tvpSvFit#

Estimate a Time-Varying Parameter VAR with Stochastic Volatility (Primiceri 2005).

Format#

result = tvpSvFit(y[, ctl])#

Parameters:

y (matrix or dataframe) – T x m matrix of endogenous variables.
ctl (struct tvpSvControl) – Optional. Control structure with estimation settings. Created by tvpSvControlCreate().

Returns:

result (struct tvpSvResult) – Estimation results including posterior draws of terminal B_T, SV parameters, and acceptance diagnostics.

Model#

The full Primiceri (2005) TVP-VAR-SV:

\[\begin{split}y_t &= X_t B_t + \varepsilon_t, \quad \varepsilon_t \sim N(0, \Sigma_t) \\ \Sigma_t &= U_t'^{-1} D_t U_t^{-1}, \quad D_t = \text{diag}(e^{h_{1,t}}, \ldots, e^{h_{m,t}}) \\ B_t &= B_{t-1} + \eta_t, \quad \eta_t \sim N(0, Q_B) \\ u_t &= u_{t-1} + \zeta_t, \quad \zeta_t \sim N(0, Q_U) \\ h_{i,t} &= \mu_i + \phi_i(h_{i,t-1} - \mu_i) + \nu_{i,t}, \quad \nu_{i,t} \sim N(0, \sigma^2_i)\end{split}\]

Three time-varying components: coefficients B_t (random walk), Cholesky off-diagonals U_t (random walk), and log-volatilities h_t (AR(1) stochastic volatility).

Example#

new;
library timeseries;

// US macro data
fname = getGAUSSHome("pkgs/timeseries/examples/data/us_macro_quarterly.csv");
y = loadd(fname, "gdp_growth + cpi_inflation + fed_funds");

// Estimate with defaults
result = tvpSvFit(y);

// Print terminal coefficients
print result.b_mean;

// With custom settings
ctl = tvpSvControlCreate();
ctl.p = 2;
ctl.n_draws = 10000;
ctl.n_burn = 10000;
result = tvpSvFit(y, ctl);

Control structure#

ctl.p	Scalar, lag order. Default = 1.
ctl.include_const	Scalar, include constant (1) or not (0). Default = 1.
ctl.n_draws	Scalar, number of posterior draws to keep. Default = 5000.
ctl.n_burn	Scalar, number of burn-in iterations to discard. Default = 5000.
ctl.n_thin	Scalar, thinning factor. Default = 1 (keep every draw).
ctl.seed	Scalar, RNG seed for reproducibility. Default = 42.
ctl.use_asis	Scalar, enable ASIS interweaving for SV (recommended). Default = 1.
ctl.q_b_shape	Scalar, IG prior shape for B drift covariance Q_B diagonal elements. Q_B,jj ~ IG(shape, scale). Default = 6.0.
ctl.q_b_scale	Scalar, IG prior scale for Q_B. Default = 0.01 (slow drift).
ctl.q_u_shape	Scalar, IG prior shape for U drift covariance Q_U diagonal elements. Default = 6.0.
ctl.q_u_scale	Scalar, IG prior scale for Q_U. Default = 0.01.
ctl.p0_b_kappa	Scalar, diffuse initialization scale for B FFBS. P_0 = kappa * I. Default = 10.0.
ctl.p0_u_kappa	Scalar, diffuse initialization scale for U FFBS. Default = 10.0.
ctl.u_bandwidth	Scalar, band-limited drifting U. 0 = full (default), k > 0 = first k off-diagonals per column. See `bvarSvControl.u_bandwidth` for details.
ctl.xreg	Matrix, exogenous regressors (T x K). Empty = none.
ctl.quiet	Scalar, set to 1 to suppress printed output. Default = 0.

Result structure#

result.m	Scalar, number of endogenous variables.
result.p	Scalar, lag order.
result.n_obs	Scalar, effective sample size (T - p).
result.n_draws	Scalar, number of posterior draws kept.
result.var_names	String array, variable names (from dataframe column names, or empty).
result.b_mean	K x m matrix, posterior mean of terminal B_T (coefficients at the last observation).
result.b_sd	K x m matrix, posterior standard deviation of terminal B_T.
result.sv_mu	m x 1 vector, posterior mean of SV level parameters μ_i.
result.sv_phi	m x 1 vector, posterior mean of SV persistence parameters φ_i.
result.sv_sigma2	m x 1 vector, posterior mean of SV innovation variance σ²_i.
result.phi_accept_rate	m x 1 vector, Metropolis-Hastings acceptance rate for φ per equation. Healthy range: 20-60%.

Remarks#

The sampler uses equation-by-equation Carter-Kohn FFBS for B_t (Primiceri’s original approach), reducing the state dimension from K*m to K per equation.
The observation variance for each equation comes from the stochastic volatility h_{eq,t}, not a constant Sigma.
Q_B and Q_U have conjugate diagonal Inverse-Gamma posteriors.
B_0 is initialized from OLS and redrawn each iteration from its full conditional.
SV updates use the OCSN 10-component mixture approximation with optional ASIS interweaving.
Set ctl.u_bandwidth > 0 for large systems (m > 15) to reduce U parameters.

References#

Primiceri, G. E. (2005). Time varying structural vector autoregressions and monetary policy. Review of Economic Studies, 72(3), 821-852.
Carter, C. K. & Kohn, R. (1994). On Gibbs sampling for state space models. Biometrika, 81(3), 541-553.
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.