1 Introduction
Empirical macroeconomics has seen an upsurge of interest in modeling the tails of predictive distributions. A recent influential paper is Adrian et al. (ABG, 2019), hereafter ABG, which investigated the impact of financial conditions on the conditional distribution of GDP growth and found it to be important in the lower quantiles. Both prior and subsequent to ABG, a large literature has emerged using quantile regression methods to forecast tail risks to economic growth (see, among many others, Adrian et al. (2018); Cook and Doh (2019); De Nicolò and Lucchetta (2017); Ferrara et al. (2019); Giglio et al. (2016); GonzálezRivera et al. (2019); Delle Monache et al. (2020); PlagborgMøller et al. (2020); Reichlin et al. (2020); Figueres and Jarociński (2020); and Mitchell et al. (forthcoming)). Other studies consider tail risks to other macroeconomic variables such as unemployment or inflation (e.g. Galbraith and van Norden (2019), Kiley (2018), Ghysels et al. (2018), Manzan (2015), Gaglianone and Lima (2012), Korobilis (2017), Manzan and Zerom (2013, 2015), Korobilis et al. (2021) and Pfarrhofer (2021)).
The existing literature, with few exceptions, uses quantile models for a single variable of interest. These models are specified conditional on a specific quantile and assume a linear relationship between the predictors and the quantile function of some outcome variable.^{1}^{1}1Examples of exceptions include Korobilis et al. (2021) and Pfarrhofer (2021), which assume time variation in the quantile regression coefficients. However, even these papers are singleequation and assume particular parametric forms for the timevariation. A multipleequation exception is Adrian et al. (2018), which exploits information in the term structure for an empirical application involving linear panel quantile regression. For macroeconomic data this assumption might be warranted in normal times but in turbulent times it could be that regression relationships change or turn nonlinear. Moreover, often several variables rather than a single one are of interest, and in these cases a joint model would be preferable. These observations motivate the model we develop in the present paper.
In contrast to much of the existing literature we propose a nonparametric model which involves multiple equations and allows for assessing whether the quantile response function is linear or unknown and possibly highly nonlinear. In particular, the model we propose is a multicountry, nonparametric quantile regression, which we then use to investigate growth at risk in a panel of 11 advanced economies.
The justification for adopting nonparametric methods is provided by Huber et al. (2020) and Clark et al. (2021b)
, which found Bayesian nonparametric vector autoregressions (VARs) to be able to successfully model the tails of predictive densities of macroeconomic variables in a flexible and accurate manner. These papers found that Bayesian Additive Regression Trees (BART) are an effective nonparametric method that is particularly useful in crisis times (e.g., the Financial Crisis or the Covid19 pandemic) when growth at risk issues are of particular importance. However, in normal periods, the predictive gains from using BART are more muted (and sometimes negative). In the present paper we extend the BART regression methods used in these papers to the quantile BART case. Since the predictive gains of BART vary over the business cycle, we assume that within each quantile, the response function is a convex combination of a linear model and some unknown nonlinear function, which we approximate using BART. Studies such as
Taddy and Kottas (2010) have developed other Bayesian approaches to nonparametric modelbased quantile regression.The justification for use of a multicountry model is that a panel dimension can often improve forecasts with respect to single country models; see, among many others, Bai et al. (2020) and Feldkircher et al. (2021). Moreover, and specifically for the quantile case, macroeconomic data sets are short, leading to a small number of observations in the tails of the distribution. We develop a model for the quantile that includes a factor that summarizes the available crosscountry information at that quantile. In addition, as indicated below, our Bayesian model specification has features that allow information from other countries to inform estimates for a given country. Exploiting this crosscountry information through a pooling prior improves predictive accuracy by parsimoniously including international information to inform coefficients associated with domestic quantities.
We then develop Bayesian Markov Chain Monte Carlo (MCMC) methods for estimation and forecasting with our quantile factor BART model (QFBART). These methods are scalable to large panels with a potentially large number of exogenous regressors.
