Bayesian VARs and prior calibration in times of COVID-19

2.1 Bayesian VARs to cope with the COVID-19 pandemic

This section presents the Bayesian VAR model with a generalized error structure of Chan (2020) and uses it as a framework to assess whether a heavy-tailed or heteroskedastic error structure should be preferred for modelling the COVID-19 observations.

Let y _t be a n × 1 vector of variables that is observed over the periods t = 1, …, T. Consider the following generic VAR(p) model:

where a ₀ is an n × 1 vector of intercepts and A ₁, …, A _p are n × n coefficient matrices. Let x t ′ = 1 , y t − 1 ′ , … , y t − p ′ be a 1 × k vector of an intercept and lags with k = 1 + np and A = (a ₀, A ₁, …, A _p )′ is a k × n. Rewrite the VAR in more compact form as y t ′ = x t ′ A + ϵ t , and stack the observations over t = 1, …, T, which yields

where Y, X, and E are of dimensions T × n, T × k, and T × n, respectively. In a standard VAR, the innovations ϵ ₁, …, ϵ _T are assumed to be independent and identically distributed (i.i.d.) as N(0, Σ). More compactly, rewrite E ∼ MN(0, Σ ⊗ I _T ), where Σ is an n × n covariance matrix, I _T is the identity matrix of dimension T, MN denotes the matric-variate normal distribution, ⊗ is the Kronecker product.

The Kronecker structure allows for a more general covariance structure:

where Ω is a T × T covariance matrix. Intuitively, the cross-sectional and serial covariance structures of Y are separately modelled by specifying Σ and Ω, respectively. By choosing a suitable serial covariance structure Ω, the model in (1) and (2) includes a variety of flexible error structures such as heavy-tailed and heteroskedastic innovations.

The main difference between these error structures lies in their treatment of large shocks. Under a heavy-tailed distribution, large shocks receive more probability mass in general, while with heteroskedastic errors large shocks are captured by allowing volatility to change over time. Intuitively, these error structures may limit the effect of extremely large innovations on the parameter estimates because they downscale their contribution through the (implied) serial covariance Ω, see Section 2.3 for details. This idea of downweighing observations to stabilize VAR parameter estimates in the context of the COVID-19 pandemic was first proposed by Lenza and Primiceri (2022).

From the class of heavy-tailed distributions, this paper considers the multivariate t distribution as it can accommodate substantially thicker tails than the normal distribution. Specifically, ϵ _t is assumed to be independently multivariate t-distributed with mean vector 0, scale matrix Σ and degree of freedom parameter ν, or more compactly, ϵ _t ∼ t(0, Σ, ν). As explained in Chan (2020), many non-Gaussian distributions can be used in this framework as they can be written as a scale mixture of Gaussian distributions.^[4] The t distribution can be modelled by letting Ω = diag(λ ₁, …, λ _T ), where each λ _t follows independently an inverse-gamma distribution (λ _t |ν) ∼ IG(ν/2, ν/2), see Geweke (1993). In other words, the scale mixture Gaussian representation of the t distribution reveals that the innovations ϵ _t conditional on λ _t exhibit common but independently distributed shifts of the cross-sectional covariance matrix over time. Note Bobeica and Hartwig (2023) also consider multivariate t errors to model the pandemic observations.

As heteroskedastic error structure, the common drift in stochastic volatility model in the spirit of Carriero, Clark, and Marcellino (2016) is considered with ϵ _t ∼ N(0, Σ · exp(h _t )) following Chan (2020). The log variance follows a stationary autoregressive process of order one:

with ϵ t h ∼ N 0 , σ h 2 , |ρ| < 1 and Ω = diag(exp(h ₁), exp(h ₂), …, exp(h _T )).

Note this covariance structure shares some similarities with the Lenza and Primiceri (2022) approach. Specifically, they assume that the COVID-19 observations scale the cross-sectional covariance matrix persistently by a common factor that decays over time. For the pre-pandemic sample, they assume a constant covariance matrix. In contrast, the model with common stochastic volatility allows volatility to change over the whole sample, not just in response to the pandemic.

The third error structure considered here is a combination of common drift in volatility and multivariate t innovations as in Chan (2020). Specifically, ϵ _t ∼ N(0, Σ · λ _t · exp(h _t )), where λ _t ∼ IG(ν/2, ν/2) and h _t follows an AR(1) process. This model allows for richer (implied) volatility dynamics as opposed to their single ingredients. It can capture both persistent but also abnormally large changes in volatility, thereby allowing the data to decide whether the COVID-19 observations have transient or persistent effects on volatility.^[5] Table 1 summarizes the list of competing Bayesian VARs with a generalized error structure.

