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Stochastic Volatility-in-mean VARs with Time-Varying Skewness
\pagenumbering{arabic} \affil[1]{Central Bank of Brazil } \affil[2]{Queen Mary University of London} \affil[3]{European Central Bank}
\pagenumbering{arabic} \setcounter{page}{2}
Bayesian vector autoregressions (BVARs) with stochastic volatility have become a workhorse model both for forecasting and estimating the impact of economic shocks. clark2011 shows that stochastic volatility is a crucial ingredient for BVARs designed for density forecasting. This finding has been confirmed by a large subsequent literature. As shown in Koopman_uspensky, Mumtaz_Surico2018 and carriero2016 amongst others, these models can be extended to allow stochastic volatility of the disturbances to affect the level of endogenous variables. These BVARs with stochastic volatility in mean have been used to estimate the effects of uncertainty shocks. \newline In contrast to these models with time-varying second moments, considerably less attention has been devoted to BVARs where the distribution of the error term also features skewness. KARLSSON2023104580 introduce skewed and heavy-tailed distributions in BVARs and show that allowing for skewness improves forecasting performance. Galdon_Ortega model structural disturbances in a VAR allowing for time-varying skewness, and find that this feature is pervasive in Euro-Area data. A similar VAR model is also considered by Renzetti who report evidence for the presence of time-varying skewness in recursively identified shocks using US data.\footnote{ISERINGHAUSEN2020275 introduces a univariate stochastic volatility model with time-varying skewness to model exchange rate returns. ISERINGHAUSEN2024229 extends this model to a panel setting. } \newline One feature that is common in these papers is that the time-variation in skewness is treated as an exogenous process. However, in a recent contribution kostas_skew show that a measure of the skewness of macroeconomic variables has a dynamic relationship with real activity and inflation, contradicting this assumption. \newline In this paper, we propose a BVAR that features time-varying volatility and skewness. Unlike existing contributions, lagged shock volatility and skewness can affect the endogenous variables. \newline We provide a Gibbs sampling algorithm to approximate the posterior distribution of the parameters. The model is applied to US and UK data where we consider the density forecasting performance of the proposed model relative to a BVAR that only features stochastic volatility but no time-varying skewness and to a BVAR with no dynamic relationship between the endogenous variables and skewness. \newline Results show that allowing skewness in mean enhances the forecasting performance of BVARs. In the US, gains are most evident for the Gross National Product price deflator (GNP deflator) and, during periods of economic stress, for the Real Gross National Product (GNP). For the UK, the benefits are even more pronounced, with consistent improvements across variables and horizons. Finally, tail risk measures show that the proposed framework provides a sharper characterisation of downside and upside risks. \newline The paper is organised as follows. The model is described in Section (ref) with the estimation algorithm summarised in Section (ref). Section (ref) describes the data. Section (ref) presents the empirical results while Section (ref) concludes.
