Time-Varying Parameter VAR

Univariate Precursor: Chow (1984)

The time-varying parameter vector autoregression (TVP-VAR) is the multivariate generalization of the random coefficient regression of Chow (1984). In the univariate version $y_t = x_t'\beta_t + \varepsilon_t$ with $\beta_t = M\beta_{t-1} + \eta_t$ , filtered and smoothed estimates of $\beta_t$ follow from the Kalman recursion (or equivalently from generalized least squares (GLS)). The hyperparameters $\sigma^2$ , $V$ , $M$ are estimated by maximizing the Kalman innovation likelihood. See Random Coefficient Model for the full derivation.

The TVP-VAR extends this by stacking $\beta_t = \text{vec}[(A_{1t},\ldots,A_{pt},\upsilon_t)']$ — the same state-space structure with a vector measurement equation.

Definition

A Time-Varying Parameter VAR (VAR-TVP) is a VAR in which the coefficient matrices $A_{1t}, \ldots, A_{pt}$ evolve over time as a latent state process — typically a driftless random walk or a mean-reverting autoregressive (AR)(1). The model is cast in state-space form, and Bayesian inference proceeds via a Gibbs sampler that alternates between drawing the full path of latent coefficients (Carter-Kohn multimove algorithm) and drawing the static parameters from conjugate full conditionals.

Key Ideas

Coefficient time variation captures structural change and slowly drifting dynamics without imposing discrete break points.
Parsimony in the state equation error covariance $\Omega$ is critical: richer $\Omega$ specifications do not improve and often hurt forecasting performance.
The Kronecker structure $\Omega = \rho(R \otimes Q_k)$ is the best practical specification — it ties coefficient dynamics to the observation covariance, reduces the parameter count from $O(K^2)$ to $O(k^2 + n^2)$ , and collapses a key $K \times K$ matrix inversion in the Kalman filter to $k \times k$ .
The multimove (joint state path) sampler is essential; single-move samplers exhibit severe mixing problems.

How It Works

State-Space Formulation

Stack the time-varying VAR coefficients as $\beta_t = \mathrm{vec}[(A_{1t}, \ldots, A_{pt}, \upsilon_t)']$ of dimension $K = n(np+1)$ . The model is:

Measurement equation: $y_t = Z_t \beta_t + e_t, \qquad e_t \sim \mathcal{N}(0, R) \tag{1}$

where $Z_t = (I_n \otimes x_t')$ , $x_t = (y_{t-1}', \ldots, y_{t-p}', 1)'$ ( $k \times 1$ ).

Transition equation: $\beta_t - \beta_0 = \phi(\beta_{t-1} - \beta_0) + \eta_t, \qquad \eta_t \sim \mathcal{N}(0, \Omega) \tag{2}$

Setting $\phi = 1$ yields a driftless random walk; $0 < \phi < 1$ gives mean-reverting AR(1) dynamics toward prior mean $\beta_0$ (set in Minnesota-prior style).

Alternative $\Omega$ Specifications (Amisano-Federico 2004)

Model	$\Omega$ structure	Free parameters
M1 (Cogley-Sargent)	Joint $M = \begin{bmatrix}R & C \\ C' & \Omega\end{bmatrix}$ , $C \neq 0$	$\Psi \equiv M^{-1}$ ~ Wishart( $m \times m$ )
M2 ( $C = 0$ )	$R$ and $\Omega$ free, independent	$H \equiv R^{-1}$ , $\Phi \equiv \Omega^{-1}$ ~ Wishart
M3 (Kronecker)	$\Omega = \rho(R \otimes Q_k)$	$R$ , $Q_k$ ( $k \times k$ ), $\rho > 0$
M4 (Litterman-type)	$\Omega = \rho Q_0$ , $Q_0$ = Minnesota diagonal	$R$ , $\rho > 0$

In M3 all equations are treated symmetrically: the ratio of prior variances for any two coefficients is the same in every equation. This constraint dramatically reduces the parameter count.

Carter-Kohn Multimove Algorithm

The full path $\beta_1, \ldots, \beta_T$ is drawn jointly in one Gibbs block via a forward Kalman filter pass followed by a backward simulation smoother (Carter and Kohn 1994; Chib-Greenberg 1995b for the seemingly unrelated regression (SUR) precursor):

Forward Kalman pass ( $t = 1, \ldots, T$ ): $\beta_{t|t-1} = \phi \beta_{t-1|t-1} + (1-\phi)\beta_0$ $Q_{t|t-1} = \phi^2 Q_{t-1|t-1} + \Omega$ $e_{t|t-1} = y_t - Z_t \beta_{t|t-1}, \qquad f_{t|t-1} = Z_t Q_{t|t-1} Z_t' + R$ $\beta_{t|t} = \beta_{t|t-1} + Q_{t|t-1} Z_t' f_{t|t-1}^{-1} e_{t|t-1}$ $Q_{t|t} = Q_{t|t-1} - Q_{t|t-1} Z_t' f_{t|t-1}^{-1} Z_t Q_{t|t-1}$

Backward simulation pass ( $t = T, T-1, \ldots, 1$ ): $\beta_T \sim \mathcal{N}(\beta_{T|T},\, Q_{T|T})$ $\beta_t \mid \beta_{t+1} \sim \mathcal{N}\!\left(\beta_{t|t} + \phi Q_{t|t} Q_{t+1|t}^{-1}(\beta_{t+1} - \phi\beta_{t|t} - (1-\phi)\beta_0),\; Q_{t|t} - \phi^2 Q_{t|t} Q_{t+1|t}^{-1} Q_{t|t}\right)$

The dominant cost is $Q_{t+1|t}^{-1}$ , a $K \times K$ matrix inversion at each step.

