Specify priors for the steady-state BVAR model

This function prepares the priors. The Minnesota prior is used for the autoregressive parameters, and is determined by the overall tightness, cross-equation tightness, and the lag decay rate. For the steady-state parameters, a normal prior is used. For the covariance matrix of the innovations, the user can choose between Jeffreys prior or an uninformative inverse-Wishart prior. Optionally enables stochastic volatility where the covariance matrix varies over time.

Usage

priors(
  x,
  lambda_1 = 0.2,
  lambda_2 = 0.5,
  lambda_3 = 1,
  first_own_lag_prior_mean = NULL,
  theta_Psi = NULL,
  Omega_Psi = NULL,
  Jeffreys = TRUE,
  SV = FALSE,
  SV_type = NULL,
  SV_priors = NULL
)

Arguments

x

A steady-state bvar object that has been passed through setup.

lambda_1

Numeric. Overall tightness of the Minnesota prior. Default 0.2.

lambda_2

Numeric. Cross-equation tightness of the Minnesota prior. Default 0.5.

lambda_3

Numeric. Lag decay rate of the Minnesota prior. Default 1.

first_own_lag_prior_mean

Numeric vector of length k. Prior means for the first own lags of the variables. If NULL (default), a zero vector is used.

theta_Psi

Numeric vector. Prior mean vector for $\text{vec}(\Psi)$, i.e. the steady-state parameters. If NULL (default), the OLS estimates are used.

Omega_Psi

Numeric matrix. Prior covariance matrix for $\text{vec}(\Psi)$, i.e. the steady-state parameters. If NULL (default), a diagonal matrix with variances 1000 is used.

Jeffreys

Logical. If TRUE (default), uses Jeffreys prior for the innovation covariance matrix. If FALSE, uses an uninformative inverse-Wishart prior. Only considered if SV=FALSE.

SV

Logical. If TRUE, enables stochastic volatility specification. Default FALSE.

SV_type

Character. Type of stochastic volatility model. Must be either "RW" or "AR1". Required if SV = TRUE.

SV_priors

List. User-supplied stochastic volatility priors. Required when SV = TRUE. The list must contain the following named elements depending on SV_type:

For "RW": theta_A, Omega_A, mu_log_lambda_1, sigma2_log_lambda_1, alpha_phi, beta_phi.
For "AR1": theta_A, Omega_A, theta_gamma_0, Omega_gamma_0, theta_gamma_1, Omega_gamma_1, theta_log_lambda_1, Omega_log_lambda_1, V_Phi, m_Phi.

Value

The steady-state bvar object with an appended priors list containing:

theta_beta: Prior mean vector for $\text{vec}(\beta)$ constructed with the Minnesota prior
Omega_beta: Prior covariance matrix for $\text{vec}(\beta)$ constructed with the Minnesota prior
theta_Psi: Prior mean vector for $\text{vec}(\Psi)$, i.e. the steady-state parameters
Omega_Psi: Prior covariance matrix for $\text{vec}(\Psi)$, i.e. the steady-state parameters
Jeffreys: Indicator for Jeffreys prior usage
Sigma_AR: Residual variance estimates from univariate AR fits, which are used by the Minnesota prior
m: Inverse-Wishart prior degrees of freedom (if Jeffreys = FALSE)
V: Inverse-Wishart prior scale matrix (if Jeffreys = FALSE)
SV: Logical indicator for stochastic volatility specification
SV_type: Stochastic volatility specification type
SV_priors: User-supplied SV prior list (if SV = TRUE)

Details

The goal is to estimate $\beta, \Psi$, and $\Sigma_u$, so priors are needed. Following Villani (2009), prior independence between $\beta, \Psi$ and $\Sigma_u$ is assumed. For $\beta$, i.e. the autoregressive parameter matrix, the Minnesota prior is used

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

The prior means (the elements of $\theta_\beta$) are set to

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

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 level (e.g. nominal interest rate), then $\kappa=\kappa^{level}$, and typical choices for $\kappa^{level}$ are $1$ or $0.9$. Evaluating the equations at their prior means, equation $i$ becomes a random walk if $\kappa^{level}=1$ and a persistent stationary AR(1) process if $\kappa^{level}=0.9$. 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. output 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. 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 is unknown and therefore replaced 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 prior is

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

This is the core of the steady-state BVAR model. In $\theta_\Psi$, one specifies the prior beliefs about the location of the steady state, and in $\Omega_\Psi$, which is assumed to be a diagonal matrix, one specifies the degree of certainty in those prior beliefs. The prior for $\Sigma_u$ is either the usual non-informative Jeffreys prior

