Title

xtbreaklm -- Panel Lagrange-multiplier (LM) unit-root tests that
allow for structural breaks: Enders-Lee flexible-Fourier (smooth breaks) and
Lee-Tieslau two-break (sharp breaks)

Syntax

xtbreaklm varname [if] [in] [, options]

The data must be xtset as a strongly balanced panel.

type(string) fourier (Enders-Lee smooth breaks, the
default) or sharp (Lee-Tieslau two breaks)

nharm(#) cumulative Fourier frequency for type(fourier)
(0-3; default 1)

pmax(#) maximum number of augmentation lags (default 8)

ic(#) lag-selection rule: 1=AIC, 2=BIC, 3=general-to-specific
t-stat (default 3)

model(string) deterministic terms; trend (default). Both
tests are trend-based

Description

xtbreaklm implements two panel LM unit-root tests that remain valid in the
presence of structural breaks without pre-testing for the break form. The
null hypothesis is a unit root in every panel; the alternative is that at least
one panel is (trend-)stationary around its break process.

type(fourier) is the Enders-Lee flexible-Fourier panel LM test. Each unit's
trend is augmented with nharm cumulative sine/cosine terms, which approximate
an unknown number of smooth (gradual) breaks; unit-specific LM-tau statistics
are converted to p-values with the paper's response surface and pooled into the
Fisher P, standardized Pm and inverse-normal Choi Z panel
statistics.

type(sharp) is the Lee-Tieslau panel LM test with two sharp breaks in
level and trend. Each unit's two break dates are selected by grid search
(minimizing the LM-tau statistic); the group-mean statistic is standardized with
tabulated moments to give the Panel LM (IPS-type) statistic.

Both tests are single-series LM tests pooled across the panel; they do not
remove common factors (use xtpstat or xtfpanic for
cross-sectional-dependence-robust tests). xtbreaklm is part of the
xtpdroot library. See
help xtbreaklm methods for the formulas.

Options

type(string): fourier (Enders-Lee, default) selects the smooth
Fourier LM test; sharp (Lee-Tieslau) selects the two-break LM test.

nharm(#): the maximum cumulative Fourier frequency k=1..#
(only for type(fourier)). nharm(1) is the single-frequency form
recommended by Enders-Lee; nharm(2) adds the second harmonic.

pmax(#): the maximum number of lagged differences in the LM
regression. The Nazlioglu et al. (2023) replication used pmax(3); the
general-to-specific rule then trims from # downward.

ic(#): 1 AIC, 2 BIC, or 3 (default) the
general-to-specific significance rule of Hall (1994) with a 10% (t=1.645)
threshold, exactly as in the source GAUSS code.

model(string): kept for interface symmetry; both tests are estimated
with a broken linear trend (trend).

Examples

Enders-Lee smooth-break panel LM (single frequency):

. xtset country year

. xtbreaklm y, type(fourier) nharm(1) pmax(3)

Add the second harmonic:

. xtbreaklm y, type(fourier) nharm(2) pmax(3)

Lee-Tieslau two-break panel LM (per-unit break dates reported):

. xtbreaklm y, type(sharp) pmax(3)

Stored results

xtbreaklm is rclass. For type(fourier) it stores:

Scalars

r(P), r(P_p) Fisher chi-square panel statistic and p-value

r(Pm), r(Pm_p) standardized panel statistic and p-value

r(Choi), r(Choi_p) inverse-normal (Choi Z) statistic and p-value

r(nreject) number of units rejecting at 5%

r(N), r(T) panel and time dimensions

For type(sharp) it stores r(PanelLM) and r(PanelLM_p)
(the standardized group-mean statistic and its p-value) plus r(N),
r(T).

Macros

r(cmd) xtbreaklm

r(type) fourier or sharp

r(model) trend

Matrices

r(units) per-unit results: fourier = id, LM-tau,
p-value, lags, cv(10/5/1%); sharp = id, LM, break1, break2, lags

Interpreting the output

The null is a unit root in all panels. For type(fourier), P and
Pm reject for large values and Choi Z for very negative
values, in favour of stationarity in at least one panel. For type(sharp),
the Panel LM statistic rejects for very negative values. The reported
break columns are the estimated break periods (in the units of your time
variable), not indices.

References

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

Lee, J., and M. C. Strazicich. 2003. Minimum Lagrange multiplier unit root
test with two structural breaks. Review of Economics and Statistics 85:
1082-1089.

Nazlioglu, S., et al. 2023. Fourier panel analysis of nonstationarity in
idiosyncratic and common components. Econometric Reviews.

Author

Dr Merwan Roudane

merwanroudane920@gmail.com

github.com/merwanroudane

Faithful Stata port of the GAUSS routines Fourier_LM (Enders-Lee) and
LMt_2breaks (Lee-Tieslau) by S. Nazlioglu; validated byte-for-byte against
the GAUSS output on the health-expenditure panel (Fourier m=1 P=42.348, m=2
P=55.282; Lee-Tieslau Panel LM=-24.048). Part of the xtpdroot
library.