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# Bayesian Vector Autoregressions: Estimation

## Summary and Keywords

Vector autoregressions (VARs) are linear multivariate time-series models able to capture the joint dynamics of multiple time series. Bayesian inference treats the VAR parameters as random variables, and it provides a framework to estimate “posterior” probability distribution of the location of the model parameters by combining information provided by a sample of observed data and prior information derived from a variety of sources, such as other macro or micro datasets, theoretical models, other macroeconomic phenomena, or introspection.

In empirical work in economics and finance, informative prior probability distributions are often adopted. These are intended to summarize stylized representations of the data generating process. For example, “Minnesota” priors, one of the most commonly adopted macroeconomic priors for the VAR coefficients, express the belief that an independent random-walk model for each variable in the system is a reasonable “center” for the beliefs about their time-series behavior. Other commonly adopted priors, the “single-unit-root” and the “sum-of-coefficients” priors are used to enforce beliefs about relations among the VAR coefficients, such as for example the existence of co-integrating relationships among variables, or of independent unit-roots.

Priors for macroeconomic variables are often adopted as “conjugate prior distributions”—that is, distributions that yields a posterior distribution in the same family as the prior p.d.f.—in the form of Normal-Inverse-Wishart distributions that are conjugate prior for the likelihood of a VAR with normally distributed disturbances. Conjugate priors allow direct sampling from the posterior distribution and fast estimation. When this is not possible, numerical techniques such as Gibbs and Metropolis-Hastings sampling algorithms are adopted.

Bayesian techniques allow for the estimation of an ever-expanding class of sophisticated autoregressive models that includes conventional fixed-parameters VAR models; Large VARs incorporating hundreds of variables; Panel VARs, that permit analyzing the joint dynamics of multiple time series of heterogeneous and interacting units. And VAR models that relax the assumption of fixed coefficients, such as time-varying parameters, threshold, and Markov-switching VARs.

# Introduction

Vector autoregressions (VARs) are linear multivariate time-series models able to capture the joint dynamics of multiple time series. The pioneering work of Sims (1980) proposed to replace the large-scale macroeconomic models popular in the 1960s with VARs and suggested that Bayesian methods could have improved upon frequentist ones in estimating the model coefficients. Bayesian VARs (BVARs) with macroeconomic variables were first employed in forecasting by Litterman (1979) and Doan et al. (1984). Since then, VARs and BVARs have been a standard macroeconometric tool routinely used by scholars and policymakers for structural analysis, forecasting and scenario analysis in an ever-growing number of applications.

The aim of this article is to review key ideas and contributions in the BVAR literature. A companion paper provides a brief survey of applications of BVARs in economics and finance, such as forecasting, scenario analysis, and structural identification (Miranda-Agrippino & Ricco, 2018). An exhaustive survey of the literature is beyond the scope of this article due to space limitations. Readers are referred to a number of monographs and more detailed surveys available on different topics in the BVARs literature.1

Differently from frequentist statistics, Bayesian inference treats the VAR parameters as random variables and provides a framework to update probability distributions about the unobserved parameters conditional on the observed data. By providing such a framework, the Bayesian approach allows for incorporation of prior information about the model parameters into post-sample probability statements. The “prior” distributions about the location of the model parameters summarize pre-sample information available from a variety of sources, such as other macro or micro datasets, theoretical models, other macroeconomic phenomena, or introspection.

In the absence of pre-sample information, Bayesian VAR inference can be thought of as adopting “non-informative” (or “diffuse” or “flat”) priors, that express complete ignorance about the model parameters, in light of the sample evidence summarized by the likelihood function (i.e., the probability density function of the data as a function of the parameters). Often, in such a case, Bayesian probability statements about the unknown parameters (conditional on the data) are similar to classical confidence statements about the probability of random intervals around the true parameters value. For example, for a VAR with Gaussian errors and a flat prior on the model coefficients, the posterior distribution is centered at the maximum likelihood estimator (MLE), with variance given by the variance-covariance matrix of the residuals. Section "Inference in BVARs" discusses inference in BVARs and “non-informative” priors.

While non-informative priors can provide a useful benchmark, in empirical work with macroeconomic and financial variables informative priors are often adopted. In scientific data analysis, priors on the model coefficients do not incorporate the investigator’s “subjective” beliefs, instead, they summarize stylized representations of the data generating process. Conditional on a model, these widely held standardized priors aim at making the likelihood-based description of the data useful to investigators with potentially diverse prior beliefs (Sims, 2010b).2

The most commonly adopted macroeconomic priors for VARs are the so-called Minnesota priors (Litterman, 1980). They express the belief that an independent random-walk model for each variable in the system is a reasonable “center” for the beliefs about their time-series behavior. While not motivated by economic theory, they are computationally convenient priors, meant to capture commonly held beliefs about how economic time series behave. Minnesota priors can be cast in the form of a Normal-Inverse-Wishart (NIW) prior, which is the conjugate prior for the likelihood of a VAR with normally distributed disturbances (see Kadiyala & Karlsson, 1997). Conjugate priors are such that the posterior distribution belongs to the same family as the prior probability distribution. Hence, they allow for analytical tractability of the posterior and computational speed. Because the data is incorporated into the posterior distribution only through the sufficient statistics, formulas for updating the prior into the posterior are in this case conveniently simple. It is often useful to think of the parameters of a prior distribution—known as “hyperparameters”—as corresponding to having observed a certain number of “dummy” or “pseudo-” observations with properties specified by the prior beliefs on the VAR parameters. Minnesota priors can be formulated in terms of artificial data featuring pseudo observations for each of the regression coefficients and that directly assert the prior on them.

Dummy observations can also implement prior beliefs about relations among the VAR coefficients, such as, for example, co-integration among variables. In this case, commonly used priors are formulated directly as linear joint stochastic restrictions among the coefficients.3 This is, for example, the case of the “single-unit root” prior, that is centered on a region of the VAR parameter space where either there is no intercept and the system contains at least one unit root, or the system is stationary and close to its steady state at the beginning of the sample (Sims, 1993).4 Another instance in which dummy observations are used to establish relations among several coefficients is the “sum-of-coefficients” prior, which incorporates the widely shared prior beliefs that economic variables can be represented by a process with unit roots and weak cross-sectional linkages (Litterman, 1979).5 Section "Informative Priors for Reduced-Form VARs" discusses some of the priors commonly adopted in the economic literature.