In terms of empirical results, our proposed models commonly improve on the benchmark single country linear quantile model in recursive growth forecast comparisons, more so in the tails than near the center of the distribution. Importantly, estimating the weight assigned to the BART component of the model as compared to the linear component is helpful to forecast accuracy. The estimated combination weight is smaller (i.e., the model is more linear) for the 25 and 75 quantiles than in the tails, i.e., for the 5 and 95 quantiles. Moreover, some form of international information definitely pays off (either via the new pooling prior, or by outright including nondomestic series). The effects of the common (volatility) factor are also relevant, as it seems to explain a large fraction of the forecast error variance in most countries, in particular in the tails. A shock to this factor, which can be interpreted as an uncertainty shock, has different (stronger negative) effects in the left tail than the right tail of the growth distribution. In the left tail, the BART piece with an estimated weight tends to slightly mitigate the effects of the shock. Finally, a financial shock in the US spills over to other countries. There is asymmetry in the responses in the sense that a positive shock affects the growth quantiles, whereas a negative shock’s effects are not as sharp. Moreover, the effects and asymmetry are more pronounced in the 201019 period than earlier.
The remainder of the paper is structured as follows. The next section defines and motivates our QFBART model including its prior and discusses MCMC estimation. The third section contains our empirical work while the fourth section concludes the paper.
2 A multicountry nonparametric regression model
We model the joint distribution of (for simplicity, demeaned) output growth for a panel of
countries. These are stored in an dimensional vector with denoting time output growth in country . Domestic real activity might depend on the lags of as well as other, exogenous factors. These predetermined quantities are included in a dimensional vector . We adopt a notational convention where is structured such that domestic quantities for country are always ordered first, followed by all nondomestic variables. We assume that follows a quantile regression model which, for the quantile, is given by:(1) 
with denoting unknown countryspecific functions and is a dimensional vector of regression coefficients. is a quantile and countryspecific parameter that controls how much weight is placed on the nonlinear part of the model. The case would correspond to a fully nonlinear model whereas would be a (conditionally) linear quantile regression specification. Contemporaneous relations across the elements in are introduced through a static factor model with denoting the countryspecific factor loading and the corresponding international factor. Finally, follows an asymmetric Laplace distribution (ALD) scaled by a parameter with its quantile being equal to zero.^{2}^{2}2The density of the ALD is given by , where
is the check/loss function and
the usual indicator function. For details on the correspondence between the Bayesian and classical approach to inference in quantile regression, see Yu and Moyeed (2001).We assume that the latent factor is uncorrelated over time and arises from a Gaussian distribution:
with being a (logarithmic) variance that evolves according to an AR(1) process:
Here we let denote the unconditional mean, the autoregressive parameter, and the error variance of the logvolatility process.
This model possesses several features which should improve not only its predictive capabilities but also allow for additional inferential opportunities. First, the presence of the quantilespecific weights allows for databased selection of the degree of nonlinearities across quantiles. Recent literature indicates that, in the lower tails of the distribution of output growth, macroeconomic relations change and might be subject to substantial nonlinearities. While such nonlinearities may be important in the extremes of a distribution, linear models might describe the behavior well in tranquil periods of the business cycle (e.g., in the center of the distribution). Our model allows for this by setting the corresponding weights appropriately. Second, our model allows for lagged relations across countries. The key point to notice is that these dynamic interdependencies can differ across quantiles. For instance, it could be that in the presence of a global adverse economic shock, crosscountry dependencies are more important than in tranquil times. Third, the presence of a common static factor that exhibits conditional heteroscedasticity can capture contemporaneous relations across the elements in for a specific quantile. Moreover, since the factor is conditionally heteroscedastic, it can also control for sudden shifts in the conditional variance of the dependent variables. Inclusion of this stochastic volatility factor allows us to control for unobserved heterogeneity, a feature which might be extremely important during periods such as the recent Covid19 pandemic (see, e.g., the discussion in Carriero et al. (2021)).
The model in Eq. (missing) is quite general and nests several commonly used alternatives in the literature. For instance, setting for all , and setting such that it includes only the first lag of GDP plus a (lagged) measure of financial conditions yields a model very closely related to the one proposed in Adrian et al. (2019). Notice that this model essentially rules out crosscountry relations. Setting for all yields a nonparametric quantile regression model which, depending on and the choice of , allows for crosscountry relations in a flexible manner. Setting returns the linear quantile regression model but with a common volatility factor.
