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The Time-Varying Multivariate Autoregressive Index Model
\maketitle
\begin{abstract}
\noindent Many economic variables feature changes in their conditional mean
and volatility, and Time Varying Vector Autoregressive Models are often used
to handle such complexity in the data. Unfortunately, when the number of
series grows, they present increasing estimation and interpretation problems.
This paper tries to address this issue proposing a new Multivariate
Autoregressive Index model that features time varying means and
volatility.\newline Technically, we develop a new estimation methodology that
mix switching algorithms with the forgetting factors strategy of
\cite{KK2012}. This substantially reduces the computational burden and allows
to select or weight, in real time, the number of common components and other
features of the data using Dynamic Model Selection or Dynamic Model Averaging
without further computational cost.\newline Using USA macroeconomic data,
we provide a structural analysis and a forecasting exercise that demonstrates
the feasibility and usefulness of this new model.\\
\\
\textbf{Keywords}: Large datasets, Multivariate Autoregressive Index models,
Stochastic volatility, Bayesian VARs.
\end{abstract}
\newpage
\section{Introduction}
The availability of large datasets and instability of the economy has changed
the nature of economic models. Time-varying parameter models are developed to
capture the ever-changing economic environment. For example,
\cite{CS2002} use a small VAR with time-varying coefficients, that follow a random walk dynamic (TVP-VAR), to detect features such as coefficient drift of the
inflation-unemployment dynamics. Contributions on small TVP-VAR include
\cite{CMS2005}, \cite{CogleySargent(2005)}, \cite{Primiceri2005} and
\cite{dAgostinoGambettiGiannone2013}.\newline The abundance of large data sets
with macroeconomic variables has called for the development of larger TVP-VAR
models. Motivated by the need for modelling instability in large systems,
\cite{KK2013}(KK, henceforth) develop a computationally efficient estimation
methodology for Large TVP-VAR models with stochastic volatility (TVP-VAR-SV).
When a large number of predictors is included in the VAR
system, the computational burden increases significantly. KK operationalize
the model by using forgetting factors (see \citealp{RKE2010}) and
estimating the error covariance matrix dynamically using an exponentially
weighted moving average (EWMA). Recently \cite{KK2013b} extended the
methodology to time-varying parameter factor augmented VAR with
stochastic volatility (TVP-FAVAR-SV). Although
the TVP-VAR-SV are overall easier to handle than TVP-FAVAR-SV in terms of the
online estimation, it remains an open question if a small amount of common
components might efficiently summarize the variation in the data for
forecasting or economic analysis.\\
In this paper, we propose a new model
that bridges TVP-VAR-SV and TVP-FAVAR-SV, with a new estimation strategy based
on the results in KK. Specifically, to reduce the dimensionality, we draw from
the recent developments in Multivariate Index Autoregressive (MAI) models, see
\cite{CKM2016}, \cite{CGH2017}, \cite{CG2019}, and \cite{CCM2020}, among
others. The MAI model, originally introduced by \cite{Reinsel1983}, is a
bridge between reduced-rank VARs (see \citealp{CCM2015} and the references
therein) and the Dynamic Factor Model (DFM, see \citealp{SW2016},
\citealp{Lippi2019} and the references therein).
On the one hand, it reduces the dimensionality by
imposing a sort of reduced rank structure to the VAR, on the other, it allows
for identifying few linear combinations of the variables, which are labelled
as the indexes, whose lags are entirely responsible for the dynamics of the
system.\newline Although the mathematical formulation of the MAI is similar to
that of the DFM, an advantage of the former is that it does not require that
the dimension of the system diverges to infinity in estimation. Hence, the MAI
can be applied even to small or medium VARs. Moreover, the factor structure
can be tested for and not simply imposed as in the DFM, and the estimation
error of the indexes is explicitly accounted for, see \cite{CG2019} for
further details.\\
The contribution of the paper is twofold. The first
is to propose a MAI with time-varying parameters and stochastic volatility
(TVP-MAI-SV). A second contribution of the paper is to develop approximate
estimation methods for the TVP-MAI-SV which do not involve the use of Markov
chain Monte Carlo (MCMC) such as in \cite{CCM2018} and \cite{CCM2019}. To
achieve this result we propose mixing the switching algorithm, see
\cite{CGH2017}, with forgetting factors in the same spirit of \cite{KK2013b}.\\
Forgetting factors (also known as discount factors), have long been
used with state-space models, see \cite{RKE2010}. They do not require the use
of MCMC methods and be useful in economic and financial
applications, see \cite{DH2012}, and \cite{GNS2014}.\\
The new model is applied in two empirical applications.
The first one is a variance decomposition on a large dataset
composed of 215 time series.
The analysis of economic uncertainty has a long history.
A large literature investigates the relationship between uncertainty and growth
by proving that both at the macro and micro level, uncertainty moves
counter-cyclically: rising steeply in recessions and falling in booms. Evidence of counter-cyclical volatility is provided, among the others, for macro stock returns in \cite{ScW1989} for firm-level stock asset prices in \cite{Campal01}, for consumption and income in \cite{SCA2004}.
Moreover, given the increase of uncertainty after major economic and political shocks and the recent 2008 financial crisis followed by the Great Recession, the interest of economists and policymakers become markedly focused on its effects on the economy. Moving from the seminal paper of \cite{Bloom2009} that provides a structural framework to analyse the impact of uncertainty shocks, more recent literature starts analysing and
measuring the macroeconomic and financial uncertainty and its impact on macroeconomic variables (see among the others \citealp{BES2013}, \citealp{CCG2014} and \citealp{Jo2015}).\\% while \cite{AC2009} proved
However, measuring uncertainty effect on macroeconomics is difficult as most macro variables move together over the business cycle. This co-movement may face challenging identification problems which are generally overcome by estimating uncertainty in a preliminary step and then evaluating its impact on macroeconomic variables.
Uncertainty measure is included together with a small set of macroeconomic variables in a VAR model computing the responses of the macro variables to the uncertainty shock (see among others \citealp{Bloom2009}, \citealp{CCG2014}, \citealp{BB2017}, \citealp{BES2013}, \citealp{JSN2015}).
The use of small VAR models, to assess the effects of uncertainty, can make the results subject to the common omitted variable bias and non-fundamentalness of the errors, besides providing results on the impact to just a few economic indicators. Differently from the previous literature, in line with \cite{CKM2016}, we focus on measures of uncertainty based on the volatility and we apply our TVP-MAI-SV on 215 macroeconomic and financial variables.
The log predictive likelihood ($\log$ PL) is used to select among
the number of indexes that following
\cite{CKM2016}, \cite{GanBreu2010} and \cite{Gugl2016}, are classified
into 5 groups: \textit{Financial, Labour Market, Nominal, Prices, Real}.
We analyse how much volatility is explained by each index over time.\\
The second empirical application provides a forecasting exercise for three key macroeconomic variables, namely Real Gross Domestic Product (GDP), Consumer Price Index (CPI) and Effective Federal Funds Rate (IntRate). The point and density forecast evaluation show that the TVP-MAI-SV model has very promising out-of-sample properties compared with a set of univariate and multivariate competitors. In particular, the specification of the model with SV and constant parameters performs significantly well. This is in line with the results in \cite{CE2018} and \cite{CEHK2020}\\
The rest of the paper proceeds as follows. Section
\ref{sec:Model} presents the MAI model and introduces the new TVP-MAI-SV.
Section \ref{sec:BayesEstimation} discusses the new estimation approach.
Section \ref{sec:EmpApplication} contains the empirical application. Finally
Section \ref{sec:Concl} draws some conclusions. All the derivations are reported in Appendix A and B.
\section{From the MAI model to the TV-MAI \label{sec:Model}}
Let $\mathbf{y}_{t}\equiv(y_{1,t},\ldots,y_{N,t})^{\prime}$\ denote the
$N$-vector of the time series of interest. In the fixed parameter framework,
it is assumed that variables $\mathbf{y}_{t}$ are generated by a stationary VAR of
order $p$ (VAR$(p)$):
\begin{equation}
\mathbf{y}_{t}=\Phi(L)\mathbf{y}_{t-1}+\mbox{\boldmath $\varepsilon$}_{t}
,\hspace{0.3cm}t=1,2,\ldots,T,\label{VAR}
\end{equation}
where $\Phi(L)=\sum_{h=0}^{p-1}\Phi_{h}L^{h}$ and
$\mbox{\boldmath $\varepsilon$}_{t}$ are i.i.d. innovations with
$\mbox{E}(\mbox{\boldmath $\varepsilon$}_{t})=0$,
$\mbox{E}(\mbox{\boldmath $\varepsilon$}_{t}\mbox{\boldmath $\varepsilon$}_{t}
^{\prime})=\mbox{H}$ (positive definite) and finite fourth moments.\newline In
order to reduce the number of parameters of model (\ref{VAR}),
\cite{Reinsel1983} proposed to impose the following set of restrictions on the
VAR mean parameters:
\begin{equation}
\Phi(L)=\beta(L)\omega^{\prime},\label{assumption}
\end{equation}
where $\omega$ is full-rank $N\times q\ $matrix with$\ q<N,$\ $\beta
(L)=\sum_{h=0}^{p-1}\beta_{h}L^{h}$,\ and $\beta_{h}$\ is a $N\times
q\ $matrix\ for $h=1,\ldots,p$.\newline The rationale underlying assumption
(\ref{assumption}) is that the unrestricted VAR foresees $N$ linearly
independent mechanisms by which past information is transmitted to the system.
