Package 'SteadyStateBVAR'

Create a steady-state BVAR model object

Description

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.

Usage

bvar(data)

Arguments

data	A numeric matrix or time series of data where each column is a variable and each row is a time period.

Details

For a detailed theoretical introduction to the steady-state BVAR model, run the following

vignette("SteadyStateBVAR-intro")

The steady-state BVAR 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. Now

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

is the unconditional mean, or the steady state of the process. Note that the current version of this package only allows for $d_t$ to contain either a constant, a constant and a dummy variable, or a constant and a time trend. We may stack the (transposed) $\Pi_i$ matrices in the $(kp \times k)$ matrix $\beta$

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

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 $w_t'=(y_{t-1}',\dots,y_{t-p}')$ 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. The goal is to estimate the parameters $\Theta = \begin{bmatrix} \beta & \Psi & \Sigma_u \end{bmatrix}$, and as such priors are needed. Please see priors for more details.

For the innovations to the model, in the case of the homoscedastic (the original) steady-state BVAR, they are $u_t \overset{\text{iid}}{\sim} \mathrm{N_k}(0,\Sigma_u)$. However, 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 (driftless) Random Walk processes

$\gamma_{0,i}=0, \ \gamma_{1,i}=1 \ \forall i$

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 Random Walk 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.

To see examples for each type of steady-state BVAR model, simply run one of the following

• vignette("Homoscedastic-steady-state-BVAR")

• vignette("RW-stochastic-volatility-steady-state-BVAR")

• vignette("AR1-stochastic-volatility-steady-state-BVAR")

Value

A steady-state bvar object.

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.

Examples

yt <- matrix(rnorm(50), 25, 2)

bvar_obj <- bvar(data = yt)

---

Conditional forecasts from a fitted steady-state BVAR model

Description

Computes and plots conditional forecasts from a fitted steady-state bvar object. Conditions are imposed on specific variables at specific horizons using Algorithm 3.3.1 from Dieppe, Legrand, and van Roye (2016). Both conditional and unconditional forecasts are plotted for comparison. Please note that for the moment, conditional forecasting is only enabled for the homoscedastic steady-state BVAR, i.e. when SV=FALSE in priors().

Usage

conditional_forecast(
  x,
  conditions,
  pi = 0.95,
  fcst_type = c("mean", "median"),
  growth_rate_idx = NULL,
  plot_idx = NULL
)

Arguments

x	A steady-state bvar object that has been passed through fit.
conditions	A data frame with three columns: var (integer index of the conditioned variable), horizon (integer forecast horizon at which the condition is imposed), and value (numeric, the imposed condition). Note that var and horizon must be integers, as they are used for indexing.
pi	Numeric. The prediction interval width. Default 0.95, i.e. 95% prediction interval.
fcst_type	Character. Whether to use "mean" or "median" as the point forecast. Default "mean".
growth_rate_idx	Integer vector. Indices of variables of which to convert forecasts to annual growth rates $100 (\ln x_{t} - \ln x_{t-f})$, where $f$ is the frequency of the data (4 for quarterly, 12 for monthly). Only suitable for variables specified as $100 (\ln x_{t} - \ln x_{t-1})$, i.e. 100*diff(log(x)). Computed by summing up to $f$ log first differences. Default is NULL.
plot_idx	Integer vector. Indices of variables to plot. If NULL (default), all variables are plotted. Forecasts are always computed and returned for all variables, regardless of plot_idx.

Details

See Section 5.3 of Dieppe, Legrand, and van Roye (2016) for more details. Please note the limitations of this method, see the detailed discussion in Section 5.4 of Dieppe, Legrand, and van Roye (2016).

Value

Invisibly returns a list with three matrices: forecast, lower, and upper, each of dimension H x k.

References

Dieppe, A., Legrand, R., and van Roye, B. (2016). The BEAR toolbox. Working Paper Series, No. 1934. European Central Bank.