2.1 Approximating the unknown functions using BART
We treat the function as unknown and approximate it using BART (Chipman et al., 2010). Though other alternatives are possible, BART has been successfully employed in economics for forecasting financial time series in Huber and Rossini (2020), nowcasting GDP in selected European economies in Huber et al. (2020), and tail forecasting of output, inflation, and unemployment in Clark et al. (2021a). BART is a sumoftrees model that approximates by summing over many individual trees that all take a simple form and act as “weak learners.” The BART approximation for is given by:
with denoting a tree function that is determined by a tree structure and a vector of terminal node parameters . This terminal node parameter vector has dimension .
The tree structure consists of multiple decision rules that ask whether a covariate exceeds a threshold and, according to these simple binary rules, produces (disjoint) partitions of the input space. These take the form or , with denoting the element of and being a splitting/threshold value. Sequences of these decision rules lead to a terminal node coupled with a corresponding terminal node parameter in .
When is large, the BART approximation is prone to overfitting if no further regularization is introduced. Chipman et al. (2010) use regularization priors to force the trees to be simple. We achieve this through shrinkage priors on the tree structure and the terminal node parameters. Following Chipman et al. (1998), the prior on is obtained by constructing a treegenerating stochastic process. The prior comprises of three aspects. First, tree complexity ultimately depends on the depth of intermediate nodes . If is large, the tree is complex and thus might overfit the data. To force the individual trees to be simple, we assume that a given node at depth
is nonterminal with probability proportional to:
where is between and and . Notice that this probability decreases in : growing more complicated trees becomes unlikely if is already large. The amount of shrinkage is controlled by and
. These hyperparameters are often set to
and, implying that trees with two or three terminal nodes receive over 80% of total prior probability.
Chipman et al. (2010) found that, for over data sets, this choice performs well, and extensive crossvalidation for andonly improves predictive accuracy by small margins. The second and third aspect are concerned with how decision rules are constructed. To this end, we use discrete uniformly distributed priors to select the variables showing up in the decision rule as well as a uniform prior over the splitting/threshold values.
The second source of shrinkage is a Gaussian shrinkage prior on , the element of . Chipman et al. (2010) recommend scaling the prior using the range of the data. More specifically, let and denote the minimum and maximum of the observed data in country . The corresponding Gaussian prior is then given by:
with being a prior scaling parameter, typically set equal to . The prior implies that if the number of trees
is large, the prior variance decreases and the amount explained by a single tree is decreased. This is consistent with the notion that each tree acts as a “weak learner,” explaining only a small share of variation in the response variable, but the ensemble model provides sufficient flexibility to capture even complicated conditional mean relations. Another feature, noted by
Huber et al. (2020), is that the prior variance increases in the range of the data. Hence, if outliers arise, the prior becomes increasingly loose and allows for more flexibility in terms of capturing observations far outside the range of past data.
The priors on the tree structures and the terminal node parameters constitute the main ingredients of BART. Since our model also features a linear part and additional coefficients, we also need to specify priors on and . We discuss them in the next subsection.
2.2 Priors on the remaining coefficients of the model
On the coefficients we use a horseshoetype prior on each element :
where
denotes a halfCauchy distribution,
is a coefficient and quantilespecific scaling parameter, and is a global shrinkage parameter that is common to all coefficients. Notice that the presence of introduces dependencies across coefficients (including across countries) and across quantiles. The key advantage is that the presence of the local shrinkage parameters allows the detection of signals (i.e., nonzero or heterogeneous over the cross section) even if is close to zero.The prior mean pools information over the cross section. In our hierarchical specification, it is estimated from the data using a Gaussian prior for the domestic variables, and deterministically set to zero for nondomestic quantities:
The parameter is the prior variance of the common mean, which we set to a weakly informative value of for the empirical application. We refer to this prior as the pooled horseshoe (HSP), while setting the common mean to a zero vector of size yields the conventional horseshoe (HS) that we consider as an alternative.
For the factor loadings , we use a set of independent Gaussian priors for all :
Note that is a scalar and, hence, we use this relatively noninformative prior rather than a prior such as the HS which is used to avoid overparameterization as might occur with high dimensional parameters.
On the weights we consider a Uniform prior:
(2) 
This prior introduces no particular prior information on the amount of nonlinearities. If we wish to be informative on we can also use a Beta prior and specify the hyperparameters appropriately.