However, since it is commonly believed that few common shocks generate most of
macroeconomic fluctuations, it is reasonable to assume that there is a reduced
number of channels through which variables are influenced by their past. In
other words, this is exactly what Equation (\ref{assumption}) states (see
\citealp{CKM2016} and \citealp{CG2019} for further details).\newline Notice that
Assumption in Equation (\ref{assumption}) is equivalent to postulating the
following structure for series $\mathbf{y}_{t}$:
\begin{equation}
\mathbf{y}_{t}=\beta(L)\mathbf{f}_{t-1}+\mbox{\boldmath $\varepsilon$}_{t}
,\label{MAI}
\end{equation}
where $\mathbf{f}_{t}=\omega^{\prime}\mathbf{y}_{t}$. \cite{Reinsel1983}
defines the $q$-dimensional series $\mathbf{f}_{t}=(f_{1,t},\ldots,f_{q,t})$
as the index variables and labels equation (\ref{MAI}) as the MAI
model.\newline An interesting property of the MAI is that the indexes
themselves have a VAR$(p)$ representation. Indeed, if we premultiply by
$\omega^{\prime}$ both sides of equation (\ref{MAI}) we get
\[
\mathbf{f}_{t}=\alpha(L)\mathbf{f}_{t-1}+\mbox{\boldmath $\epsilon$}_{t},
\]
where $\alpha(L)=\omega^{\prime}\beta(L)$ and $\mbox{\boldmath $\epsilon$}_{t}
=\omega^{\prime}\mbox{\boldmath $\varepsilon$}_{t}$. This feature is in sharp
contrast with reduced rank VAR models, where linear combinations of the
variables do not generally admit a finite order VAR representation, see
\cite{CHP2009}, and highlights the analogy between the role of the indexes in
the MAI and the factors in DFMs.\newline Recently, there has been a
renewed interest in the MAI, \cite{CKM2016} derived classical and Bayesian
estimation of large MAIs and applied this modelling for structural analysis,
\cite{CGH2017} proposed a multivariate realized volatility model that is
endowed with an index structure. \cite{CG2019} extended the model by allowing
for individual AR structures, and \cite{CCM2020} offered a MAI with stochastic
volatility and provide MCMC estimation.\newline We extend the traditional MAI
model allowing the variation in both the mean and variance equation, the
TVP-MAI-SV takes the form:
\begin{equation}
\mathbf{y}_{t}={\textstyle\sum\limits_{j=1}^{q}}
\mbox{\boldmath $\beta$}(L)_{j,t}f_{j,t-1}+\mbox{\boldmath $\varepsilon$}_{t}
,\hspace{0.5cm}\mbox{\boldmath $\varepsilon$}_{t}\sim \mathcal{N}(0,\mbox{H}_{t}
),\label{eq:TVIAAR}
\end{equation}
where $f_{j,t}=\sum_{k=1}^{N}\omega_{k,j}y_{k,t}$, and
$\mbox{\boldmath $\beta$}(L)_{j,t}$ is a polynomial $N-$vector of time varying
coefficients that evolve as random walks for $j=1,\ldots,q$. \ Notice that,
similarly in the literature on the TVP-FAVAR, we assume that the loadings
of the indexes vary over time whereas the weights $\omega$\ remain stable.
Finally the $\mbox{\boldmath $\varepsilon$}_{t}$ follows a multivariate
stochastic volatility model given by $\mbox{H}_{t}$.\newline The model given
in Equation \eqref{eq:TVIAAR} is difficult to estimate with already existing
methods, and to tackle this task we develop a new hybrid algorithm that is
described below.
\section{Estimation \label{sec:BayesEstimation}}
The estimation of model in Equation \eqref{eq:TVIAAR} is based in a fast
two-step algorithm which vastly reduces the computational burden. Subsection
\ref{subsec:Bayes} presents the state space representation of TVP-MAI-SV and
briefly explains the related estimation issues. Subsection \ref{subsec:Hybrid}
presents the new hybrid switching algorithm used to estimate the TVP-MAI-SV.
Subsection \ref{subsec:Predictive} describes model selection. All the
derivations are reported in Appendix A.
\subsection{Bayesian Estimation \label{subsec:Bayes}}
The model in Equation \eqref{eq:TVIAAR} can be casted in state space form as follows:
\begin{equation}
\begin{split}
\mathbf{y}_{t} & =\mathbf{Z}_{t}\mbox{\boldmath $\beta$}_{t}
+\mbox{\boldmath $\varepsilon$}_{t},\hspace{0.5cm}
\mbox{\boldmath $\varepsilon$}_{t}\sim \mathcal{N}(0,\mathbf{H}_{t}),\\
\mbox{\boldmath $\beta$}_{t} & =\mbox{\boldmath $\beta$}_{t-1}
+\mbox{\boldmath $\eta$}_{t},\hspace{0.5cm}\mbox{\boldmath $\eta$}_{t}
\sim \mathcal{N}(0,\mbox{Q}),
\end{split}
\label{eq:ModelloSSTV}
\end{equation}
where $\mathbf{y}_{t}$ is the vector of observed time series at time $t$,
$\mathbf{Z}_{t}=I_N \otimes (f_{1,t},\ldots, f_{q,t})$ is the stack of
all the indexes depending on the unknown $\mbox{\boldmath $\omega$}$,
$\mbox{\boldmath $\beta$}_{t}=(\beta'_{1,t},\ldots,\beta'_{q,t})'$
is an $Nq\times1$ vector containing the time varying $\mbox{\boldmath $\beta$}$s
(states), which are assumed to follow a random walk dynamic. Finally the
errors, $\mbox{\boldmath $\varepsilon$}_{t}$ and $\mbox{\boldmath $\eta$}_{t}$
are assumed to be
mutually independent at all leads and lags and $\mathbf{H}_{t}$ features
stochastic volatility.\\
The model in equation
\eqref{eq:ModelloSSTV} is used in a number of recent paper, see among others
\cite{Primiceri2005}, \cite{KoopLeonStrachan2009} and \cite{KK2012}.
Traditionally, it is estimated with both classical and Bayesian approaches. In
the first case, the likelihood is efficiently calculated with the Kalman
filter (KF) routine, see \cite{DurbinKoopman2001}, and the time-varying
parameters are automatically filtered as latent state variables, once that
$\mbox{H}_{t}$ and $\mbox{Q}$ are estimated.\newline The Bayesian estimation
method, which requires simulation based methods, such as the MCMC, involves
the specification of $\mbox{H}_{t}$ and $\mbox{Q}$ together with the initial
condition, $\mbox{\boldmath $\beta$}_{0\mid0}$, of the model parameters,
for an introduction see \cite{Koop2003}. Although Bayesian algorithms are reliable
in this context, as recently discussed in \cite{CCM2018}, they become
computational intensive as the number of parameters increases and they become
unfeasible when a large amount of models (e.g. different number of factors)
have to be estimated.\newline To solve this issue we propose a new hybrid
estimation technique, that uses the discount factor methodology proposed by
\cite{RKE2010} and \cite{KK2012} (see Appendix A), to estimate
$\mbox{\boldmath $\beta$}_{t}$ and $\mbox{H}_{t}$, and a switching algorithm
to estimate $\mbox{\boldmath $\omega$}$.