Apart from the COVID-19 related macroeconomic disruptions, there is a vast literature documenting a departure from classical Gaussian errors in a wide range of macroeconometric models. In the context of VARs, Christiano (2007) presents evidence that even before the Global Financial Crisis VAR residuals are not normally distributed. Moreover, Chiu et al. (2017), Clark and Ravazzolo (2015) and Chan (2020) document that the data favors stochastic volatility and t-distributed errors over Gaussian errors in a VAR model and that these features improve forecast accuracy. In addition, Cúrdia et al. (2014) and Chib and Ramamurthy (2014) provide evidence that structural shocks in DSGE models exhibit heavy tails.^[6] Antolín-Díaz et al. (2020) show that accounting for both shifts in variances and large shocks improves nowcast accuracy of a dynamic factor model relative to a model with classical Gaussian errors.

Though stochastic volatility is often found to be a more important feature than t-distributed errors for macroeconomic time series, several studies document that inference and forecast accuracy may hinge on allowing for heavy-tailed errors.^[7] For instance, ignoring fat tails may lead to misleading inference about the historical evolution of volatility, especially during the Global Financial Crisis, which is interpreted as a rare event rather than a persistent increase in macroeconomic volatility when allowing for both features.

The Bayesian VAR models that combine stochastic volatility with t-distributed errors of Clark and Ravazzolo (2015) and Chiu, Mumtaz, and Pinter (2017) or with outlier-adjusted errors of Carriero et al. (2022) differ from the models discussed above by their assumption that time-varying volatility and heavy-tailedness is variable-specific rather than common across all variables. Though an individual error distribution may be more realistic and less restrictive, the modelling approach pursued by these papers is prone to be sensitive to the variable ordering due to the triangular factorization of the time-varying covariance matrix in the spirit of Cogley and Sargent (2005). Specifically, this variable ordering sensitivity may give rise to ambiguous conclusions and is especially relevant when volatility clusters idiosyncratically, see Hartwig (2020). It can also affect conditional projections in an economically significant way, see Arias, Rubio-Ramírez, and Shin (2022).

2.2 Prior calibration

Besides the error structure in a Bayesian VAR, another source of parameter instability may be due to an unusually strong and potentially unintended re-calibration of the prior distribution during the pandemic. Specifically, many prior distributions rely on simple variable-specific volatility estimates based on the full sample and as such are sensitive to extreme observations as well. The Minnesota prior is just one example of such a commonly used prior distribution in the context of Bayesian VARs. This section discusses the calibration sensitivity in detail and proposes outlier-robust calibration strategies for the Minnesota prior, which can also be used for calibrating other scale-dependent prior distributions.^[8]

For all considered Bayesian VAR models, this paper assumes a normal-inverse-Wishart prior for the VAR coefficients A and cross-sectional covariance matrix Σ, i.e. Σ ∼ IW(S ₀, ν ₀), and (A|Σ) ∼ MN(A ₀, Σ ⊗ V _a ). As a naïve benchmark prior, a weakly informative but sample-independent calibration is used to specify the normal-inverse-Wishart prior. The hyperparameters of the inverse-Wishart prior are set to ν ₀ = 3 + n and S ₀ = I _n . The prior mean A ₀ is centered at zero and the covariance matrix, V a W , is assumed to be diagonal with ith diagonal element v _a,ii set as:

where κ ₂ = 10 controls the overall strength of shrinkage for the intercepts.

To maintain the Kronecker structure of the prior, this paper adopts a Minnesota prior without cross-variable shrinkage, see Carriero, Clark, and Marcellino (2016) and Chan (2020).^[9] The Minnesota prior is implemented as a normal-inverse-Wishart prior using the same notation as above. For the inverse-Wishart prior, the same sample-independent calibration is used to isolate the effects of a sample-dependent calibration to the prior covariance matrix V a M N of the VAR coefficients only.^[10] V a M N is assumed to be diagonal with ith diagonal element v _a,ii set as:

where s ̂ r , m is the scale estimate of variable r and calibration strategy m, specified below, and κ _1,m controls the degree of shrinkage of the VAR coefficients. For comparability, κ _1,m is calibrated for each method m to yield the maximum marginal likelihood of the BVAR- N model at pre-pandemic times, see Section 3.2.

In practice, s ̂ r , m is calibrated to be the variable-specific sample volatility of the residual from an AR(p) model with Gaussian and homoskedastic errors, see Litterman (1986). Particularly, the root mean squared deviation (RMSD) is used as the sample-specific volatility estimate of variable r:

where x _r is the AR(p) residual of variable r in the standard specification.^[11]

There are two main reasons why the Minnesota prior is based on variable-specific volatility estimates. First, the prior is not scale-invariant, i.e. it matters whether the data is entered in decimals or percentage points. Second, a variable-specific volatility estimate is necessary to express the degree of shrinkage relative to the variability of the underlying stochastic process which is similar in spirit to a ridge regression.