The hyperparameters can be either fixed using prior information (and sometimes “unorthodoxly” using sample information) or associated with hyperprior distributions that express beliefs about their values. A Bayesian model with more than one level of priors is called a hierarchical Bayes model. In empirical macroeconomic modeling, the hyperparameters associated with the informativeness of the prior beliefs (i.e., the tightness of the prior distribution) are usually left to the investigator’s judgment. In order to select a value for these hyperparameters, the VAR literature has adopted mostly heuristic methodologies that minimize pre-specified loss functions over a pre-sample (e.g., the out-of-sample mean squared forecast error in Litterman, 1979, or the in-sample fit in Bańbura et al., 2010). Conversely, Giannone et al. (2015) specified hyperprior distributions and chose the hyperparameters that maximize their posterior probability distribution conditional on the data. Section "Hyperpriors and Hierarchical Modeling" discusses hierarchical modeling and common approaches to choose hyperparameters not specified by prior information.

In section "Time-Varying Parameter, State-Dependent, Stochastic Volatility VARs" we discuss Bayesian inference in VAR models that relax the assumption of fixed coefficients in order to capture changes in the time-series dynamics of macroeconomic and financial variables, such as VARs with autoregressive coefficients, Threshold, and Markov Switching VARs.

Finally, in section "Bayesian Panel VARs" we discuss Panel Bayesian VARs that generalize VAR models by describing the joint dynamics of multiple time series of potentially heterogenous and interacting units—as, for example, the economies of several countries, regions, or sectors.

# Inference in BVARs

Vector autoregressions (VARs) are linear stochastic models that describe the joint dynamics of multiple time series. Let yt be an n × 1 random vector that takes values in $ℝn$. The evolution of yt—the endogenous variables—is described by a system of p-th order difference equations—the VAR(p):

$Display mathematics$
(1)

In Eq. (1), Aj, j = 1, . . . , p are n × n matrices of autoregressive coefficients, c is a vector of n intercepts, and ut is an n-dimensional vector of one-step-ahead forecast errors, or reduced-form innovations. The vector of stochastic innovations, ut, is an independent and identically distributed random variable for each t. The distribution from which ut is drawn determines the distribution of yt, conditional on its past y1–p:t–1 ≡ {y1–p,…,y0,…, yt–2,yt–1}. The standard assumption in the macroeconometric literature is that errors are Gaussian

$Display mathematics$
(2)

This implies that also the conditional distribution of yt is normal.6

Bayesian inference on the model in Eq. (1) amounts to updating prior beliefs about the VAR parameters, that are seen as stochastic variables, after having observed a sample y1–p:t ≡ {y1–p,..., y0,…, yt–2, yt}. Prior beliefs about the VAR coefficients are summarized by a probability density function (p.d.f.), and updated using Bayes’ Law

$Display mathematics$
(3)

where we define A ≡ [A1,…, Ap, c]′ as a k × n matrix, with k = np +1. The joint posterior distribution of the VAR(p) coefficients $p(A,Σ|y1−p:t)$ incorporates the information contained in the prior distribution p(A, Σ‎)—summarizing the initial information about the model parameters –, and the sample information summarized by p(y1–p:t|A, Σ‎). Viewed as a function of the parameters, the sample information is the likelihood function.7 The posterior distribution summarizes all the information available and is used to conduct inference on the VAR parameters.

Given the autoregressive structure of the model, and the i.i.d. innovations, the (conditional) likelihood function of the sample observations y1:T—conditional on A, Σ‎ and on the first p observations y1–p:0 –, can be written as the product of the conditional distribution of each observation

$Display mathematics$
(4)

Under the assumption of Gaussian errors, the conditional likelihood of the VAR in Eq. (1) is

$Display mathematics$
(5)

where $xt′≡[yt−1′…yt−p′1]$.

The likelihood in Eq. (5) can be written in compact form, by using the seemingly unrelated regression (SUR) representation of the VAR

$Display mathematics$
(6)

where the T × n matrices y and u and the T × k matrix x are defined as

$Display mathematics$
(7)

Using this notation and standard properties of the trace operator, the conditional likelihood function can be equivalently expressed as

$Display mathematics$
(8)

where $A^$ is the maximum-likelihood estimator (MLE) of A, and $S^$ the matrix of sums of squared residuals, that is

$Display mathematics$
(9)

The likelihood can also be written in terms of the vectorized representation of the VAR

$Display mathematics$
(10)

where yvec(y) and uvec(u) are Tn × 1 vectors, and α≡ vec(A) is nk × 1. In this vectorized notation the likelihood function is written as

$Display mathematics$
(11)

where, consistently, $α^≡vec(A^)$ is nk × 1. Detailed derivations for the multivariate Gaussian linear regression model can be found in Zellner (1971).

Given the likelihood function, Eq. (3) is used to update the prior information regarding the VAR parameters. An interesting case arises when we assume the absence of any information on the location of the model parameters. This setting can be formalized by assuming that α‎ and Σ‎ are independently distributed, that is

$Display mathematics$
(12)

with prior p.d.f.

$Display mathematics$
(13)

These priors are known as diffuse or Jeffreys prior (Geisser, 1965; Tiao & Zellner, 1964). Jeffreys priors are proportional to the square root of the determinant of the Fisher information matrix, and are derived from the Jeffreys “invariance principle,” meaning that the prior is invariant to re-parameterization (see Zellner, 1971).8

Given this set of priors, it is straightforward to derive the posterior distribution of the VAR parameters as

$Display mathematics$
(14)

where the proportionality factor has been dropped for convenience.