We model skewness in the disturbances of the VAR model by using the multivariate skew-normal distribution proposed by Arellano-Valle. Arellano-Valle show that if a vector $E$ has a skew-normal distribution with location vector $\mu$, scale matrix $\Sigma$ and a (diagonal) skewness matrix $\Delta$, it can be written as:
where $X_{0}$ and $X_{1}$ are independent and $X_{0} \sim N(0,1)$ and $X_{1} \sim N(\mu,\Sigma)$. Thus, if skewness (i.e. diagonal elements of $\Delta$) is equal to zero, the distribution collapses to the multivariate normal.\newline With this definition in hand, we consider the following state-space model:
where $Y_{t}$ denotes the vector of $N$ endogenous variables included in the VAR. The error term of the observation equation ((ref)) follows a skewed normal distribution and is defined as $E_{t}=\tilde{d}_{t}\odot \tau_{T}+e_{t} $ with $\tau_{t}=|\Theta_{t}|, \Theta_{t} \sim N(0,1)$ and $e_{t}$ is a multivariate normal as described below. Note that $A$ is a lower triangular matrix with ones on the main diagonal. \newline The state variables are denoted by $\underbrace{\beta_{t}}_{K\times 1}=
$ and $\underbrace{\tau_{t}}_{N\times 1}$ . The stochastic volatilities are denoted by $\tilde{h}_{t}=[h_{1t},h_{2t},\dots,h_{N,t}]$ and $H_{t}=diag\left( \exp \left(\tilde{h}_{t}\right) \right) $. Time-varying skewness is denoted by $\tilde{d_{t}}=[d_{1t},d_{2t},\dots,d_{N,t}]$. These state variables evolve according to equation \ref{eq1}. As evident, $\tilde{h}_{t}$ and $\tilde{d_{t}}$ are allowed to have a lagged impact on the endogenous variables in equation \ref{eq2}. \newline The disturbances $\varepsilon _{t}=\left(
\right) $ are distributed normally $N\left(
,
\right)$
We use a Gibbs sampling algorithm to approximate the posterior distributions of the model parameters. The algorithm, which is based on Mumtaz_Surico2018 is described in detail in the technical appendix. Here, we present a summary of the conditional posterior distributions. \paragraph{Coefficients of the VAR in the transition equation ((ref)) and observation equation ((ref)):} We exploit the fact that conditional on the state vector, the transition equation is a VAR and the conditional posterior is normal. \newline Conditional on the states, we rewrite the VAR in ((ref)) as:
The residuals of this VAR are Gaussian but with a known form of heteroscedasticity with $var(\tilde{e}_{t})=\Sigma_{t}=A^{-1}H_{t}A^{-1\prime}$. We use the Kalman filter to calculate the mean and variance of the conditionally Gaussian posterior.
\paragraph{Elements of $A$ and $Q$:} Conditional on the states and VAR parameters, the residuals $V_{t}$ can be calculated where $AV_{t}=E_{t}$. As shown in cogley-sargent-05, the elements of the $A$ matrix are coefficients in regressions among these residuals and conditional on the skewness parameters $\tilde{d}_t$ and $\tau_{t}$ the conditional posterior is standard. The conditional posterior of $Q$ is inverse Wishart and is easily sampled.
\paragraph{ States $\tilde{h}_{t} ,\tilde{d_{t}}, \tau_{t}$ :} Conditional on the coefficients and fixed variances, the model has the non-linear state-space representation depicted in equations ((ref)) and ((ref)). As in mumtaz2018, we draw from the conditional posterior of the state vector using the particle Gibbs of RePEc:bla:jorssb:v:72:y:2010:i:3:p:269-342 and JMLR:v15:lindsten14a.
We estimate the model using quarterly data on real activity growth (GNP), inflation (GNP deflator) and the spread between the corporate bond yield and a long-term government bond yield, included as a measure of financial stress. We use data for the US and the UK. \newline For the US, the sample runs from 1875Q1 to 2023Q4. The series are constructed by splicing pre-1947 data from the NBER's \hyperlink{https://www.nber.org/research/data/tables-american-business-cycle}{tables from the "The American Business Cycle."} with more recent data from \hyperlink{https://fred.stlouisfed.org/}{FRED} and \hyperlink{https://globalfinancialdata.com/}{Global Financial Database (GFD)}. The measure of real activity for the US is real GNP growth, while inflation is constructed using the GNP deflator. We use the variables RGNP72 and GNPD72 from the NBER pre-1947, splicing them with GNPC1 and GNPDEF from FRED. Similarly, the corporate bond yield is a spliced series using CORPYIELD from the NBER pre-1947 and BAA from FRED post 1947. We obtain the 10-year government bond yield from GFD (IGUSA10d). \newline The full sample for the UK is 1920Q2 to 2023Q4. Data on GDP growth and CPI inflation are obtained from the Bank of England's database \hyperlink{https://www.bankofengland.co.uk/-/media/boe/files/statistics/research-datasets/a-millennium-of-macroeconomic-data-for-the-uk.xlsx}{A Millennium of Macroeconomic data} and the Office of National Statistics (ONS). Real GDP is available at a quarterly frequency from 1920-1938, but only annually over the period 1939-1954 from the Bank of England database. We linearly interpolate the GDP data over the Second World War (WWII) period and splice it with the quarterly series. The post-1955 real GDP data are taken from the ONS with code ABMI. CPI data from 1920-1916 are obtained from the Bank of England database and combined with the series with code D7BT from the ONS for the subsequent period. UK spreads are corporate bond yields from the Bank of England database spliced with data from GFD, minus 10 year yield from GFD.