\subsection{Hybrid algorithm for TVP-MAI-SV \label{subsec:Hybrid}}
The model in equation \eqref{eq:ModelloSSTV}, has both static
($\mbox{\boldmath $\omega$}$) and dynamic parameters
($\mbox{\boldmath $\beta$}_{t}$). Following \cite{CGH2017} and \cite{KK2013b}
we combines the ideas of variance discounting methods with the switching
algorithm in order to obtain analytical results for the posteriors of the
``indexes" ($\mbox{\boldmath $\beta$}_{t}$) as well as the static parameters
($\mbox{\boldmath $\omega$}$). The model also features stochastic volatility
and the algorithm needs to take it into account in the $\mathbf{H}_{t}$. The
$\mathbf{H}_{t}$ can be easily estimated using the EWMA filter.\newline The
estimation starts from the same algorithm described in \cite{CGH2017} and
introduces the time-varying parameters as follows:
\begin{enumerate}
\item[1)] Given (initial) estimates of $\mbox{\boldmath $\omega$}$ and
$\mathbf{H}_{t}$, run the KF for model reported in equation \eqref{eq:ModelloSSTV}
to get the optimal estimates of the latent states $\hat
{\mbox{\boldmath $\beta$}}_{t}=(\hat{\beta}_{1,t},\ldots,\hat{\beta}_{Nq,t})$,
see Appendix A;
\item[2)] Given the $\hat{\mbox{\boldmath $\beta$}}_{t}$ and the
$\mbox{\boldmath $\omega$}$, extract $\mathbf{H}_{t}$ using an EWMA estimator
for the measurement error covariance matrix:
\[
\hat{\mathbf{H}}_{t} = \kappa\hat{\mathbf{H}}_{t-1} + (1-\kappa)\hat
{\mbox{\boldmath $\varepsilon$}}_{t}\hat{\mbox{\boldmath $\varepsilon$}}
_{t}^{^{\prime}},
\]
where $\hat{\mbox{\boldmath $\varepsilon$}}_{t} = \mathbf{y}_{t} -
\mbox{\boldmath $\beta$}_{t}\mathbf{Z}_{t}$ is given as output from the Kalman
filter. The EWMA decay factor $\kappa$ requires to be selected, we discuss
this issue in Section \ref{sec:EmpApplication}.
\item[3)] Premultiply by $\hat{\mathbf{H}}_{t}^{-1/2}$ and apply the
$\mathrm{Vec}$ operator to both the sides of Equation (\ref{eq:TVIAAR}), then
use the property $\mathrm{Vec}(\mbox{A} \mbox{B} \mbox{C})=(\mbox{C}^{\prime
}\otimes\mbox{A})\mathrm{Vec}(\mbox{B})$ to get:
\begin{equation}
\mathrm{Vec}(\hat{\mathbf{H}}_{t}^{-1/2}\mathbf{y}_{t})={\textstyle\sum
\limits_{h=1}^{p-1}}(\mathbf{y}_{t-h}^{\prime}\otimes\hat{\mathbf{H}}_{t}
^{-1/2}\hat{\beta}_{h,t})\mathrm{Vec}(\mbox{\boldmath $\omega$}^{\prime
})+\mathrm{Vec}(\hat{\mathbf{H}} _{t}^{-1/2}\mbox{\boldmath $\varepsilon$}_{t}
). \label{eq:Vec-model}
\end{equation}
Given the previously obtained estimates of $\mbox{\boldmath $\hat{\beta}$}_{t}
$ and $\hat{\mathbf{H}}_{t}$, estimate $\mathrm{Vec}
(\mbox{\boldmath $\omega$}^{\prime})$\ with OLS in equation
(\ref{eq:Vec-model}).
\item[4)] Repeat steps 1, 2 and 3 till numerical convergence occurs.
\end{enumerate}
Few comments are in order. The above algorithm offers several
advantages over the available alternatives, including computational
simplicity, no need of a normalization condition on the parameters
$\mbox{\boldmath $\omega$}$, over-identifying restrictions can be easily
imposed on $\mbox{\boldmath $\omega$}$ and optimization is explicit at each
step. In order to speed up numerical convergence, it is important to take
proper choices regarding various hyperparameters and initial conditions. As in
\cite{KK2013b}, we choose fairly non-informative priors. The initial
conditions for the time-varying parameters $\mbox{\boldmath $\beta$}_{t}$ and
the time-varying covariance $\mathbf{H}_{t}$ are set as follows:
$\mbox{\boldmath $\beta$}_{0}\sim\mbox{N}(0,4)$ and $\mathbf{H}_{0}
=\mbox{I}_{n}$.
Finally the initial conditions for $\mbox{\boldmath $\omega$}$ are obtained
from the eigenvectors that are associated with the first $q$ principal
components of series $\mathbf{y}_{t}$.
\subsection{Dynamic model averaging and Dynamic model selection for the TVP-MAI-SV \label{subsec:Predictive}}
The model discussed in Section \ref{subsec:Bayes} is just one of the
$\mathcal{M}$ possible models. Depending on the selection of several parameters (number of
factors, forgetting factor, decay factor $\lambda$, time varying parameters
$\mbox{\boldmath $\beta$}$), many models can be switched over time. Considering the
$\mathcal{M}$ possible models equation \eqref{eq:ModelloSSTV} can be written as follows:
\begin{equation}
\begin{split}
\mathbf{y}_{t} & = \mathbf{Z}_{t}^{(k)}\mbox{\boldmath $\beta$}_{t}^{(k)} +
\mbox{\boldmath $\mbox{\boldmath $\varepsilon$}$}_{t}^{(k)}, \hspace{0.5cm }
\mbox{\boldmath $\mbox{\boldmath $\varepsilon$}$}_{t}^{(k)}\sim \mathcal{N}(0, \mathbf{H}_{t}^{(k)}),\\
\mbox{\boldmath $\beta$}_{t}^{(k)} & = \mbox{\boldmath $\beta$}_{t-1}^{(k)}
+ \mbox{\boldmath $\eta$}_{t}^{(k)}, \hspace{1.1cm}\mbox{\boldmath $\eta$}_{t}^{(k)}
\sim \mathcal{N}(0, \mbox{Q}^{(k)}),
\end{split}
\label{eq:PDModelloSSTV}
\end{equation}
where $\mathbf{Z}_{t}^{(k)}$ and $\mbox{\boldmath $\beta$}_{t}^{(k)}$ are, respectively, one of the possible set of indexes and
time-varying parameters.\newline Looking at equation \eqref{eq:PDModelloSSTV} is clear that
there are potentially a large number of models at each time point $t$. When
faced with multiple models, it is common to use model selection or model
averaging techniques, that in our framework have to be dynamic. More
specifically in a model selection exercise, we want to allow for the selected
model to change over time, thus doing Dynamic Model Selection (DMS).
In a model averaging exercise, we
want to allow for the weights used in the averaging process to change over
time, thus leading to Dynamic Model Averaging (DMA).
In this paper, we do both using the same approach
of \cite{RKE2010}, see Appendix A.\\
In DMS and DMA the main objective
is to calculate $\pi_{t|t-1,j}$ which is the probability that model $j$
applies at time $t$, given information through time $t-1$. Once $\pi
_{t|t-1,j}$ for $j= 1, \dots, J$ are obtained they can either be used to do
model averaging or model selection. \newline DMS arises if, at each point in
time, the model with the highest value for $\pi_{t|t-1,j}$ is used. Note that
$\pi_{t|t-1,j}$ will vary over time and, hence, the selected model may switch
over time. DMA arises if model averaging is done in period $t$ using
$\pi_{t|t-1,j}$ for $j=1, \dots, J$ as weights. The contribution of
\cite{RKE2010} is to develop a fast recursive algorithm for calculating
$\pi_{t|t-1,j}$ that, given an initial condition: $\pi_{0|0,j}$ for $j=1,
\ldots, J$, derives a model prediction equation using the forgetting factor
$\alpha$:
\[
\pi_{t|t-1,j} = \dfrac{\pi_{t-1|t-1,j}^{\alpha}}{\sum_{j=1}^{J}\pi
_{t-1|t-1,j}^{\alpha}},
\]
and a model updating equation of
\[
\pi_{t|t,j} = \dfrac{\pi_{t|t-1,j}f_{j}(\mbox{y}_{t}|\mbox{y}_{1:t-1})}
{\sum_{j=1}^{J}\pi_{t|t-1,j}f_{j}(\mbox{y}_{t}|\mbox{y}_{1:t-1})},
\]
where $f_{j}(\mbox{y}_{t}|\mbox{y}_{1:t-1})$ is the predictive likelihood of
model $j$. The $0<\alpha\leq1$ is a forgetting factor that tunes the frequency
of switch between models occurred over time. Low values of $\alpha$
corresponds to a rapid switch, high values give the opposite. Naturally $\alpha=
1$ brings to the conventional Bayesian Model Averaging (BMA).\newline Finally,
the initial condition is set to the equal probability $\pi_{0|0,j}=1/J,
\hspace{0.1cm} \forall j$.
\section{Empirical application \label{sec:EmpApplication}}
We use the new TVP-MAI-SV to carry out both a structural and a forecasting
exercise.\newline Subsection \ref{subsec:DataDescription} presents the dataset
considered in our study. Structural Analysis is described in subsection
\ref{subsec:VarDec}. While, subsection \ref{subsec:Forecast} discusses the
forecasting exercise.
\newpage
\subsection{Data description \label{subsec:DataDescription}}
The quarterly data used in the paper are download from Fred-Database and they run from 1960:1 to 2019:4. Following \cite{CKM2016}, \cite{GanBreu2010} and \cite{Gugl2016} and references therein, we classify the series into 5 groups: Real (RI),
Nominal (NI), Labour Market (LMI), Prices (PI) and Financial (FI) indexes.