The remaining coefficients of the model relate to the error term. Kozumi and Kobayashi (2011) write the ALD using a scalelocation mixture of Gaussians:
with , and . On the scale parameter we use an inverse Gamma prior:
with the relatively noninformative choices of and .
This completes the prior setup. In the next subsection we briefly discuss the Markov chain Monte Carlo (MCMC) algorithm used to carry out estimation and inference.
2.3 Full conditional posterior simulation
We use Markov Chain Monte Carlo (MCMC) techniques to obtain a draw from the joint posterior of the latent quantities and coefficients of the model. Specifically, the following steps of the algorithm are carried out for each equation (i.e., country) and quantile :

Sampling from and . The full conditional posterior of the tree structures takes no wellknown form. Chipman et al. (2010) propose a Bayesian backfitting strategy to set up a Metropolis Hastings (MH) algorithm to sample the trees individually, conditionally on the other trees. This step is carried out marginally of
. The terminal node parameters can then, under our conjugate prior, be simulated from a set of independent Gaussian distributions that take a wellknown form.

Sampling
. The regression coefficients are, conditional on the remaining parameters and latent states, simulated from a multivariate Gaussian posterior distribution with known moments:
is a dimensional response vector with element given by , is a matrix with typical row , and is a prior variance matrix with main diagonal element . The prior mean collects the estimated common means in the corresponding position of the domestic variables with the remaining elements being zero for the HSP prior, while for the conventional HS prior.

Sampling from . The posterior distribution for the nonzero elements for of the prior mean for HSP is , with moments

Sampling from : The full conditional posterior of takes no wellknown form. Since the support of is bounded and the target density univariate, we adopt a slice sampler (see, e.g., Neal (2003)) that is straightforward to implement and mixes well.

Sampling from . The factor loadings are obtained by simulating from univariate Gaussian conditional posterior distributions:
denotes the response vector with typical element given by , and has typical element .

Sampling from . Kozumi and Kobayashi (2011) show that the conditional posterior of the scaling parameter
follows an inverse Gamma distribution:
with , , and .

Sampling from . For each , we simulate from a generalized inverse Gaussian (GIG) posterior distribution:
where and .^{3}^{3}3
The generalized inverse Gaussian (GIG) distribution is parameterized such that a random variable
has probability density function
. 
Sampling from . The local, coefficientspecific scaling parameters are simulated using the scheme outlined in Makalic and Schmidt (2015). Conditional on auxiliary shrinkage parameters and , the posterior of is inverse Gamma distributed:
These steps relate to the quantities we have to simulate for each country (or equation) and quantile. Next we turn to the quantities that we simulate per quantile and thus pool over countries.

Sampling from . For each , we simulate from a sequence of independent Gaussian posterior distributions as follows:
whereby is a vector with , , and .

Sampling from and . We sample the full history of logvolatilities and the parameters of the state evolution equation using the efficient sampler proposed in Kastner and FrühwirthSchnatter (2014). This algorithm samples the logvolatilities, conditional on everything else, all without a loop.
The final step refers to the global shrinkage parameter of the horseshoe prior. This step pools information across all equations and quantiles. Since we rely on auxiliary random variables to obtain a wellknown full conditional posterior distribution, we first simulate from and then from .

Sampling from and . The conditional posteriors of the global shrinkage parameter and the auxiliary global parameter are, respectively, inverse Gamma distributed:
This completes our MCMC algorithm. In all our empirical work we repeat the different steps times and discard the first draws as burnin. One key advantage of the present algorithm is that it is scalable to larger data sets (i.e., including more countries, additional endogenous variables, or more covariates) because, conditional on the factors and , the different posterior quantities are independent across equations and quantiles.
3 Empirical results
In this section we first investigate whether our modeling approach improves upon a set of simpler, nested alternatives by means of a forecasting horse race. We then focus on international growth at risk dynamics in two ways. First, we analyze how GDP growth reacts to changes in the common factor. Afterwards, we focus on how a shock to US financial conditions spills over to the other economies in our sample.