We begin by presenting the estimated state variables from the model estimated on the full sample of data for each country.
Figure (ref) displays the (square root of) the estimated stochastic volatility. The largest peaks in US GNP volatility occurred at the end of the First World War (WWI), during the Great Depression and then the WWII and most recently during the Covid-19 crisis. Volatility of the US inflation shock was highest during the pre-1980 period, with the largest peak during the WWI. Volatility of the US spread shock peaked during the Great Depression and then during the Great Financial Crisis (GFC) and the Covid-19 period. Note that, unlike the macro variables, the spread disturbance appears to be more volatile in the post-1980 period. For the UK, the volatility of the GDP shock during the Covid period dwarfs the variance estimated in the previous years. The volatility of the UK inflation shock follows a pattern similar to that of the US and was at its highest in the pre-WWII period and during the 1970s and early 1980s. The volatility of the spread was higher, on average, after 1970 with peaks in the mid-1970s, mid-1980s, early 1990s and during the GFC.
The top left panel of Figure (ref) shows that estimated skewness of output shocks in the US largely fluctuates around zero with (imprecisely estimated) negative movements during the Great Depression, the mid-1950s, the early 1980s and the GFC. Inflation shock skewness was positive and statistically different from zero after WWI, the early 1940s and the mid 1950s. Inflation shock skewness remained above zero during the mid-1970s and the early 1980s and showed an increase during the post-Covid period. The skewness of the spread shock is estimated to be positive during the Great Depression and the GFC.
It is interesting to note that the skewness of the UK output shock shows more variation than the associated volatility over the full sample. The shock is negatively skewed during the large recession of the mid-1940s, the early 1970s and 1980s, in 1990s and then during the GFC. The large positive spike in 2020Q3, picks up the rebound in GDP growth after the initial decline associated with the Covid-19 crisis that began in 2020Q1. The skewness of the inflation shock in the UK shows a persistent increase during the 1970s, the early 1980s and 1990s and then after the Covid-19 period in 2022. Spread shocks were positively skewed over most of the pre-WWII period, and the same feature is evident after 1998.
Before moving on to the forecasting exercise, we present impulse response functions to a skewness shock. This structural analysis contributes to a deeper understanding of how skewness influences the economy. To identify the shocks, we adopt a simple recursive scheme in which we use a Cholesky decomposition of $Q$ to compute the impulse responses to shocks to the transition equation. Impulse responses are calculated using Monte Carlo integration as described in koop-pesaran-potter-96 and are conditioned on the last data point in the sample. Figure (ref) shows the responses to shocks to volatilities and skewnesses for the US, and Figure (ref) shows these responses for the UK.
In the US, stochastic volatility shocks primarily influence other stochastic volatilities, with the exception of spread volatility, which affects spread skewness and the GNP deflator. In contrast, skewness shocks exhibit a broader impact, affecting both volatilities and the main variables. For example, an increase in the GNP skewness increases the growth of GNP. GNP deflator skewness shocks, on the other hand, lead to a decline in GNP growth, while driving up both the GNP deflator and the spread. Finally, spread skewness causes a reduction in both GNP growth and its deflator and an increase in spread. It is worth noting that, among these, the shocks to the GNP deflator and spread skewness exhibit the most pronounced effects.