By construction, however, this classical prior calibration strategy is sensitive to extreme observations. Because of this, volatility estimates of some variables may be strongly inflated once the pandemic observations are included in the calibration sample, leading to a tighter prior for the affected variables as compared to the pre-pandemic calibration. This mechanical revision of the prior distribution of the VAR coefficients may be unattractive for several reasons. First, an extremely tight prior for some variables may alter the dynamic propagation of shocks by muting their transmission and breaking links between variables. Second, it is questionable whether a few extreme observations should receive so much weight in revising the prior distribution of the VAR dynamics. Third, it is not clear if and how quickly macroeconomic transmission channels change in response to the pandemic. Therefore, it might be more attractive to adopt a less sensitive prior calibration to not a-priori mute the information in the data through an excessively tight prior.

The unique character of the COVID-19 crisis allows the researcher to easily pin down such a prior by cutting the calibration sample at pre-pandemic times, see, for instance, Schorfheide and Song (2021) and Lenza and Primiceri (2022). However, a drawback of this approach is that it discards all future and potentially useful information from the new normal once macroeconomic variation stabilizes. It also raises the question as to how new observations should be treated when specifying a prior distribution for empirical applications going forward. Moreover, an ad-hoc fine-tuning of the prior calibration sample might also limit the comparability across empirical studies as the appropriate timing to include new observations may vary across applications, countries and the judgement of researchers.

For this reason, this paper searches for outlier-robust calibration strategies that can process all information and yield a prior distribution that is comparable to the standard method in absence of extreme observations. To meet these requirement, the search is guided by looking for robust estimators that yield asymptotically the same point estimate as a mean regression when the data is normally distributed. There are two key elements that may lead to a revision of the variable-specific volatility estimate: (1) the volatility estimator itself and (2) the input series x _r used for the volatility estimator. To investigate the importance of these sources, this paper considers three robust volatility estimators and combines them with two alternative input series to calibrate the Minnesota prior.

The scaled median absolute deviation (MAD) is considered as a first alternative to the RMSD because it is robust to extreme observations, simple to implement and yields asymptotically the same point estimate of volatility when the data is normally distributed, see Rousseeuw and Croux (1993):

where b = 1.4826 is a scaling parameter.

However, a drawback of the MAD is that it takes a symmetric view point on dispersion and has a rather low Gaussian efficiency, see Rousseeuw and Croux (1993). As an alternative, the authors propose the S _n and Q _n statistic as a more efficient and general alternative to the MAD. Specifically, these volatility estimators remain valid even when the input series x _r follows an asymmetric or a heavy-tailed distribution. Therefore, these statistics are considered as further alternatives to the RMSD.

The S _n statistic measures dispersion as the typical distance between observations as opposed to the typical distance from the central value like the MAD:^[12]

where c = 1.1926 is a scaling parameter.

The Q _n statistic is defined as the kth order statistic of the T 2 interpoint distances:

where d = 2.219 is a scaling parameter and k = ( T / 2 ) + 1 2 .^[13]

The second ingredient to make the prior calibration robust to extreme observations is to use a less sensitive input series than the residual of a standard Gaussian AR(p) model (oReg). Here, the first difference of the time series (FD) and the residual of a median AR(p) model (qReg) are considered as an alternative. The median AR(p) model is estimated using the quantile regression framework of Koenker and Bassett (1978).

The advantage of using a first-differenced time series over an AR(p) residual is that the former remains unchanged, while the latter may be revised due to updated model parameters. This invariance property makes the first-differenced time series an ideal input series to assess the resilience of alternative volatility estimators during the pandemic. However, a drawback is that the associated volatility estimate is generally larger but not proportionally larger across the n equations because of different lag properties of the individual variables. Consequently, the prior distribution based on this input series might be somewhat different and the amount of shrinkage may be suboptimal for a VAR model.^[14]

Relative to the first-differenced time series, the residual of a median AR(p) and mean AR(p) regression tend to be more similar as they take the lag structure into account. For Gaussian data, both estimation methods produce asymptotically the same coefficient estimate as the (conditional) median and mean of a normally distributed variable is identical. However, the median regression has the advantage of being more robust and efficient than the mean regression when the error distribution is heavy-tailed, see Koenker and Bassett (1978). Thus, the median AR(p) residual is expected to be less sensitive during the pandemic. Table 2 summarizes all considered prior specifications.

To complete the prior specification, this paper follows Chan (2020) and assumes a uniform prior on (2, 100) for the degree of freedom parameter, i.e. ν ∼ U(2, 100). This prior implies that degrees of freedom may become sufficiently large to approximate Gaussian errors. For stochastic volatility, independent priors for σ h 2 and ρ: σ h 2 ∼ I G ( ν h 0 , S h 0 ) and ρ ∼ N(ρ ₀, V _ρ )1(|ρ| < 1) are assumed, where ν _h0 = 5, S _h0 = 0.04, ρ ₀ = 0.9, and V _ρ = 0.2². These values imply that the prior mean of σ h 2 is 0.1 and ρ is centered at 0.9. Further, an independent prior for the parameter blocks (A, Σ) and Ω is assumed, i.e., p(A, Σ, Ω) = p(A, Σ)p(Ω).

2.3 Bayesian estimation

The Bayesian VAR with covariance structure (2) can be easily estimated under a normal-inverse-Wishart prior for (A, Σ) and an independent prior for the parameter blocks (A, Σ) and Ω, see Chan (2020).