A detailed description of all the 215 series as listed in each group are
reported from Table \ref{t:fc1} to Table \ref{t:fc6} in Appendix B. All the
variables are transformed to achieve stationarity and then
standardized as specified in \cite{MCNg2020}.
\newline For the structural exercise we use all the 215 series,
for the forecasting exercise we follow KK and we use a selection of 25 series. Those are
highlighted in bold in the Tables \ref{t:fc1} to \ref{t:fc6} of Appendix B.
\subsection{Variance Decomposition \label{subsec:VarDec}}
Following \cite{CCM2020} we carry out a variance
decomposition analysis using the full dataset
described in Appendix B. Before using the TVP-MAI-SV we proceed
to select some of its key features such as: the number of indexes ranging from 1 to 5; different values for $\lambda = \{0.96, 0.97, \ldots, 1\}$; and $\kappa = \{0.94, 0.96, \ldots, 1\}$. This provides a total of 80 alternative specifications.
Regarding the lag length we select 4 lags, see \cite{KK2013} and \cite{KMV2019b}. We estimate all these specifications
and rank them according to the $\log$ PL, as discussed in \cite{BCRvD2016} and
computed as discussed in Appendix A.\\
Table \ref{tab:modelsel} provides results for the best 5 specifications together with the worst 5 specifications we found over the total 80 specifications we searched over. The table
contains the number of factors and the values of $\lambda$ and $\kappa$ that
uniquely identifies a specification. Looking at the table the best specification
contains 5 indexes, $\lambda = 0.99$ and a $\kappa = 0.94$. This parameter combination
features a $\log$ PL equal to 488.1631.
\begin{table}[h]
\caption{The Table reports: the model
ranking (Ranking); the number of indexes (\#Factors); the shrinkage parameter for the
time-varying parameters ($\lambda$); the decay factor for the EWMA estimator ($\kappa$); and
the log-predictive likelihood for each model ($\log$ PL). The
second, third and forth columns contain the factor,$\lambda$-$\kappa$ combination that
uniquely identifies a specification.}
\label{tab:modelsel}
\centering{
\begin{tabular}[c]{c|c|c|c|r}
\toprule
Ranking & \# Factors & $\lambda$ & $\kappa$ & $\log$ PL\\
\midrule
1 & 5 & 0.99 & 0.94 & 488.1631 \\
2 & 4 & 1.00 & 0.94 & 487.7487 \\
3 & 2 & 0.97 & 0.94 & 481.1135 \\
4 & 3 & 1.00 & 0.94 & 478.4866 \\
5 & 2 & 0.99 & 0.94 & 478.8624 \\
\dots & \dots & \dots & \dots & \dots \\
76 & 4 & 0.97 & 1.00 & -242.8570 \\
77 & 2 & 0.97 & 1.00 & -243.4300 \\
78 & 3 & 0.96 & 1.00 & -246.4240 \\
79 & 4 & 0.96 & 1.00 & -250.2354 \\
80 & 5 & 0.96 & 1.00 & -258.7879 \\
\midrule
\end{tabular}}
\end{table}
\begin{figure}[h]
\caption{Estimated Indexes. NBER recession are reported with grey
vertical bar. The name of the factor is also reported.}
\label{fig:Factors}
\vspace{-0.2cm}
\hspace{-0.5cm}
\includegraphics[width=16.5cm,height=10cm]{Fattori.png}
\end{figure}
Figure \ref{fig:Factors} reports the five estimated indexes.\\
Generally speaking, indexes capture accurately both the Oil crisis in 1970s and the Great Financial Crisis of 2008. In more detail, it is evident the effect of the first oil Crisis in 1973 in which prices increased 400\%, is well captured by the pick of the related Price Index (PI) at the beginning of the recession period, followed by a decreasing effect in Real Index (RI), Nominal Index (NI) and above all Labour Market Index (LMI).
Differently, at the beginning of 1979, the second shock evolved more slowly, as producers, led by the Organization of Petroleum Exporting Countries (OPEC), affirmed the concept of setting oil prices and establishing production quotas.
Consequently, if we see an increase in the prices factor, the decreasing effect on the other components arrives later on in 80s. It is interesting that the financial crisis is well displayed by all the factors. Indeed, we appreciate an increase in the financial market associated with a large decrease of both real, nominal factors and an increase in prices. While we notice a slower decrease in the labour trend whose fully recovery seems to come long after.\\
The RI has a large volatility spike after the 1970s, observed in correspondence with the last Financial Crisis. The NI shows a smaller degree of time variation than the RI. The FI shows that financial
uncertainty spikes during the recession with the Great Financial Crisis that
dominates the others.
\begin{figure}[!h]
\vspace{-3cm} \centering
\subfloat[Real Gross Domestical Product]{\includegraphics[width=0.48\textwidth]{Volatility_Real.png}}\quad
\subfloat[TB-3M]{\includegraphics[width=0.48\textwidth]{Volatility_Nominal.png}}\newline
\subfloat[Nonfarm Payrolex]{\includegraphics[width=0.48\textwidth]{Volatility_Labour.png}}\quad
\subfloat[CPI Inflation]{\includegraphics[width=0.48\textwidth]{Volatility_Prices.png}}\newline
\subfloat[FedFunds]{\includegraphics[width=0.48\textwidth]{Volatility_Financial.png}}\quad
\subfloat[Fanchart of all the 217 serie.]{\includegraphics[width=0.48\textwidth]{FanChartPaperRev2.png}}\newline
\caption{Volatility shares (\%): Common (red), Idiosyncratic (blue).
The NBER recessions are reported with grey vertical lines. Each plot correspond
to the first series reported in Tables \ref{t:fc1} to Table \ref{t:fc6} in
Appendix B. (a) Real Gross Domestic Product, first series in Table \ref{t:fc1}
Appendix B. (b) 3-Month Treasury Bill first series in Table \ref{t:fc3}
Appendix B. (c) Nonfarm payroll first series in Table \ref{t:fc4} Appendix B.
(d) Consumer price index first series in Table \ref{t:fc5} Appendix B. (e)
Federal Funds rate first series in Table \ref{t:fc6} Appendix B. (f) Fanchart
of all the 217 series, with the mean as black(dotted) line.}
\label{fig:VarianceDecomposition}
\end{figure}
To analyse the explained volatility of each component we
report in Figure \ref{fig:VarianceDecomposition} the time-varying percentage
shares of explain volatility by the idiosyncratic and common components. The
Figure reports just the first series of Tables \ref{t:fc1} to \ref{t:fc5} in
Appendix B.\newline Starting from the RI and the corresponding series GDP, it
seems that the common component of volatility has increased its importance
during 2000 with a big increase after the Great Recession. The fraction of
volatility explained by the common component is much higher for the Nonfarm
payroll, first series of LMI.\\
Moving to the FI and for the first series
that is the Federal funds rate, the common component is quite high during the
considered period. Interesting the percentage of common component change a lot
with big increase in correspondence of the economic crisis.
\subsection{Forecasting exercise \label{subsec:Forecast}}
This section provides the out-of-sample performance of the TVP-MAI-SV against a set of standard
competitors.\newline In this exercise we consider the 25 major quarterly U.S.
macroeconomic variables as discussed in KK. The series are reported in bold
from Table \ref{t:fc1} to Table \ref{t:fc5} in Appendix B. We focus on
empirical results relating to three variables: CPI inflation, GDP growth and
the Federal Funds rate and refer to these as the main variables.\newline The
forecasting exercise is performed using an expanding window with an initial
estimation sample runs from 1960:Q3 to 1972:Q4. The model is then recursively
estimated on a forecast windows of 183 quarterly vintages (forecasting windows
start from 1973:Q1 through 2020:Q1).\\
Imposing simple restriction, the
TVP-MAI-SV of equation \eqref{eq:ModelloSSTV}, encompasses some multivariate models:
\begin{itemize}
\item[1)] The original MAI model, as in \cite{Reinsel1983} and \cite{CKM2016}
when $\mbox{\boldmath $\beta$}_{t} = \mbox{\boldmath $\beta$}_{t-1}$ and
$\mathbf{H}_{t}$ are time invariant ($\mbox{Q} = 0$, $\kappa= 1$ and
$\mathbf{H}_{t}$ set to the OLS estimates $\forall t$).
\item[2)] The MAI-SV similar to \cite{CCM2018} when
$\mbox{\boldmath $\beta$}_{t} = \mbox{\boldmath $\beta$}_{t-1}$ is time
invariant ($\mbox{Q}=0$) but $\kappa= 1$ and $\mathbf{H}_{t}$ evolves over time.
\item[3)] The TVP-MAI model without stochastic volatility when
$\mbox{\boldmath $\beta$}_{t}$ is time varying ($\mbox{Q}\neq0$) but
$\mathbf{H}_{t}$ is fixed and set to the OLS estimates.