Examples

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

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p=1, deterministic = "constant")

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)
                   
bvar_obj <- fit(bvar_obj,
                H = 8,
                iter = 200,
                warmup = 50,
                chains = 1,
                cores = 1)
                
conditions <- data.frame(var = rep(2,8),
                         horizon = rep(1:8),
                         value   = rep(1,8))
                         
(cond_fcst <- conditional_forecast(bvar_obj,
                                  conditions,
                                  pi=0.68,
                                  fcst_type = "mean"))

---

Estimate the steady-state BVAR model using Stan

Description

Estimates the steady-state BVAR model using the No-U-Turn sampler (a variant of Hamiltonian Monte Carlo) via Stan. Also generates draws from the joint predictive distribution. Uses the data, setup, and priors stored in the steady-state bvar object.

Usage

fit(x, H = 1, d_pred = NULL, ...)

Arguments

x	A steady-state bvar object that has been passed through setup and priors.
H	Positive Integer. Forecast horizon. Default is 1.
d_pred	Matrix of size $H \times q$. Future values of the deterministic variables $d_t$. Default is NULL (d_pred is automatically created). If $d_t$ contains only a constant, $d_{t+h}=1 \ \forall \ h$. If $d_t$ contains a constant and a dummy, it is assumed that the dummy (and the constant) stays at its last observed value for all future forecast periods. However, if this is not the intention, the user may supply d_pred themselves. If $d_t$ contains a constant and a trend, the trend is extrapolated from its last observed value (i.e. $trend_{T+h}=trend_{T}+h$). Naturally, the constant is again equal to one for all future periods.
...	Additional arguments passed directly to the rstan function sampling

Details

The function selects the appropriate precompiled Stan model based on settings from priors:

• steady_state_bvar_homoscedastic_jeffreys_prior

• steady_state_bvar_homoscedastic_inverse_wishart_prior

• steady_state_bvar_RW_stochastic_volatility

• steady_state_bvar_AR1_stochastic_volatility

The function estimates the following parameter matrices/vectors (see bvar for details):

• beta: $kp \times k$ VAR coefficient matrix

• Psi: $k \times q$ steady-state parameter matrix

• Sigma_u: innovation covariance matrix ($k \times k$ for homoscedastic, $T \times k \times k$ for stochastic volatility)

• If Random Walk stochastic volatility:

  • A: $k \times k$ lower triangular matrix with ones on the diagonal that describes the contemporaneous interaction of the endogenous variables

  • phi: $k$-dimensional vector of log volatility innovation variances

• If AR1 stochastic volatility:

  • A: $k \times k$ lower triangular matrix with ones on the diagonal that describes the contemporaneous interaction of the endogenous variables

  • gamma_0: $k$-dimensional vector of log volatility intercepts

  • gamma_1: $k$-dimensional vector of log volatility slopes

  • Phi: $k \times k$ log volatility innovation covariance matrix

Value

A fitted steady-state bvar object with:

• x$fit$stan: An object of class stanfit

• x$fit$posterior_means: List of elementwise posterior mean estimates

• x$fit$posterior_medians: List of elementwise posterior median estimates

Examples

#homoscedastic with Jeffreys prior, d_t = constant

yt <- matrix(rnorm(20), 10, 2)

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p = 1, deterministic = "constant")

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)
                   

bvar_obj <- fit(bvar_obj,
                H = 8,
                iter = 200,
                warmup = 50,
                chains = 1,
                cores = 1)
                
                
#homoscedastic with inverse-Wishart prior, d_t = constant and dummy

yt <- matrix(rnorm(20), 10, 2)

bvar_obj <- bvar(data = yt)

dummy_variable <- c(rep(1,5), rep(0,5))

bvar_obj <- setup(bvar_obj, p = 1,
                  deterministic = "constant_and_dummy",
                  dummy = dummy_variable)
                  
k <- bvar_obj$setup$k
q <- bvar_obj$setup$q

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, k*q),
                   Omega_Psi = diag(0.1, k*q, k*q),
                   Jeffreys = FALSE) #inverse-Wishart


bvar_obj <- fit(bvar_obj,
                H = 8,
                iter = 200,
                warmup = 50,
                chains = 1,
                cores = 1)
 