Posterior draws can be obtained by sequentially sampling from (1) p(A, Σ|Y, Ω) and (2) p(Ω|Y, A, Σ). Depending on the covariance structure Ω, additional blocks might be needed to sample some extra hierarchical parameters. In particular, the BVAR with t-distributed errors requires additional sampling of the degree of freedom parameter ν, while ρ , σ h 2 need to be sampled for the BVAR with common stochastic volatility. Following Chan (2020), these parameters are fitted using univariate time series models.

To understand how (extreme) observations are treated in this model, it is useful to investigate the conditional posterior p(A, Σ|Y, Ω). As the prior p(A, Σ) is natural conjugate, the conditional posterior p(A, Σ|Y, Ω) is still normal-inverse-Wishart distributed:

where K A = V A − 1 + X ′ Ω − 1 X , A ̂ = K A − 1 V A − 1 A 0 + X ′ Ω − 1 Y , S ̂ = S 0 + A 0 ′ V A − 1 A 0 + Y ′ Ω − 1 Y − A ̂ ′ K A − 1 A ̂ .

Hence, (A, Σ|Y, Ω) can be sampled in two steps. First, sample Σ marginally from ( Σ | Y , Ω ) ∼ I W ( S ̂ , v 0 + T ) . Then, given the Σ drawn, sample A from

Note under a noninformative prior for the VAR coefficients, i.e. when the inverse of V _A goes to zero, the conditional posterior mean of A converges to the generalized least squared estimator for some conditionally known Ω (weighting matrix).^[15] Thus, the data (Y, X) is not equally informative for the parameters (A, Σ), but their contribution is weighted by Ω. Under t-distributed errors and common stochastic volatility, Ω is a diagonal matrix with generic elements ω t 2 , t = 1 , … , T . In period t, the observations (y _t , x _t ) are weighted by the (implicit) volatility ω t − 1 . Specifically, when ω _t is large, then (y _t , x _t ) is less informative for (A, Σ). Intuitively, the common volatility estimate ω _t becomes large if some of the forecast errors ϵ _t are unusually large, either driven by some variable r or a set of variables.

3.1 Data

This section assesses the impact of the COVID-19 observations on these more flexible Bayesian VARs and various prior calibration strategies by considering a medium and large VAR model for inference and forecasting at quarterly frequency, see Table 3.^[16] The medium VAR consists of six core U.S. macroeconomic variables similar to Lenza and Primiceri (2022).^[17] The large VAR adds 14 additional variables typically used in forecasting exercises such as Bańbura, Giannone, and Reichlin (2010), Carriero, Clark, and Marcellino (2015) and Chan (2020). Both models are estimated over an expanding window from 1988:Q4 to 2019:Q4 until 2022:Q1. As in Lenza and Primiceri (2022), the sample is not extended before 1988:Q4 as inflation has been reacting less to real activity since the 1990s (Del Negro et al. 2020).

Figure 1 shows an excerpt of the data for the variables in the medium VAR from 2006:Q1 until 2022:Q1. From March 2020 onwards, COVID-19 started to spread through the U.S. and led to a widespread shutdown of the U.S. economy. This triggered an unparalleled decline in labor market indicators like employment but also in consumption and output in the second quarter of 2020. With lockdown measures easing over the summer, the U.S. economy rebounded strongly during the third quarter but lost momentum due to a second wave of infections in the fourth quarter. Surprisingly, CPI and PCE core inflation did not particularly react to the economic turmoil at first. In 2021, the economy recovered steadily but inflation started to accelerate substantially which took many economists by surprise. In fact, inflation turned out to be more persistent than initially expected as supply could not catch up with demand due to the strong re-opening of the economy and sustained supply bottlenecks in the global economy. This raises the question whether standard econometric models are still useful in tracking and forecasting these developments and what specifications work best.

Figure 1:

Time-series plots for key U.S. macroeconomic variables.

3.2 (Re)-calibration of the Minnesota prior

A careful calibration of the Minnesota prior – especially in turbulent times – is one important aspect for modelling and forecasting macroeconomic time series with a Bayesian VAR. To study what the proposed calibration strategies entail for the Minnesota prior, the variable-specific volatility estimates of various estimation strategies is compared first. Then, the implications for the optimal degree of parameter shrinkage as well as the sensitivity during the COVID-19 pandemic is assessed.

Table 4(a) reports variable-specific volatility estimates for various calibration strategies of the Minnesota prior at pre-pandemic times (2019:Q4) in absolute and relative terms to RMSD(oReg). Three facts stand out. First, Panel (a) shows that the volatility estimates are generally larger but not proportionally larger when the first difference is used as an input series due to different serial correlation patterns, see column RMSD(FD). For instance, the volatility estimate of the unemployment rate is about 1.4 times higher than the RMSD(oReg), while this factor is about 4.1 for the PCE core price index.