3.1 Data overview, competing models and forecasting design
Our sample runs from 1975Q1 to 2020Q4. We use annualized quarterly growth rates of GDP data from the Main Economic Indicators (MEI) database, maintained by the OECD, and the composite indicator of systemic stress (CISS) by the European Central Bank (ECB). For data availability reasons we include Austria (AT), Denmark (DK), Finland (FI), France (FR), Germany (DE), Italy (IT), Netherlands (NL), Spain (ES), Sweden (SE), United Kingdom (UK), and the United States (US).
We estimate the models for . For each model we consider two different choices for the covariates. The first, which we label CISS, includes the CISS and a single lag of in , implying that . The second includes crosscountry information in by including the first lag of GDP growth and the CISS of all countries; hence, . The latter is referred to as CISSCC to indicate that the information set includes crosscountry data.
Since our model is quite flexible and nests several competing models, we also include a range of restricted variants of the general model outlined in Section 2. First, we obtain the ABG model by using the CISS covariates and setting and . We use frequentist methods to carry out estimation so as to be the same as ABG, while we estimate all other models using Bayesian methods with either the HS or HSP prior (so, for example, we will consider both CISSCCHS and CISSCCHSP specifications). ABG will serve as our benchmark model to which we compare all other specifications. We then add features to this benchmark. We begin by remaining linear () but adding the international factor to the ABG model by letting in order to investigate whether it plays an empirically important role. All subsequent models also let . We next investigate nonlinearities by setting and thus obtain a multicountry quantile BART model with a common international factor. Finally our most flexible model allows for to be estimated from the data. An overview of all model specifications is provided in Table 1.
We compute pseudo outofsample forecasts based on a holdout from 1990Q1 to 2020Q4 (so the initial training sample comprises quarters). We compute Quantile Scores (QS, for quantiles ) and quantileweighted cumulative ranked probability scores (qwCRPS, see Gneiting and Ranjan (2011)) with five weighting schemes (“none” refers to no weighting, i.e., conventional CRPS; both tails “tails;” left tail, “left;” right tail, “right;” and “center”). We compute direct forecasts for .
Data  Prior  Weights  Factor 

CISS (domestic)  HS (shrinkage to zero)  (parametric)  (independence) 
CISSCC (crosscountry)  HSP (pooling crosssection)  (nonparametric)  (dependence) 
(estimated) 

Notes: “Data” refers to the information set for individual country models. “Prior” indicates the prior on the parametric part of the model; we consider the conventional horseshoe prior (HS) shrinking towards zero and the pooled horseshoe (HSP) prior pushing the model towards crosssectional homogeneity. “Weights” refers to the specification of the conditional quantile function: parametric, nonparametric, or whether we estimate weights on the parametric and nonparametric part. “Factor” indicates whether an international factor modeling the crosssectional covariance structure within quantiles is present. We consider all possible combinations of these specification choices.
3.2 Tail forecasting results
Table 2 reports the forecast comparison of the various models based on the relative qwCRPS. Each cell in the heatmap shows the qwCRPS relative to the ABG benchmark model. Numbers smaller than one indicate outperformance (green colored) visávis the ABG model whereas numbers exceeding one suggest a weaker performance (red colored) than the benchmark.
Four main comments can be made. First, and focusing on aggregate results across countries, our proposed models commonly improve on the benchmark ABG model. The gains are about 20 percent when looking at the standard CRPS, decrease to about 10 to 15 percent in the left tail, and increase to about 30 percent in the right tail (based on additional results reported in the Appendix). Overall, these results suggest that at each quantile, and particularly in the right tail, it pays off to allow for nonlinearities and for crosscountry relations.
Second, while there are small differences between setting (linear quantile) or (BART quantile), there is often some benefit to estimating the weight , in turn allowing for both linear and BART pieces in the model. The key advantage of estimating is that it combines the best of both worlds and thus translates into a model that is strongly nonlinear and nonparametric in the tails and close to a linear quantile regression model in the center of the distribution. Such a behavior is beneficial if loss functions which evaluate the full predictive distribution are used.
Third, the HSP prior, that includes pooling, is typically better than HS, but the differences shrink or are eliminated once crosscountry information is included in the model. It is noteworthy that once we use a pooling prior the predictive benefit of adding crosscountry information directly diminishes sharply. This points towards the fact that, through pooling, our approach successfully picks up crosssectional information in a very parsimonious manner.
