simulate.md

VECM simulator

pyardl.simulate

!!! note "Looking for dynamic simulations of a fitted model?" This page is about generating data for Monte Carlo studies. To run a fitted ARDL forward under a counterfactual shock — the interpretation layer of Jordan and Philips (2018) — see Dynamic simulation.

Every Monte Carlo study in this library draws its data from one generator. Before that, each specification carried its own — which is how two studies end up disagreeing for a reason nobody can locate: not because the estimators differ, but because the data did.

Δz_t = Π z_{t-1} + Σ Γ_i Δz_{t-i} + d_t + ε_t,    Π = α β'

with z_t = (y_t, x_1t, …, x_kt)'.

vecm_ardl(n_obs, alpha, beta, gammas=(), case=3, sigma=None, ...)

import numpy as np
from pyardl.simulate import vecm_ardl, degenerate_system

alpha, beta = degenerate_system(None, k=1, speed=-0.5)   # cointegration
sim = vecm_ardl(300, alpha=alpha, beta=beta,
                sigma=np.array([[1.0, 0.6], [0.6, 1.0]]), seed=42)

sim.y        # the dependent variable
sim.x        # the regressors
sim.rank     # 1
sim.lam      # the true λ
sim.seed     # 42, recorded even when drawn from entropy

Writing Π = α β' is what makes the rank chosen rather than hoped for: beta holds the long-run relations, one per column, and alpha the speed at which each equation adjusts to them.

rank reports the rank of Π, not the number of columns you passed. A zero alpha creates no relation, and reporting the column count there would claim a relation the data do not contain.

The canonical systems

degenerate_system(kind, k, speed) returns the (alpha, beta) pair for each case the three-test framework has to tell apart:

kind	System	Π[0] for k=1
None	genuine cointegration	[-0.4, 0.4]
1	type 1: λ ≠ 0, γ = 0	[-0.4, 0.0]
2	type 2: γ ≠ 0, λ = 0	[0.0, -0.4]

For no cointegration at all there is nothing to build: pass a zero system to vecm_ardl directly.

What is guaranteed, and what is not

Guaranteed: what comes back is what the parameters describe. Shapes are checked, and a deterministic term the requested case does not carry is refused rather than absorbed — passing a trend to case 3 is a specification error, not a detail.

Not guaranteed: stability. An explosive system is a legitimate thing to simulate — to check that a test never calls it cointegration, for instance — so no stationarity condition is imposed behind your back.

Why sigma deserves attention

With a diagonal sigma, the conditional and unconditional models coincide: Δx_t carries no information about Δy_t beyond the lags. Any study meant to distinguish them needs correlated innovations, which is also the realistic case — that correlation is what weak exogeneity is about.

How the simulator is tested

Not on what it prints. On what estimators recover from it:

• the Johansen test (spec 07) finds the injected rank;
• the three-test classification (spec 15) never calls an injected degeneracy cointegration;
• a zero-Π system is detected as cointegrated no more often than Johansen's own false-detection rate (OBS-10) — the generator invents no relation.

That is the only check that catches an error in the construction of the DGP itself.

References

• Bertelli, S., Vacca, G. & Zoia, M. (2022). Bootstrap cointegration tests in ARDL models. Economic Modelling, 116, 105987.
• Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica, 59(6), 1551-1580.