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Methodology and equations -- CS-NARDL

Unifying principle

Every estimator in this package is the nonlinear (asymmetric) extension of a canonical
CCE/ARDL method. The asymmetric decomposition of Shin, Yu and Greenwood-Nimmo (2014) is
applied before the chosen engine is called, so the seven estimators provided by
xtcsnardl are precisely:

Engine Nonlinear extension of

pmg / mg / dfe Panel ARDL (Pesaran-Shin-Smith 1999)

engine(csardl) CS-ARDL (Chudik-Pesaran 2015)

engine(csdl) CS-DL (Chudik-Pesaran 2015)

engine(dcce) Dynamic CCE (Chudik-Pesaran 2015)

engine(cce) Static CCE (Pesaran 2006)

The theoretical justification rests on the two foundational papers of the nonlinear panel CSD
literature: Kapetanios, Mitchell and Shin (2014) formalise the nonlinear panel data model
with interactive (factor) errors and establish that consistent estimation requires the proxy
set to include nonlinear transforms of the cross-sectional averages. Hacioglu-Hoke and
Kapetanios (2020) then sharpen this into an explicit CCE correction: the standard Pesaran
(2006) proxy set must be augmented with cross-sectional averages of the
nonlinear-transformed regressors -- in CS-NARDL, the positive and negative partial sums
x+ and x-. This is done automatically by every engine in xtcsnardl.

Contents

1. Big picture: three layers of generalisation

2. Asymmetric decomposition (NARDL)

3. CCE and dynamic CCE (Pesaran 2006; Chudik-Pesaran 2015)

4. Nonlinear CCE (Kapetanios-Mitchell-Shin 2014 & Hacioglu-Hoke-Kapetanios 2020)

5. The CS-NARDL model

6. Error-correction reparameterisation (estimated form)

7. Relation to CS-DL and CS-ARDL

8. Estimation (PMG / MG / DFE) and identification

9. Tests for asymmetry

10. Asymmetric long-run and dynamic multipliers

11. CSD diagnostics

12. How to read CS-NARDL output

1. Big picture: three layers of generalisation

The CS-NARDL nests three orthogonal extensions of the classical panel ARDL:

Layer A -- Asymmetry. Drop the implicit linearity restriction
{&beta}+ = {&beta}- on regressors that may impact y differently when they
rise vs when they fall. Implemented as cumulative partial sums after
Shin, Yu and Greenwood-Nimmo (2014).

Layer B -- Cross-section dependence (linear CCE). Replace the strict
cross-sectional independence assumption with an interactive factor structure and
proxy the unobserved factors by cross-sectional averages (CSA), per
Pesaran (2006) and
Chudik and Pesaran (2015).

Layer C -- Nonlinear CCE. When the conditional mean is nonlinear
(here: positive/negative cumulative sums), the Pesaran proxy set must be augmented
with CSA of the nonlinear-transformed regressors to preserve consistency, per
Hacioglu-Hoke and Kapetanios (2020).

xtcsnardl implements all three layers simultaneously.

2. Asymmetric decomposition (NARDL)

Shin, Yu and Greenwood-Nimmo (2014) modify the Pesaran-Shin-Smith (1999) ARDL by replacing
selected regressors with their positive and negative cumulative partial sums. Let
{&Delta}xit = xit - xi,t-1. Define

x+it = {&Sigma}s=1..t max({&Delta}xis, 0) -- cumulative positive shocks

x-it = {&Sigma}s=1..t min({&Delta}xis, 0) -- cumulative negative shocks

By construction xit = xi0 + x+it + x-it, so the
decomposition reparametrises the path of x without information loss. Replacing
{&beta}xit in the cointegrating vector with {&beta}+x+it +
{&beta}-x-it allows the long-run elasticity to differ between rises and
falls. The linear ARDL is the nested null {&beta}+ = {&beta}-; rejecting this
restriction is the test of long-run asymmetry (Table 5).

The partial-sum variables generated by xtcsnardl are named
varname_pos and varname_neg, and they are retained in the dataset after estimation.
You can recover them for diagnostics, plotting, or future regressions.