Finally, there is some heterogeneity across the countries under analysis. In particular, for Spain, France, Italy, and the UK the results are broadly in line with those mentioned above. In contrast, for Austria and Sweden estimating the weight yields little gains (setting is often best), and for the other countries it is overall difficult to beat the benchmark.
3.3 Estimated weights over time
In the previous subsection we have shown that our proposed framework yields forecast distributions which are often more precise than the ones obtained from the ABG benchmark and simpler nested alternatives. One key advantage of the model is that it allows for different weights across countries and quantiles and this improves forecasts when the full predictive density is evaluated. In this subsection, we investigate whether our intuition that nonlinearities are relevant in the tails while linear models are adequate in the center of the distribution is supported by our model.
Figure 1 reports the estimated weight over our holdout period. Darkblue cells indicate a weight close to one while gray shaded cells imply a weight close to zero. We focus on two models, the CISS and CISSCC models coupled with the pooled Horseshoe prior (HSP).^{4}^{4}4The results for the remaining specifications look similar and may be found in the Appendix.
It turns out that for most countries, our conjecture is confirmed. That is, we observe weights which approach unity if we move out in the tails (i.e., the model becomes more nonlinear). When we focus on the center of the distribution, the combination weights approach zero (i.e., the model is linear). Comparing the right and left tails reveals that the estimated weight is often larger for the 5 percent quantiles than the 95 percent quantiles. This indicates that nonlinearites are important when our focus is on modeling sharp upswings in GDP growth but become even more important when interest centers on capturing downturns in GDP growth that are extreme (i.e., below the 5 percent quantile).
When we compare the model which does not utilize crosscountry information (CISS) to the one which explicitly includes crosssectional data (CISSCC), we find only modest differences in combination weights. These differences are mainly related to somewhat smaller weights on the BART specification in the upper tail of the distribution (for some selected countries such as AT, DK, the UK, and the US), but the main finding that, in the center of the distribution, our model assigns no weight to the nonlinear model still holds.
Zooming into the different results reveals that most countries share the general dynamics described in the previous paragraphs (i.e., close to one in the tails and in the center of the distribution). One exception to this broadbased finding is France, which displays larger combination weights across all quantiles. In addition, there exists temporal heterogeneity. For instance, in several countries we observe that, after the global financial crisis (and sometimes slightly earlier), combination weights decrease markedly in the upper tails of the distribution.
3.4 The role of the common volatility factor
In this subsection, we investigate the common volatility factor across quantiles. In a first step, we assess the relevance of the common factor volatility specification by considering time averages of variance decompositions. These are computed, by taking the Gaussian representation of the ALD (the distribution of in Equation (1)), as follows:
with denoting the variance of . This decomposition provides information on the share of variation in the shocks (conditional on the quantile) that is explained through the common factor (similar to Stock and Watson (2005)).
Table 3 reports time averages of variance decompositions resulting from the CCHSP model. Interestingly, for most countries the commonality is larger and more substantial in the tails than at the center of the distribution, and a bit larger in the right than in the left tail. These larger contributions in extreme periods can be traced back to the fact that several of the recessions in our holdout period can be viewed as shocks with a pronounced global dimension (such as the global financial crisis or the Covid19 pandemic) and the factor is picking this up.
Across countries, we find a considerable degree of homogeneity within country groups. For instance, Finland, Denmark, and Sweden feature commonalities that are very pronounced in the tails but decline once we approach the center of the distribution both from left and right. The US and the UK share a rather similar pattern in terms of commonalities (high shares in the tails and for the median, smaller shares for the quantiles in between).