\end{itemize}
In addition to the models discussed previously, Table
\ref{tab:Multivariatemodels} reports all the competitive models considered in
our forecasting analyses.\newline\begin{table}[h]
\caption{The table reports all the models considered in the forecasting
exercise plus the benchmark model. The first column is the abbreviation of the
model. The second column provides a brief description of each model.}
\label{tab:Multivariatemodels}
{\tiny \setlength\tabcolsep{3.0pt}{}
\begin{tabular}
[c]{l|l}
\toprule \textit{Abbreviation} & \textit{Full Description}\\
\midrule $\mathcal{M}_{1}$ & TVP-MAI-SV. Number of indexes and the optimal
value of the $\lambda$ and $\kappa$\\
& are selected using DMA as outlined in \cite{KK2013}. In this model $\alpha=
0.99$.\\
\midrule $\mathcal{M}_{2}$ & TVP-MAI-SV. Number of indexes and the optimal
value of the $\lambda$ and $\kappa$\\
& are selected using DMS as outlined in \cite{KK2013}. In this model $\alpha=
0.99$.\\
\midrule $\mathcal{M}_{3}$ & MAI-SV, with fix $\beta_{t}$($\lambda= 1$).
Number of indexes and the optimal value of $\kappa$\\
& are selected using DMA as outlined in \cite{KK2013}. In this model $\alpha=
0.99$.\\
\midrule $\mathcal{M}_{4}$ & MAI-SV, with fix $\beta_{t}$($\lambda= 1$).
Number of indexes and the optimal value of $\kappa$\\
& are selected using DMS as outlined in \cite{KK2013}. In this model $\alpha=
0.99$.\\
\midrule $\mathcal{M}_{5}$ & TVP-MAI, with fix $\mbox{H}$ ($\kappa= 1$).
Number of indexes and the optimal value of the $\lambda$\\
& are selected using DMA as outlined in \cite{KK2013}. In this model $\alpha=
0.99$.\\
\midrule $\mathcal{M}_{6}$ & TVP-MAI, with fix $\mbox{H}$ ($\kappa= 1$).
Number of indexes and the optimal value of the $\lambda$\\
& are selected using DMS as outlined in \cite{KK2013}. In this model $\alpha=
0.99$.\\
\midrule $\mathcal{M}_{7}$ & MAI. Number of indexes are selected using DMA as
outlined in \cite{KK2013}. In this model $\alpha= 0.99$.\\
\midrule $\mathcal{M}_{8}$ & MAI. Number of indexes are selected using DMS as
outlined in \cite{KK2013}. In this model $\alpha= 0.99$.\\
\midrule $\mathcal{M}_{9}$ & Random Walk.\\
\midrule $\mathcal{M}_{10}$ & Vector Autoregressive(1) estimated using the
OLS.\\
\midrule $\mathcal{M}_{11}$ & Vector Autoregressive(4) estimated using the
OLS.\\
\midrule $\mathcal{M}_{12}$ & Dynamic Factor Model\\
\midrule $\mathcal{M}_{13}$ & TVP-VAR-SV with 4 lags and stochastic
volatility. Optimal value of the shrinkage parameter is selected\\
& using DMS as outlined in \cite{KK2013}. In this model $\lambda= 0.99$,
$\kappa= 0.98$ and $\alpha= 0.99$. \textbf{Benchmark model}\\
\midrule $\mathcal{M}_{14}$ & TVP-VAR-SV with 4 lags and stochastic
volatility. Optimal value of the shrinkage parameter is selected\\
& using DMS as outlined in \cite{KK2013}. In this model $\lambda$ is
dynamically selected\\
& and $\kappa= 0.98$ and $\alpha= 0.99$.\\
\midrule $\mathcal{M}_{15}$ & TVP-FAVAR-SV with 4 lags and 4 indexes as in
\cite{KK2013b}.\\
\bottomrule
\end{tabular}
}\end{table}The benchmark model is a TVP-VAR with four lags ($\mathcal{M}
_{15}$), following KK, the optimal Minnesota shrinkage coefficient ($\gamma$)
is set to 0.005. We also include VAR with 1 and 4 lags estimated using OLS, a
DFM and a random walk process(RW).\\
The TVP-MAI-SV require to specify
the number of indexes, and the values of $\lambda$, $\kappa$ and $\alpha$. We
consider the $q = \{1, 2, 3\}$ indexes and a range of values for the forgetting
factor, $\lambda\in\{0.97, 0.98, 0.99, 1\}$ covering from rapid coefficient
change to no change. For the decay factor, we consider the grid of values
$\kappa\in\{0.96, 0.98, 1\}$. The number of indexes and the values of
$\lambda$, and $\kappa$ are selected dynamically using DMS or DMA, see Table
\ref{tab:Multivariatemodels}. We fix the $\alpha$ to 0.99, other values are
possible and the results are available from the authors upon request.\newline We
assess the performance of our forecasts accuracy in term of point and density
forecast following the evaluation framework of \cite{CEHK2020}. Let $y_{t}
^{*}$ denote the random variables being forecast and $\hat{y}_{t}$ be their
realizations and the root mean squared forecast error (RMSFE) and mean
absolute forecast error (MAFE) given by:
\begin{equation}
\begin{split}
\mbox{RMSFE}_{j, h}^{k} & = \sqrt{\frac{\sum_{t=t_{0}}^{T-h}y_{t+h}^{*}
-\hat{y}_{j, t+h}^{k}}{T - h - t_{0} + 1}},\\
\mbox{MAFE}_{j, h}^{k} & = \frac{\sum_{t=t_{0}}^{T-h} \left| y_{t+h}^{*} -
\hat{y}_{j, t+h}^{k}\right| }{T - h - t_{0} + 1}.
\end{split}
\end{equation}
where $k = \{\mathcal{M}_{1}, \ldots, \mathcal{M}_{15}\}$ is the model set, $h =
\{1, \ldots, H\}$ are the forecasting step ahead and $j = \{1, \ldots, 3\}$
are the main variables.\newline To evaluate the density forecasts, we use
the average log-predictive likelihood (ALPL) as described in \cite{Korobilis2019}
and \cite{CEHK2020} as a broadest measure of density accuracy, see also
\cite{Geweke2005}:
\begin{equation}
\mbox{ALPL}_{j, h}^{k} = \frac{\log p_{t+h} (y_{t+h}^{*} - \hat{y}_{j,
t+h}^{k}) }{T - h - t_{0} + 1}.\\
\end{equation}
Table \ref{tab:GDP} to Table \ref{tab:FedFunds} report the ratios of each
model's RMSFE and MAFE with respect to the benchmark model. Entries smaller
(bigger) than 1 indicate that the given model yields forecasts that are more
(less) accurate than those from the baseline. The Tables also report the ALPL
relative to the benchmark model. Values of ALPL higher (lower) than 1 signify
better (worse) performance than the benchmark.
