*! version 1.0.0 05jun2026

Title

xtpfnardl -- Fourier-augmented panel nonlinear ARDL (Panel Fourier NARDL)

Syntax

xtpfnardl
depvar asymvars
[if] [in]
[, options]

where asymvars are the regressors to be decomposed into positive and
negative partial sums (the asymmetric regressors).

Model
lin(varlist) linear (symmetric) control variables, entered without decomposition

model(type) pmg (default), csardl, mg, dfe

crlags(#) lags of cross-section averages; default crlags(3)

nocross omit cross-sectional averages

Lag / Fourier
plags(#) / qlags(#) short-run lags; default plags(1) qlags(1)

maxk(#) max Fourier frequency searched; default maxk(2)

k(#) fix the Fourier frequency

fractional search frequencies in 0.1 steps

nofourier standard panel NARDL (no Fourier terms)

trend include a linear trend

Reporting
hausman report the PMG-vs-MG long-run poolability (Hausman) test

mhorizon(#) horizon for the cumulative dynamic multipliers; default mhorizon(20)

replace overwrite existing v_pos/v_neg variables

level(#) confidence level

nograph suppress graphs

graphprefix(string) prefix for exported .png files

nodiag suppress diagnostics

Data must be xtset. The engine is xtdcce2.

Description

xtpfnardl estimates the Fourier-augmented panel nonlinear ARDL, i.e.
the asymmetric (NARDL) counterpart of xtpfardl. Each variable in
asymvars is decomposed into cumulative positive and negative partial sums
(Shin, Yu & Greenwood-Nimmo 2014), allowing increases and decreases to exert
distinct long- and short-run effects, while Enders-Lee Fourier terms absorb
unknown smooth structural breaks. This is the FPMG-NARDL estimator used to test
whether detected asymmetries are genuine or merely spurious artefacts
of unmodelled structural change.

For each decomposed variable it reports the long-run (+)/(-)
coefficients, the error-correction speed, the short-run dynamics, and formal
Wald asymmetry tests (long-run H0: {&theta}+ = {&theta}-
and short-run H0: {&gamma}+ = {&gamma}-). For each decomposed
variable it draws two figures: the cumulative dynamic multipliers with
confidence-interval bands (converging to {&beta}+ and {&beta}-), and the
asymmetry path m+(h){&minus}m-(h) with its CI band, which converges
to the long-run asymmetry {&beta}+{&minus}{&beta}- (zero under symmetry).
With hausman it also reports the PMG-vs-MG poolability test.

Options

lin(varlist) lists symmetric controls that enter the model linearly
(e.g. GDP, FDI, trade openness), without +/- decomposition.

model(type) selects the estimator. pmg (default) pools the
long-run +/- coefficients while keeping the adjustment and short-run
dynamics heterogeneous, as in the source paper. csardl/mg make all
coefficients heterogeneous (with/without cross-section averages); dfe
pools everything.

replace is required to re-run on data where the partial-sum
variables v_pos / v_neg already exist (they are left in
memory for inspection).

The remaining options (maxk, k, fractional,
nofourier, crlags, nocross, plags, qlags,
trend) behave exactly as in xtpfardl.

Methodology

Each asymmetric regressor X is split into

X+(i,t) = {&Sigma}s max({&Delta}X(i,s),0), X-(i,t) = {&Sigma}s min({&Delta}X(i,s),0),

and the Fourier-augmented asymmetric UECM is estimated:

D.Y = {&mu} + {&rho}·L.Y + {&theta}+·L.X+ + {&theta}-·L.X-
+ {&Sigma}{&psi}D.Y + {&Sigma}({&gamma}+D.X+ + {&gamma}-D.X-)
+ {&lambda}0 sin(2{&pi}kt/T) + {&lambda}1 cos(2{&pi}kt/T) + e.

Long-run effects are {&beta}+ = -{&theta}+/{&rho} and {&beta}- =
-{&theta}-/{&rho} (delta method). Asymmetry is assessed by Wald tests; if the
long-run asymmetry survives the Fourier augmentation, it is attributed to genuine
nonlinearity rather than smooth structural change.

Examples

. xtset country year

Baseline FPMG-NARDL (paper specification): REC on three uncertainty
indicators (decomposed) with GDP, FDI, TO as linear controls

. xtpfnardl rec cpu eru gpr, lin(gdp fdi to) model(pmg) maxk(2)

Heterogeneous (mean-group) NARDL, fixed frequency

. xtpfnardl rec cpu eru gpr, lin(gdp fdi to) model(mg) k(1) replace

Robustness: drop Fourier terms to see whether asymmetry was spurious

. xtpfnardl rec cpu eru gpr, lin(gdp fdi to) nofourier replace

Stored results

xtpfnardl stores in e() (plus the xtdcce2 engine results):

Scalars

e(N_g), e(kstar), e(plags), e(qlags) groups, frequency, lags

e(ect_b), e(ect_se) speed of adjustment

Macros

e(asymvars), e(posvars), e(negvars), e(linvars) variable lists

e(lrnames) long-run coefficient names

Matrices

e(lr_b), e(lr_V) long-run estimates and VCV

e(asym) per-variable Wald asymmetry tests (LR W, p, SR W, p)

e(kgrid) frequency grid and SSR

References

Bildirici, M., and F. Kay{&iacute}k{&ccedil}{&inodot}. 2022. Renewable energy and current
account balance nexus. Environmental Science and Pollution Research 29:
48759-48768. doi:10.1007/s11356-022-19286-9.

Bildirici, M., and F. {&Ccedil}oban Kay{&iacute}k{&ccedil}{&inodot}. 2024. Energy consumption,
energy intensity, economic growth, FDI, urbanization, PM2.5 concentrations
nexus. Environment, Development and Sustainability 26: 5047-5065.
doi:10.1007/s10668-023-02923-9.

Arif, M. A., and F. Furuoka. 2026. Genuine or spurious asymmetry?
Disentangling uncertainty effects on renewable energy consumption using
Fourier-augmented panel NARDL. SSRN Working Paper 6719166.
papers.ssrn.com/sol3/papers.cfm?abstract_id=6719166.

Shin, Y., B. Yu, and M. Greenwood-Nimmo. 2014. Modelling asymmetric
cointegration and dynamic multipliers in a nonlinear ARDL framework. In
Festschrift in Honor of Peter Schmidt, 281-314. Springer.

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

Enders, W., and J. Lee. 2012. The flexible Fourier form and Dickey-Fuller
type unit root tests. Economics Letters 117: 196-199.

Also see

Help: xtpfardl (symmetric Fourier panel ARDL),
xtfdh (Fourier-DH causality),
fbnardl (time-series Fourier NARDL),
pnardl, xtdcce2

Author: Dr. Merwan Roudane

Email: merwanroudane920@gmail.com

GitHub: github.com/merwanroudane