Steady-state BVAR models

Introduction

The steady-state BVAR (Villani, 2009) is simply a BVAR rewritten in mean-adjusted form. The benefit of the mean-adjusted parametrization is that it allows the user to specify prior beliefs about the unconditional mean, or steady state of the VAR system. The model has proven very useful for forecasting of macroeconomic variables, and is routinely used in many central banks and other financial institutions (Gustafsson and Villani, 2025). The steady-state BVAR(\(p\)) model is

\[ y_t = \Psi d_t + \Pi_1(y_{t-1}-\Psi d_{t-1})+\dots+\Pi_p(y_{t-p}-\Psi d_{t-p})+u_t \]

where \(y_t\) is a \(k\)-dimensional vector of endogenous variables (time series) at time \(t\), and \(d_t\) is a \(q\)-dimensional vector of deterministic (exogenous) variables at time \(t\), and the (reduced-form) innovations are \(u_t \overset{\text{iid}}{\sim} \mathrm{N_k}(0,\Sigma_u)\). Here \(\Pi_\ell\) for \(\ell=1,\dots,p\) is a \((k \times k)\) autoregressive parameter matrix, and \(\Psi\) is the \((k \times q)\) steady-state parameter matrix. Now

\[ \mathrm{E}(y_t)=\mu_t=\Psi d_t \]

is the unconditional mean, or the steady state, of the process. Since long-horizon forecasts from stationary VARs converge to the unconditional mean (steady state), it is naturally very important from a forecasting perspective to obtain precise inference on \(\Psi\). Note that the current version of this package allows for \(d_t\) to contain either just a constant, a constant and a dummy variable, or a constant and a time trend. Now, we may stack the (transposed) \(\Pi_i\) matrices in the \((kp \times k)\) matrix \(\beta\)

\[ \beta= \begin{bmatrix} \Pi'_1 \\ \vdots \\ \Pi'_p \end{bmatrix} \]

We can then rewrite the model 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 \(w_t'=(y_{t-1}',\dots,y_{t-p}')\) 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. Now, our goal is to estimate the parameters \(\Theta = \begin{bmatrix} \beta & \Psi & \Sigma_u \end{bmatrix}\), and as such priors are needed. Following Villani (2009), prior independence between \(\beta, \Psi\) and \(\Sigma_u\) is assumed. For \(\beta\), the Minnesota prior is used

\[ \mathrm{vec}(\beta) \sim \mathrm{N}_{kpk} (\theta_\beta,\Omega_\beta) \]

The prior means (the elements of \(\theta_\beta\)) are specified according to

\[ \begin{aligned} \mathrm{E}\left(\Pi_{\ell}^{(i,j)}\right)&= \begin{cases} \kappa & \text{if } \ell = 1 \ \text{and} \ i = j \\ 0 & \text{otherwise} \end{cases}\\ \kappa&= \begin{cases} \kappa^{levels} & \text{if} \ \text{variable} \ i \ \text{is} \ \text{in} \ \text{levels} \\ \kappa^{\Delta} & \text{if} \ \text{variable} \ i \ \text{is} \ \text{differenced} \end{cases}\\ \end{aligned} \]

Here, the autoregressive coefficient \(\Pi_{\ell}^{(i,j)}\) is element \(\left(i,j\right)\) of \(\Pi_{\ell}\) for \(\ell=1,\dots,p\). As such, the Minnesota prior sets all prior means for the elements in \(\beta\) to \(0\), except for the elements that relate to the first own lags of the variables, which are set to \(\kappa\). If variable \(i\) is in levels (e.g. nominal interest rate), then \(\kappa=\kappa^{levels}\), and typical choices for \(\kappa^{levels}\) are \(1\) or \(0.9\). Evaluating the equations at their prior means, equation \(i\) becomes a random walk if \(\kappa^{levels}=1\) and a persistent stationary AR(1) process if \(\kappa^{levels}=0.9\). Since the steady state only exists if the process is stationary, \(0.9\) is recommended for the steady-state BVAR. If variable \(i\) is in difference (e.g. GDP growth), then \(\kappa=\kappa^{\Delta}\), and the most common choice for \(\kappa^{\Delta}\) is \(0\), i.e. equation \(i\) becomes (when evaluating it at its prior means) a random walk expressed in first differences (a white noise process). If a differenced variable still shows some degree of persistence (can be examined with an ACF plot), a suitable value for \(\kappa^{\Delta}\) can be (for example) \(0.6\) instead of \(0\). Moving on to the prior variances, \(\Omega_\beta\) is a diagonal matrix containing the prior variances for the elements in \(\beta\). They are specified as

\[ \mathrm{Var}\left(\Pi_{\ell}^{(i,j)}\right)= \begin{cases} \left(\frac{\lambda_1}{\ell^{\lambda_3}}\right)^2 & \text{if } i = j \\ \left(\frac{\lambda_1 \lambda_2\sigma_i}{\ell^{\lambda_3}\sigma_j}\right)^2& \text{if } i \neq j \end{cases} \]

Here \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\) are scalar hyperparameters known as the overall tightness, the cross-equation tightness and the lag decay rate. Furthermore, \(\sigma_i^2\) is the \((i,i)\):th element of \(\Sigma_u\), which we do not know and therefore replace with an estimate. In this package, it is replaced by the least squares residual variance from a univariate autoregression for variable \(i\) with \(p\) lags (including the constant and dummy/trend variable if applicable). Moving on to \(\Psi\), the steady-state parameter matrix, the (steady-state) prior is

\[ \mathrm{vec}(\Psi) \sim \mathrm{N}_{kq}(\theta_\Psi,\Omega_\Psi) \]