#RW stochastic volatility

yt <- matrix(rnorm(20), 10, 2)

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p=1, deterministic = "constant")

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),
theta_log_lambda_1   =  rep(0, k),
Omega_log_lambda_1   =  diag(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)

bvar_obj <- fit(bvar_obj,
                H = 8,
                iter = 200,
                warmup = 50,
                chains = 1,
                cores = 1,
                control = list(max_treedepth = 12, adapt_delta = 0.85)
                )


#AR1 stochastic volatility

yt <- matrix(rnorm(20), 10, 2)

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p=1, deterministic = "constant")

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)

bvar_obj <- fit(bvar_obj,
                H = 8,
                iter = 200,
                warmup = 50,
                chains = 1,
                cores = 1,
                control = list(max_treedepth = 12, adapt_delta = 0.85)
                )

---

Forecast from a fitted steady-state BVAR model

Description

Computes and plots forecasts from a fitted steady-state bvar object. Draws from the joint predictive distribution are used to construct point forecasts and prediction intervals. Optionally converts monthly or quarterly growth rate forecasts to annual growth rates for variables specified as 100*diff(log(x)). Optionally overlays the posterior steady states.

Usage

forecast(
  x,
  pi = 0.95,
  fcst_type = c("mean", "median"),
  growth_rate_idx = NULL,
  plot_idx = NULL,
  show_all = TRUE,
  ss = TRUE,
  ss_type = c("mean", "median"),
  ss_ci = 0.95
)

Arguments

x	A steady-state bvar object that has been passed through fit.
pi	Numeric. The prediction interval width. Default 0.95, i.e. 95% prediction interval.
fcst_type	Character. Whether to use "mean" or "median" as the point forecast. Default "mean".
growth_rate_idx	Integer vector. Indices of variables of which to convert forecasts to annual growth rates $100 (\ln x_{t} - \ln x_{t-f})$, where $f$ is the frequency of the data (4 for quarterly, 12 for monthly). Only suitable for variables specified as $100 (\ln x_{t} - \ln x_{t-1})$, i.e. 100*diff(log(x)). Computed by summing up to $f$ log first differences. Default is NULL.
plot_idx	Integer vector. Indices of variables to plot. If NULL (default), all variables are plotted. Forecasts are always computed and returned for all variables, regardless of plot_idx.
show_all	Logical. If TRUE (default), the full history is shown. If FALSE, the last eight years of history are shown.
ss	Logical. If TRUE, overlays the posterior steady state $\mu_t = \Psi d_t$. Default TRUE. For variables in growth_rate_idx, the posterior steady-state is annualized.
ss_type	Character. Whether to use "mean" or "median" of the posterior as the steady-state point estimate. Default "mean".
ss_ci	Numeric. The posterior credible interval width for the steady state. Default 0.95, i.e. 95% credible interval.

Value

Invisibly returns a list with three matrices: forecast, lower, and upper, each of dimension H x k where H is the forecast horizon and k is the number of variables.

Examples

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

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p=1, deterministic = "constant")

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)
                   
bvar_obj <- fit(bvar_obj,
                H = 8,
                iter = 200,
                warmup = 50,
                chains = 1,
                cores = 1)
                
forecast(bvar_obj,
         fcst_type = "mean",
         pi = 0.90,
         show_all = TRUE,
         ss = TRUE,
         ss_type = "mean",
         ss_ci = 0.95)

---

Impulse Response Functions for a fitted steady-state BVAR model

Description

Computes and plots impulse response functions (IRFs) from a fitted steady-state bvar object. Supports both orthogonalized (OIRF) and generalized (GIRF) impulse responses, with optional conversion to annual growth rates.

