Methods and formulas

Overview

All three procedures test H0: no cointegration over the whole sample, against the
alternative that a stable cointegrating relation holds over a subset of the sample.
Let the cointegrating regression be y_t = d_t'g + b'x_t + e_t with deterministic
kernel d_t (none/const/trend). What differs is how the sample is
partitioned, how b is estimated, and which residual statistic is extremised.

Kim (2003) - weighted-LS infimum tests

For a trial noncointegration interval N_T (complement C_T) the cointegrating vector
is estimated by weighted least squares with weight w_t = 1 on C_T and 0 on N_T
(eq 3.1-3.2); this is the only weighting that gives a consistent b under the
alternative (Note 2: the N_T "variance" is O_p(T), so its reciprocal weight
vanishes). On the C_T residuals e_t an AR(1) e_t = rho e_ + v_t is fitted
using within-C_T consecutive pairs, and:

Zrho(C_T) = Tc(rho^-1) - (1/2)(Tc^2 s2rho/s^2)(lam^2 - g0) (eq 3.3)

Zt(C_T) = sqrt(g0/lam^2) t_rho - (1/2)(lam^2-g0)/lam (Tc s_rho/s) (eq 3.4)

where g0 is the short-run variance and lam^2 the Bartlett long-run variance of v_t.
An ADF variant (eq 3.5-3.7) regresses De_t on p lagged differences and the weighted
error-correction term w_t(rho-1)e_. The reported statistics are the
infimum over admissible segmentations (eq 3.13-3.15, Theorem 2), restricted to
noncointegration length <= l-bar (Lemma 1; the command's trimbar()). Critical
values are Kim's Tables 1 (Zrho*/ADFrho*) and 2 (Zt*/ADFt*) for Case I/II/III and
n = 1..6. The noncointegration interval is located by the extremum estimator
Lambda_T(tau) of eq (3.16)-(3.17) and, equivalently, by the inf-statistic
segmentation (eq 3.18).

Davidson & Monticini (2010) - subsample minima

On each subsample [T*lam1, T*lam2] the data are put in (subsample) mean-deviation
form, b is re-estimated (eq 3.4), residuals z_t are formed (eq 3.2), and a
Dickey-Fuller t-statistic (eq 3.1) or its Phillips-Perron correction (eq 3.5-3.6)
is computed. The tests take the minimum over a family of subsamples:

QS, QS* split-sample halves (eq 3.9-3.10);

QI(lam0) incremental forward + backward, min length lam0 (eq 3.11);

QR, QR* rolling windows of length lam0 (eq 3.12-3.13).

Critical values are Table 1 (1-2 regressors, +/- trend, T = 1000). DF and PP share
the same limiting distribution (eq 4.7-4.8, Theorem 4.1), so both use the same
critical values. Consistency requires one tested subset to lie inside a
cointegrated subset (Theorem 4.2).

Martins & Rodrigues (2021) - residual sup-Wald

Full-sample OLS residuals e_t (eq 2.1) enter the ADF regression

De_t = c_j + gamma_j e_ + sum_i pi_i De_ + a_t (eq 3.1)

with regime-specific (c_j, gamma_j) and common short-run dynamics pi_i. For m
breaks the Wald-type statistic contrasts the restricted SSR0 (c_j = gamma_j = 0,
lags kept) with the unrestricted SSR over the m+1 segments,

F_k(tau,m) = [(T-m-2*dB-p)/(m+2*dB)] (SSR0-SSR_k)/SSR_k (m even)

F_k(tau,m) = [(T-m-1-p)/(m+1)] (SSR0-SSR_k)/SSR_k (m odd)

(eq 3.2), k = A, B and dB = 1 for k = B. Under F_A the odd-indexed regimes are
restricted (unit root) and the even ones free (stationary); under F_B the roles are
reversed. Because F is monotone decreasing in the segment SSR, sup F_k equals F_k
at the SSR-minimising partition, found here by Bai-Perron dynamic programming with
the segment cost alternating between the restricted (lags-only) and free
([1, e_, lags]) fits according to the regime parity. Then

W(m) = max( sup F_A, sup F_B ) (eq 3.4)

Wmax = max over m = 1..m-bar of W(m) (eq 3.5)

Critical values are Table 1 (K+1 = 2..6, no det / intercept / intercept+trend,
T = 1000). Serial correlation is absorbed by the ADF lags (adflags()); the
long-run-variance nuisance correction of Remark A.1 is not applied.

The complete step -> equation map is also in the package file COMPAT_MAP.md.

References

Davidson, J., and A. Monticini. 2010. Comput. Stat. Data Anal. 54: 2498-2511.

Kim, J.-Y. 2003. Econometric Theory 19: 620-639.

Martins, L. F., and P. M. M. Rodrigues. 2021. Empirical Economics 63: 567-600.

Dr Merwan Roudane | merwanroudane920@gmail.com |
github.com/merwanroudane