3. CCE and Dynamic CCE

The linear CCE-MG estimator of Pesaran (2006) models the error structure of a heterogeneous
panel as

yit = ai + {&beta}i'xit + uit,
uit = {&gamma}i'ft + {&epsilon}it,
xit = {&Lambda}i'ft + vit

where ft is a vector of unobserved common factors. The
cross-sectional averages z-t = (y-t, x-t)' are then a
linear function of ft (up to O(N-1/2)), so including them in the regression
removes the factor-induced endogeneity:

yit = ai + {&beta}i'xit + {&delta}i'z-t + eit.

Dynamic CCE (Chudik and Pesaran 2015) extends this to ARDL-type dynamics by adding pT
lags of z-t, where the optimal lag-length is

pT = {&lfloor}T1/3{&rfloor}

under standard assumptions. This is the default rule used by xtcsnardl via the
cr_lags(#) option -- set to cr_lags(-1) for the floor(T1/3) heuristic
or override with any non-negative integer.

4. Nonlinear CCE (Kapetanios-Mitchell-Shin 2014 and Hacioglu-Hoke & Kapetanios 2020)

The nonlinear panel CSD framework rests on two foundational papers. xtcsnardl
uses both, and the methodology page describes their distinct contributions explicitly.

Step A. The nonlinear panel model with CSD (Kapetanios, Mitchell and Shin 2014).
KMS introduce a general nonlinear panel data model in which the dependent variable is a
nonlinear function of the regressors and a vector of unobserved common factors. Their
contribution is two-fold. First, they formalise a panel with heterogeneous coefficients,
nonlinear conditional mean and an interactive (factor) error structure of the form

yit = mi(xit, {&theta}i) + {&lambda}i'ft + {&epsilon}it,
xit = {&Pi}i'ft + vit.

Second, they study the asymptotic identification of {&theta}i when N, T -> infinity and
propose a sieve-CCE strategy: factors are proxied not only by linear cross-sectional averages
but also by nonlinear transforms of them. KMS (2014) thus establish the principle that
nonlinear panel CSD models require nonlinear CSA proxies.

Step B. CCE corrections for nonlinear conditional mean (Hacioglu-Hoke and Kapetanios 2020).
HHK refine and operationalise the KMS programme for nonlinear conditional-mean models of the
form

yit = g(xit, {&gamma}0)'{&beta}i + uit,
uit = {&lambda}i'ft + {&epsilon}it.

They show that when g is nonlinear, the cross-sectional average y-t contains
X-({&gamma})t = (1/N){&Sigma}i g(xit, {&gamma}), which is no longer
a linear function of the factors. The Pesaran (2006) rank condition fails, so the naive proxy
set Z- = (y-, x-) does not consistently estimate {&beta}.

HHK (2020, Theorem 2) prove that consistency is restored by augmenting the proxy set with the
CSA of the nonlinear-transformed regressors: Z-{&gamma} = (z-, X-({&gamma})).
They derive consistent pooled and Mean Group estimators with sqrt(N)-asymptotic normality
under standard assumptions.

Application to CS-NARDL. In our setting the nonlinear transformation g is precisely
the Shin-Yu-Greenwood-Nimmo positive/negative partial-sum decomposition. Combining
KMS (2014) and HHK (2020) the prescription becomes:

Take CSA of every x+ and every x-, in addition to CSA of y and the linear
controls -- and add Chudik-Pesaran (2015) lags of these CSA series.

This is exactly what xtcsnardl does by default. KMS (2014) supplies the framework
(why we need nonlinear CSA at all); HHK (2020) supplies the operational rule
(which nonlinear CSA series to add and how to derive consistent estimators). The
csavars(varlist) option lets you override the proxy set when you have a theoretical
reason; the default proxy set is jointly KMS- and HHK-compliant.

5. The CS-NARDL model

For panel i = 1, ..., N and period t = 1, ..., Ti, let yit be the dependent
variable, xit a k-vector of asymmetric regressors and cit an m-vector of symmetric
controls. The CS-NARDL data-generating process is

yit = ai + {&beta}+'x+it + {&beta}-'x-it +
{&beta}c'cit + uit, (Long-run)

uit = {&lambda}i'ft + {&epsilon}it, (Common factors)

xit = {&Pi}i'ft + vit. (Reduced form for x)

{&beta}+, {&beta}- and {&beta}c are the long-run elasticities of interest.
{&epsilon}it is idiosyncratic and {&lambda}i'ft captures common shocks
(global cycles, energy prices, policy waves, ...).