From the joint posterior in Eq. (14) one can readily deduce the form of the posterior for α‎, conditional on Σ‎ and the observed sample. Also, the posterior can be integrated over α‎ to obtain the marginal posterior for Σ‎. Therefore, it is possible to conveniently write the posterior distribution of the parameters as

$Display mathematics$
(15)

Where

$Display mathematics$
(16)

$Display mathematics$
(17)

Hence, given the diffuse priors on α‎ and Σ‎, the posterior for the autoregressive coefficients is centered at the MLE, with posterior variance $Σ ⊗(x'x)−1$.9 Interestingly, in this standard normal multivariate linear regression model, Bayesian probability statements about the parameters (given the data) have the same form as the frequentist pre-sample probability statements about the parameters’ estimator (see also Sims, 2010b). This is a more general property, in fact, Kwan (1998) has shown that under widely applicable regularity conditions an estimator $α^T$ for which

$Display mathematics$

allows, with high accuracy, to approximate the distribution of $T(α−α^T)|α^$ as $N(0,Σ)$ in large samples. Hence, it is often possible to interpret (1 – ρ‎) approximate confidence sets generated from the frequentist asymptotic approximate distribution as if they were sets in the parameter space with posterior probability (1 – ρ‎).

In potentially mis-specified models for which linear regression coefficients are the object of interest, Muller (2013) proposes to adopt an artificial Gaussian posterior centered at the MLE but with a sandwich estimator for the covariance matrix. In fact, in the case of a mis-specified model, the shape of the likelihood (the posterior) is asymptotically Gaussian and centered at the MLE but of a different variance than the asymptotically normal sampling distribution of the MLE. This argument can be seen as a “flipping” of the frequentist asymptotic statement that supports the use of a sandwich estimator for the covariance matrix in mis-specified models, in line with the results in Kwan (1998).10

$Display mathematics$
(18)

An important case in which frequentist pre-sample probability statements and Bayesian post-sample probability statements about parameters diverge, is the case of time-series regression models with unit roots. In such cases, while the frequentist distribution of the estimator is skewed asymptotically, the likelihood, and hence the posterior p.d.f., remain unaffected (see Sims & Uhlig, 1991; Kim, 1994).

# Informative Priors for Reduced-Form VARs

Informative prior probability distributions incorporate information about the VAR parameters that is available before some sample is observed. Such prior information can be contained in samples of past data—from the same or a related system –, or can be elicited from introspection, casual observation, and theoretical models. The first case is sometimes referred to as a “data-based” prior, while the second as a “nondata-based” prior.

An important case arises when the prior probability distribution yields a posterior distribution for the parameters in the same family as the prior p.d.f. In this case the prior is called a “natural conjugate prior” for the likelihood function (Raiffa & Schlaifer, 1961). In general, it has been shown that exponential distributions are the only class of distributions that admit a natural conjugate prior, due to these having a fixed number of sufficient statistics that does not increase as the sample size T increases (see e.g., Gelman et al., 2013). Because the data is incorporated into the posterior distribution only through the sufficient statistics, formulas for updating the prior into the posterior are in these cases conveniently simple.

Prior distributions can be expressed in terms of coefficients, known as hyperparameters, whose functions are sufficient statistics for the model parameters. It is often useful to think of the hyperparameters of a conjugate prior distribution as corresponding to having observed a certain number of pseudo-observations with properties specified by the priors on the parameters. In general, for nearly all conjugate prior distributions, the hyperparameters can be interpreted in terms of “dummy” or pseudo-observations. The basic idea is to add to the observed sample extra “data” that express prior beliefs about the hyperparameters. The prior then takes the form of the likelihood function of these dummy observations. Hyperparameters can be either fixed using prior information, or associated to hyperprior distributions that express beliefs about their values. A Bayesian model with more than one level of priors is called a “hierarchical Bayes model.” In this section we review some of the most commonly used priors for VARs with macroeconomic and financial variables, while we discuss the choice of the hyperpriors and hierarchical modeling in section "Hyperpriors and Hierarchical Modeling."

## Natural Conjugate Normal-Inverse Wishart Priors

The Normal-Inverse Wishart (NIW) conjugate priors, part of the exponential family, are commonly used prior distributions for (A, Σ‎) in VARs with Gaussian errors. These assume a multivariate normal distribution for the regression coefficients, and an Inverse Wishart specification for the covariance matrix of the error term, and can be written as

$Display mathematics$
(19)

$Display mathematics$
(20)

where $(S_,d_,α_,Ω_)$ are the priors’ hyperparameters; $d_$ and $S_$ denote, respectively, the degrees of freedom and the scale of the prior Inverse-Wishart distribution for the variance-covariance matrix of the residuals. $α_$ is the prior mean of the VAR coefficients, and $Ω_$ acts as a prior on the variance-covariance matrix of the dummy regressors.11 The posterior distribution can be analytically derived and is given by

$Display mathematics$
(21)

$Display mathematics$
(22)

Where

$Display mathematics$
(23)

$Display mathematics$
(24)

$Display mathematics$
(25)

Comparing Eqs. (16)–(17) to Eqs. (19)–(20), it is evident that informative priors can be thought of as equivalent to having observed dummy observations (yd, xd) of size Td, such that

$Display mathematics$
(26)

$Display mathematics$
(27)

$Display mathematics$
(28)

$Display mathematics$
(29)

This idea was first proposed for a classical estimator for stochastically restricted coefficients by Theil (1963). Once a set of pseudo-observations able to match the wished hyperparameters is found, the posterior can be equivalently estimated using the extended samples $y*=[y′,yd'],x*=[x′,xd']′$ of size $T*=T+Td$ obtaining

$Display mathematics$
(30)

$Display mathematics$
(31)

Indeed, it is easy to verify that the posterior moments obtained with the starred variables coincide with those in Eqs. (21)–(22). The posterior estimator efficiently combines sample and prior information using their precisions as weights in the spirit of the mixed estimation of Theil and Goldberger (1961). Posterior inference can be conducted via direct sampling.