Usage

IRF(
  x,
  H = 20,
  response = NULL,
  impulse = NULL,
  type = c("median", "mean"),
  method = c("OIRF", "GIRF"),
  ci = 0.95,
  t = NULL,
  growth_rate_idx = NULL
)

Arguments

x	A steady-state bvar object that has been passed through fit.
H	Integer. The forecast horizon for the IRF. Default 20.
response	Integer vector. Indices of the response variables to plot. If NULL (default), all responses are plotted.
impulse	Integer vector. Indices of the impulse variables to plot. If NULL (default), all impulses are plotted.
type	Character. Whether to use "median" or "mean" as the point estimate. Default "median".
method	Character. The IRF method: "OIRF" for orthogonalized or "GIRF" for generalized impulse responses. Default "OIRF".
ci	Numeric. The credible interval width. Default 0.95, i.e. 95%.
t	Integer. Time index for the covariance matrix when using stochastic volatility models. If NULL (default), the last time t is used.
growth_rate_idx	Integer vector. Indices of variables for which the impulse response is converted from a quarterly or monthly log first difference to an annual growth rate response, i.e. $100 (\ln x_{t} - \ln x_{t-f})$, where $f$ is the frequency of the data (4 for quarterly, 12 for monthly). Only suitable for variables specified as $100 (\ln x_{t} - \ln x_{t-1})$, i.e. 100*diff(log(x)). Computed by summing up to $f$ periods of the impulse response. Default is NULL.

Value

Invisibly returns a list with three arrays: the point estimate IRF, lower, and upper credible bounds, each of dimension k x k x (H+1).

Examples

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

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p=1, deterministic = "constant")

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)
                   
bvar_obj <- fit(bvar_obj,
                H = 8,
                iter = 200,
                warmup = 50,
                chains = 1,
                cores = 1)
                
IRF(bvar_obj)

---

Koop and Korobilis (2010) dataset

Description

Quarterly US macroeconomic data (1953Q1-2006Q3) from Koop and Korobilis (2010)

Usage

KoopKorobilis2010

Format

KoopKorobilis2010

A multivariate time series object with 215 rows and 3 columns:

delta pi

inflation rate (annual percentage change in a chain-weighted GDP price index)

u

unemployment rate (seasonally adjusted civilian unemployment rate, all civilian workers aged 16 years or older)

r

interest rate (yield on the three-month Treasury bill rate)

Source

Koop, G. and Korobilis, D. (2010). Bayesian multivariate time series methods for empirical macroeconomics. Foundations and Trends in Econometrics, 3(4), pp. 267-358.

---

Prior Probability Interval for a Normal Distribution

Description

Calculates the mean and variance of a normal prior probability interval. Given a lower and upper bound of a prior probability interval this function recovers the implied normal prior parameters. Useful for specifying informative priors on the steady-state parameters (elements of $\Psi$).

Usage

ppi(l, u, interval = 0.95, annualized_growthrate = FALSE, freq = NULL)

Arguments

l	Numeric. The lower bound of the prior probability interval.
u	Numeric. The upper bound of the prior probability interval.
interval	Numeric. The prior probability mass within the interval. Default 0.95, i.e. 95%.
annualized_growthrate	Logical. If TRUE, treats l and u as bounds of the annualized steady-state growth rate and calculates the implied mean and variance on the corresponding quarterly/monthly scale. Useful if you are working with a variable specified as 100*diff(log(x)). Default FALSE.
freq	Integer. The data frequency (e.g. 4 for quarterly, 12 for monthly). Only used when annualized_growthrate = TRUE. Default NULL.

Details

Consider a CPI variable CPI <- data$CPI, observed at quarterly frequency. In the model, quarter-on-quarter inflation is used x <- 100*diff(log(CPI)). Suppose the prior belief is that annualized steady-state inflation lies between 1.7 and 2.3 with 95% probability (mean 2). On the quarterly scale used by x, this corresponds to a 95% interval of 0.425 to 0.575 (mean 0.5). Since it is typically more natural to elicit a prior on the annualized scale, the annualized_growthrate argument performs this conversion. See vignette("Homoscedastic-steady-state-BVAR") for usage in practice.

Value

A list with two elements: mean and var, giving the mean and variance of the implied normal distribution.