\begin{sidewaystable}
\caption{Point and density forecast results for GDP. Root Mean Squared Forecast Error (RMSFE) upper panel, Median Absolute Forecast Error (MAFE),
middle panel, Average Log Predictive Likelihood (ALPL) bottom panel. Results are reported relative to the benchmark specification
($M_{13}$) for which the values is equal to 1, RMSFE-MAFE lower (higher) than 1 signify
better (worse) performance than the benchmark. ALPL higher (lower) then 1 signify better (worse) performance. The description of the model is
reported in Table \ref{tab:Multivariatemodels}. $---$ indicates divergence of the forecast.}
\label{tab:GDP}
\centering
{\scriptsize {
\begin{tabular}
[c]{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
\toprule
\multicolumn{15}{c}{\textit{Mean Square Forecast Error (MSFE)}}\\
\midrule
$H$ & $\mathcal{M}_{1}$ & $\mathcal{M}_{2}$ & $\mathcal{M}_{3}$ & $\mathcal{M}_{4}$ & $\mathcal{M}_{5}$ & $\mathcal{M}_{6}$ & $\mathcal{M}_{7}$ & $\mathcal{M}_{8}$ & $\mathcal{M}_{9}$ &
$\mathcal{M}_{10}$ & $\mathcal{M}_{11}$ &
$\mathcal{M}_{12}$ & $\mathcal{M}_{13}$ & $\mathcal{M}_{14}$ & $\mathcal{M}_{15}$ \\
\midrule
1 & 0.5989 & 0.6178 & 0.5892 & 0.5890 & 0.7296 & 0.7154 & 0.6322 & 0.6429 & 0.8208 & 1.0254 & --- & 0.7875 & 1.0000 & 0.9621 & 0.8183 \\
2 & 0.8072 & 0.8249 & 0.8071 & 0.8144 & --- & --- & 0.7610 & 0.7687 & 1.0886 & 1.6143 & --- & 1.0306 & 1.0000 & 0.6719 & 0.8359 \\
3 & 0.8600 & 0.8550 & 0.8761 & 0.8743 & --- & --- & 0.9973 & 1.0003 & 1.4003 & --- & --- & 1.0625 & 1.0000 & 0.9011 & 1.2530 \\
4 & 0.7182 & 0.7177 & 0.7308 & 0.7319 & --- & --- & 0.7338 & 0.7429 & 1.3066 & --- & --- & 0.7727 & 1.0000 & 0.8035 & 0.7715 \\
5 & 1.0631 & 1.0512 & 1.0708 & 1.0689 & --- & --- & 1.1704 & 1.1830 & --- & --- & --- & 1.1073 & 1.0000 & 0.9719 & 1.1174 \\
6 & 1.0669 & 1.0605 & 1.0625 & 1.0613 & --- & --- & 1.1639 & 1.1810 & --- & --- & --- & 1.0810 & 1.0000 & 0.9716 & 1.0828 \\
7 & 1.0644 & 1.0581 & 1.0619 & 1.0610 & --- & --- & 1.0745 & 1.0781 & --- & --- & --- & 1.0817 & 1.0000 & 0.9738 & 1.0960 \\
8 & 1.0607 & 1.0566 & 1.0535 & 1.0535 & --- & --- & 1.0864 & 1.0955 & --- & --- & --- & 1.0806 & 1.0000 & 0.9716 & 1.1063 \\
\midrule
\multicolumn{15}{c}{\textit{Mean Absolute Forecast Error (MAFE)}}\\
\midrule
H & $\mathcal{M}_{1}$ & $\mathcal{M}_{2}$ & $\mathcal{M}_{3}$ & $\mathcal{M}_{4}$ & $\mathcal{M}_{5}$ & $\mathcal{M}_{6}$ & $\mathcal{M}_{7}$ & $\mathcal{M}_{8}$ & $\mathcal{M}_{9}$ &
$\mathcal{M}_{10}$ & $\mathcal{M}_{11}$ &
$\mathcal{M}_{12}$ & $\mathcal{M}_{13}$ & $\mathcal{M}_{14}$ & $\mathcal{M}_{15}$ \\
\midrule
1 & 0.7548 & 0.7619 & 0.7570 & 0.7587 & 0.7605 & 0.7516 & 0.7452 & 0.7558 & 0.9040 & 0.7966 & --- & 0.7948 & 1.0000 & 0.9898 & 0.8505 \\
2 & 0.8596 & 0.8707 & 0.8622 & 0.8648 & 0.9452 & 0.9485 & 0.8468 & 0.8514 & 0.9964 & 0.9609 & --- & 0.9345 & 1.0000 & 0.8203 & 0.9067 \\
3 & 0.8783 & 0.8795 & 0.8910 & 0.8901 & 1.1748 & 1.1708 & 0.9266 & 0.9296 & 1.1909 & 1.2577 & --- & 0.9486 & 1.0000 & 0.9363 & 0.9600 \\
4 & 0.8187 & 0.8199 & 0.8276 & 0.8283 & 1.3873 & 1.3912 & 0.8298 & 0.8321 & 1.1560 & 1.3600 & --- & 0.8291 & 1.0000 & 0.8865 & 0.8613 \\
5 & 1.0076 & 1.0024 & 1.0085 & 1.0075 & --- & --- & 1.0399 & 1.0425 & 1.4274 & --- & --- & 1.0126 & 1.0000 & 0.9654 & 1.0453 \\
6 & 1.0083 & 1.0051 & 1.0035 & 1.0035 & --- & --- & 1.0343 & 1.0363 & 1.4354 & --- & --- & 1.0063 & 1.0000 & 0.9629 & 1.0227 \\
7 & 1.0048 & 1.0012 & 1.0019 & 1.0015 & --- & --- & 1.0121 & 1.0130 & 1.5180 & --- & --- & 1.0245 & 1.0000 & 0.9640 & 1.0284 \\
8 & 0.9993 & 0.9957 & 0.9955 & 0.9953 & --- & --- & 0.9994 & 1.0042 & 1.5703 & --- & --- & 1.0126 & 1.0000 & 0.9589 & 1.0390 \\
\midrule
\multicolumn{15}{c}{\textit{Average Log Predictive Likelihood (ALPL)}}\\
\midrule
\multicolumn{2}{c}{} & H & $\mathcal{M}_{1}$ & $\mathcal{M}_{2}$ & $\mathcal{M}_{3}$ & $\mathcal{M}_{4}$ & $\mathcal{M}_{5}$ & $\mathcal{M}_{6}$ & $\mathcal{M}_{7}$ & $\mathcal{M}_{8}$ & $\mathcal{M}_{13}$ & $\mathcal{M}_{14}$ & $\mathcal{M}_{15}$ & \multicolumn{2}{c}{} \\
\midrule
\multicolumn{2}{c}{} & 1 & 1.7176 & 1.7277 & 1.6432 & 1.6455 & 1.3031 & 1.3031 & 1.3031 & 1.3031 & 1.0000 & 1.0046 & 1.4581 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 2 & 1.6248 & 1.6415 & 1.5459 & 1.5478 & 1.2565 & 1.2573 & 1.2575 & 1.2577 & 1.0000 & 0.9844 & 1.4621 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 3 & 1.5443 & 1.5535 & 1.4806 & 1.4803 & 1.2468 & 1.2477 & 1.2508 & 1.2515 & 1.0000 & 0.9807 & 1.4666 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 4 & 1.5332 & 1.5407 & 1.4712 & 1.4711 & 1.2321 & 1.2339 & 1.2368 & 1.2374 & 1.0000 & 0.9831 & 1.4566 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 5 & 1.5347 & 1.5430 & 1.4726 & 1.4725 & 1.2289 & 1.2306 & 1.2341 & 1.2349 & 1.0000 & 0.9821 & 1.4598 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 6 & 1.5264 & 1.5343 & 1.4634 & 1.4632 & 1.2206 & 1.2226 & 1.2253 & 1.2262 & 1.0000 & 0.9805 & 1.4576 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 7 & 1.5239 & 1.5319 & 1.4608 & 1.4606 & 1.2176 & 1.2197 & 1.2226 & 1.2235 & 1.0000 & 0.9798 & 1.4563 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 8 & 1.5143 & 1.5223 & 1.4514 & 1.4512 & 1.2087 & 1.2109 & 1.2139 & 1.2148 & 1.0000 & 0.9782 & 1.4474 & \multicolumn{2}{c}{} \\
\bottomrule
\end{tabular}
}}
\end{sidewaystable}
\begin{sidewaystable}
\caption{Point and density forecast results for CPI Inflation. Root Mean Squared Forecast Error (RMSFE) upper panel, Median Absolute Forecast Error (MAFE),
middle panel, Average Log Predictive Likelihood (ALPL) bottom panel. Results are reported relative to the benchmark specification
($M_{13}$) for which the values is equal to 1, RMSFE-MAFE lower (higher) than 1 signify
better (worse) performance than the benchmark. ALPL higher (lower) then 1 signify better (worse) performance. The description of the model is
reported in Table \ref{tab:Multivariatemodels}. $---$ indicates divergence of the forecast.}
\label{tab:CPI}
\centering
{\scriptsize {
\begin{tabular}
[c]{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
\toprule
\multicolumn{15}{c}{\textit{Mean Square Forecast Error (MSFE)}}\\
\midrule
$H$ & $\mathcal{M}_{1}$ & $\mathcal{M}_{2}$ & $\mathcal{M}_{3}$ & $\mathcal{M}_{4}$ & $\mathcal{M}_{5}$ & $\mathcal{M}_{6}$ & $\mathcal{M}_{7}$ & $\mathcal{M}_{8}$ & $\mathcal{M}_{9}$ &
$\mathcal{M}_{10}$ & $\mathcal{M}_{11}$ &