Algorithm 1: Direct Monte Carlo Sampling From Posterior of VAR Parameters

For s = 1, . . .,nsim:

1. 1. Draw Σ‎(s) from the Inverse-Wishart distribution $Σ|y~ℐW(S*,T*+d_)$.

2. 2. Draw A(s) from the Normal distribution of $A(s)|Σ(s),y~N(α*,Σ(s)⊗(x*′x*)−1)$.

When it is not possible to sample directly from the posterior distribution, as in this case, Markov chain Monte Carlo (MCMC) algorithms are usually adopted (see, e.g., Chib, 2001).12

An important feature of the NIW priors in Eqs. (19)–(20) is the Kronecker factorization that appears in the Gaussian prior for α‎. As discussed in the previous section, because the same set of regressors appears in each equation, homoskedastic VARs can be written as SUR models. This symmetry across equations means that homoskedastic VAR models have a Kronecker factorization in the likelihood, which in turn implies that estimation can be broken into n separate least-squares calculations, each only of dimension np + 1. The symmetry in the likelihood can be inherited by the posterior, if the prior adopted also features a Kronecker structure as in Eq. (20). This is a desirable property that guarantees tractability of the posterior p.d.f. and computational speed. However, such a specification can result in unappealing restrictions and may not fit the actual prior beliefs one has—see discussions in Kadiyala and Karlsson (1997) and Sims and Zha (1998). In fact, it forces symmetry across equations, because the coefficients of each equation have the same prior variance matrix (up to a scale factor given by the elements of Σ‎). There may be situations in which theory suggests “asymmetric restrictions” may be desirable instead, for example, money neutrality implies that the money supply does not Granger-cause real output.13 Also, the Kronecker structure implies that prior beliefs must be correlated across the equations of the reduced form representation of the VAR, with a correlation structure that is proportional to that of the disturbances.

## Minnesota Prior

In macroeconomic and financial applications, the parameters of the NIW prior in Eqs. (19)–(20) are often chosen so that prior expectations and variances of A coincide with the so-called Minnesota prior, which was originally proposed in Litterman (1980, 1986).14 The basic intuition behind this prior is that the behavior of most macroeconomic variables is well approximated by a random walk with drift. Hence, it “centers” the distribution of the coefficients in A at a value that implies a random-walk behavior for all the elements in yt

$Display mathematics$
(32)

While not motivated by economic theory, these are computationally convenient priors, meant to capture commonly held beliefs about how economic time series behave.

The Minnesota prior assumes the coefficients A1,. . . , Ap to be a priori independent and normally distributed, with the following moments

$Display mathematics$
(33)

In Eq. (33), $(Aℓ)ij$ denotes the coefficient of variable j in equation i at lag $ℓ$. In the original formulation of the prior δi = 1, in accordance with Eq. (32). The random-walk assumption, however, may not be appropriate if the variables in yt were characterized by substantial mean-reversion. For stationary series, or series that have been transformed to achieve stationarity, Bańbura et al. (2010) center the distribution around zero (i.e., δi = 0). The prior also assumes that lags of other variables are less informative than own lags, and that most recent lags of a variable tend to be more informative than more distant lags. This intuition is formalized with $f(ℓ)$. A common choice for this function is a harmonic lag decay—that is $f(ℓ)=ℓλ2$, a special case of which is $f(ℓ)=ℓ−$, where the severity of the lag decay is regulated by the hyperparameter λ2. The factor $Σij/ωj2$ accounts for the different scales of variables i and j. The hyperparameters $ωj2$ are often fixed using sample information, for example from univariate regressions of each variable onto its own lags.

Importantly, λ1 is a hyperparameter that controls the overall tightness of the random walk prior. If λ1 = 0 the prior information dominates, and the VAR reduces to a vector of univariate models. Conversely, as $λ1→∞$ the prior becomes less informative, and the posterior mostly mirrors sample information. We discuss the choice of the free hyperparameters in section "Hyperpriors and Hierarchical Modeling."

The Minnesota prior can be implemented using dummy observations. Priors on the A coefficients are implemented via the following pseudo-observations

$Display mathematics$
(34)

where $Jp=diag([1λ2,2λ2,…,pλ2])$ with geometric lag decay.15 To provide intuition on how the prior is implemented using artificial observations, we consider the simplified case of a n = 2, p = 2 VAR for the pseudo-observations. The first n rows of Eq. (34) impose priors on A1; that is, on the coefficients of the first lag. In the n = 2, p =2 case one obtains,

$Display mathematics$
(35)

that implies, for example, the following equations for the elements (1,1) and (1, 2) of A1

$Display mathematics$

Similar restrictions are obtained for the elements the elements (2,1) and (2, 2) of A1. The following (n – 1)p rows in Eq. (34) implement priors on the coefficients of the other lags. In fact, we readily obtain

$Display mathematics$
(36)

which for example implies the following restriction for the element (1,1) of A2

$Display mathematics$

Similar restrictions obtain for the other elements of A2. Priors beliefs on the residual covariance matrix Σ‎ can instead be implemented by the following block of dummies

$Display mathematics$
(37)

In the n = 2, p = 2 case, they correspond to appending to the VAR equations λ3 replications of

$Display mathematics$
(39)

λ3 is the hyperparameter that determines the tightness of the prior on Σ‎. To understand how this works, it is sufficient to consider that given λ3 artificial observations zi, with $zi~N(0,σz2)$, an estimator for the covariance is given by $λ3−1∑i=1λ3zi2$.

