\name{szbvar}
\alias{szbvar}
%- Also NEED an '\alias' for EACH other topic documented here.
\title{ Reduced form Sims-Zha Bayesian VAR model estimation}
\description{
  Estimation of the Bayesian VAR model for just identified VARs
  described in Sims and Zha (1998)
}
\usage{
szbvar(Y, p, z = NULL, lambda0, lambda1, lambda3, lambda4, lambda5,
       mu5, mu6, nu = ncol(Y)+1, qm = 4, prior = 0,
       posterior.fit = FALSE)
}
%- maybe also 'usage' for other objects documented here.
\arguments{
  \item{Y}{ \eqn{T \times m}{T x m} multiple time
    series object created with \code{ts()}.}
  \item{p}{ Lag length }
  \item{z}{ \eqn{T \times k}{T x k} matrix of exogenous variables.  Can
    be \code{z = NULL} if there are none. }
  \item{lambda0}{ \eqn{[0,1]}, Overall tightness of the prior (discounting
  of prior scale). }
  \item{lambda1}{ \eqn{[0,1]}, Standard deviation or tightness of the prior around the AR(1) parameters. }
  \item{lambda3}{ Lag decay (\eqn{>0}, with 1=harmonic) }
  \item{lambda4}{ Standard deviation or tightness around the intercept \eqn{>0} }
  \item{lambda5}{ Standard deviation or tightness around the exogneous
    variable coefficients \eqn{>0}}
  \item{mu5}{ Sum of coefficients prior weight \eqn{\ge0}.  Larger
    values imply difference stationarity.}
  \item{mu6}{ Dummy initial observations or drift prior \eqn{\ge0}.
    Larger values allow for common trends.}
  \item{nu}{ Prior degrees of freedom,  \eqn{m+1}}
  \item{qm}{ Frequency of the data for lag decay equivalence.  Default
    is 4, and a value of 12 will match the lag decay of monthly to
    quarterly data.  Other values have the same effect as "4"}
  \item{prior}{ One of three values: 0 = Normal-Wishart prior, 1 =
    Normal-flat prior, 2 = flat-flat prior (i.e., akin to MLE)}
  \item{posterior.fit}{ logical, F = do not estimate log-posterior fit measures,
    T = estimate log-posterior fit measures.}
}
\details{
  This function estimates the Bayesian VAR (BVAR) model described by
  Sims and Zha (1998).  This BVAR model is based a specification of the
  dynamic simultaneous equation representation of the model.  The prior
  is constructed for the structural parameters.  The basic SVAR model
  used here is documented in \code{\link{szbsvar}}.
  
  The prior covariance matrix of the errors, \eqn{\bar{S}_i}{S(i)}, is
  initially estimated using a VAR(p) model via OLS, with an intercept
  and no demeaning of the data.

}
\value{
  Returns a list of multiple elements.  This is a workhorse function to
  get the estimates, so nothing is displayed to the screen.  The
  elements of the list are intended as inputs for the various
  post-estimation functions (e.g., impulse response analyses,
  forecasting, decompositions of forecast error variance, etc.)

  Returns a list of the class "BVAR" with the following elements:

  \item{intercept}{\eqn{m \times 1}{m x 1} row vector of the \eqn{m} intercepts}
  \item{ar.coefs}{ \eqn{m \times m \times p}{m x m x p} array of the AR
    coefficients.  The first \eqn{m \times m}{m x m} array is for lag 1, the p'th array for lag p.}
  \item{exog.coefs}{\eqn{k \times m}{k x m} matrix of the coefficients for any
    exogenous variables}
  \item{Bhat }{ \eqn{(mp + k + 1) \times m}{(mp + k + 1) x m} matrix of the coefficients, where the
    columns correspond to the variables in the VAR}
  \item{vcv}{ \eqn{m \times m}{m x m} matrix of the maximum likelihood estimate of the
    residual covariance}
  \item{vcv.Bh}{Posterior estimate of the parameter covariance that is
    conformable with Bhat. }
  \item{mean.S}{ \eqn{m \times m}{m x m} matrix of the posterior residual covariance.}
  \item{St }{ \eqn{m \times m}{m x m} matrix of the degrees of freedom times the posterior
    residual covariance.}
  \item{hstar}{\eqn{(mp + k + 1) \times (mp + k + 1)}{(mp + k + 1) x (mp
      + k + 1)} prior precision plus right hand side
    variables crossproduct.}
  \item{hstarinv}{\eqn{(mp + k + 1) \times (mp + k + 1)}{(mp + k + 1) x
      (mp + k + 1)} prior covariance crossproduct
    \code{solve(hstar)}}
  \item{H0}{\eqn{(mp + k + 1) \times (mp + k + 1)}{(mp + k + 1) x
      (mp + k + 1)} prior precision for the parameters}
  \item{S0}{\eqn{m \times m}{m x m} prior error covariance}
  \item{residuals }{ \eqn{(T-p) \times m}{(T-p) x m} matrix of the residuals}
  \item{X}{ \eqn{T \times (mp + 1 + k)}{T x (mp + k + 1)} matrix of
    right hand side variables for the estimation of BVAR}
  \item{Y}{ \eqn{T \times m}{T x m} matrix of the left hand side variables for the estimation of BVAR}
  \item{y}{ \eqn{T \times m}{T x m} input data in \code{dat}}
  \item{z}{ \eqn{T \times k}{T xk} exogenous variables matrix}
  \item{p}{Lag length}
  \item{num.exog}{Number of exogenous variables}
  \item{qm}{Value of parameter to match quarterly to monthly lag decay
    (4 or 12)}
  \item{prior.type}{Numeric code for prior type: 0 = Normal-Wishart, 1
    = Normal-Flat, 2 = Flat-Flat (approximate MLE)}
  \item{prior }{List of the prior parameter:
    c(lambda0,lambda1,lambda3,lambda4,lambda5, mu5, mu6, nu)}
  \item{marg.llf}{Value of the in-sample marginal log-likelihood for
    the data, if \code{posterior.fit=T}}
  \item{marg.post}{Value of the in-sample marginal log posterior of the
    data, if \code{posterior.fit=T}}
  \item{coef.post}{Value of the marginal log posterior estimate of the
    coefficients, if \code{posterior.fit=T}}
}
\references{ Sims, C.A. and Tao Zha. 1998. "Bayesian Methods for Dynamic
  Multivariate Models." \emph{International Economic Review}. 39(4):949-968.
  
  Brandt, Patrick T. and John R. Freeman. 2006. "Advances in Bayesian
  Time Series Modeling and the Study of Politics: Theory Testing,
  Forecasting, and Policy Analysis". \emph{Political Analysis}.
  
}
\author{ Patrick T. Brandt, based on code from Robertson and Tallman and
Sims and Zha.}
\note{ This is a work horse function.  You will probably want to use
  other functions to summarize and report the BVAR results.}


\seealso{ \code{\link{reduced.form.var}} \code{\link{szbsvar}} }
\examples{
  \dontrun{
    data(IsraelPalestineConflict)
    varnames <- colnames(IsraelPalestineConflict)

    fit.BVAR <- szbvar(IsraelPalestineConflict, p=6, z=NULL,
                           lambda0=0.6, lambda1=0.1,
                            lambda3=2, lambda4=0.25, lambda5=0, mu5=0,
                            mu6=0, nu=3, qm=4,
                            prior=0, posterior.fit=FALSE)

    # Draw from the posterior pdf of the impulse responses.
    posterior.impulses <- mc.irf(fit.BVAR, nsteps=10, draws=5000)

    # Plot the responses
    plot(posterior.impulses, method=c("Sims-Zha2"), component=1,
         probs=c(0.16,0.84), varnames=varnames) 
}

}
\keyword{ ts}
\keyword{ models}