In the UK, stochastic volatilities are also important drivers of skewness. However, as in the US, skewness shocks have a more pervasive influence. Specifically, GNP skewness increases GNP growth while decreasing the GNP deflator. The skewness of the GNP deflator increases both the growth of GNPs and its deflator, whereas the spread skewness reduces the GNP deflator and increases the spread level. Taken together, the evidence from both countries indicates that skewness has a substantial impact on core economic indicators. This underscores the importance of time-varying skewness in mean to help explain the dynamics of real economic activity, inflation, and corporate bond spreads. Consequently, accounting for this feature may also enhance the forecasting performance of BVARs -- a hypothesis we explore further in the next subsection.
Our main application is a pseudo-real-time density forecasting experiment. For each country, we use a training sample that runs from the start of the available sample to 1974Q4. We then estimate the forecasting models recursively with the final estimation sample ending in 2021Q4. At each iteration we construct the predictive density up to eight quarters ahead and evaluate the forecasting performance. For output and inflation, which are modeled in first log-differences, we look at cumulative growth rates.
We first consider the accuracy of point forecasts (defined as posterior means), using root mean squared errors (RMSEs). We then evaluate density forecasts based on log scores and continuous ranked probability scores (CRPS), as in clark2015 and carriero2024addressing, among many others. We focus on weighted log scores and weighted CRPS. Weighted scoring rules extend the standard log scores and CRPS by applying a weighting function to emphasise specific regions of the predictive distribution (Amisano-Giacomini-07, Gneiting2011). In particular, we target the evaluation of forecast accuracy at the tails of the distribution, by assigning higher weights to these areas.\footnote{More details as well as standard log scores and CRPS, and log scores and CRPS focusing on other areas are presented in the appendix.} Statistical significance is assessed based on giacomini2006tests's tests. Nonetheless, p-values are only indicative as the models are estimated recursively, rather than using a fixed rolling window.
We compare the forecasting performance of the proposed model (which we denote as $VAR^{\sigma,\kappa}$) with two alternatives, which are restricted versions of the proposed model.\footnote{$\sigma$ denotes stochastic volatility, and $\kappa$ denotes time-varying skewness.} First, we restrict the coefficients $d_{j},b_{l}$ and $a_{l}$ in equations (ref) and (ref) to be equal to zero, thus assuming no dynamic relationship between the endogenous variables and the states. Thus, this model relates to BOTELHO2024104849 who do not allow an in-mean effect.\footnote{Note, however, that BOTELHO2024104849 do allow risk factors to influence the time-variation in skewness.} We denote this model as $VAR^{\sigma,\kappa}_{restricted}$. Second, we consider a VAR model where the disturbances are assumed to be conditional Gaussian with time-varying volatility only with skewness assumed to be zero (denoted as $VAR^{\sigma}$). Like the proposed model, the stochastic volatility is allowed to have a dynamic relationship with the endogenous variables $Y$.
Table (ref) presents the average forecast performance relative to the competing models over the entire evaluation period. In terms of point forecast accuracy, $VAR^{\sigma,\kappa}$ tends to be marginally more accurate than $VAR^{\sigma}$ for GNP and the GNP deflator. When compared to $VAR^{\sigma,\kappa}_{restricted}$, improvements in RMSE are modest and primarily concentrated in GNP forecasts at horizons 2--7, with reductions of 1--4%. Against $VAR^{\sigma}$, the proposed model shows more substantial gains, achieving RMSE reductions of 3--4% for GNP at most horizons and 2--4% for the GNP deflator across all forecast horizons.