Finally, uninformative priors for the intercept are often implemented with the following set of pseudo-observations

$Display mathematics$

where $ϵ$ is a hyperparameter usually set to a very small number.16

## Priors for VAR With Unit Roots and Trends

Sims (1996, 2000) observed that flat-prior VARs, or more generally estimation methods that condition on initial values, tend to attribute an implausibly large share of the variation in observed time series to deterministic—and hence entirely predictable—components. The issue stems from the fact that ML and OLS estimators that condition on the initial observations and treat them as non-stochastic do not apply any penalization to parameters values that imply that these observations are distant from the variables’ steady state (or their trend if non-stationary). As a consequence, complex transient dynamics from the initial conditions to the steady state are treated as plausible and can explain an “implausibly” large share of the low-frequency variation of the data. This typically translates into poor out-of-sample forecasts. To understand the intuition, consider the univariate model

$Display mathematics$
(40)

Iterating Eq. (40) backward yields

$Display mathematics$
(41)

which, if |a| < 1, reduces to

$Display mathematics$
(42)

The first term in square brackets in Eq. (41) is the deterministic component: the evolution of yt from the initial conditions y0, absent any shocks. The second term instead captures the stochastic evolution of yt due to the shocks realized between [0, t – 1]. c/(1 – a) in Eq. (42) is the unconditional mean of yt. If yt is close to non-stationary—i.e., $a≃1−$, the MLE estimator of the unconditional mean of yt may be very far from y0, and the “reversion to the mean” from y0 is then used to fit the data (see Eq. 42).

One way to deal with this issue is to use the unconditional likelihood, by explicitly incorporating the density of the initial observations in the inference. However, because most macroeconomic time series are effectively nonstationary, it is not obvious how the density of the initial observations should be specified.17 Another approach, following Sims and Zha (1998), is that of Sims (2000), which would be instead to specify priors that downplay the importance of the initial observations and hence reduce the explanatory power of the deterministic component.

These types of priors, implemented through artificial observations, aim to reduce the importance that the deterministic component has in explaining a large share of the in-sample variation of the data, eventually improving forecasting performances out-of-sample (see Sims, 1996; Sims & Zha, 1998, for a richer discussion on this point).18

The “co-persistence” (or “single-unit-root” or “dummy initial observation”) prior (Sims, 1993) reflects the belief that when all lagged yt’s are at some level $y¯0$, yt tends to persist at that level. It is implemented using the following artificial observation

$Display mathematics$

where $y¯0,i,i=1,…,n$ are the average of the initial values of each variable, and usually set to be equal to the average of the first p observations in the sample, and k = np + 1. Writing down the implied system of equations $yd(4)=Axd(4)+ud(4)$ one obtains the following stochastic restriction on the VAR coefficients

$Display mathematics$
(44)

where $In−A(1)=(In−A1−…−Ap)$. The hyperparameter λ4 controls the tightness of this stochastic constraint. The prior is uninformative for $λ4→∞$. Conversely, as $λ4→0$ the model tends to a form where either there is at least one explosive common unit root and the constant c is equal to zero ($y¯0$ is the eigenvector of the unit root), or the VAR is stationary, c is different from zero, and the initial conditions are close to the implied unconditional mean $(y¯0=[In−A(1)]−1c)$. In the stationary form, this prior does not rule out cointegrated models. This prior induces prior correlation among all the VAR coefficients in each equation, including the constant.19

The “sums-of-coefficients” (or “no-cointegration”) prior (Doan et al., 1984), captures the belief that when the average lagged values of a variable yj,t is at some level $y¯0,j$, then $y¯0,j$ is likely to be a good forecast of yj,t. It also implies that knowing the average of lagged values of variable j does not help in predicting a variable ij. This prior is implemented using n artificial observations, one for each variable in yt

$Display mathematics$
(45)

The prior implied by these dummy observations is centered at 1 for the sum of coefficients on own lags for each variable, and at 0 for the sum of coefficients on other variables’ lags. It also introduces correlation among the coefficients of each variable in each equation. In fact, it is easy to show that equation by equation this prior implies the stochastic constraint

$Display mathematics$
(46)

where $(Aℓ)jj$ denotes the coefficient of variable j in equation j at lag $ℓ$. The hyperparameter λ5 controls the variance of these prior beliefs. As $λ5→∞$ the prior becomes uninformative, while $λ5→0$ implies that each variable is an independent unit-root process, and there are no co-integration relationships.20

The Bayesian analysis of cointegrated VARs is an active area of research, (a detailed survey is in Koop et al. 2006).21 Giannone et al. (2016) elicited theory-based priors for the long run of persistent variables that shrink toward a random walk those linear combination of variables that are likely to have a unit root. Conversely, combinations that are likely to be stationary (i.e., cointegrating relationships among variables) are shrunk toward stationary processes. Operationally, this is achieved by rewriting the VAR in Eq. (1) as

$Display mathematics$
(47)

where $Π=A1+…+Ap−In,Pj=−(Aj+1+…+Ap)$, and F is any invertible n- dimensional matrix. The problem is then specified as setting a prior for $Π˜≡ΠF−1$, conditional on a specific choice of F. F defines the relevant linear combinations of the variables in yt that macroeconomic theory suggests to be a priori stationary or otherwise.

Another alternative is in Villani (2009). Here the VAR is written as

$Display mathematics$
(48)

for Math Keying.

where ρ0 and ρ1 are n × 1 vectors. The first term, ρ0 + ρ1t, captures a linear deterministic trend of yt, whereas the law of motion of $y~t$ captures stochastic fluctuations around the deterministic trend, which can be either stationary or non-stationary. This alternative specification allows to separate beliefs about the deterministic trend component from beliefs about the persistence of fluctuations around this trend. Let A = [A1, . . ., Ap]′ and $ρ=[ρ0′,ρ1′]′$. It can be shown that if the prior distribution of ρ‎ conditional on A and Σ‎ is Normal, the (conditional) posterior distribution of ρ‎ is also Normal (see also Del Negro & Schorfheide, 2011, for details). Hence, posterior inference can be implemented via Gibbs sampling.

## Priors From Structural Models

DeJong et al. (1993), Ingram and Whiteman (1994), and Del Negro and Schorfheide (2004) have proposed the use of priors for VARs that are derived from Dynamic Stochastic General Equilibrium (DSGE) models. This approach bridges VARs and DSGEs by constructing families of prior distributions informed by the restrictions that a DSGE-model implies on the VAR coefficients. This modeling approach is sometimes referred to as DSGE-VAR. Ingram and Whiteman (1994) derived prior information from the basic stochastic growth model of King et al. (1988) and reported that a BVAR based on the Real Business Cycle model prior outperforms a BVAR with a Litterman prior in forecasting real economic activity. Del Negro and Schorfheide (2004) extend and generalize this approach and show how to conduct policy simulations within this framework.