This is the core of the steady-state BVAR model. In \(\theta_\Psi\), we specify our prior beliefs about the location of the steady state, and in \(\Omega_\Psi\), which we assume to be a diagonal matrix, we specify our degree of certainty in those prior beliefs. Too see how to specify steady-state priors in practice, please see the other vignettes. Now regarding the reduced-form innovations \(u_t\), this package supports the homoscedastic steady-state BVAR (Villani, 2009) and steady-state BVARs with stochastic volatility, either Random Walk (Clark, 2011) or AR(1) stochastic volatility. In the homoscedastic case, we have

\[ u_t \overset{\text{iid}}{\sim} \mathrm{N_k}(0,\Sigma_u) \] where the prior for \(\Sigma_u\) is the usual non-informative Jeffreys prior

\[ p(\Sigma_u) \propto\left|\Sigma_u \right|^{-(k+1)/2} \]

Alternatively, a proper inverse-Wishart prior can be used (Karlsson, 2013)

\[ \Sigma_u \sim \mathrm{IW}(V,m) \] where \(V\) is the scale matrix and \(m\) is the number of degrees of freedom. For stochastic volatility, the reduced-form innovations are instead specified as

\[ \begin{aligned} u_t &= A^{-1} \Lambda^{0.5}_t \epsilon_t \\ \epsilon_t &\overset{\text{iid}}{\sim} \mathrm{N}(0, \mathrm{I}_k) \end{aligned} \]

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. Please note that the \(\lambda\)’s here (volatilities) have nothing to do with the \(\lambda\)’s (hyperparameters) from the Minnesota prior. Now, for the AR(1) 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 Random Walk (RW) stochastic volatility specification, the log volatilities follow driftless Random Walk processes, i.e. we restrict the above by

\[ \gamma_{0,i}=0, \ \gamma_{1,i}=1 \ \forall i \] The innovations to the log volatilities follow in the AR(1) 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, we have

\[ \begin{aligned} u_t &\sim \mathrm{N_k}(0,\Sigma_{u,t}) \\ \Sigma_{u,t} &= A^{-1} \Lambda_t (A^{-1})' \end{aligned} \]

For the Random Walk stochastic volatility specification, the following priors are available

\[ \begin{aligned}a &\sim \mathrm{N}(\theta_A, \Omega_A) \\ \ln \lambda_{1} &\sim \mathrm{N}(\theta_{\ln \lambda_{1}}, \Omega_{\ln \lambda_{1}}) \\ \phi_i &\sim \mathrm{IG}(\alpha_{\phi_i},\beta_{\phi_i})\end{aligned} \]

Here \(a\) is a \(k(k-1)/2\)-dimensional vector that collects the free parameters in \(A\) in row-major order, and \(\ln \lambda_1\) is a \(k\)-dimensional vector containing the time \(t=1\) values (initial conditions) of \(\ln \lambda_{t}\). We assume that \(\Omega_A\) and \(\Omega_{\ln \lambda_{1}}\) are diagonal matrices. Furthermore, \(\phi_i\) for \(i=1,\dots,k\) are the log volatility innovation variances. For the AR(1) stochastic volatility specification, the following priors are available

\[ \begin{aligned} a &\sim \mathrm{N}(\theta_A, \Omega_A) \\ \gamma_{0} &\sim \mathrm{N}(\theta_{\gamma_0}, \Omega_{\gamma_0}) \\ \gamma_{1} &\sim \mathrm{N}(\theta_{\gamma_1}, \Omega_{\gamma_1}) \ I(|\gamma_{1,i}| < 1)\\ \ln \lambda_{1} &\sim \mathrm{N}(\theta_{\ln \lambda_{1}}, \Omega_{\ln \lambda_{1}}) \\ \Phi &\sim \mathrm{IW}(V_{\Phi},m_{\Phi}) \end{aligned} \]

Here \(a\) is again the \(k(k-1)/2\)-dimensional vector that collects the free parameters in \(A\) in row-major order, and \(\ln \lambda_1\) is a \(k\)-dimensional vector containing the time \(t=1\) values (initial conditions) of \(\ln \lambda_t\). Furthermore, \(\gamma_{0}\) is a \(k\)-dimensional vector of log volatility intercepts, \(\gamma_{1}\) is a \(k\)-dimensional vector of log volatility slopes, and \(\Phi\) is the \(k \times k\) log volatility innovation covariance matrix. We assume that \(\Omega_A\), \(\Omega_{\gamma_0}\), \(\Omega_{\gamma_1}\), and \(\Omega_{\ln \lambda_{1}}\) are diagonal matrices. Note that the prior for \(\gamma_1\) is truncated normal.

To see how to estimate (and forecast, produce IRFs etc.) the homoscedastic steady-state BVAR, or any of the two heteroscedastic steady-state BVARs, please see the vignettes

vignette("Homoscedastic-steady-state-BVAR")
vignette("RW-stochastic-volatility-steady-state-BVAR")
vignette("AR1-stochastic-volatility-steady-state-BVAR")

Steady-state BVAR models

Introduction

References