Allowing stochastic volatilities and skewness to affect the endogenous variables plays a more significant role for metrics based on the full distributions. For weighted log scores emphasising both tails, the results reveal heterogeneous performance across variables and horizons. Relative to $VAR^{\sigma,\kappa}_{restricted}$, the proposed model delivers improvements for GNP deflator forecasts, with gains ranging from 0.1% to 0.8% across horizons, becoming statistically significant at longer horizons (H6--H8). Against $VAR^{\sigma}$, the GNP deflator shows consistent and statistically significant improvements, with gains of 0.3--1.3%, particularly strong at shorter horizons where significance reaches the 1% level.
The weighted CRPS results reinforce these findings. Compared to $VAR^{\sigma,\kappa}_{restricted}$, the proposed model achieves CRPS reductions of 6--14% for the GNP deflator at longer horizons (H6--H8), with statistical significance. Against $VAR^{\sigma}$, improvements are even more pronounced, with CRPS reductions of 8--18% for the GNP deflator and 6--16% for the spread variable, many statistically significant at the 1% level. Overall, scoring rules reveal that the proposed model consistently outperforms the alternatives for GNP deflator forecasts.
The analysis over time in Table (ref) shows several periods when the proposed model outperforms the alternatives for GNP and the spread forecasts as well. During the Great Inflation period, $VAR^{\sigma,\kappa}$ shows a strong performance across all variables, with cumulative log score gains of 9.8 for GNP, 69.0 for the GNP deflator, and 108.0 for the spread against $VAR^{\sigma,\kappa}_{restricted}$. The Covid-19 period shows particularly notable improvements for GNP, with cumulative log score gains of 1108.2 relative to $VAR^{\sigma,\kappa}_{restricted}$, averaging 221.6 per quarter. The most recent period (since 2021Q3) shows consistent improvements across all variables relative to $VAR^{\sigma,\kappa}_{restricted}$, with statistically significant gains.
A key takeaway is that skewness is a valuable asset for density forecasting across variables and horizons. The results demonstrate that incorporating time-varying skewness with dynamic linkages to macroeconomic fundamentals enhances forecast accuracy, particularly for density forecasts and during periods of economic stress.
Table (ref) presents the forecast performance for the UK throughout the evaluation period. In contrast to the US results, the UK shows more substantial and consistent improvements in point forecast accuracy. Compared to $VAR^{\sigma,\kappa}_{restricted}$, the proposed model achieves RMSE reductions of 4--13% for GNP across all horizons, 5--12% for the GNP deflator (with larger improvements at longer horizons), and modest improvements of 1--2% for the spread variable. Against $VAR^{\sigma}$, the gains are even more pronounced, with RMSE reductions of 8--15% for GNP (becoming statistically significant at horizons 6--8), 2--7% for the GNP deflator, and 1--3% for the spread across all forecast horizons.
The density forecast metrics reveal that incorporating time-varying skewness with dynamic linkages yields substantial benefits for the UK economy. For weighted log scores emphasising both tails, the GNP deflator exhibits consistent and statistically significant improvements relative to $VAR^{\sigma,\kappa}_{restricted}$, with gains of 1.6--2.5% across horizons 2--8, significant at the 5--10% level. Against $VAR^{\sigma}$, the improvements are similar, ranging from 1.2--3.2%, with statistical significance at multiple horizons. The GNP variable shows more heterogeneous performance, with notable improvements at horizons 3, 5, 6, and particularly at horizon 7 (23.3% gain relative to $VAR^{\sigma,\kappa}_{restricted}$).
The weighted CRPS results strongly favour the proposed model for the GNP deflator, demonstrating improvements of 12--28% relative to $VAR^{\sigma,\kappa}_{restricted}$ across horizons, with many results statistically significant. The most substantial gains appear at longer horizons (H6--H8), where CRPS reductions reach 23--28%. Against $VAR^{\sigma}$, the GNP deflator improvements range from 7--16%, with statistical significance at horizon 2. For GNP, the proposed model achieves CRPS reductions of 2--11% relative to $VAR^{\sigma,\kappa}_{restricted}$ at most horizons, and 2--12% against $VAR^{\sigma}$ at shorter to medium horizons.