Schematically, the exercises can be thought of as follows. First, time series are simulated from a DSGE model. Second, a VAR is estimated from these simulated data. Population moments of the simulated data computed from the DSGE model solution are considered in place of sample moments. Since the DSGE model depends on unknown structural parameters, hierarchical prior modeling is adopted by specifying a distribution on the DSGE model parameters. A tightness parameter controls the weight of the DSGE model prior relative to the weight of the actual sample. Finally, Markov Chain Monte Carlo methods are used to generate draws from the joint posterior distribution of the VAR and DSGE model parameters.

## Priors for Model Selection

It is standard practice in VAR models to pre-select the relevant variables to be included in the system (and with how many lags). This procedure may be thought of as having dogmatic priors about which variables have non-zero coefficients in the system. The challenge is in selecting among an expansive set of potential models. Indeed, for a VAR with n endogenous variables, q additional potentially exogenous variables including a constant, and p lags, there are 2(q+pn)n+n(n–1)/2 possible models.

Jarociński and Maćkowiak (2017) proposed to select the variables to be included in the system by systematically assessing the posterior probability of “Granger causal priority” (Sims, 2010a) in a BVAR with conjugate priors. Granger causal priority answers questions of the form “Is variable z relevant for variable x, after controlling for other variables in the system?” The authors provide a closed form expression for the posterior probability of Granger causal priority and suggest that variables associated with high Granger causal priority probabilities can be omitted from a VAR with the variables of interest.

Alternatively, one can adopt priors that support model selection and enforce sparsity. A variety of techniques, including double exponential (Laplace) prior, spike-and-slab prior (among others) have been adopted to handle this issue. Some recent theoretical and empirical contributions on this topic are in Mitchell and Beauchamp (1988), George et al. (2008), Koop (2013), Korobilis (2013), Bhattacharya et al. (2015), Griffin and Brown (2010, 2017), Giannone et al. (2017), and Huber and Feldkircher (2017).

# Hyperpriors and Hierarchical Modeling

As seen in the previous section, the informativeness of prior beliefs on the VAR parameters often depends on a set of free hyperparameters. Let λ‎ ≡ [λ1, λ2,. . .] denote the vector collecting all the hyperparameters not fixed using (pre)sample information, and θ‎ denote all the VAR parameters (i.e., A and Σ‎). The prior distribution of θ‎ is thus effectively pλ‎(θ‎). Choosing a value for λ‎ alters the tightness of the prior distribution and hence determines how strictly the prior is enforced on the data.

In order to set the informativeness of the prior distribution of the VAR coefficients, the literature has initially used mostly heuristic methodologies. Litterman (1980) and Doan et al. (1984), for example, choose a value for the hyperparameters that maximizes the out-of-sample forecasting performance over a pre-sample. Conversely, Bańbura et al. (2010) proposed to choose the shrinkage parameters that yield a desired in-sample fit in order to control for overfitting. Subsequent studies have then either used these as “default” values or adopted either one of these criteria. Robertson and Tallman (1999), Wright (2009), and Giannone et al. (2014) opt for the first, while for example, Giannone et al. (2008), Bloor and Matheson (2011), Carriero et al. (2009), and Koop (2013) follow Bańbura et al. (2010).

In VARs, Giannone et al. (2015) observed that, from a purely Bayesian perspective, choosing λ‎ is conceptually identical to conducting inference on any other unknown parameter of the model. Specifically, the model is interpreted as a hierarchical one (Berger, 1985; Koop, 2003) and λ‎ can be chosen as the maximizer of

$Display mathematics$
(49)

This method is also known in the literature as the Maximum Likelihood Type II (ML-II) approach to prior selection (Berger, 1985; Canova, 2007). In Eq. (49), p(λ‎|y) is the posterior distribution of λ‎ conditional on the data, and p(λ‎) denotes a prior probability density specified on the hyperparameters themselves and also known as the hyperprior distribution. In such hierarchical model, the prior distribution for the VAR coefficients is treated as a conditional prior, that is pλ‎(θ‎) is replaced by p(θ‎|λ‎). In the case of a NIW family of distributions, the prior structure becomes p(α‎|Σ‎, λ‎)p(Σ‎|λ‎)p(λ‎). p(y|λ‎, y1–p:0) is the marginal likelihood (ML) and is obtained as the density of the data as a function of λ‎, after integrating out all the VAR parameters. Conveniently, with conjugate priors the ML is available in closed form.

Conversely, the joint posterior of α‎, Σ‎ and λ‎ is not available in closed form. However, with NIW priors for θ‎, Giannone et al. (2015) set up the following Metropolis-Hasting sampler for the joint distribution

Algorithm 2: MCMC Sampler for a VAR with Hierarchical Prior.

For s = 1,. . .,nsim:

1. 1. Draw a candidate vector λ* from the random walk distribution $λ*~N(λs−1,κH−1)$, where H is the Hessian of the negative of the log-posterior at the peak for λ‎, and κ‎ is a tuning constant. Choose

$Display mathematics$

2. 2. Draw Σ‎(s) form the full conditional posterior Σ‎|y, λ(s) in Eq. (21).

3. 3. Draw A(s) from the full conditional posterior A(s)|y, Σ‎(s), λ(s) in Eq. (22).

In a similar fashion, Belmonte et al. (2014) applied a hierarchical structure to time-varying parameters (TVP) models and specify priors for Bayesian Lasso shrinkage parameters to determine whether coefficients in a forecasting model for inflation are zero, constant, or time-varying in a data driven way.

Carriero et al. (2015) evaluated the forecasting performance of BVARs where tightness hyperparameters are chosen as the maximisers of Eq. (49) or rather set to default values and find that the former route yields modest but statistically significant gains in forecasting accuracy particularly at short horizons.