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

or a proper uninformative inverse-Wishart prior

$$\Sigma_u \sim \mathrm{IW}(V,m)$$

where $V$ is the scale matrix and $m$ is the number of degrees of freedom. An uninformative prior is specified by setting $V=(m-k-1)\hat{\Sigma}_u$ where $\hat{\Sigma}_u$ is the least squares estimate from the VAR($p$) (including the constant and dummy/trend variable if applicable), and $m=k+2$. For the stochastic volatility specifications, the innovation covariance matrix is now time-varying $\Sigma_{u,t}$. Therefore, stochastic volatility priors are needed, see bvar for more details. For the Random Walk ("RW") stochastic volatility specification, the following priors are used

$$\begin{aligned}a &\sim \mathrm{N}(\theta_A, \Omega_A) \\ \ln \lambda_{i,1} &\sim \mathrm{N}(\mu_{\ln \lambda_{i,1}}, \sigma^2_{\ln \lambda_{i,1}}) \\ \phi_i &\sim \mathrm{IG}(\alpha_{\phi_i},\beta_{\phi_i})\end{aligned}$$

Here $a$ is a $k(k-1)/2$-dimensional vector that collects the free parameters in $A$ in row-major order, and $\ln \lambda_{i,1}$ are the time $t=1$ values (initial conditions) of $\ln \lambda_{i,t}$ for $i=1,\dots,k$. Furthermore, $\phi_i$ for $i=1,\dots,k$ are the log volatility innovation variances. For the AR(1) ("AR1") stochastic volatility specification, the following priors are used

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

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.

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 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.

Villani, M. (2009). Steady-state priors for vector autoregressions. Journal of Applied Econometrics, 24(4), pp. 630–650.

Examples

#homoscedastic with Jeffreys prior
yt <- matrix(rnorm(50), 25, 2)

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p=1)

bvar_obj <- priors(bvar_obj,
                   lambda_1 = 0.2,
                   lambda_2 = 0.5,
                   lambda_3 = 1,
                   first_own_lag_prior_mean = rep(1,2),
                   theta_Psi = rep(0, 2),
                   Omega_Psi = diag(0.1, 2, 2),
                   Jeffreys = TRUE,
                   SV = FALSE,
                   SV_type = NULL,
                   SV_priors = NULL)
                   
#RW stochastic volatility
yt <- matrix(rnorm(50), 25, 2)

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p=1)

k <- bvar_obj$setup$k
n_free_params_A <- bvar_obj$setup$n_free_params_A

SV_priors_RW <- list(
theta_A              =  rep(0, n_free_params_A),
Omega_A              =  diag(1000, n_free_params_A),
mu_log_lambda_1      =  rep(0, k),
sigma2_log_lambda_1  =  rep(1000, k),
alpha_phi            =  rep(5, k),
beta_phi             = (rep(5, k) - 1) * rep(0.1, k)
)

bvar_obj <- priors(bvar_obj,
                   lambda_1 = 0.2,
                   lambda_2 = 0.5,
                   lambda_3 = 1,
                   first_own_lag_prior_mean = rep(1,2),
                   theta_Psi = rep(0, 2),
                   Omega_Psi = diag(0.1, 2, 2),
                   SV = TRUE,
                   SV_type = "RW",
                   SV_priors = SV_priors_RW)
                   
#AR1 stochastic volatility
yt <- matrix(rnorm(50), 25, 2)

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p=1)

k <- bvar_obj$setup$k
n_free_params_A <- bvar_obj$setup$n_free_params_A

SV_priors_AR1 <- list(
theta_A               =  rep(0, n_free_params_A),
Omega_A               =  diag(1000, n_free_params_A),
theta_gamma_0         =  rep(0.1, k),
Omega_gamma_0         =  diag(1000, k),
theta_gamma_1         =  rep(0.9, k),
Omega_gamma_1         =  diag(10, k),
theta_log_lambda_1    =  rep(0.1, k)/(1-rep(0.9, k)),
Omega_log_lambda_1    =  diag(1000, k),
V_Phi                 = (10 - k - 1) * diag(k),
m_Phi                 =  10
)

bvar_obj <- priors(bvar_obj,
                   lambda_1 = 0.2,
                   lambda_2 = 0.5,
                   lambda_3 = 1,
                   first_own_lag_prior_mean = rep(1,2),
                   theta_Psi = rep(0, 2),
                   Omega_Psi = diag(0.1, 2, 2),
                   SV = TRUE,
                   SV_type = "AR1",
                   SV_priors = SV_priors_AR1)