Factor Augmented Vector Autoregressions, Panel VARs, and Global VARs

Abstract

This chapter provides a thorough introduction to panel, global, and factor augmented vector autoregressive models. These models are typically used to capture interactions across units (i.e., countries) and variable types. Since including a large number of countries and/or variables increases the dimension of the models, all three approaches aim to decrease the dimensionality of the parameter space. After introducing each model, we briefly discuss key specification issues. A running toy example serves to highlight this point and outlines key differences across the different models. To illustrate the merits of the competing approaches, we perform a forecasting exercise and show that it pays off to introduce cross-sectional information in terms of forecasting key macroeconomic quantities.

Appendix A: Details on Prior Specification

Priors for the BVAR and FAVAR Coefficients

The Minnesota prior pushes the system of equations towards a multivariate random walk, featuring cross-variable and cross-equation shrinkage. We follow a data-driven approach to select the amount of shrinkage applied, by imposing Gamma distributed priors on the hyperparameter governing how tight the prior is on the own lags of a variable, and the hyperparameter related to shrinkage of the lags of other variables in the system. Different ranges of the endogenous variables are reflected in the prior by scaling it based on standard deviations obtained from univariate autoregressive processes of order one for all series.

A similar Minnesota prior is used for estimating the state-equation of the country-specific FAVAR models. We consider extracting one factor (FAVAR-F1) and three factors (FAVAR-F3) for the forecasting exercise, respectively. For the factors and factor loadings we employ standard Gaussian priors and simulate the related quantities using the forward filtering backward sampling algorithm by Chris and Kohn (1994) and Frühwirth-Schnatter (1994). The errors in the measurement equation are assumed to have zero mean with a diagonal variance-covariance matrix. On the corresponding diagonal elements, we impose independent weakly informative inverse Gamma priors.

Variance Estimation for BVARs and FAVARs

Turning to modeling the variance-covariance matrix of the VAR processes that allows for heteroscedasticity, we employ analogous stochastic volatility specifications for the BVAR, BVAR-SC, FAVAR-F1, and FAVAR-F3. A complete treatment of stochastic volatility models is out of scope of this chapter, the interested reader is referred to Jacquier, Polson, and Rossi (2002). Let Σ _t be a generic variance-covariance matrix applicable to all specifications. This matrix may be decomposed into

$$\displaystyle \begin{aligned} \boldsymbol{\Sigma}_t = \boldsymbol{H}^{-1}\boldsymbol{S}_t(\boldsymbol{H}^{-1})', \end{aligned}$$

with H ⁻¹ denoting a square lower triangular matrix with ones on the main diagonal of appropriate dimension. Time variation stems from the elements of S _t, a diagonal matrix with characteristic elements s _it. A stochastic volatility specification results assuming the logarithm of s _it for all i follows an AR(1) process

$$\displaystyle \begin{aligned} \ln (s_{it}) = \mu_{i} + \rho_{i} \left(\ln (s_{it}) - \mu_{i}\right) + \nu_{it}, \quad \nu_{it} \sim \mathcal{N}(0,\varsigma^{2}_{i}),{} \end{aligned} $$

(3.8)

where μ _i, ρ _i, and ς ² denote the unconditional mean, persistence parameter, and innovation variance of this state equation. For the purposes of this forecasting exercise, we rely on the R-package stochvol for estimation and use its default prior settings (Kastner, 2016). For the free elements of H ⁻¹, weakly informative independent Gaussian priors with zero mean are employed. Combining the likelihood with the respective priors yields conditional posterior distributions to be used in a Gibbs sampler with most of the involved quantities being of standard form (for detailed information, see for instance Koop, 2003).

Priors for the PVAR and GVAR Coefficients

The priors and model specifics for the PVAR and the GVAR specifications are designed to account for dynamic interdependencies using a shrinkage priors and static interdependencies via factor stochastic volatility. Moreover, we introduce a prior to be used for extracting information across countries, reflecting cross-sectional homogeneity. For the domestic coefficients of the PVAR variants (PVAR and PVAR-DI) and for all coefficients of GVAR-CP, we stack the coefficients specific to country i in a column vector a _i. We assume the country-specific coefficients to be homogenous across cross-sectional observations with deviations governed by an error w _i, cast in regression form as