# Time-Varying Parameter, State-Dependent, Stochastic Volatility VARs

Models that allow parameters to change over time are increasingly popular in empirical research, in recognition of the fact that they can capture structural changes in the economy. In fact, it seems to be a common belief that the properties of many (if not most) macroeconomic time series have changed over time and can change across regimes or phases of the business cycle. Model parameters either change frequently and gradually over time according to a multivariate autoregressive process—as for example in time-varying parameters VARs (TVP-VARs) –, or they change abruptly and infrequently in, for instance, Markov-switching or structural-break models.

## Time-Varying Parameters VAR (TVP-VAR)

Time-varying parameters VARs differ from fixed-coefficient VARs in that they allow the parameters of the model to vary over time, according to a specified law of motion.22 TVP-VARs often include also stochastic volatility (SV), which allows for time variation in the variance of the stochastic disturbances.23 Doan et al. (1984) were the first to show how estimation of a TVP-VAR with Litterman priors could be conducted by casting the VAR in state space form and using Kalman filtering techniques. This same specification is in Sims (1993). Bayesian time-varying parameter VARs have become popular in empirical macroeconomics following the work of Cogley and Sargent (2002, 2005) and Primiceri (2005) who provided the foundations for Bayesian inference in these models, and used then innovations in MCMC algorithms to improve on their computational feasibility.

The basic TVP-VAR is of the form

$Display mathematics$
(50)

where the constant coefficients of Eq. (1) are replaced by the time-varying Aj,t. Eq. (50) can be rewritten in compact form as

$Display mathematics$
(51)

where xt is defined as in Eq. (5), and At = [A1,t,…, Ap,t, ct]′. It is common to assume that the coefficients follow a random-walk process

$Display mathematics$
(52)

where αtvec(At). The covariance matrix $ϒ$ is usually restricted to be diagonal, and the innovations ςt to be uncorrelated with ut, with ut distributed as in Eq. (61). The law of motion for αt in Eq. (52)—that is, the state equation –, implies that $αt+1|αt,ϒ~N(αt,ϒ)$, which can be used as a prior distribution for αt+1. Hence, the prior for all the states (i.e., $αt∀t$) is a product of normal distributions. For the initial vector of the VAR coefficients Cogley and Sargent (2002, 2005) use a prior of the form $α1~N(α_1|0,ϒ_1|0)$, where $α_1|0$ and $ϒ_1|0$ are set by estimating a fixed-coefficient VAR with a flat prior on a pre-sample.24 If the Gaussian prior for the states is complemented with IW priors for Stochastic volatility in Bayesian VARs was initially introduced in Uhlig (1997), both Σ‎ and Υ‎, then sampling from the joint posterior is possible with a Gibbs sampling algorithm

Algorithm 3: Gibbs Sampling From Posterior of TVP-VAR Parameters

Select starting values for Σ‎(0) and ϒ‎(0). For s = 1,. . . ., nsim:

1. 1. Draw $αT(s)$ from the full conditional posterior

$Display mathematics$

obtained from the Kalman filter. For t = T – 1,. . . ., 1 draw $αT(s)$ from the full conditional posterior

$Display mathematics$

obtained from a simulation smoother.

2. 2. Draw ϒ‎(s) from

$Display mathematics$

3. 3. Draw Σ‎(s) from

$Display mathematics$

When stochastic volatility is added to the framework, the VAR innovations are assumed to be still normally distributed but with variance that evolves over time (see Cogley & Sargent, 2002, 2005; Primiceri, 2005)

$Display mathematics$
(53)

where K is a lower-triangular matrix with ones on the main diagonal, and Ξ‎t a diagonal matrix with elements evolving following a geometric random-walk process

$Display mathematics$
(54)

The prior distributions for ϒ‎ and $ση,j2j=1,…,n$ can be used to express beliefs about the magnitude of the period-to-period drift in the VAR coefficients, and the changes in the volatility of the VAR innovations respectively. In practice, these priors are chosen to ensure that innovations to the parameters are small enough that the short- and medium-run dynamics of yt are not swamped by the random-walk behavior of At and Ξ‎t Primiceri (2005) extends the above TVP-VAR by also allowing the nonzero off-diagonal elements of the contemporaneous covariance matrix K to evolve as random-walk processes (i.e., K is replaced by Kt to allow for an arbitrary time-varying correlation structure). A Gibbs sampler to draw from the posterior distribution of the parameters is in Primiceri (2005).

## Markov Switching, Threshold, and Smooth Transition VARs

Contrary to the drifting coefficients models discussed in the previous section, Markov switching (MS) VARs are designed to capture abrupt changes in the dynamics of yt.25 These can be viewed as models that allow for at least one structural break to occur within the sample, with the timing of the break being unknown. They are of the form

$Display mathematics$
(55)

where xt is defined as in Eq. (5). The matrix of autoregressive coefficients A(st) and the variance of the error term Σ‎(st) are a function of a discrete m-state Markov process st with fixed transition probabilities

$Display mathematics$
(56)

If πii = 1 for some $i∈[1,…,m]$, then $Si$ is an absorbing state from which the system is not allowed to move away. Suppose m = 2, and that both A(st) and Σ‎(st) change simultaneously when switching from $S1$ to $S2$ and vice versa. If a NIW prior is specified for A(st) and Σ‎(st), and π11 and π22 have independent Beta prior distributions, a Gibbs sampler can be used to sample from the posterior (see, e.g., Del Negro & Schorfheide, 2011).

A MS-VAR with non-recurrent states is called a “change-point” model (see Chib, 1998; Bauwens & Rombouts, 2012). Generalizing the specification to allow for more states, with the appropriate transition probabilities, allows to adapt the change-point model to the case of several structural breaks (see also Koop & Potter, 2007, 2009; Liu et al., 2017, for models where the number of change points is unknown). Important extensions regard the transmission of structural shocks in the presence of structural breaks and in a time-varying coefficient environment discussed in, for example, Sims (2006) and Koop et al. (2011) who also allow for cointegration.

