.1.0)
multip(#) compute dynamic multipliers for +/- shocks over # periods

asytable display asymmetry comparison table (LR and SR)

irfshock(#) simulate IRF for positive vs negative shocks over # periods

panelcoef display per-panel ECT coefficients, half-life, and convergence

graph generate publication-quality graphs

You must xtset your data before using pnardl; see tsset.

Description

pnardl implements the Panel Nonlinear ARDL (Panel NARDL) methodology for
estimating asymmetric long-run and short-run relationships in panel data. It
automates the complete Panel NARDL workflow:

1. Decomposition: Splits specified variables into positive and negative
cumulative partial sums for asymmetric effect analysis.

2. Estimation: Estimates Pooled Mean Group (PMG), Mean Group (MG), or Dynamic
Fixed Effects (DFE) models via xtpmg.

3. Model Selection: Optionally performs Hausman test to choose between
MG and PMG estimators.

4. Asymmetry Testing: Performs Wald tests for both long-run and short-run
asymmetry for each decomposed variable.

The methodology is based on Shin, Yu, and Greenwood-Nimmo (2014), who extended
the linear ARDL cointegration framework of Pesaran, Shin, and Smith (2001) to
allow for asymmetric effects. The Panel NARDL approach, as applied by
Salisu and Isah (2017), combines this nonlinear decomposition with the panel
estimators of Pesaran, Shin, and Smith (1999).

Options

Required

lr(varlist) specifies the variables for the long-run cointegrating vector.
The first variable must be the lagged dependent variable. Variables listed in
asymmetric() will be automatically replaced with their positive and negative
partial sums in the long-run equation.

asymmetric(varlist) specifies the variables to be decomposed into positive
and negative cumulative partial sums. For each variable x, two new variables
are created: x_pos (positive partial sum) and x_neg (negative partial sum).

Model

pmg, mg, dfe select the estimation method. pmg (default) constrains
long-run coefficients to be homogeneous across panels. mg allows full heterogeneity.
dfe constrains all parameters except intercepts.

hausman performs a Hausman test comparing MG and PMG estimates to guide model selection.
Significant test results favor MG; insignificant results favor PMG.

noasymtest suppresses the automatic Wald tests for long-run and short-run asymmetry.

replace overwrites existing decomposed variables (x_pos, x_neg) and
the error-correction term in the dataset.

Methodology

The Panel NARDL model extends the standard panel ARDL framework by decomposing
an independent variable x into positive and negative cumulative partial sums:

x_pos(t) = sum(j=1 to t) max(delta_x(j), 0)

x_neg(t) = sum(j=1 to t) min(delta_x(j), 0)

These partial sums replace the original variable in the ARDL specification,
allowing distinct long-run and short-run coefficients for positive and negative
changes. The error-correction model becomes:

d.y(it) = phi(i) * [y(it-1) - beta_pos * x_pos(it) - beta_neg * x_neg(it)]
+ gamma_pos * d.x_pos(it) + gamma_neg * d.x_neg(it) + e(it)

Long-run asymmetry: H0: beta_pos = beta_neg (tested via Wald test)

Short-run asymmetry: H0: gamma_pos = gamma_neg (tested via Wald test)

See Salisu and Isah (2017) for a complete application of this methodology to
the oil price-stock market nexus, demonstrating how positive and negative oil
price shocks have asymmetric effects across both developed and developing
economies.

Stored Results

pnardl stores all xtpmg results in e(), plus:

Macros

e(cmd) pnardl

e(model) estimation model (PMG, MG, or DFE)

e(asymmetric) names of asymmetrically decomposed variables

e(pos_vars) names of positive partial sum variables

e(neg_vars) names of negative partial sum variables

Examples

Basic Panel NARDL estimation (PMG, default):

. xtset id year

. pnardl d.y d.x1 d.x2, lr(l.y x1 x2) asymmetric(x1) replace

Mean Group estimator with Hausman test:

. pnardl d.y d.x1 d.x2, lr(l.y x1 x2) asymmetric(x1) mg hausman replace

Multiple asymmetric variables:

. pnardl d.y d.x1 d.x2, lr(l.y x1 x2) asymmetric(x1 x2) replace

Suppress asymmetry tests:

. pnardl d.y d.x1 d.x2, lr(l.y x1 x2) asymmetric(x1) noasymtest replace

Application (Salisu and Isah, 2017):

. * Oil price-stock market nexus

. pnardl d.stock d.oil d.controls, lr(l.stock oil controls) asymmetric(oil) pmg hausman replace

Diagnostics and Graphs (New in 1.1.0):

Asymmetry comparison table:

. pnardl d.y d.x1 d.x2, lr(l.y x1 x2) asymmetric(x1) asytable replace

Per-panel coefficients:

. pnardl d.y d.x1 d.x2, lr(l.y x1 x2) asymmetric(x1) panelcoef replace

Dynamic multipliers (20 periods):

. pnardl d.y d.x1 d.x2, lr(l.y x1 x2) asymmetric(x1) multip(20) replace

IRF for positive vs negative shocks:

. pnardl d.y d.x1 d.x2, lr(l.y x1 x2) asymmetric(x1) irfshock(15) replace

All diagnostics with graphs:

. pnardl d.y d.x1 d.x2, lr(l.y x1 x2) asymmetric(x1) asytable panelcoef multip(20) irfshock(20) graph full replace

Exporting graphs:

. graph export pnardl_ect.png, name(pnardl_ect) replace width(1200)

. graph export pnardl_asym_lr.png, name(pnardl_asym_lr) replace width(1200)

. graph export pnardl_multiplier.png, name(pnardl_multiplier) replace width(1200)

References

Salisu, A.A. and Isah, K.O. 2017.
Revisiting the oil price and stock market nexus: A nonlinear Panel ARDL approach.
Economic Modelling 66: 258-271.
doi:10.1016/j.econmod.2017.07.010

Shin, Y., Yu, B. and Greenwood-Nimmo, M. 2014.
Modelling Asymmetric Cointegration and Dynamic Multipliers in a Nonlinear
ARDL Framework. In: Sickles, R., Horrace, W. (eds) Festschrift in Honor of Peter Schmidt.
Springer, New York. pp. 281-314.

Pesaran, M.H., Shin, Y. and Smith, R.P. 1999.
Pooled mean group estimation of dynamic heterogeneous panels.
Journal of the American Statistical Association 94: 621-634.

Pesaran, M.H., Shin, Y. and Smith, R.J. 2001.
Bounds testing approaches to the analysis of level relationships.
Journal of Applied Econometrics 16: 289-326.

Blackburne, E.F. III and M.W. Frank. 2007.
Estimation of nonstationary heterogeneous panels.
Stata Journal 7(2): 197-208.

Author

Dr Merwan Roudane

merwanroudane920@gmail.com

Also see

xtpmg, xtreg, tsset, hausman