Initialises a steady-state bvar object. This is the starting point for all models in SteadyStateBVAR.
After creation, pass the object sequentially to setup, priors,
and fit to build and estimate the model.
Details
The model takes the form
$$y_t = \Psi d_t + \Pi_1(y_{t-1}-\Psi d_{t-1})+\dots+\Pi_p(y_{t-p}-\Psi d_{t-p})+u_t$$
where \(y_t\) is an \(k\)-dimensional vector of endogenous variables at time \(t\), and \(d_t\) is a \(q\)-dimensional vector of deterministic (exogenous) variables at time \(t\). Here \(\Pi_\ell\) for \(\ell=1,\dots,p\) is a \((k \times k)\) matrix of autoregressive parameters, and \(\Psi\) is a \((k \times q)\) matrix of steady-state parameters. Note that \(\mathrm{E}(y_t)=\mu_t=\Psi d_t\) is the unconditional mean, or the steady state of the process. One can stack the (transposed) \(\Pi_i\) matrices in the \((kp \times k)\) matrix \(\beta\) $$\beta=\begin{bmatrix}\Pi'_1 \\ \vdots \\\Pi'_p\end{bmatrix}$$ Then the model can be rewritten as a nonlinear regression (Karlsson, 2013) $$y_t' =d_t'\Psi' + \left[w_t'-q_t'(I_p \otimes \Psi') \right]\beta +u_t'$$ where where \(w_t'=(y_{t-1}',\dots,y_{t-p}')\) is a \(kp\)-dimensional vector of lagged endogenous variables and \(q_t'=(d_{t-1}',\dots,d_{t-p}')\) is a \(qp\)-dimensional vector of lagged deterministic (exogenous) variables, \(I_p\) is the \((p \times p)\) identity matrix and \(\otimes\) denotes the Kronecker product. This is how the likelihood is written in the Stan code. The goal is to estimate \(\beta, \Psi\), and \(\Sigma_u\).
For the innovations to the model, in the case of the homoscedastic steady-state BVAR, they are \(u_t \overset{\text{iid}}{\sim} \mathrm{N_k}(0,\Sigma_u)\). However, for models with stochastic volatility, there is instead a time-varying covariance matrix \(u_t \sim \mathrm{N_k}(0,\Sigma_{u,t})\). The innovations then take the form
$$\begin{aligned} u_t &= A^{-1} \Lambda^{0.5}_t \epsilon_t \\ \epsilon_t &\overset{\text{iid}}{\sim} \mathrm{N}(0, \mathrm{I}_k)\end{aligned}$$
where \(A\) is a lower triangular matrix with ones on the diagonal that describes the contemporaneous interaction of the endogenous variables, and
$$\Lambda_t = \mathrm{diag}(\lambda_{1,t},\dots,\lambda_{k,t})$$
contains the time-varying volatilities.
For the AR1 stochastic volatility specification, the log volatilities follow AR(1) processes
$$\ln \lambda_{i,t} = \gamma_{0,i} + \gamma_{1,i} \ln \lambda_{i,t-1} + \nu_{i,t}, \ i=1,\dots,k$$
where the log volatility AR(1) processes are restricted to the stationary region, i.e. \(|\gamma_{1,i}|<1 \ \forall i\).
For the RW stochastic volatility specification, the log volatilities follow Random Walk processes
$$\gamma_{0,i}=0, \ \gamma_{1,i}=1 \ \forall i$$
The innovations to the log volatilities follow in the AR1 case
$$\nu_{t} = (\nu_{1,t},\dots,\nu_{k,t})'\overset{\text{iid}}{\sim} \mathrm{N}(0, \Phi)$$
where \(\Phi\) is not diagonal and as such the innovations to the log
volatilities are allowed to be correlated across variables. For the RW case, \(\Phi\) is diagonal
with variances \(\phi_i\) for \(i=1,\dots,k\).
Note that under both stochastic volatility specifications, the time-varying covariance matrix is
$$\Sigma_{u,t} = A^{-1} \Lambda_t (A^{-1})'$$
For details on the homoscedastic steady-state BVAR model, see Villani (2009). For details on the Random Walk stochastic volatility steady-state BVAR model, see Clark (2011). See Carriero, Clark, and Marcellino (2024) for the above-mentioned AR(1) stochastic volatility specification applied to a conventional BVAR.
References
Carriero, A., Clark, T. E., and Marcellino, M. (2024). Capturing macro-economic tail risks with Bayesian vector autoregressions. Journal of Money, Credit and Banking, 56(5), pp. 1099–1127.
Clark, T. E. (2011). Real-time density forecasts from Bayesian vector autoregressions with stochastic volatility. Journal of Business & Economic Statistics, 29(3), pp. 327–341.
Karlsson, S. (2013). Forecasting with Bayesian vector autoregression. In: Elliott, G. and Timmermann, A. (eds), Handbook of Economic Forecasting. Elsevier B.V., Vol. 2, Part B, pp. 791–897.
Villani, M. (2009). Steady-state priors for vector autoregressions. Journal of Applied Econometrics, 24(4), pp. 630–650.