Examples

ppi(l = 1.7, u = 2.3, interval = 0.95)
ppi(l = 1.7, u = 2.3, interval = 0.95, annualized_growthrate = TRUE, freq = 4)

---

Specify priors for the steady-state BVAR model

Description

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. Regarding the covariance matrix of the reduced-form innovations, for the homoscedastic model, the user can choose between Jeffreys prior or an uninformative inverse-Wishart prior. The user may also choose a stochastic volatility specification for the innovations, 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. Assumed to be diagonal. If NULL (default), the identity matrix 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, theta_log_lambda_1, Omega_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.

Details

The 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 $\beta$, which contains the autoregressive parameters, 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

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

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 differenced (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 is unknown and therefore needs to be 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 (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 $\theta_\Psi$, 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. Too see how to specify steady-state priors in practice, please see the package vignettes. 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 $m=k+2$. However, if we opt to use stochastic volatility, the innovation covariance matrix is time-varying $\Sigma_{u,t}$, instead of constant $\Sigma_{u}$. Therefore, stochastic volatility priors are needed. See bvar for more details on the stochastic volatility specifications. Please note that $\lambda$ below (volatilities) has nothing to do with the $\lambda$ from the Minnesota prior (hyperparameters). Now, for the Random Walk ("RW") 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) ("AR1") 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.

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.

To see examples of how to specify the priors, simply view the relevant vignette

• vignette("Homoscedastic-steady-state-BVAR")

• vignette("RW-stochastic-volatility-steady-state-BVAR")

• vignette("AR1-stochastic-volatility-steady-state-BVAR")

Value

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

theta_beta	Prior mean vector for $\text{vec}(\beta)$, i.e. the autoregressive parameters, constructed with the Minnesota prior
Omega_beta	Prior covariance matrix for $\text{vec}(\beta)$, i.e. the autoregressive parameters, 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 and SV = FALSE)
V	Inverse-Wishart prior scale matrix (if Jeffreys = FALSE and SV = FALSE)
SV	Logical indicator for stochastic volatility specification
SV_type	Stochastic volatility specification type
SV_priors	User-supplied SV prior list (if SV = TRUE)

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),
theta_log_lambda_1   =  rep(0, k),
Omega_log_lambda_1   =  diag(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)

---

Restrict VAR coefficients to zero

Description

Applies zero restrictions to the VAR coefficient matrix $\beta$ by setting the corresponding prior variances in Omega_beta to a value near zero (prior means are zero by default in the Minnesota prior).

Usage

restrict_beta(x, restriction_matrix)

Arguments

x	A steady-state bvar object that has been passed through priors.
restriction_matrix	A numeric matrix of dimension k*p x k where entries of 0 indicate coefficients to be restricted to zero and entries of 1 indicate unrestricted coefficients.

Details

The steady-state BVAR 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. One can stack the (transposed) $\Pi_i$ matrices in the $(kp \times k)$ matrix $\beta$

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

This function puts zero restrictions on the autoregressive parameters/coefficients, i.e the elements of $\beta$.

Value

The steady-state bvar object with the restriction matrix stored in setup and the prior covariance matrix Omega_beta updated accordingly.

Examples

yt <- matrix(rnorm(50), 25, 2)
colnames(yt) <- c("y1", "y2")

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p=2)

bvar_obj <- priors(bvar_obj,
                   theta_Psi = rep(0, 2),
                   Omega_Psi = diag(0.1, 2, 2))
                   
p <- bvar_obj$setup$p
k <- bvar_obj$setup$k

restriction_matrix <- matrix(1,k*p,k)

#restrict beta so y2 does not granger-cause y1

restriction_matrix[2, 1] <- 0
restriction_matrix[4, 1] <- 0

bvar_obj <- restrict_beta(bvar_obj, restriction_matrix)

---

Set up the steady-state BVAR model

Description

Prepares the components needed for prior specification and estimation. Also computes OLS estimates.