Quantile ()  ALL  AT  DE  DK  ES  FI  FR  IT  NL  SE  UK  US 

0.05  0.83  0.85  0.84  0.79  0.83  0.77  0.86  0.84  0.84  0.86  0.87  0.82 
0.1  0.73  0.77  0.72  0.63  0.77  0.61  0.80  0.76  0.73  0.76  0.77  0.75 
0.25  0.55  0.63  0.59  0.42  0.61  0.38  0.56  0.61  0.56  0.53  0.61  0.58 
0.5  0.67  0.83  0.68  0.30  0.95  0.21  0.98  0.90  0.56  0.40  0.89  0.71 
0.75  0.70  0.79  0.76  0.50  0.77  0.48  0.80  0.81  0.69  0.59  0.78  0.69 
0.9  0.79  0.83  0.85  0.68  0.81  0.65  0.85  0.84  0.79  0.75  0.82  0.77 
0.95  0.89  0.91  0.91  0.82  0.90  0.82  0.92  0.91  0.89  0.87  0.92  0.87 
The heterogeneity across quantiles in the role of the volatility factor is further supported by Figure 2, which reports estimates of the factors (upper panels) and associated logvolatility per quantile (lower panels). In the upper panel, we observe that especially in the tails the factor moves sharply during global events such as the global financial crisis and the Covid19 pandemic. To a somewhat smaller extent the results also suggest declines in the beginning of the 1990s and the early 2000s. When we focus attention on the 50 percent quantile we find strikingly different results. In the center of the distribution, the factor is small and very close to zero throughout the sample. During the pandemic we find a strong pronounced decrease in 2020:Q2, which was triggered by an unprecedented downturn in real activity globally but also a strong increase in 2020:Q3 (which was accompanied with sharply increasing GDP growth rates throughout all our countries).
Turning to the evolution of the logvolatilities in the lower panel generally yields consistent insights with the findings discussed for the level of the factor. The logvolatility spikes during recessions (i.e., in the early 1990s, 20082009, and 2020), and for the level of the logvolatility is much smaller than for the other quantiles but then exceeds the increases in volatility observed for the other quantiles of the distribution. This finding also sheds light on why the amount of variation explained through the factor for most countries is lowest but still sizable in the 50 percent quantile. In most periods, the volatility factor is small (around to on the logscale) but then during the pandemic it rapidly increases and reaches values of around 5 on the logscale. This suggests that in tranquil periods, the factor only explains little variation in the shocks but in recessions (or turbulent times) this share increases appreciably and approaches 1.
3.5 Generalized impulse responses to a global business cycle shock
The discussion on the qualitative and quantitative properties of the estimated factor provides evidence that it can be interpreted as a global business cycle shock since, depending on the quantile adopted, it closely tracks events such as global recessions. Following Stock and Watson (2005), we now consider how changes to the factor, labeled factor shocks, impact GDP growth across countries and quantiles.
The posterior quantiles of the generalized impulse response functions (GIRFs) for a common factor shock as estimated with the CISSCCHSP model are reported in Figure 3. This figure includes the GIRFs for our model with estimated (gray shaded areas) and for (solid blue lines).
A first interesting finding is that, for all countries, a factor shock has different effects in the left tail than the right tail. In both tails, growth is negatively affected, confirming that higher volatility/uncertainty is detrimental for growth, but the size of the effect (and persistence of the negative effect) is much larger in the left tail. Moreover, notice that for the right tail we also observe an overshoot in real activity in response to an adverse business cycle shock.
Second, when we consider the left tail, the BART piece with an estimated weight tends to mitigate the effects of the shock. This is most likely driven by the fact that, if we rule out nonlinearities, there is more to be explained through the factor model and this might translate into factor dynamics which not only pick up business cycle shocks but also soak up information left in the error term potentially arising from ignoring nonlinear dynamics between GDP growth and the CISS.
A third striking pattern is the pronounced degree of crosscountry heterogeneity in the 5 percent quantile (and, to a somewhat lesser extent, in the 10 percent quantile). When our focus is on the left tail, we observe that France, Italy, the UK, and Spain exhibit sharp declines in GDP growth. Once we consider higher quantiles the GIRFs become much more similar across countries. For instance, we find only modest differences if we focus on .
To sum up, we find that the countries in our sample display pronounced reactions to an international business cycle shock. These reactions differ not only across quantiles but also across countries.
3.6 Generalized impulse responses to a US financial conditions shock
The previous subsection emphasized that our latent factor can be interpreted as a global business cycle shock. In this subsection we will instead focus attention on the international effects of a shock to US financial conditions and whether the real effects of such a shock differ from the ones arising from changes in .