6. Error-correction reparameterisation (estimated form)

The full CS-NARDL ECM (the form actually estimated by xtcsnardl via xtpmg) is

{&Delta}yit
= {&phi}i [ yi,t-1 - {&beta}+'x+i,t-1 - {&beta}-'x-i,t-1 - {&beta}c'ci,t-1

- {&Sigma}k=0p_T {&psi}k'z-t-k ]
+ {&Sigma}j=1p {&gamma}ij {&Delta}yi,t-j

+ {&Sigma}j=0q-1 ({&omega}+ij'{&Delta}x+i,t-j + {&omega}-ij'{&Delta}x-i,t-j)
+ {&Sigma}j=0q-1 {&delta}ij'{&Delta}ci,t-j
+ {&eta}i'{&Delta}z-t
+ {&epsilon}it.

Reading off the parts:

{&phi}i < 0 is the speed of adjustment of unit i to its long-run
equilibrium. Strict {&phi}i < 0 is the cointegration condition.

The square-bracketed term in {&Delta}y is the cointegrating residual
ECi,t-1. Its coefficients {&beta}+, {&beta}- are the long-run elasticities;
{&psi}k are nuisance loadings on CSA-lags.

{&omega}+ij, {&omega}-ij are the short-run asymmetric
coefficients.

{&eta}i'{&Delta}z-t is the short-run CSA term (additional
nuisance, contemporaneous correction for common shocks in the {&Delta}-equation).

7. Relation to CS-DL and CS-ARDL

CS-DL (Cross-Section Distributed Lag, Chudik and Pesaran 2015) writes

yit = ai + {&beta}i'xit + {&Sigma}k=0p_T {&psi}k'z-t-k + eit

with no lagged y on the right-hand side. It estimates the long-run elasticity directly
but provides no information on the short-run dynamics. The nonlinear CS-DL is the same
equation after replacing {&beta}i'xit with {&beta}+i'x+it +
{&beta}-i'x-it.

CS-ARDL (Chudik, Mohaddes, Pesaran and Raissi 2017) writes the ARDL(p,q) form

yit = ai + {&Sigma}j=1p {&phi}ijyi,t-j
+ {&Sigma}j=0q-1 {&beta}ij'xi,t-j
+ {&Sigma}k=0p_T {&psi}k'z-t-k + eit

and recovers the long-run elasticities through the standard ARDL transformation
{&beta}LR = {&Sigma}j{&beta}ij / (1 - {&Sigma}j{&phi}ij). The
nonlinear CS-ARDL replaces x with x+, x-.

CS-NARDL = nonlinear CS-ARDL in error-correction form. The estimated equation in section
6 is the algebraic reparameterisation of the CS-ARDL with
the partial-sum decomposition substituted in. This unifies short-run and long-run
asymmetric inference in a single regression and exposes the speed of adjustment {&phi}i
as an estimable parameter.

8. Estimation (PMG / MG / DFE)

xtcsnardl delegates estimation to xtpmg. Three flavours are available:

PMG (default). Long-run coefficients {&beta}+, {&beta}-, {&beta}c and
all CSA loadings are pooled (equal across panels); the speed of adjustment {&phi}i,
the short-run dynamics and the intercept are panel-specific. Estimation is by maximum
likelihood as in Pesaran, Shin and Smith (1999), implemented in xtpmg via Newton-Raphson.
PMG is the workhorse of the empirical CS-NARDL literature (e.g. Mehta & Derbeneva 2024;
Wang et al. 2022).

MG. All slopes are heterogeneous; the reported estimates are simple cross-section
averages with Pesaran-Smith (1995) standard errors. Use MG when the Hausman test rejects
long-run pooling.

DFE. All slopes are pooled; only panel-specific intercepts remain. Useful for very
short T and as a baseline.

Identification. Under Pesaran (2006, Assumption A3) and Hacioglu-Hoke & Kapetanios
(2020, Theorem 2):

N, T -> {&infin} with T/N -> {&kappa} 1;

the factor loadings are i.i.d. and independent of the errors;

the CSA proxy set Z-{&gamma} = (y-, x-+, x--, c-)
contains at least as many series as the number of factors (rank condition);

the cointegrating relationship holds in each panel.

The PMG estimator is then sqrt(N)-consistent and asymptotically normal.