Second, volatility estimates are similar when the AR(p) residual of the mean or median regression is considered, see column RMSD(qReg). This is because both regression techniques produce comparable coefficient estimates in the pre-pandemic sample.

Third, the RMSD generally yields higher volatility estimates than the robust volatility estimators. Overall, the robust estimators produce fairly similar estimates. Moreover, a larger difference between RMSD and robust estimators suggests that the input series for variable r exhibits some non-normal behavior. This difference is generally larger when the first difference is used as the input series. Therefore, the S _n and Q _n statistic should be preferred over the MAD in these cases as they are more efficient and may be more informative due to their ability to account for skewness in the distribution.

What are the implications of these characteristics on κ 1 , m * , the optimal degree of shrinkage for the VAR coefficients? Table 4(b) shows that κ 1 , m * is generally larger for the RMSD estimator and first-differenced input series as compared to analogous robust metrics in both BVARs.^{[18] , [19]} This difference can be rationalized by noting from (5) that if the ratio of variable-specific volatility estimates of method j and i is constant for all r variables then the optimal κ _1,j and κ _1,i are proportional:

where σ ̂ r , ⋅ is the variable r specific volatility estimate of method j and i respectively.

The second row in each subtable of Panel (b) shows that the volatility-adjusted hyperparameter κ ̂ 1 , m * is fairly close to its optimal value κ 1 , m * . This indicates that the bulk of the variation can be explained by adjusting κ 1 , m * for an average level shift between volatility estimates (SHIFT). This approximation becomes more accurate the more proportional the variable-specific volatility estimates are across calibration methods. The degree of proportionality may be measured by the standard deviation of relative volatility estimates (STD). The STD statistic shows that residual based strategies yield more proportional volatility estimates than the first-differenced one, and, hence a more comparable prior calibration. Taken together, this is an encouraging empirical fact as established values for κ 1 , m * in various empirical applications may be easily mapped by (10) into an alternative calibration strategy without necessarily re-optimizing this tuning parameter.

Next, the influence of the pandemic observations on the Minnesota prior is discussed. Table 5 (a) reports the ratio of volatility estimates from 2022:Q1 to 2019:Q4 and (b) presents statistics summarizing the average change of the ratio of volatility (MEAN) and dispersion of the ratio of volatility measured against an unchanged ratio (DEV).

The table shows that the prior based on the standard Minnesota calibration strategy is substantially altered as some variable-specific volatility estimates are strongly inflated, see column RMSD(oReg). Overall, the prior distribution becomes disproportionately tighter for the VAR coefficients associated with indicators of real activity and labor market conditions (except hourly earnings), while it is hardly affected for price and financial variables (except real M2 stock). More specifically, the volatility estimate of employment is over 6 times larger than its pre-pandemic value, for instance. This means that the updated parameters set an extremely tight prior for the own and cross-equation dynamic lags of employment, roughly 42(≈6.46²) times tighter than its pre-pandemic value.

To further illustrate the impact of a prior re-calibration on the VAR dynamics, Figure 2 shows impulse response functions to a one standard deviation shock in employment of the Gaussian BVAR estimated until pre-pandemic times but calibrating the standard Minnesota prior with data until 2019:Q4, 2020:Q4 and 2022:Q1, respectively.^[20] The figure shows that the revised prior distribution has a decisive impact on the posterior dynamic propagation of the employment shock in both the medium and large VAR.^[21] It a-priori mutes the internal propagation of the shock by shrinking cross-dynamic relations to zero and thereby breaks historical links between variables such as employment and prices. Thus, the COVID-19 shock not only has a decisive impact on the estimation stage of a standard Bayesian VAR but also on the prior calibration stage, which is an important ingredient for modelling macroeconomic transmission channels during the pandemic.

Figure 2:

Impact of prior re-calibration on VAR dynamics. Impulse response functions to a one standard deviation shock in employment of the BVAR- N estimated until 2019:Q4 with RMSD(oReg) Minnesota prior calibrated on samples until 2019:Q4, 2020:Q4 and 2022:Q1, respectively. The thick lines are posterior median estimates and the colored area depicts the 16%–84% credible interval.

In contrast, volatility estimates based on robust scale estimators (MAD, S _n , Q _n ) coupled with robust input series (first difference or median AR(p) residual) are hardly affected by the pandemic observations, see Table 5(a) and (b). Thus, the Minnesota prior based on a two-dimensional robustified calibration strategy remains comparable before and throughout the pandemic and also does not a-priori mute internal propagation of shocks in a VAR when the prior is revised during the pandemic.

3.3 Model and prior comparison

To discriminate between these Bayesian VAR models and prior specifications, the marginal likelihood is used as a formal Bayesian model selection criterion following Chan (2020).^[22] The marginal likelihood under model M _k,m is defined as

where p(y|θ _k , M _k,m ) is the likelihood function, p(θ _k |M _k,m ) is the prior distribution, θ _k is a model-specific parameter vector and m is a prior calibration. For the BVAR- N , the analytical formula is used to compute the marginal likelihood, while for the other BVAR models no closed form exists and Chib’s method (Chib 1995) is used instead.

