Title

xtfaclm methods -- Formulas for the factor-augmented two-break panel
LM test

Iterated break and factor estimation

Let Dy_it be the first differences. Starting from Dy, the procedure
iterates:

1. Breaks. For each unit the two break dates that minimise the
residual sum of squares of Dy_it on the (differenced) two-break deterministic
terms are found by grid search; the residuals r_it and fitted values are
stored.

2. Factors. The number of common factors is chosen by the Bai-Ng
(2002) criterion and nf factors F_t with loadings L_i are extracted
from r by principal components.

3. De-factoring. The common component F L' is removed from Dy
and the loop repeats until the idiosyncratic variance changes by less than 0.001.

The converged idiosyncratic differences are e_it = Dy_it - (fitted
break term) - F_t L_i'.

Two-break LM regression with factors

For each unit the LM score is the partial sum S_t = sum_ e_is,
rescaled within the three regimes defined by the two breaks (the Lee-Strazicich
transformation). The augmented LM regression is

Dy_t = phi S_(t-1) + a' Dz_t + sum_j b_j D S_(t-j)
+ h' F_t + u_t,

where z collects the trend and the two break dummies (level breaks) or the
trend, break dummies and broken trends (level & trend breaks), and F_t are
the estimated common factors. The statistic is the t-ratio on phi. The
lag j is chosen by AIC, BIC or a general-to-specific t-rule.

Response-surface p-value and panel pooling

Because the two-break LM null distribution depends on the break fractions and the
number of augmentation lags, the p-value is read from an embedded MacKinnon-type
response surface (a cubic-in-quantile interpolation of tabulated critical values,
indexed by the break configuration, T and the lag length). The unit p-values
pv_i are combined into

P = -2 sum_i ln(pv_i) ~ chi2(2N), Pm = (P - 2N)/sqrt(4N).

A note on lag selection

The AIC and BIC criteria depend only on the regression sum of squares and are
invariant to the (arbitrary) sign and rotation of the estimated common factors, so
they reproduce the source routine exactly. The general-to-specific t-rule, as
written in the original code, evaluates the t-statistic of the last common
factor rather than the last lagged difference; that statistic is not
identified in sign or rotation, so the selected lag can vary with the linear-
algebra library. xtfaclm therefore defaults to BIC.

Author

Dr Merwan Roudane

merwanroudane920@gmail.com

github.com/merwanroudane