9. Tests for asymmetry

For each variable in asymmetric(), xtcsnardl reports two Wald tests
(asymptotically {&chi}2(1)):

Long-run asymmetry. H0: {&beta}+ = {&beta}- against
{&beta}+ {&ne} {&beta}-. Rejection means the cumulative effect of a unit
rise differs in magnitude from the cumulative effect of a unit fall.

Short-run asymmetry. H0: {&gamma}+ = {&gamma}-. Rejection means the
immediate response (within the period) differs between rises and falls.

A common pattern in the empirical literature is to find long-run asymmetry without
short-run asymmetry -- markets respond symmetrically to rises and falls in the
short run but accumulate the imbalance over the cointegrating horizon.

10. Asymmetric long-run and dynamic multipliers

Long-run asymmetric multipliers. Read directly from the cointegrating vector:

m+({&infin}) = {&beta}+, m-({&infin}) = {&beta}-,
Asymmetry = {&beta}+ - {&beta}-.

Cumulative dynamic multipliers. For horizon h = 0, 1, 2, ..., xtcsnardl computes

m+(h) = {&Sigma}k=0h {&part}yi,t+k / {&part}x+i,t,
m-(h) = {&Sigma}k=0h {&part}yi,t+k / {&part}x-i,t.

Using the AR(1) approximation of the cointegrating recursion under PMG (Shin-Yu-Greenwood-Nimmo
2014, eq. 18), the trajectories satisfy

m+(h+1) = m+(h) + {&phi}*(m+(h) - {&beta}+), m+(0) = 1

m-(h+1) = m-(h) + {&phi}*(m-(h) - {&beta}-), m-(0) = -1

with {&phi}* the cross-section average of {&phi}i over convergent panels. Both
trajectories converge to their long-run targets at rate -{&phi}*; the asymmetry
curve m+(h) - m-(h) measures the cumulative imbalance.

The half-life of disequilibrium is

HL = ln(2) / |{&phi}*|.

A typical convergent CS-NARDL has |{&phi}*| = 0.3 and HL = 2.3 periods.

11. Cross-sectional dependence diagnostics

After estimation xtcsnardl runs the Pesaran (2004) CD test on the residuals:

CD = sqrt(2T / (N(N-1))) * {&Sigma}i<j {&rho}-ij -> N(0,1)

under H0: residual cross-sectional independence. It also reports the average pairwise
correlation {&rho}-bar and the Pesaran (2015) absolute correlation measure
|{&rho}-bar|.

CD p-value > 0.10 -> CSA augmentation appears sufficient.

CD p-value < 0.05 -> residual CSD remains; increase cr_lags() or extend
csavars().

Use nocdtest to suppress this diagnostic. For independent confirmation, run
xtcd2 on the residuals manually.

12. How to read CS-NARDL output

A typical "good" CS-NARDL printout looks like this:

Table 1 -- {&beta}+ and {&beta}- are both significant but different
in magnitude or sign. Their CIs do not overlap.

Table 2 -- {&phi} is negative and significant, half-life under 5 periods, class
strong or moderate.

Table 3 -- short-run coefficients smaller in magnitude than long-run ones (full
adjustment takes several periods).

Table 5 -- Long-run Wald test rejects (asymmetry confirmed); short-run Wald test
may or may not reject.

Table 10 -- CD test does not reject (residuals are weakly dependent after
CSA augmentation).

Red flags to watch for:

{&phi} positive or {&phi} -2 -> no convergence; model is misspecified
or no cointegration.

Standard errors of {&beta}+/{&beta}- larger than the point estimates
-> weak identification, often from too many CSA lags or too few panels.

CD test rejects -> residual common factors; CSA augmentation insufficient.

Hausman test rejects PMG and the per-panel {&phi}i are widely dispersed
-> switch to MG.

Further reading

Worked examples - five complete specifications with output and
interpretation.

Post-estimation - e() returns, custom Wald tests,
prediction, manual multipliers.

Graphs - publication-quality plots.

Main reference - syntax and options.

Author

Dr Merwan Roudane

merwanroudane920@gmail.com

xtcsnardl v1.0.0, 28 May 2026

Also see

Related Stata packages: xtpmg pnardl xtdcce2 xtcspqardl xtcd2 xtcse2 xtbreak