Kronecker Structure Simplification

Under M3, $Q_{t|t} = H^{-1} \otimes Q_{t|t,k}$ where $Q_{t|t,k}$ is $k \times k$ . All Kalman recursions reduce to operations on this small matrix: $Q_{t|t-1,k} = \phi^2 Q_{t-1|t-1,k} + \rho Q_k, \qquad f_{t|t-1,k} = x_t' Q_{t|t-1,k} x_t + 1$

The $K \times K$ inversion becomes a scalar inversion ( $f_{t|t-1,k}^{-1}$ ), reducing per-iteration cost from $O(K^3)$ to $O(k^3 + n^3)$ .

Multivariate Stochastic Volatility (Cogley-Morozov-Sargent 2003)

The CMS (2003) extension replaces the constant observation covariance $R$ with a time-varying matrix $R_t$ factored as:

$R_t = B^{-1} H_t B'^{-1}$

where $B$ is a fixed lower-triangular matrix (with ones on the diagonal) and $H_t = \mathrm{diag}(h_{1t}, \ldots, h_{nt})$ is diagonal with geometric random walk log-variances:

$\log h_{it} = \log h_{i,t-1} + \eta_{it}, \qquad \eta_{it} \sim \mathcal{N}(0, \sigma_i^2)$

This parameterisation separates contemporaneous cross-equation correlation (captured by $B$ ) from time-varying scale (captured by $H_t$ ). The log-variance draws $\{h_{it}\}$ cannot be obtained from a conjugate conditional and require a single-move Metropolis step with a log-normal proposal centred at the previous draw (following the Jacquier-Polson-Rossi (JPR) 1994 approach for univariate stochastic volatility (SV)).

Five-Block Metropolis-within-Gibbs (CMS 2003)

The full sampler cycles through:

Block	Parameter	Sampler
1	$\theta_T = (\theta_1, \ldots, \theta_T)$	Carter-Kohn forward filter backward sampler (FFBS) (rejection-sample for stability)
2	$Q$ (coefficient drift covariance)	Inverse-Wishart conjugate
3	$\beta$ (columns of lower-triangular $B$ )	Normal seemingly unrelated regression (SUR) conjugate (independent across columns)
4	$\sigma^2 = (\sigma_1^2, \ldots, \sigma_n^2)$	Inverse-Gamma conjugate
5	$\{h_{it}\}$ for each $i$	Single-move Metropolis, log-normal proposal

The stability constraint on $\theta_t$ (all VAR eigenvalues inside the unit circle at every $t$ ) is enforced by rejection sampling within the Carter-Kohn block: proposed draws that violate stability are discarded and resampled.

Full Conditionals for Static Parameters

Given $\beta_T$ from the state block, each parameter draws from a conjugate distribution:

Parameter	Full conditional
$H \equiv R^{-1}$	$\mathrm{Wishart}(T + \nu_0,\, [(e^m e^{m'}) + H_0^{-1}]^{-1})$
$\Phi_k \equiv Q_k^{-1}$ (M3)	$\mathrm{Wishart}(Tn + \nu_0,\, [\sum_t \Upsilon_t \Upsilon_t' + \Phi_{0,k}^{-1}]^{-1})$
$h \equiv \rho^{-1}$ (M3, M4)	$\mathrm{Gamma}(\tfrac{TK}{2} + \upsilon_0,\, h_1^{-1})$
$\phi$ (M4 random walk)	$\mathrm{TruncNormal}_{[0,1]}(\mu_1, \sigma_1^2)$ — accept-reject

Why It Matters

TVP-VARs consistently outperform fixed-coefficient Bayesian VARs (BVARs) for inflation forecasting at short and medium horizons, especially when structural change is present.
The Kronecker M3 model is the recommended practical specification: parsimonious, computationally efficient via the Kalman simplification, and best out-of-sample.
The Carter-Kohn multimove algorithm is the workhorse for all state-space Bayesian inference; its backward simulation step produces independent draws from the joint state path posterior.
The CMS (2003) UK application demonstrates that multivariate stochastic volatility (SV) is essential for credible fan charts: UK inflation core swept 3% → 13% → 2.2% over 1957–2002, persistence shifted from white noise (1960s) to an 8-year-cycle peak (1975) to high-frequency dominance after 1992, and innovation variance fell 75–80% — none of which a fixed- $R$ model can accommodate.

Open Questions

The mean-reversion hyperparameter $\phi$ has a slowly-mixing Gibbs chain; Frühwirth-Schnatter (2004) reparameterizations may improve sampling.
High-dimensional TVP-VARs ( $n > 6$ ) still face large state vectors despite the Kronecker simplification; sparse factor-structure approximations are an active research area.
Selecting lag length $p$ with time-varying coefficients is non-trivial: standard information criteria derived for fixed-coefficient models are not valid.