Figures 4 and 5 report the posterior median of the cumulative (oneyear ahead) generalized impulse response functions of GDP by quantile () to a , , , , and insample standard deviation US financial conditions shock, based on the CISSCCHSP model with estimated weight and either with (Figure 4) or without (Figure 5) the common factor in the model’s innovation component. Recall that the CISS is defined so that higher values represent tighter financial conditions; a positive shock may be expected to reduce GDP growth.
As the model is nonlinear, the sign and size of the shocks can matter to determine the effects (i.e., contrary to the linear case, the effects are not proportional to the size of the shock, or symmetric). Hence, we have recomputed the model for various subsamples to analyze how different global business cycle conditions impact the estimates of the GIRFs.
Comparing the two figures shows that the factor volatility has little effect on the results, which is not surprising as it should not affect much the point estimates of the GIRFs (rather their precision). In both cases, a financial shock in the US spills over to other countries. There is asymmetry in the sense that a positive shock affects the growth quantiles, whereas a negative shock’s effects are not as sharp. Moreover, the effects and asymmetry are sharper in the 20102019 period than earlier.
Analyzing crosscountry differences provides additional interesting insights. For some countries (DE, FR, DK, and ES), we find substantial time variation in the GIRFs. Prior to 2010, the corresponding heatmaps feature a great deal of gray colored cells, implying no reactions at all. After the global financial crisis, USCISS shocks have pronounced effects for these countries that are mostly located in the left tail of the distribution of GDP growth. For other countries (IT, UK, US, and NL), we find less evidence in favor of timevariation in the GIRFs. In these countries, positive (negative) shocks to the USCISS have negative (positive) effects on GDP growth for . Notice, however, that when we consider , the effect of a CISS shock seems to reverse sign; a positive shock to the CISS has positive effects on GDP growth if it is already historically high and a negative shock triggers a decline in GDP growth.
Figures 6 and 7 are similar to Figures 4 and 5 but now the weight is set to zero, so that the quantile part is linear. Here, too, the common (volatility) factor component does not seem to have an obvious effect on the results. Instead, the effects of financial conditions are now more linear; positive and negative shocks to financial conditions in the US have similar effects, of opposite sign. In addition, the pattern is clearer in the data since 2007 than before. The effects also look smaller and more stable over time than in the nonlinear quantile specification, and in general they are more marked at the lower quantiles. These marked differences in the empirical findings highlight the importance of allowing for nonlinearities also in the context of quantile regressions.
4 Conclusions
In this paper we propose a nonparametric quantile panel regression model which assumes that the conditional mean is a convex combination of a linear and an unknown nonlinear function. We learn the unknown functions using BART, a successful tool closely related to random forests. To decide on how much weight the BART piece should receive in the
quantile, we estimate it alongside the remaining model parameters. This nonparametric feature enhances model flexibility, especially in the tails. Using crosssectional information, in addition, enables us to improve predictive accuracy. This is achieved by proposing a novel pooling prior as well as introducing crosscountry information directly. To carry out estimation and inference we design a scalable MCMC algorithm and apply the model to investigate ”growth at risk” using an international panel of 11 countries.In terms of empirical results, our proposed models commonly improve on the benchmark single country linear quantile model in recursive growth forecast comparisons, more so in the tails than near the center of the distribution and in particular when estimating the weight , in turn allowing for both linear and BART pieces in the model. The estimated combination weight is smaller (i.e., the model is more linear) for the 25 and 75 percent quantiles than in the tails, i.e., for the 5 and 95 percent quantiles. Moreover, some form of international information definitely pays off (either via the new pooling prior, or by outright including nondomestic series). The effects of the common (volatility) factor are also relevant, as it seems to explain a large fraction of the forecast error variance in most countries, in particular in the tails. A shock to this factor, which can be interpreted as an uncertainty shock, has different (stronger negative) effects in the left tail than the right tail of the growth distribution. In the left tail, the BART piece with an estimated weight tends to mitigate a bit the effects of the shock. Finally, a financial shock in the US spills over to other countries. There is asymmetry in the responses in the sense that a positive shock affects the growth quantiles, whereas a negative shock’s effects are not as sharp. Moreover, the effects and asymmetry are sharper in the 201020 period than earlier. The responses are instead much more proportional and symmetric in the linear model, highlighting the importance of allowing for nonlinearities in the specification of quantile regressions.
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