$$\displaystyle \begin{aligned} \boldsymbol{a}_i = \boldsymbol{a} + \boldsymbol{w}_i, \quad \boldsymbol{w}_i \sim \mathcal{N}(\boldsymbol{0},\boldsymbol{V}), \end{aligned}$$

with variance-covariance matrix V assumed to be diagonal with characteristic elements v _j. Notice that v _j governs the degree of heterogeneity of coefficients across countries. The specification above may be written in terms of the prior distribution on the country-specific coefficients as $\boldsymbol {a}_i\sim \mathcal {N}(\boldsymbol {a},\boldsymbol {V})$. As priors on the common mean we use $\boldsymbol {a}\sim \mathcal {N}(\boldsymbol {0},10\times \boldsymbol {I})$, and for $v_j\sim \mathcal {G}(0.01,0.01)$.

For the non-domestic coefficients of PVAR-DI and for all coefficients of GVAR, we use a normal-gamma shrinkage (NG, Griffin & Brown, 2010) prior. This prior is among the class of absolutely continuous global-local shrinkage priors and mimics the discrete stochastic search variable selection (SSVS, George & McCulloch, 1993; George et al., 2008) prior discussed in Chap. 4 of this volume. We choose the NG prior rather than the SSVS prior due to its advantageous empirical properties in high-dimensional model spaces (for an application in the VAR context that we base the hyperparameter values on, see Huber & Feldkircher, 2019). The global parameter of the hierarchical prior setup strongly pushes all coefficients towards zero, while local parameters allow for a priori non-zero idiosyncratic coefficients if suggested by the data likelihood. Intuitively, this prior allows for stochastic selection of inclusion and exclusion of VAR coefficients. The specification PVAR rules out all dynamic interdependencies a priori.

Factor Stochastic Volatility for Variance Estimation

Rather than decomposing the variance-covariance matrix as in the context of the BVAR and FAVAR specifications, we structure the stacked K-dimensional error vector ε _t for all countries in the PVAR and GVAR case as follows. We specify

$$\displaystyle \begin{aligned} \boldsymbol{\varepsilon}_t = \boldsymbol{\Lambda} \boldsymbol{F}_t + \boldsymbol{\eta}_t, \quad \boldsymbol{F}_t\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{H}_t), \quad \boldsymbol{\eta}_t\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{\Omega}_t), \end{aligned}$$

with F _t denoting a set of Q latent dynamic factors following a Gaussian distribution with zero mean and time-varying diagonal variance-covariance matrix . Λ is a K × Q matrix of factor loadings, linking the unobserved low-dimensional factors to the high dimensional error term ε _t. The vector η _t is a K-vector of idiosyncratic errors that is normally distributed with zero mean and diagonal variance covariance matrix . This setup implies that $\text{E}(\boldsymbol {\varepsilon }_t \boldsymbol {\varepsilon }_t^{\prime }) = \boldsymbol {\Lambda }\boldsymbol {H}_t\boldsymbol {\Lambda }^{\prime } + \boldsymbol {\Omega }_t$. By the assumed diagonal structure of Ω _t, this translates to the covariances being driven by the respective factor loadings in Λ. Static interdependencies can, for instance, be tested by imposing a suitable shrinkage prior on the elements of this matrix.

It remains to specify the law of motion on the respective logarithm of the diagonal elements of H _t and Σ _t. Here, we assume

$$\displaystyle \begin{aligned} \ln(h_{jt}) &= \mu_{hj} + \rho_{hj}(\ln(h_{jt}) - \mu_{h}) + \nu_{hjt}, \quad \nu_{hjt}\sim\mathcal{N}(0,\varsigma^{2}_{hj})\\ \ln(\omega_{lt}) &= \mu_{\omega l} + \rho_{\omega l} (\ln(\omega_{lt}) - \mu_{\omega l}) + \nu_{\omega lt}, \quad \nu_{\omega lt}\sim\mathcal{N}(0,\varsigma^{2}_{\omega l}), \end{aligned} $$

for and , with the specification and parameters to be understood analogous to Eq. (3.8). Again we employ the R-package stochvol for estimation and use its default prior settings (Kastner, 2016) for the AR(1) state equations. For the elements of the factor loadings matrix, we impose independent normally distributed priors with zero mean and unit variance. The latent factors are simulated using a forward filter backward sampling algorithm (Chris & Kohn, 1994; Frühwirth-Schnatter, 1994) similar to the one employed for sampling the factors in the context of the FAVAR specifications.