Usage

setup(
  x,
  p = 1,
  deterministic = c("constant", "constant_and_dummy", "constant_and_trend"),
  dummy = NULL
)

Arguments

x	A steady-state bvar object created by bvar.
p	Integer. The lag order of the VAR. Default 1.
deterministic	Character. The deterministic components to include in $d_t$. One of "constant" (default), "constant_and_dummy", or "constant_and_trend".
dummy	Numeric vector of a dummy variable. Only used when deterministic = "constant_and_dummy". Default NULL.

Value

The steady-state bvar object with a setup list containing the required components for prior specification and estimation, and also the OLS estimates.

Examples

yt <- matrix(rnorm(50), 25, 2)

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj)

---

Plot steady-state priors

Description

Produces time series plots of the data along with the implied prior for the steady state $\mu_t = \Psi d_t$.

Usage

steady_state_priors_plot(
  x,
  interval = 0.95,
  growth_rate_idx = NULL,
  plot_idx = NULL
)

Arguments

x	A steady-state bvar object that has been passed through priors.
interval	Numeric. The prior interval width. Default 0.95, i.e. a 95% prior interval for the steady state.
growth_rate_idx	Integer vector. Indices of variables specified as $100 (\ln x_{t} - \ln x_{t-1})$, i.e. 100*diff(log(x)), for which the historical series is converted to year-over-year (annual) growth scale $100 (\ln x_{t} - \ln x_{t-f})$ and the steady-state prior is converted to the annualized (not annual) growth scale $f100(\ln x_{t} - \ln x_{t-1})$ where $f$ = frequency. Default is NULL.
plot_idx	Integer vector. Indices of variables to plot. If NULL (default), all variables are plotted.

Details

The implied prior for the steady state $\mu_t = \Psi d_t$ is based on the prior for $\Psi$

$\mathrm{vec}(\Psi) \sim \mathrm{N}(\theta_\Psi, \Omega_\Psi)$

which is specified in priors. Note that it is assumed that $\Omega_\Psi$ is diagonal.

Value

Invisibly returns a list with three matrices, lower, mean, and upper, each of dimension T x k giving the steady-state prior bounds and mean over the historical sample. For growth_rate_idx columns, these are on the annualized scale.

Examples

yt <- matrix(rnorm(50), 25, 2)

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p = 1, deterministic = "constant")

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)

steady_state_priors_plot(bvar_obj)

---

Plot stochastic volatility estimates and forecasts

Description

Produces time series plots of posterior stochastic volatility estimates over the estimation sample together with predictive paths over the forecast horizon for a fitted steady-state bvar object with stochastic volatility.

Usage

stochastic_volatility_plot(
  x,
  ci = 0.95,
  vol = "log_lambda",
  stat = "mean",
  plot_idx = NULL,
  xlim = NULL,
  ylim = NULL
)

Arguments

x	A steady-state bvar object that has been passed through fit with stochastic volatility enabled.
ci	Numeric. Interval level. Default is 0.95.
vol	Character. Volatility representation: "log_lambda" (default) or "sd".
stat	Character. Posterior summary statistic to use as point estimate: "mean" (default) or "median".
plot_idx	Integer vector. Indices of variables to plot. If NULL (default), all variables are plotted.
xlim	Numeric vector of length 2. Optional x-axis limits.
ylim	Numeric vector of length 2. Optional y-axis limits.

Details

The function supports two volatility representations:

• "log_lambda": log volatilities

• "sd": implied standard deviations of the reduced-form innovations $u_{t}$

For each selected variable, the function plots the posterior point estimate path and credible bands over the estimation sample, and the predictive point estimate path and prediction intervals over the forecast horizon.

