cultivars.multivariate.nonlinear.time_varying

Time-varying-parameter VARs: drift instead of switching, posteriors instead of dates.

Where the regime families let parameters jump between a few states, here they drift: the stacked coefficient vector follows a random walk, and under stochastic volatility so do the log variances,

\[y_t = X_t' \beta_t + u_t, \qquad \beta_t = \beta_{t-1} + \eta_t, \quad \eta_t \sim N(0, Q), \qquad u_t \sim N(0, \Sigma_t), \quad \Sigma_t = A^{-1} H_t A^{-\top}, \quad h_t = h_{t-1} + \zeta_t,\]

so every date has its own VAR and its own covariance (the homoskedastic model holds \(\Sigma_t = \Sigma\)). This is the Cogley-Sargent / Primiceri (2005) workhorse for questions like whether monetary policy transmission itself has changed – questions a constant-parameter model cannot even pose.

Two commitments shape the surface. First, estimation is Bayesian by necessity, not preference: an unrestricted coefficient path has more parameters than observations, and what disciplines it is the prior on how fast parameters may drift. Following Primiceri, a training sample is split off the front, its OLS estimates calibrate the coefficient prior and the drift scale k_drift (his \(k_Q\)) and the volatility scale k_vol (his \(k_W\)), and it is then discarded rather than used twice. Those two scales are the model’s findings in disguise: at the defaults of 0.01 a coefficient that drifts by several tenths, or a volatility that triples, over a few hundred observations is reported as nearly flat, so the results compare each path’s movement with its own band and the user is expected to vary the scales. The sampler is Gibbs on the package’s own state-space substrate: every coefficient-path draw is one Durbin-Koopman simulation-smoother call, the covariance blocks are conjugate inverse-Wisharts, and the stochastic-volatility block is Kim-Shephard-Chib, with the log-volatility innovation covariance \(W\) diagonal as in Cogley and Sargent rather than Primiceri’s full matrix. Second, under SV the innovation covariance is parameterized through a constant unit-lower-triangular \(A\) in the order of names, which is why the SV model’s structural interpretation is settled at estimation time (see TVPSVAR) and why its sigma_u is the time-averaged covariance rather than any one date’s.

Being posteriors, the results report uncertainty rather than test statistics: coefficient and volatility paths come with posterior bands, and there is deliberately no likelihood, information criterion, or parameter count on these results – a Gibbs posterior has none to report, and numbers shaped like them would be wrong. Structural responses at a date freeze that date’s posterior means and are read as the transmission prevailing there.

Layout. TVPVAR and TVPVARSV validate on _TimeVaryingVectorAutoRegressionModel in _internals, which resolves the training split, builds the Kronecker-stacked observation matrices, calibrates the priors by OLS on the training rows, and in _fit_tvp runs the sampler: a _LinearGaussianStateSpace from the state-space layer supplies the simulation smoother, _draw_inverse_wishart and _draw_triangular_volatility_block from _core._samplers the covariance and volatility blocks, and a _TimeVaryingFit is packed. TVPVARResult and TVPVARSVResult take their summary from _SummaryMixin and their chain diagnostics from _ConvergenceMixin; TVPSVAR reads either into a TVPSVARResult through companion_matrix(). The constant-coefficient posteriors these paths relax are in bayesian and volatility; the models whose parameters switch rather than drift are the rest of this package and markov_switching.

References

Primiceri, G. E. (2005). Time varying structural vector autoregressions and monetary policy. Review of Economic Studies, 72(3), 821-852.

Del Negro, M., & Primiceri, G. E. (2015). Time varying structural vector autoregressions and monetary policy: A corrigendum. Review of Economic Studies, 82(4), 1342-1345.

Cogley, T., & Sargent, T. J. (2005). Drifts and volatilities: Monetary policies and outcomes in the post WWII US. Review of Economic Dynamics, 8(2), 262-302.

Kim, S., Shephard, N., & Chib, S. (1998). Stochastic volatility: Likelihood inference and comparison with ARCH models. Review of Economic Studies, 65(3), 361-393.

Durbin, J., & Koopman, S. J. (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89(3), 603-616.

Example

Drift and volatility change in one system: the own-lag coefficient of y1 climbs over the sample while the shocks grow louder. With the scales loosened the SV fit sees both, and the structural impact of the first shock at the end of the sample exceeds its impact at the start:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> n = 200
>>> y = np.zeros((n, 2))
>>> for t in range(1, n):
...     a = 0.2 + 0.6 * t / n
...     scale = 0.5 + 1.0 * t / n
...     y[t, 0] = a * y[t - 1, 0] + scale * rng.standard_normal()
...     y[t, 1] = 0.4 * y[t - 1, 1] + 0.3 * y[t - 1, 0] + rng.standard_normal()
>>> res = TVPVARSV(y, order=1, training=40).fit(
...     n_draws=60, n_burn=20, k_drift=0.2, k_vol=0.1, seed=0
... )
>>> path = res.coefficient_path("y1", "y1.L1")[:, 1]
>>> vol = res.volatility_path("y1")[:, 1]
>>> bool(path[-1] - path[0] > 0.2), bool(vol[-1] / vol[0] > 1.5)
(True, True)
>>> svar = TVPSVAR(res).identify()
>>> bool(svar.impact(res.nobs - 1)[0, 0] > svar.impact(0)[0, 0])
True
>>> svar.irf(4, t=res.nobs - 1).shape
(5, 2, 2)

Classes

TVPSVAR	Recursive structural interpretation of a fitted TVP-VAR, per date.
TVPVAR	Time-varying-parameter VAR with constant innovation covariance.
TVPVARSV	Time-varying-parameter VAR with stochastic volatility, after Primiceri (2005).
TVPSVARResult	A structurally interpreted TVP-VAR: one recursive declaration, a path of answers.
TVPVARResult	A fitted time-varying-parameter VAR, homoskedastic innovations.
TVPVARSVResult	A fitted TVP-VAR with stochastic volatility, after Primiceri (2005).