Title

xtpdcause methods -- Formulas for the factor-corrected panel
Granger-causality test

Factor correction

Let y_it = (d_it, c_it) be the bivariate series (dependent,
causing) for unit i. The common-factor structure is estimated from the
standardized first differences Dz_it = Dy_it/sd(Dy_it),
stacked as a (T-1) x 2N matrix DZ.

PANIC. The common factors F and loadings L are the
principal components of DZ: DZ = F L' + e. The number of
factors is the Bai-Ng (2002) IC2 minimizer over 0..kmax. The de-factored
differences e = DZ - F L' are cumulated back to levels.

PANIC-CA. A single factor is proxied by the cross-section average
Fbar_t = mean over the 2N columns of DZ; loadings are the OLS
projection of each column on Fbar, and the de-factored differences are
cumulated.

none. No factor removal; the VAR is run on the raw levels with an
intercept.

Lag-augmented VAR and the Wald test

For each unit the (factor-corrected) bivariate series is fitted as a VAR of order
p + dmax, where p is chosen by AIC or BIC (maximum pmax) and
dmax extra lags are appended. Writing the depvar equation as

d_t = sum_(j=1..p+dmax) a_j d_(t-j) + sum_(j=1..p+dmax) b_j c_(t-j) +
u_t,

the Wald statistic tests H0: b_1 = ... = b_p = 0 (the first p
lags of causevar; the dmax augmenting lags are left unrestricted). With
RSS_r and RSS_ur the restricted and unrestricted residual sums of
squares of the depvar equation,

W = p x [(RSS_r - RSS_ur)/p] /
[RSS_ur/(T - k)] ~ chi2(p),

which is asymptotically chi-square with p degrees of freedom irrespective of
the integration order of the data (Toda-Yamamoto).

Panel pooling

Let pv_i = 1 - F_chi2(W_i; p_i) be the unit p-values. The panel
statistics are the Fisher combinations

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

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

Holm's step-down procedure is applied to the pv_i to give family-wise
error-controlled individual p-values, identifying the panels responsible for a
panel rejection.

Author

Dr Merwan Roudane

merwanroudane920@gmail.com

github.com/merwanroudane