Specifically, if the marginal likelihood under model M _i,s is larger than under M _j,q , then the data is more likely under model M _i,s as compared to M _j,q . Given all models are a-priori equally likely, the weight of evidence between two models can be measured by the Bayes factor defined as the ratio of marginal likelihoods, see Chan (2017). Table 6 presents log Bayes factors against the BVAR- N with weakly informative prior before and during the pandemic for the medium VAR, for the large VAR see Table A.1 in Appendix A.^[23]

Before the pandemic, all Minnesota priors are decisively favored over the naïve benchmark prior, see Panel (a). For instance, the average log Bayes factor is 72 in the medium BVAR which means the BVARs with Minnesota prior are 1.85 × 10³¹ more likely than those with a naïve benchmark prior. This overwhelming support for shrinking more distant VAR coefficients to zero is not surprising as more distant lags are relatively less informative for current dynamics. Moreover, the sample fit of the standard RMSD(oReg) and robustified calibration strategies is broadly similar. The figures based on AR(p) residuals are almost identical across volatility estimators, while those based on the first difference are somewhat smaller. Thus, two-dimensional robustified calibration strategies do not sacrifice in-sample fit for robustness.

Across all prior specifications, the BVAR-t-CSV is the best performing model albeit the Bayes factor is just immaterially larger than that of the BVAR- N -CSV model. Both models have, however, a substantial margin over the BVAR-t model, while the latter clearly outperforms the standard Gaussian BVAR- N model. Thus, both error extensions are favored over standard Gaussian errors by the data. But on a single ingredient basis, stochastic volatility is a more important feature than a heavy-tailed error distribution. This is line with the findings of Cúrdia, Del Negro, and Greenwald (2014), Chiu, Mumtaz, and Pinter (2017) and Chan (2020) who also look at the U.S. but consider a different data set and an earlier period.

How does the pandemic affect the model fit across these BVARs? Panel (b) shows that while the overall ranking across BVARs remains unchanged, the relative distance between Bayes factor with respect to the best performing model changes markedly before and during the pandemic. For the naïve benchmark prior, the Bayes factor falls by about 200 and 290 log points in the medium and large BVAR- N , respectively.^[24] This indicates that the classical Bayesian VAR is poorly equipped to deal with these extreme observations. Moreover, the Bayes factor increases by 2.91 log points for BVAR-t, while it falls by 4.28 for the BVAR- N -CSV in the medium VAR. This relative change is even more pronounced in the large VAR – with an increase of 9.9 and a decrease of 13.82 log points, respectively. This suggests that the heavy-tailed error extension might be better equipped to capture the extreme variation during the pandemic than the common stochastic volatility specification.

Turning to prior sensitivity, the Minnesota calibrations based on the AR(p) residual and RMSD volatility estimator yield a substantially lower in-sample fit as compared to the robust calibration strategies. For instance, the Bayes factor between MAD(qReg) and RMSD(oReg) of the BVAR-t-CSV is 37.72 in the medium and 115.43 in the large model, suggesting that the robust prior calibration is decisively preferred. In addition, the naïve benchmark prior in the medium BVAR with Gaussian and t-distributed errors is now decisively favored over these non-robust Minnesota calibrations. Furthermore, Bayes factors of robust calibration strategies are very similar across specifications. Thus, the more general S _n or Q _n statistic of Rousseeuw and Croux (1993) do not provide a substantial improvement over the scaled MAD in terms of model fit.

To complement this analysis and evaluate the impact of prior revisions on model fit in and over time, Figure 3 shows Bayes factors of the BVARs estimated with re-calibrated prior against a 2019:Q4 fixed prior obtained over an expanding estimation window from 2019:Q4 to 2022:Q1. The figure shows that re-calibrating the Minnesota prior with the standard calibration strategy (red thick colored line) has a decisive negative effect on the overall in-sample fit starting in 2020:Q2. In contrast, a fully robust calibration hardly affects the Bayes factor and thus yields about the same model fit as fixing the prior.^[25]

Figure 3:

The costs of re-calibrating the prior distribution. Log Bayes factor between BVARs with re-calibrated and 2019:Q4 fixed prior distribution estimated over an expanding sample from 2019:Q4 to 2022:Q1.

In the remainder of the paper, empirical results are presented for the MAD(qReg) Minnesota prior. This calibration is chosen as the preferred fully robust alternative to the RMSD(oReg) for three reasons. First, it is resilient to extreme observations – not only those related to the pandemic. Second, it yields asymptotically the same point estimates as the RMSD(oReg) when the data is normally distributed and not contaminated by outliers. Third, it is simpler to implement than similar strategies with the S _n and Q _n statistic.