$\mathcal{M}_{12}$ & $\mathcal{M}_{13}$ & $\mathcal{M}_{14}$ & $\mathcal{M}_{15}$ \\
\midrule
1 & 1.0061 & 0.9992 & 1.0415 & 1.0408 & --- & --- & 1.3206 & 1.3265 & --- & 1.6301 & --- & 0.8885 & 1.0000 & 0.9971 & 1.0621 \\
2 & 0.8491 & 0.8644 & 0.8311 & 0.8330 & --- & --- & 0.9159 & 0.9197 & --- & 1.1349 & --- & 1.0564 & 1.0000 & 0.9398 & 0.7965 \\
3 & 0.8939 & 0.9081 & 0.8812 & 0.8801 & --- & --- & 0.9126 & 0.9143 & 1.4230 & 1.4269 & --- & 0.8852 & 1.0000 & 0.9248 & 0.9016 \\
4 & 0.9667 & 0.9712 & 0.9639 & 0.9648 & --- & --- & 1.0104 & 1.0107 & --- & 1.4161 & --- & 0.9749 & 1.0000 & 0.9557 & 0.9908 \\
5 & 1.0088 & 1.0114 & 1.0023 & 1.0034 & --- & --- & 0.9809 & 0.9770 & --- & --- & --- & 1.0586 & 1.0000 & 0.9986 & 1.0296 \\
6 & 0.9899 & 0.9923 & 0.9909 & 0.9908 & --- & --- & 1.0002 & 0.9998 & --- & --- & --- & 1.0180 & 1.0000 & 0.9995 & 1.0184 \\
7 & 0.9944 & 0.9941 & 0.9964 & 0.9965 & --- & --- & 1.0099 & 1.0094 & --- & --- & --- & 1.0131 & 1.0000 & 0.9991 & 1.0029 \\
8 & 1.0010 & 0.9999 & 0.9995 & 0.9996 & --- & --- & 1.0070 & 1.0066 & --- & --- & --- & 0.9936 & 1.0000 & 0.9989 & 1.0069 \\
\midrule
\multicolumn{15}{c}{\textit{Mean Absolute Forecast Error (MAFE)}}\\
\midrule
$H$ & $\mathcal{M}_{1}$ & $\mathcal{M}_{2}$ & $\mathcal{M}_{3}$ & $\mathcal{M}_{4}$ & $\mathcal{M}_{5}$ & $\mathcal{M}_{6}$ & $\mathcal{M}_{7}$ & $\mathcal{M}_{8}$ & $\mathcal{M}_{9}$ &
$\mathcal{M}_{10}$ & $\mathcal{M}_{11}$ &
$\mathcal{M}_{12}$ & $\mathcal{M}_{13}$ & $\mathcal{M}_{14}$ & $\mathcal{M}_{15}$ \\
\midrule
1 & 0.9731 & 0.9670 & 0.9942 & 0.9945 & 1.1331 & 1.1567 & 1.0499 & 1.0485 & 1.6029 & 1.0888 & --- & 0.9390 & 1.0000 & 0.9959 & 1.0039\\
2 & 0.9131 & 0.9233 & 0.9013 & 0.9027 & 1.1032 & 1.1091 & 0.9513 & 0.9564 & 1.4702 & 0.9717 & --- & 1.0166 & 1.0000 & 0.9722 & 0.8789\\
3 & 0.9201 & 0.9278 & 0.9128 & 0.9133 & 1.3179 & 1.3263 & 0.9349 & 0.9382 & 1.1703 & 1.0347 & --- & 0.9298 & 1.0000 & 0.9727 & 0.9459\\
4 & 0.9644 & 0.9675 & 0.9641 & 0.9642 & 1.8109 & 1.8308 & 1.0004 & 1.0019 & 1.4349 & 1.0776 & --- & 0.9780 & 1.0000 & 0.9781 & 0.9875\\
5 & 0.9985 & 0.9992 & 0.9939 & 0.9945 & --- & --- & 0.9813 & 0.9794 & 1.5745 & 1.2265 & --- & 1.0306 & 1.0000 & 0.9995 & 1.0213\\
6 & 0.9877 & 0.9899 & 0.9864 & 0.9864 & --- & --- & 0.9958 & 0.9954 & 1.3528 & 1.1740 & --- & 1.0076 & 1.0000 & 0.9969 & 1.0027\\
7 & 0.9886 & 0.9898 & 0.9903 & 0.9903 & --- & --- & 1.0076 & 1.0082 & 1.4098 & 1.5174 & --- & 1.0070 & 1.0000 & 0.9961 & 1.0055\\
8 & 0.9912 & 0.9904 & 0.9898 & 0.9898 & --- & --- & 0.9962 & 0.9965 & 1.6238 & 1.4867 & --- & 0.9883 & 1.0000 & 0.9937 & 0.9818\\
\midrule
\multicolumn{15}{c}{\textit{Average Log Predictive Likelihood (ALPL)}}\\
\midrule
\multicolumn{2}{c}{} & $H$ & $\mathcal{M}_{1}$ & $\mathcal{M}_{2}$ & $\mathcal{M}_{3}$ & $\mathcal{M}_{4}$ & $\mathcal{M}_{5}$ & $\mathcal{M}_{6}$ & $\mathcal{M}_{7}$ & $\mathcal{M}_{8}$ & $\mathcal{M}_{13}$ & $\mathcal{M}_{14}$ & $\mathcal{M}_{15}$ & \multicolumn{2}{c}{} \\
\midrule
\multicolumn{2}{c}{} & 1 & 1.2543 & 1.2477 & 1.2441 & 1.2442 & 1.2681 & 1.2681 & 1.2681 & 1.2681 & 1.0000 & 1.0304 & 1.3009 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 2 & 1.1629 & 1.1525 & 1.1585 & 1.1553 & 1.2024 & 1.2002 & 1.1874 & 1.1871 & 1.0000 & 1.0189 & 1.3135 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 3 & 1.1704 & 1.1599 & 1.1652 & 1.1620 & 1.1837 & 1.1825 & 1.1846 & 1.1845 & 1.0000 & 1.0211 & 1.3233 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 4 & 1.1614 & 1.1475 & 1.1590 & 1.1555 & 1.1676 & 1.1669 & 1.1696 & 1.1696 & 1.0000 & 1.0155 & 1.3208 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 5 & 1.1446 & 1.1332 & 1.1441 & 1.1408 & 1.1342 & 1.1327 & 1.1363 & 1.1364 & 1.0000 & 1.0121 & 1.3106 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 6 & 1.1508 & 1.1398 & 1.1525 & 1.1491 & 1.1391 & 1.1372 & 1.1434 & 1.1435 & 1.0000 & 1.0124 & 1.3228 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 7 & 1.1443 & 1.1338 & 1.1469 & 1.1435 & 1.1321 & 1.1302 & 1.1372 & 1.1373 & 1.0000 & 1.0122 & 1.3175 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 8 & 1.1395 & 1.1291 & 1.1427 & 1.1393 & 1.1260 & 1.1240 & 1.1316 & 1.1317 & 1.0000 & 1.0121 & 1.3130 & \multicolumn{2}{c}{} \\
\bottomrule
\end{tabular}
}}
\end{sidewaystable}
\begin{sidewaystable}
\caption{Point and density forecast results for Federal Funds Rate. Root Mean Squared Forecast Error (RMSFE) upper panel, Median Absolute Forecast Error (MAFE),
middle panel, Average Log Predictive Likelihood (ALPL) bottom panel. Results are reported relative to the benchmark specification
($M_{13}$) for which the values is equal to 1, RMSFE-MAFE lower (higher) than 1 signify
better (worse) performance than the benchmark. ALPL higher (lower) then 1 signify better (worse) performance. The description of the model is
reported in Table \ref{tab:Multivariatemodels}. $---$ indicates divergence of the forecast.}
\label{tab:FedFunds}
\centering
{\scriptsize {
\begin{tabular}
[c]{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
\toprule
\multicolumn{14}{c}{\textit{Mean Square Forecast Error (MSFE)}}\\
\midrule
$H$ & $\mathcal{M}_{1}$ & $\mathcal{M}_{2}$ & $\mathcal{M}_{3}$ & $\mathcal{M}_{4}$ & $\mathcal{M}_{5}$ & $\mathcal{M}_{6}$ & $\mathcal{M}_{7}$ & $\mathcal{M}_{8}$ & $\mathcal{M}_{9}$ &
$\mathcal{M}_{10}$ & $\mathcal{M}_{11}$ &
$\mathcal{M}_{12}$ & $\mathcal{M}_{13}$ & $\mathcal{M}_{14}$ & $\mathcal{M}_{15}$ \\
\midrule
1 & 1.0169 & 1.0492 & 0.9920 & 0.9975 & 0.9432 & 0.8942 & 1.2031 & 1.2534 & 1.6167 & 1.4448 & --- & --- & 1.0000 & 1.0257 & 1.0797 \\
2 & 1.0618 & 1.0669 & 1.0672 & 1.0686 & 1.5283 & 1.5521 & 1.2092 & 1.2049 & 2.2307 & 1.6105 & --- & 1.2776 & 1.0000 & 1.0070 & 1.1305 \\
3 & 0.9844 & 0.9845 & 0.9880 & 0.9865 & --- & --- & 1.2535 & 1.2648 & 1.8044 & 1.6272 & --- & 1.2062 & 1.0000 & 1.0036 & 1.1549 \\
4 & 1.0058 & 1.0084 & 1.0063 & 1.0103 & --- & --- & 1.0891 & 1.0907 & 1.7889 & --- & --- & 1.1566 & 1.0000 & 0.9999 & 1.2485 \\
5 & 0.9807 & 0.9824 & 0.9878 & 0.9873 & --- & --- & 1.0565 & 1.0628 & 1.6379 & --- & --- & 1.1301 & 1.0000 & 0.9958 & 1.0943 \\
6 & 0.9935 & 0.9957 & 0.9969 & 0.9967 & --- & --- & 1.0894 & 1.0908 & --- & --- & --- & 1.0798 & 1.0000 & 0.9901 & 1.1044 \\
7 & 1.0107 & 1.0082 & 1.0066 & 1.0059 & --- & --- & 1.0662 & 1.0675 & --- & --- & --- & 1.0645 & 1.0000 & 0.9906 & 0.9893 \\
8 & 1.0005 & 1.0009 & 0.9973 & 0.9993 & --- & --- & 1.0577 & 1.0610 & --- & --- & --- & 1.0392 & 1.0000 & 0.9927 & 1.0508 \\
