Title

xtbreaklm methods -- Formulas for the panel LM tests with breaks

Enders-Lee flexible-Fourier panel LM

For each unit the LM detrending regression uses a broken trend approximated by a
Fourier expansion. Let z_t = (t, sin(2pi k t/T), cos(2pi k
t/T)) for k=1..m. The score (LM) series is

S_t = y_t - psi - z_t' delta,

where delta is estimated from the differenced regression Dy_t on
Dz_t and psi = y_1 - z_1' delta. The test statistic is the
t-ratio tau on phi in

Dy_t = phi S_(t-1) + Dz_t' gamma + sum_j c_j D S_(t-j)
+ e_t.

The number of lags is chosen by AIC, BIC or the general-to-specific t-rule
(t=1.645). Each tau_i is mapped to a p-value p_i through the MacKinnon-
type response surface of the Fourier LM distribution (indexed by m and the
selected lag length), and the p-values are pooled:

P = -2 sum_i ln(p_i) ~ chi2(2N)

Pm = (P - 2N) / sqrt(4N) ~ N(0,1)

Choi Z = sum_i { Phi^{-1}(p_i)} / sqrt(N)
~ N(0,1).

Lee-Tieslau two-break panel LM

Each unit is tested for a unit root allowing two sharp breaks in level and
trend. With break dates (TB1,TB2) the deterministic block is

z_t = (t, DU1_t, DT1_t, DU2_t, DT2_t,
D1_t, D2_t),

where DU are level dummies, DT broken trends and D one-time
impulse dummies. The LM score series S_t is formed as above, rescaled within
each regime (the Lee-Strazicich transformation), and the augmented LM regression
gives the t-ratio tau. The two break dates are chosen by grid search over
15%-85% of the sample to minimize tau. The panel statistic is the
standardized cross-unit average

Panel LM = sqrt(N) ( mean_i tau_i - E ) / sqrt(V),

where (E,V) are the mean and variance of the individual LM statistic under the
null, read from the tabulated two-trend-break moments as a function of the number
of lags and the time dimension T. Under the null Panel LM ~ N(0,1) and
rejects for large negative values.

Notes on replication

The augmentation-lag maximum pmax(3) reproduces the panel results reported
in Nazlioglu et al. (2023). Because the general-to-specific rule trims downward
from pmax, changing pmax() changes the selected lag and hence the
statistic, exactly as in the original GAUSS routines.

Author

Dr Merwan Roudane

merwanroudane920@gmail.com

github.com/merwanroudane