3.4 Macroeconomic tail risk and volatility

Model diagnostics strongly reject Gaussianity, but how do the pandemic observations affect macroeconomic tail risk and volatility? This section discusses the implications and adds the recently proposed explicit common volatility BVAR model of Lenza and Primiceri (2022) as another benchmark (denoted as BVAR-LP).^[26]

Figure 4 shows how macroeconomic tail risk and volatility are affected by the pandemic observations.^[27] Recall, the lower the degree of freedom parameter ν, the stronger the t distribution departs from Gaussianity and the heavier the tails. Panel (a) shows that the posterior of ν peaks around 10 before the pandemic in all t error models, indicating already considerable tail risk.^[28] Once the pandemic observations are included, the posterior distribution becomes much tighter and peaks around 3.5 in the medium (red area) and 5 in the large model (blue area), respectively. For the BVAR-t, this is not surprising as this is the model’s main channel to accommodate more extreme innovations. However, the much sharper identification in the BVAR-t-CSV means that the tail risk increased strongly besides changes in macroeconomic volatility.

Figure 4:

Macroeconomic tail risk and volatility during the pandemic. BVARs with MAD(qReg) Minnesota prior estimated until 2019:Q4 and 2022:Q1, respectively. The thick line is the posterior median and the dashed lines are the 16%–84% credible interval.

The nature of volatility during the pandemic hinges on the error structure and dimension of the VAR models, see Panel (b) and (c). Both the BVAR- N -CSV and BVAR-LP interpret the pandemic shock as an enormous increase in macroeconomic volatility in 2020:Q2, about 35 standard deviations in the BVAR-LP and half this size in the BVAR- N -CSV. After the initial burst, the volatility profiles start to diverge. Volatility remains high for an extended period in the stochastic volatility specification, while in the BVAR-LP the profiles differ across VAR dimensions. In the medium model, volatility decays slowly while in the large model volatility jumps back immediately to a more moderate level.^[29] Thus, the error structure and cross-sectional information plays an important role for identifying the nature of volatility with these VARs featuring no tail risk.

In contrast, the BVAR-t-CSV – which allows for both tail risk and stochastic volatility – assigns a completely different interpretation to the COVID-19 shock. The common stochastic volatility series is almost entirely flat during the pandemic period. This means that the model interprets the pandemic as a rare event and not as a (persistent) increase in macroeconomic volatility.^[30] Since fat tails are an important feature of the data before and during the pandemic, ignoring it may lead to misleading inference about the nature of volatility and other objects of interest. This property was also previously pointed out by Jacquier, Polson, and Rossi (2004), Cúrdia, Del Negro, and Greenwald (2014), Chiu, Mumtaz, and Pinter (2017) and Chan (2020) for different data sets and sample periods.

3.5 Model (in)stability and forecasting

Having discussed the implications of the COVID-19 shock for tail risk and volatility, this section focuses on model (in)stability and forecast properties. Figure 5 shows scatter plots of the posterior mean VAR coefficients and residual correlation matrix obtained in 2019:Q4 against 2022:Q1. Panel (a) shows that the VAR coefficients in the BVAR- N are substantially revised with intercepts and coefficients measuring (cross)-dynamics of employment, the unemployment rate, consumption and output being particularly affected. For instance, the first-order autoregressive coefficient of employment changes from 1.28 in 2019:Q4 to −0.12 in 2022:Q1. In contrast, the VAR parameters in the more flexible medium BVARs are hardly influenced by the extreme observations. This robustness also extends to the large BVARs which however exhibit some revision in the VAR parameters, but nothing extraordinary as compared to the BVAR- N .

Figure 5:

Change in common VAR parameters. Common VAR parameters in 2019:Q4 (x-axis) against 2022:Q1 (y-axis) for various BVARs with MAD(qReg) Minnesota prior.

The pandemic observations induce a drastic and long-lasting change in the residual correlation structure in both BVAR- N models, see Panel (b). For instance, the average correlation between the residual of the unemployment rate and those of employment, consumption and output changes from −0.37 in 2019:Q4 to −0.91 in 2022:Q1 in the medium BVAR. This abrupt revision is solely driven by the very synchronized co-movement of these variables in response to the COVID-19 shock. The residual correlations in the BVARs with a more flexible error structure, however, are only mildly influenced by the pandemic observations as they downweigh the information from these extreme realizations. Therefore, inference based on the traditional Gaussian BVAR may be overshadowed by the extreme size of the COVID-19 shock for an extended period of time as its impact on the residual covariance matrix cannot wash out easily at quarterly frequency.

Turning to forecast properties, parameter revisions and prior re-calibrations also have a decisive impact on unconditional projections. Focusing first on the impact of parameter revisions, Figure 6 shows unconditional projections starting in 2021:Q1 for employment, GDP and CPI inflation based on BVARs with MAD(qReg) Minnesota prior and estimated over different samples, stopping in 2019:Q4, 2020:Q4 and 2022:Q1.^[31] This exercise reveals several important insights into how well these models are able to digest the COVID-19 shock and whether they can put this information to good use in hindsight.