The period-by-period analysis in Table (ref) reveals important patterns across different economic episodes. During the stagflation period (1975Q2--1979Q4), the proposed model shows strong performance for the GNP deflator, with cumulative log score gains of 29.9 and 21.1 against the two competitors, both statistically significant. The ERM crisis period demonstrates substantial improvements across variables, with cumulative log score gains of 13.3 for GNP, 8.3 for the GNP deflator, and strong CRPS performance against both competitors.
The Global Financial Crisis period reveals particularly striking results, with cumulative log score gains of 171.2 for GNP relative to $VAR^{\sigma,\kappa}_{restricted}$, averaging 7.1 per quarter. The GNP deflator also shows consistent improvements during this period, with gains of 5.8 and 25.4 against the two competitors. The Covid-19 period mirrors this pattern, with dramatic improvements for GNP (cumulative log score of 287.0, averaging 43.8 per quarter) and consistent gains for the GNP deflator. The most recent period (since 2021Q3) shows statistically significant improvements for both the GNP deflator (cumulative log score of 54.4) and the spread variable (72.3), highlighting the continued relevance of time-varying skewness in the current economic environment.
Overall, UK results provide strong evidence that incorporating time-varying skewness with dynamic linkages to macroeconomic fundamentals substantially improves forecast accuracy. The improvements are more consistent and pronounced than in the US case, particularly for point forecasts and density forecasts of the GNP deflator. Once more, the model performs especially well during periods of economic stress and structural change, such as the ERM crisis, GFC, and Covid-19 pandemic, underscoring the value of capturing asymmetries in the predictive distribution during turbulent times.
In this section, we use the proposed model to construct measures of tail risk for the endogenous variables. Following CaldaraMumtazZhong2024, we construct the forecast distribution for the endogenous variables at each point in time using the full-sample estimates. The $5^{th}$ and $95^{th}$ percentiles of the forecast distribution capture the variation in the left and the right tail over time. Figure (ref) displays these percentiles estimated using the proposed model and compares them to those obtained from the stochastic volatility in mean model without skewness.
A number of key differences in the estimated risk measures are apparent. The forecast distribution from the stochastic volatility in mean model is, in general, more dispersed than the estimate from the proposed model. As a consequence, the stochastic volatility model points to higher uncertainty rather than highlighting tail risk. This is especially apparent when considering the forecast distribution in the post-second world war period for both countries. The proposed model consistently points to higher right tail risk during the 1970s, the early 1980s/1990s and the post-Covid period. In contrast, the stochastic volatility model also assigns substantially more mass to downside risk during these periods. For example, in March 2021, the proposed model assigns a probability of 0.7 to annualized inflation in the US exceeding 5% next quarter. In contrast, this probability is estimated to be 0.5 using the stochastic volatility model.
This paper proposes a BVAR that incorporates both stochastic volatility-in-mean and time-varying skewness, allowing these features to directly affect macroeconomic variables. Our results, based on US and UK data, demonstrate that time-varying skewness plays a crucial role in shaping the dynamics of output, inflation, and financial spreads. Impulse responses show that skewness shocks significantly impact output, inflation and spreads in both the US and the UK, often with larger effects than volatility shocks.
The forecasting evaluation further underscores the value of incorporating time-varying skewness. In the US, gains are most evident for the GNP deflator and, during periods of economic stress, for the GNP. For the UK, the benefits are even more pronounced, with consistent improvements across variables and horizons. Finally, tail risk measures show that the proposed framework provides a sharper characterisation of downside and upside risks, offering a useful tool for policy analysis. Overall, this paper contributes to the growing literature emphasising the importance of non-Gaussian features in macroeconomic models and shows that time-varying skewness is a key dimension for understanding and forecasting macro-financial dynamics.