In threshold VARs (TVARs), the coefficients of the model change across regimes when an observable variable exceeds a given threshold value. Bayesian inference in TVAR models is discussed in detail in Geweke and Terui (1993) and Chen and Lee (1995). A TVAR with two regimes can be written as

$Display mathematics$
(57)

where A and A* are n × k matrices that collect the autoregressive coefficients of the two regimes, Θ‎(•) is a Heaviside step function, which is a discontinuous function whose value is zero for a negative argument, and one for a positive argument, τt–d is threshold variable at lag d, and τ‎ is a potentially unobserved threshold value. The system in Eq. (57) can be easily generalized to allow for multiple regimes. TVARs have been applied to several problems in the economic literature (see, e.g., Koop & Potter, 1999; Ricco et al., 2016; Alessandri & Mumtaz, 2017).

If the coefficients gradually migrate to the new state(s), the model is called a smooth-transition VAR (STVAR). A STVAR model with two regimes can be written as

$Display mathematics$
(58)

where A*, A, and xt are defined as in Eq. (57). The function G(wt; ϑ‎, w) governs the transition across states, and is a function of the observable variable wt, and of the parameters ϑ‎ and w. In an exponential smooth-transition (EST) VAR, typically

$Display mathematics$
(59)

where ϑ‎ > 0 determines the speed of transition across regimes, w can be thought of as a threshold value, and σw is the sample standard deviation of wt. The higher ϑ‎ the more abrupt the transition, the more the model collapses into a fixed threshold VAR. Among others, Gefang and Strachan (2009) and Gefang (2012) apply Bayesian techniques to estimate smooth-transition VAR models.

# Bayesian Panel VARs

Panel VARs generalize VAR models by describing the joint dynamics of multiple time series of heterogenous and interacting units—as, for example, the economies of several countries, regions, or sectors. Thorough reviews are in Canova and Ciccarelli (2013) and in Dieppe, van Roye, and Legrand (2016).

A panel VAR describes the the evolution of yt,i—the vector of n × 1 endogenous variables of each unit $i∈[1,...N]−$ by a system of p-th order VARs

$Display mathematics$
(60)

where wt is a vector of m exogenous controls. The innovations are generally assumed to be i.i.d. and Gaussian

$Display mathematics$
(61)

while being possibly correlated across units.

Stacking over the N units, the model assumes the form of a VAR(p) with exogenous controls

$Display mathematics$
(62)

In Eq. (63)

$Display mathematics$
(63)

moreover

$Display mathematics$

and

$Display mathematics$

While in Eq. (63) the system appears as a standard VAR, its panel structure is captured by three properties: (i) dynamic interdependencies—the dynamics of the variables in each unit depend on the lagged values of the other endogenous variables in the unit and possibly all other units: $Aℓ,jk≠0$ for jk; (ii) static interdependencies—the innovations ut,i can be correlated across units: Σ‎ij• ≠ 0 for ij; (iii) cross-unit (sub-sectional) heterogeneity—the VAR coefficients and residual variances can be unit-specific: $Aℓ,ik≠Aℓ,jk,Hi≠Hj$ and $Σii≠Σjj$ for $i≠j$.26

If all of these properties are present in the data, and relationships do not exist among the coefficients, the system is a VAR with a large cross-section (i.e., a Large VAR) and can be estimated with standard macroeconomic priors such as, for example, the Minnesota priors of section "Informative Priors for Reduced-Form VARs."27 However, it is often possible to assume that some of these properties are relevant to system of interest.

If the units do not have dynamic or static interdependencies (i.e., $Aℓ,jk=0$ and Σ‎jk = 0 for jk) and the dynamic coefficients are homogenous across units (i.e., $Aℓ,jj=A¯ℓ$, $Hj=H¯$, and $Σjj=Σ¯∀j$), then

$Display mathematics$
(64)

and the system simplifies into a single pooled VAR, with only n × (np + m) coefficients to be estimated.28 By stacking the observations first over different units and then over different times, the system can be cast in the standard SUR representation (Eq. 6) and estimated with standard priors (e.g., Normal-Inverse Wishart priors) and techniques.

If the dynamic coefficients are heterogenous across units but no dynamic or static interdependencies are present, then the system breaks up into N independent VARs, with the following SUR representation

$Display mathematics$
(65)

A random coefficients model for Eq. (65) assumes that the coefficients from each unit can be thought of as random draws from a common distribution. For example,

$Display mathematics$
(66)

where αi ≡ vec(Ai). Eq. (66) can be thought of as an exchangeable Bayesian prior on the units’ coefficients—the unit indices i are uninformative, in the sense that they can be exchanged without any loss of information. This approach was proposed by Zellner and Hong (1989) who used a “Minnesota” type prior with fixed and known residual covariance matrix, a diagonal Σ‎a with overall tightness hyperparameter λa, and a plug-in pooled estimator for a. Jarociñski (2010) proposes instead a fully Bayesian model in which all the parameters are treated as random variables, and a sophisticated hierarchical prior approach is adopted. Estimation can be archived using a Gibbs sampler.

If dynamic interdependencies are allowed, the estimation problem becomes more complex. Canova and Ciccarelli (2004, 2009) have suggested solutions based on different Bayesian and cross-sectional shrinkage techniques that can deal with the issue of parameters proliferation that arises in these cases. The approach works by assuming a factor structure for the matrix of coefficients and can be estimated with standard Bayesian priors and a Gibbs sampler. This structural factor approach is flexible and can also be used to estimate Panel VARs with dynamic coefficients that evolve over time, as done in, for example, Ciccarelli et al. (2012) and Canova and Ciccarelli (2013).

# Acknowledgments

We thank the editors of the Oxford Research Encyclopedia of Economics and Finance and an anonymous referee for useful comments and suggestions. We are also grateful to Fabio Canova, Andrea Carriero, Matteo Ciccarelli, Domenico Giannone, Marek Jarociński, Marco del Negro, Massimiliano Marcellino, Giorgio Primiceri, Lucrezia Reichlin, and Frank Shorfheide for helpful comments and discussions. The views expressed in this paper are those of the authors and do not necessarily reflect those of the Bank of England or any of its committees.

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