Figure 6:

Impact of re-estimated parameters on unconditional forecasts starting 2021:Q1. BVARs with MAD(qReg) Minnesota prior estimated until 2019:Q4, 2020:Q4 and 2022:Q1, respectively. The thick line is the posterior median and the colored area depicts the 16%–84% credible interval.

First, parameter instability may cause serious problems for forecasting with Gaussian VARs. Looking at the predictions for employment, for instance, the updated parameters in 2020:Q4 trigger oscillations in the projections of the medium-sized model (blue dotted line).^[32] Though these oscillating dynamics eventually wash out as more information become available (see predictions with 2022:Q1 parameters, green dashed line), the extreme observations also alter long-run forecasts of the variables. For instance, gross domestic product is projected to stagnate in the long run, while all of the more flexible BVARs suggest a continuation of the pre-pandemic growth pace. This complements the findings in Bobeica and Hartwig (2023) and Carriero et al. (2022) and Lenza and Primiceri (2022) and Schorfheide and Song (2021) by the fact that the COVID-19 observations may not only distort the short-run dynamics but also have a persistent impact on long-term predictions.

Second, the more flexible BVARs are robust to the COVID-19 shock. They produce stable forecasts in the pandemic period and are able to track the transmission of the shock through the U.S. economy by orderly revising the VAR structure. At first, the projections in both medium and large VARs are hardly affected by the incorporation of four additional quarters, see projections with 2019:Q4 parameters (red line) versus 2020:Q4 parameters (blue dotted line). However, as time passes, the VAR better understands the nature of the COVID-19 shock and unconditional projections are more visibly revised (green dashed line). In fact, the large VAR is able to closely track the actual developments of all six core U.S. macroeconomic variables whereas the medium model misses some key patterns (e.g. underestimation of inflationary pressure). Thus, this analysis suggests that the COVID-19 shock might have altered macroeconomics dynamics and that extreme observation-robust BVARs are well equipped to track these changes if they feature a sufficiently large cross-sectional information set. Furthermore, traditional Gaussian VARs cannot easily cope with such extreme observations and induce non-sensible parameter revisions instead.

Third, the COVID-19 shock may substantially inflate density forecasts intervals when the VAR only allows for common time-varying volatility and lacks a sufficiently large cross-sectional dimension. Specifically, forecasts based on the medium BVAR- N -CSV and BVAR-LP may exhibit extremely wide forecast intervals whereas the predictive intervals of the large models are comparable to those featuring t errors. This complements the evidence in Carriero et al. (2022) by the fact that though predictive intervals based on variable-specific volatility structure may become excessively wide after the pandemic shock, the same is not necessarily true for a common volatility structure.

The re-calibration of the standard Minnesota prior also affects forecast properties and reveals a trade-off for the prediction of real and nominal variables.^[33] To illustrate this, Figure 7 shows unconditional projections stating in 2021:Q1 for employment, GDP and CPI inflation based on BVARs with the fully robust MAD(qReg) and standard RMSD(oReg) Minnesota prior calibration estimated until 2020:Q4.^[34] As discussed in Section 3.2, the RMSD(oReg) calibration sets a very tight prior on the VAR coefficients for most real variables. For forecasting purposes, this re-calibration may be beneficial when predicting real variables at the beginning of the pandemic as the dampened internal propagation of shocks results in less overshooting predictions. However, the same channel prevents the large VAR from tracking the mounting inflationary pressure in 2021, making it a less attractive choice for understanding and predicting current inflation dynamics.

Figure 7:

Impact of re-calibrated priors on unconditional forecasts starting 2021:Q1. BVARs with MAD(qReg) and RMSD(oReg) prior, respectively, estimated until 2020:Q4. The thick line is the posterior median and the colored area depicts the 16%–84% credible interval.

The previously discussed forecast properties are supported by a very short forecast evaluation from 2020:Q1 until 2022:Q1, which should be viewed as suggestive evidence. Table 7 shows root mean squared forecast errors (RMSFE) of a pseudo-out-of-sample forecast evaluation of unconditional forecasts for the one-quarter-ahead and one-year-ahead horizon.^[35] This exercise underlines the importance of using robust BVARs and carefully calibrating the prior distribution during the pandemic. It shows that the robust BVARs coupled with a large cross-sectional information set is the group of superior forecasting models. However, there is no clear superior forecasting approach among the robust BVARs as the predictive accuracy is broadly similar. Rather, the optimal BVAR depends on the variable and forecast horizon of interest as well as the chosen prior calibration. Nevertheless, the large BVAR-LP yields the most accurate forecast overall.

Though, this forecast evaluation is somewhat inconclusive about the best error extension and prior calibration for a BVAR during the pandemic period, it can still provide some valuable lessons for forecasters. Going forward, the robust BVARs with robust prior calibration may have an edge over the standard method in forecasting both real and nominal variables at the current juncture. This is because the VAR forecasting information set does no longer include the very large shocks since the start of the pandemic, which led to an overshooting forecast with the robust prior. Therefore, it may become more important to exploit cross-sectional information rather than a-priori muting transmission of real variables in order to obtain accurate forecasts in the future.