\midrule
\multicolumn{14}{c}{\textit{Mean Absolute Forecast Error (MAFE)}}\\
\midrule
$H$ & $\mathcal{M}_{1}$ & $\mathcal{M}_{2}$ & $\mathcal{M}_{3}$ & $\mathcal{M}_{4}$ & $\mathcal{M}_{5}$ & $\mathcal{M}_{6}$ & $\mathcal{M}_{7}$ & $\mathcal{M}_{8}$ & $\mathcal{M}_{9}$ &
$\mathcal{M}_{10}$ & $\mathcal{M}_{11}$ &
$\mathcal{M}_{12}$ & $\mathcal{M}_{13}$ & $\mathcal{M}_{14}$ & $\mathcal{M}_{15}$ \\
\midrule
1 & 0.9849 & 1.0078 & 0.9626 & 0.9679 & 1.1551 & 1.1323 & 1.2447 & 1.2623 & 1.0871 & 1.2772 & --- & 1.6658 & 1.0000 & 1.0376 & 1.0137 \\
2 & 1.0021 & 1.0082 & 1.0000 & 1.0013 & 1.2102 & 1.2188 & 1.2064 & 1.2076 & 1.4070 & 1.3413 & --- & 1.2075 & 1.0000 & 1.0369 & 1.1817 \\
3 & 0.9946 & 0.9957 & 0.9893 & 0.9890 & 1.3053 & 1.3146 & 1.2017 & 1.2126 & 1.3457 & 1.2529 & --- & 1.1237 & 1.0000 & 1.0325 & 1.0834 \\
4 & 0.9819 & 0.9818 & 0.9783 & 0.9793 & 1.5724 & 1.5881 & 1.1179 & 1.1264 & 1.3241 & 1.3810 & --- & 1.0762 & 1.0000 & 1.0282 & 1.0904 \\
5 & 0.9721 & 0.9724 & 0.9659 & 0.9652 & 1.9735 & 1.9804 & 1.0762 & 1.0817 & 1.3743 & 1.3477 & --- & 1.0754 & 1.0000 & 1.0220 & 1.0249 \\
6 & 0.9789 & 0.9750 & 0.9715 & 0.9716 & --- & --- & 1.0784 & 1.0827 & 1.5579 & 1.5628 & --- & 1.0099 & 1.0000 & 1.0202 & 1.0444 \\
7 & 0.9795 & 0.9754 & 0.9718 & 0.9717 & --- & --- & 1.0756 & 1.0768 & 1.6735 & 1.6391 & --- & 1.0108 & 1.0000 & 1.0189 & 0.9944 \\
8 & 0.9715 & 0.9689 & 0.9672 & 0.9675 & --- & --- & 1.0400 & 1.0400 & 1.5535 & --- & --- & 0.9869 & 1.0000 & 1.0182 & 1.0296 \\
\toprule
\multicolumn{14}{c}{\textit{Average Log Predictive Likelihood (ALPL)}}\\
\midrule
\multicolumn{2}{c}{} & $H$ & $\mathcal{M}_{1}$ & $\mathcal{M}_{2}$ & $\mathcal{M}_{3}$ & $\mathcal{M}_{4}$ & $\mathcal{M}_{5}$ & $\mathcal{M}_{6}$ & $\mathcal{M}_{7}$ & $\mathcal{M}_{8}$ & $\mathcal{M}_{13}$ & $\mathcal{M}_{14}$ & $\mathcal{M}_{15}$ & \multicolumn{2}{c}{} \\
\midrule
\multicolumn{2}{c}{} & 1 & 1.4291 & 1.4162 & 1.3453 & 1.3500 & 0.9299 & 0.9409 & 0.9169 & 0.9176 & 1.0000 & 1.0412 & 1.0197 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 2 & 1.3384 & 1.3295 & 1.2789 & 1.2839 & 0.9266 & 0.9297 & 0.9159 & 0.9177 & 1.0000 & 1.0100 & 0.9865 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 3 & 1.2732 & 1.2668 & 1.2203 & 1.2235 & 0.9336 & 0.9341 & 0.9168 & 0.9171 & 1.0000 & 1.0055 & 1.0096 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 4 & 1.2434 & 1.2339 & 1.2038 & 1.2063 & 0.9162 & 0.9166 & 0.9153 & 0.9167 & 1.0000 & 1.0038 & 1.0127 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 5 & 1.2209 & 1.2116 & 1.1935 & 1.1965 & 0.9207 & 0.9209 & 0.9155 & 0.9162 & 1.0000 & 1.0027 & 1.0282 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 6 & 1.2157 & 1.2065 & 1.1896 & 1.1920 & 0.9109 & 0.9061 & 0.9145 & 0.9160 & 1.0000 & 1.0035 & 1.0208 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 7 & 1.2187 & 1.2072 & 1.1935 & 1.1961 & 0.9170 & 0.9151 & 0.9155 & 0.9167 & 1.0000 & 1.0037 & 1.0357 & \multicolumn{2}{c}{} \\
\multicolumn{2}{c}{} & 8 & 1.2271 & 1.2171 & 1.1989 & 1.2013 & 0.9221 & 0.9190 & 0.9235 & 0.9249 & 1.0000 & 1.0032 & 1.0415 & \multicolumn{2}{c}{} \\
\bottomrule
\end{tabular}
}}
\end{sidewaystable}
\noindent With three different variables, eight different forecast horizons
and two different forecast metrics, virtually every model can be found to do
well for some cases, but several observations can be made.\newline First,
TVP-MAI-SV is one of the best models for all the main variables, improving upon
its counterparts (MAI, TVP-MAI, MAI-SV) especially at the short horizon.\newline Second, an important
pattern emerges from the Tables. Adding SV to the MAI improve significantly
the point and density forecast performances, this finding is in line with
\cite{CCM2019}. On all Tables the $\mathcal{M}_{7}$ and $\mathcal{M}_{8}$
models show worse performance than models $\mathcal{M}_{1}$, $\mathcal{M}_{2}
$, $\mathcal{M}_{3}$ and $\mathcal{M}_{4}$. Moreover the $\mathcal{M}_{5}$ and
$\mathcal{M}_{6}$ show the poor performance of TVP-MAI homoskedastic
models.Tables \ref{tab:GDP}-\ref{tab:FedFunds} are a clear demonstration of
the importance of allowing for heteroskedastic errors to get good
RMSFE, MAFE and predictive likelihood.\newline One final point regards the
usefulness of adding time-varying parameters in the MAI-SV. The Tables show
that the two models have comparable point forecast performance, the TVP-MAI-SV
has always better predictive likelihood.\newline Overall, the results show that
the TVP-MAI-SV guarantees safe forecasting compared with the other
competitors such as the TVP-VAR-SV (see KK) and more similar models like the
TVP-FAVAR-SV as described in \cite{KK2013b}.
\section{Conclusions \label{sec:Concl}}
Many economic variables features changing mean and volatility, TVP-VAR with stochastic
volatility are commonly used to model those features. Starting from the recent MAI literature
the paper introduces the TVP-MAI-SV that can handle large datasets. The paper
introduces a new estimation methodology that substantially reduces the
computational burden, and allows to select in real time, the number of
indexes and other features of the data using DMS and DMA without further
computational cost.\\
The paper proposes two empirical applications. The first provides a measure of
uncertainty using the TVP-MAI-SV. We overcame endogeneity problems coming from the co-movement of macroeconomic variables by extracting a set of common unobservable factors representing the underlying aggregate
uncertainty affecting the levels of the component variables.
We get a rich set of volatility
dynamics whose path provides interesting results in terms of common and idiosyncratic volatility changes.\\
The new switching algorithm, reduces the computational burden, allowing to apply to large databases (in our application we consider 215 series). From an economic point of view
we were able to accurately capture both the Oil Crisis as well as the the Great Recession associated
with much larger uncertainty shocks but no major changes in their effects on the economy.\\
The second empirical application shows the out-of-sample forecasting performance of the TVP-MAI-SV. Using both point and density forecast, we found that the TVP-MAI-SV model has good forecasting performance compared to a set
of multivariate and univariate competitors.