Value

An invisible list with:

• estimated: List of posterior summaries for the estimation sample:

  • point: $T \times k$ matrix of posterior point estimates (mean or median depending on stat)

  • lower: $T \times k$ matrix of lower credible bounds

  • upper: $T \times k$ matrix of upper credible bounds

• predicted: List of posterior summaries for the forecast horizon:

  • point: $(T+H) \times k$ matrix of predictive point estimates (mean or median depending on stat)

  • lower: $(T+H) \times k$ matrix of lower bound of prediction interval

  • upper: $(T+H) \times k$ matrix of upper bound of prediction interval

Examples

yt <- matrix(rnorm(50), 25, 2)

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p = 1, deterministic = "constant")

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),
theta_log_lambda_1   =  rep(0, k),
Omega_log_lambda_1   =  diag(1000, k),
alpha_phi            =  rep(5, k),
beta_phi             = (rep(5, k) - 1) * rep(0.1, k)
)

bvar_obj <- priors(bvar_obj,
                   theta_Psi = rep(0, 2),
                   Omega_Psi = diag(0.1, 2, 2),
                   SV = TRUE,
                   SV_type = "RW",
                   SV_priors = SV_priors_RW)

bvar_obj <- fit(bvar_obj,
                H = 8,
                iter = 200,
                warmup = 50,
                chains = 1,
                cores = 1,
                verbose = FALSE)

stochastic_volatility_plot(bvar_obj, ci = 0.95)

---

Summarize a fitted steady-state BVAR model

Description

Computes and prints elementwise posterior summaries from a fitted steady-state bvar object. The printed output depends on whether the model is homoscedastic or includes stochastic volatility (RW or AR1 specification).

Usage

## S3 method for class 'bvar'
summary(object, pars = NULL, stat = "mean", t = NULL, ...)

Arguments

object	A steady-state bvar object that has been passed through fit.
pars	Character vector of parameter names to include. If NULL (default), all available parameters are displayed.
stat	Character. Posterior summary statistic to display: "mean" (default) or "median".
t	Integer. Time index for the innovation covariance matrix if stochastic volatility was estimated. If NULL, the latest available t is used.
...	Additional arguments (currently unused).

Details

The function summarizes, by calculating elementwise posterior means/medians, the following estimated parameter matrices/vectors:

• beta: $kp \times k$ VAR coefficient matrix

• Psi: $k \times q$ steady-state parameter matrix

• Sigma_u: innovation covariance matrix ($k \times k$ for homoscedastic, $T \times k \times k$ for stochastic volatility)

• If Random Walk stochastic volatility:

  • A: $k \times k$ lower triangular matrix with ones on the diagonal that describes the contemporaneous interaction of the endogenous variables

  • phi: $k$-dimensional vector of log volatility innovation variances

• If AR1 stochastic volatility:

  • A: $k \times k$ lower triangular matrix with ones on the diagonal that describes the contemporaneous interaction of the endogenous variables

  • gamma_0: $k$-dimensional vector of log volatility intercepts

  • gamma_1: $k$-dimensional vector of log volatility slopes

  • Phi: $k \times k$ log volatility innovation covariance matrix

Value

Returns the input object invisibly.

Examples

yt <- matrix(rnorm(20), 10, 2)
bvar_obj <- bvar(data = yt)
bvar_obj <- setup(bvar_obj, p = 1, deterministic = "constant")
bvar_obj <- priors(bvar_obj,
                   theta_Psi = rep(0, 2),
                   Omega_Psi = diag(0.1, 2, 2))

bvar_obj <- fit(bvar_obj,
                H = 1,
                iter = 100,
                warmup = 50,
                chains = 1,
                cores = 1)

summary(bvar_obj)

---

Villani (2009) dataset

Description

Swedish macroeconomic data (1980Q1–2005Q4) from Section 4.1 in Villani (2009)

Usage

Villani2009

Format

Villani2009

A multivariate time series object with 104 rows and 7 columns:

delta y_f

foreign GDP growth

pi_f

foreign CPI inflation

i_f

foreign 3-month interest rate

delta y

domestic GDP growth

pi

domestic CPI inflation

i

domestic 3-month interest rate

q

level of the real exchange rate defined as $q = s + p_f - p$, where $p_f$ and $p$ are the foreign and domestic CPI levels in logs and $s$ is the log of the trade-weighted nominal exchange rate

Source

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

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