do it best , economy and management.
Author : Daniel tulips liu . Copyright © tulipsliu.
Also , Copyright © Sebastian Ankargren , Yukai Yang

First ,Abstract

Abstract Time series are often sampled at different frequencies, which leads to mixed-frequency data. Mixed frequencies are often neglected in applications as high-frequency series are aggregated to lower frequencies. In the mfbvar package, we introduce the possibility to estimate Bayesian vector autoregressive (VAR) models when the set of included time series consists of monthly and quarterly variables. TWe provide a user-friendly interface for model estimation and forecasting. The capabilities of the package are illustrated in an application.
Keywords: vector autoregression, steady-state prior

1. Introduction

Vector autoregressive (VAR) models constitute an important tool for multivariate time series analysis. They are, in their original form, easy to fit and to use and have hence been used for various types of policy analyses as well as for forecasting purposes. A major obstacle in applied VAR modeling is the curse of dimensionality: the number of parameters grows quadratically in the number of variables, and having several hundred or even thousands of parameters is not uncommon. Thus, VAR models estimated by maximum likelihood are usually associated with bad precision.

2. Mixed-Frequency models

Suppose that the system evolves at the monthly frequency. Let $x_t$ be an $n\times 1$ monthly process. Decompose $x_t=(x_{m, t}^\top, x_{q, t}^\top)^\top$ into $n_m$ monthly variables, and a $n_q$-dimensional latent process for the quarterly observations. By letting $y_t=(y_{m, t}^\top, y_{q, t}^\top)^\top$ denote observations, it is implied that $y_{m, t}=x_{m, t}$ as the monthly part is always observed. For the remaining quarterly variables, we instead observe a weighted average of $x_q$. There are two common aggregations used in the literature: intra-quarterly averaging and triangular aggregation. The former assumes the relation between observed and latent variables to be
As the system is assumed to evolve at the monthly frequency, we specify a VAR($p$) model for $x_t$:
$x_t=\phi+\Phi_1 x_{t-1}+\cdots+\Phi_p x_{t-p}+\epsilon_t, \quad \epsilon_t \sim \operatorname{N}(0, \Sigma).\tag{1.1}$
The VAR($p$) model can be written in companion form, where we let $z_t=(x_t^\top, x_{t-1}^\top, \dots, x_{t-p+1}^\top)^\top$. Thus, we obtain
$z_t=\pi+\Pi z_{t-1}+u_t, \quad u_t \sim \operatorname{N}(0, \Omega), \tag{1.2}$
where $\pi$, $\Pi$ and $\Omega$ are the corresponding companion form matrices constructed from $(\phi, \Phi_1, \dots, \Phi_p, \Sigma)$;
It is now possible to specify the observation equation as
$y_t = M_t\Lambda z_t \tag{1.3}$
where $M_t$ is a deterministic selection matrix and $\Lambda$ an aggregation matrix based on the weighting scheme employed. The $M_t$ yields a time-varying observation vector by selecting rows corresponding to variables which are observed, whereas $\Lambda$ aggregates the underlying latent process.

2.1 The steady-state prior proposed by \cite{Villani2009} reformulates \eqref{eq:original} to be on the mean-adjusted form
$\Phi(L)(x_t-\Psi d_t)=\epsilon_t,$
where $\Phi(L)=(I_{n}-\Phi_1 L -\cdots - \Phi_p L^p)$ is an invertible lag polynomial. The intercept $\phi$ in \eqref{eq:original} can be replaced by the more general deterministic term $\Phi_0d_t$, where $\Phi_0$ is $n\times m$ and $d_t$ is $m\times 1$. The steady-state parameters $\Psi$ in \eqref{eq:meanadj} relate to $\Phi_0$ through $\Psi=[\Phi(L)]^{-1}\Phi_0$. By the reformulation, we obtain parameters $\Psi$ that immediately yield the unconditional mean of $x_t$—the steady state. The rationale is that while it is potentially difficult to express prior beliefs about $\Phi_0$, eliciting prior beliefs about $\Psi$ is often easier.
$\begin{aligned} \psi_j|\omega_{\psi ,j}&\sim \operatorname{N}(\underline{\psi}_j, \omega_{\psi, j})\\ \omega_{\psi, j}|\phi_\psi, \lambda_\psi &\sim \operatorname{G}(\phi_\psi, 0.5\phi_\psi\lambda_\psi)\\ \phi_\psi &\sim \operatorname{Exp}(1)\\ \lambda_\psi &\sim \operatorname{G}(c_0, c_1)\\ j&=1, \dots, nm \end{aligned}\tag{1.3}$

where $\operatorname{G}(a,b)$ denotes the gamma distribution with shape-rate parametrization, and $\operatorname{Exp}(c)$ denotes the exponential distribution.
The common stochastic volatility specification presented by \cite{Carriero2016} assumes that the covariance structure in the model is constant over time, but adds a factor that enables time-dependent scaling of the error covariance matrix. More specifically, it is assumed that
\begin{align}
\VAR(\epsilon_t|f_t, \Sigma)=f_t\Sigma,
\end{align}
where $f_t$ is a scalar, $\Sigma$ is inverse Wishart as in \eqref{eq:iw}, and
\begin{equation}
\begin{aligned}
\log f_t&=\rho\log f_{t-1}+v_t\
v_t&\sim \operatorname{N}(0, \sigma^2)\
\rho&\sim \operatorname{N}(\underline{\mu}\rho, \underline{\Omega}\rho; |\rho|<1)\
\sigma^2&\sim \operatorname{IG}(\underline{d}\cdot\underline{\sigma}^2, , \underline{d}),
\end{aligned}\label{eq:csv}
\end{equation}
where $\operatorname{N}(a, b; |x|&lt;c)$ denotes the truncated normal distribution with support $(-c, c)$, and $\operatorname{IG}(a,b)$ is the inverse gamma distribution with parameters $(a, b)$.