The Time-Varying Multivariate Autoregressive Index Model

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abstractMany 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 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.\\ \\ Keywords: Large datasets, Multivariate Autoregressive Index models, Stochastic volatility, Bayesian VARs.

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, 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 CMS2005, CogleySargent(2005), Primiceri2005 and 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, 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 RKE2010) and estimating the error covariance matrix dynamically using an exponentially weighted moving average (EWMA). Recently 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 CKM2016, CGH2017, CG2019, and CCM2020, among others. The MAI model, originally introduced by Reinsel1983, is a bridge between reduced-rank VARs (see CCM2015 and the references therein) and the Dynamic Factor Model (DFM, see SW2016, 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 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 CCM2018 and CCM2019. To achieve this result we propose mixing the switching algorithm, see CGH2017, with forgetting factors in the same spirit of KK2013b.\\ Forgetting factors (also known as discount factors), have long been used with state-space models, see RKE2010. They do not require the use of MCMC methods and be useful in economic and financial applications, see DH2012, and 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 ScW1989 for firm-level stock asset prices in Campal01, for consumption and income in 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 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 BES2013, CCG2014 and Jo2015).\% while 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 Bloom2009, CCG2014, BB2017, BES2013, 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 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 CKM2016, GanBreu2010 and Gugl2016, are classified into 5 groups: 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 CE2018 and CEHK2020\\ The rest of the paper proceeds as follows. Section (ref) presents the MAI model and introduces the new TVP-MAI-SV. Section (ref) discusses the new estimation approach. Section (ref) contains the empirical application. Finally Section (ref) draws some conclusions. All the derivations are reported in Appendix A and B.

From the MAI model to the TV-MAI

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$(p)$):

equation[equation omitted — 131 chars of source]

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)), Reinsel1983 proposed to impose the following set of restrictions on the VAR mean parameters:

equation[equation omitted — 66 chars of source]

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)) 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)) states (see CKM2016 and CG2019 for further details).\newline Notice that Assumption in Equation ((ref)) is equivalent to postulating the following structure for series $\mathbf{y}_{t}$:

equation[equation omitted — 103 chars of source]

where $\mathbf{f}_{t}=\omega^{\prime}\mathbf{y}_{t}$. Reinsel1983 defines the $q$-dimensional series $\mathbf{f}_{t}=(f_{1,t},\ldots,f_{q,t})$ as the index variables and labels equation ((ref)) 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)) 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 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, CKM2016 derived classical and Bayesian estimation of large MAIs and applied this modelling for structural analysis, CGH2017 proposed a multivariate realized volatility model that is endowed with an index structure. CG2019 extended the model by allowing for individual AR structures, and 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:

equation[equation omitted — 243 chars of source]

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 (ref) is difficult to estimate with already existing methods, and to tackle this task we develop a new hybrid algorithm that is described below.

Estimation

The estimation of model in Equation (ref) is based in a fast two-step algorithm which vastly reduces the computational burden. Subsection (ref) presents the state space representation of TVP-MAI-SV and briefly explains the related estimation issues. Subsection (ref) presents the new hybrid switching algorithm used to estimate the TVP-MAI-SV. Subsection (ref) describes model selection. All the derivations are reported in Appendix A.

Bayesian Estimation

The model in Equation (ref) can be casted in state space form as follows:

equation[equation omitted — 414 chars of source]

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 (ref) is used in a number of recent paper, see among others Primiceri2005, KoopLeonStrachan2009 and 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 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 Koop2003. Although Bayesian algorithms are reliable in this context, as recently discussed in 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 RKE2010 and KK2012 (see Appendix A), to estimate $\mbox{\boldmath $\beta$}_{t}$ and $\mbox{H}_{t}$, and a switching algorithm to estimate $\mbox{\boldmath $\omega$}$.

Hybrid algorithm for TVP-MAI-SV

The model in equation (ref), has both static ($\mbox{\boldmath $\omega$}$) and dynamic parameters ($\mbox{\boldmath $\beta$}_{t}$). Following CGH2017 and KK2013b we combines the ideas of variance discounting methods with the switching algorithm in order to obtain analytical results for the posteriors of the “indexes" ($\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 CGH2017 and introduces the time-varying parameters as follows:

enumerate• Given (initial) estimates of $\mbox{\boldmath $\omega$}$ and $\mathbf{H}_{t}$, run the KF for model reported in equation (ref) 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; • 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). • Premultiply by $\hat{\mathbf{H}}_{t}^{-1/2}$ and apply the $\mathrm{Vec}$ operator to both the sides of Equation ((ref)), 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}(\boldmath $\omega$^{\prime })+\mathrm{Vec}(\hat{\mathbf{H}} _{t}^{-1/2}\boldmath $\varepsilon$_{t} ). \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)). • Repeat steps 1, 2 and 3 till numerical convergence occurs.

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 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 $\mathbf{y}_{t}$.

Dynamic model averaging and Dynamic model selection for the TVP-MAI-SV

The model discussed in Section (ref) 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 (ref) can be written as follows:

equation[equation omitted — 525 chars of source]

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 (ref) 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 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 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 $\pi_{0|0,j}=1/J, \hspace{0.1cm} \forall j$.

Empirical application

We use the new TVP-MAI-SV to carry out both a structural and a forecasting exercise.\newline Subsection (ref) presents the dataset considered in our study. Structural Analysis is described in subsection (ref). While, subsection (ref) discusses the forecasting exercise.

Data description

The quarterly data used in the paper are download from Fred-Database and they run from 1960:1 to 2019:4. Following CKM2016, GanBreu2010 and 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) to Table (ref) in Appendix B. All the variables are transformed to achieve stationarity and then standardized as specified in 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) to (ref) of Appendix B.

Variance Decomposition

Following 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 KK2013 and KMV2019b. We estimate all these specifications and rank them according to the $\log$ PL, as discussed in BCRvD2016 and computed as discussed in Appendix A.\\ Table (ref) 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.

table[table omitted — 994 chars of source]

figure[figure omitted — 246 chars of source]

Figure (ref) 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.

figure[figure omitted — 1,335 chars of source]

To analyse the explained volatility of each component we report in Figure (ref) the time-varying percentage shares of explain volatility by the idiosyncratic and common components. The Figure reports just the first series of Tables (ref) to (ref) 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.

Forecasting exercise

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) to Table (ref) 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 (ref), encompasses some multivariate models:

itemize• The original MAI model, as in Reinsel1983 and 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 $\forall t$). • The MAI-SV similar to 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. • 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.

In addition to the models discussed previously, Table (ref) reports all the competitive models considered in our forecasting analyses.\newline

table[table omitted — 2,851 chars of source]

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). 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 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:

equation[equation omitted — 290 chars of source]

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 Korobilis2019 and CEHK2020 as a broadest measure of density accuracy, see also Geweke2005:

equation[equation omitted — 120 chars of source]

Table (ref) to Table (ref) 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.

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). $---$ indicates divergence of the forecast.} {\scriptsize { \begin{tabular} [c]{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} \toprule \multicolumn{15}{c}{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}{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}{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} }}

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). $---$ indicates divergence of the forecast.} {\scriptsize { \begin{tabular} [c]{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} \toprule \multicolumn{15}{c}{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}{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}{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} }}

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). $---$ indicates divergence of the forecast.} {\scriptsize { \begin{tabular} [c]{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} \toprule \multicolumn{14}{c}{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}{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}{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} }}

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 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)-(ref) 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 KK2013b.

Conclusions

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.