Stochastic Volatility Specifications Research Articles

Gaussian Process Vector Autoregressions and Macroeconomic Uncertainty

We develop a nonparametric multivariate time series model that remains agnostic on the precise relationship between a (possibly) large set of macroeconomic time series and their lagged values. The main building block of our model is a Gaussian process prior on the functional relationship that determines the conditional mean of the model, hence, the name of Gaussian process vector autoregression (GP-VAR). A flexible stochastic volatility specification is used to provide additional flexibility and control for heteroscedasticity. Markov chain Monte Carlo (MCMC) estimation is carried out through an efficient and scalable algorithm which can handle large models. The GP-VAR is used to analyze the effects of macroeconomic uncertainty, with a particular emphasis on time variation and asymmetries in the transmission mechanisms.

Real-time forecast of DSGE models with time-varying volatility in GARCH form

Recent research shows that time-varying volatility plays a crucial role in non-linear modeling. Contributing to this literature, we suggest an approach that allows for straightforward computation of DSGE models with time-varying volatility, where the volatility component is formulated as a GARCH process. As an application of our approach, we examine the forecasting performance of this DSGE-GARCH model using euro area real-time data. Our findings suggest that the DSGE-GARCH approach is superior in out-of-sample forecasting performance in comparison to various other benchmarks for the forecast of inflation rates, output growth and interest rates, especially in the short term. Comparing our approach to the widely used stochastic volatility specification using in-sample forecasts, we also show that the DSGE-GARCH is superior in in-sample forecast quality and computational efficiency. In addition to these results, our approach reveals interesting properties and dynamics of time-varying correlations (conditional correlations).

Capturing Macro‐Economic Tail Risks with Bayesian Vector Autoregressions

AbstractMany studies using quantile regressions (QRs) have found that downside risk to output growth varies more than upside risk. We show that Bayesian vector autoregressions (BVARs) with stochastic volatility are able to capture tail risks in forecast distributions. Even though the one‐step‐ahead conditional predictive distributions from the conventional stochastic volatility specification are symmetric, forecasts of downside risks to output growth are more variable than upside risks, and the reverse applies in the case of inflation and unemployment. Overall, BVAR models perform comparably to QR for estimating and forecasting tail risks, complementing BVARs' established performance for forecasting and structural analysis.

Accelerated American option pricing with deep neural networks

<abstract><p>Given the competitiveness of a market-making environment, the ability to speedily quote option prices consistent with an ever-changing market environment is essential. Thus, the smallest acceleration or improvement over traditional pricing methods is crucial to avoid arbitrage. We propose a method for accelerating the pricing of American options to near-instantaneous using a feed-forward neural network. This neural network is trained over the chosen (e.g., Heston) stochastic volatility specification. Such an approach facilitates parameter interpretability, as generally required by the regulators, and establishes our method in the area of eXplainable Artificial Intelligence (XAI) for finance. We show that the proposed deep explainable pricer induces a speed-accuracy trade-off compared to the typical Monte Carlo or Partial Differential Equation-based pricing methods. Moreover, the proposed approach allows for pricing derivatives with path-dependent and more complex payoffs and is, given the sufficient accuracy of computation and its tractable nature, applicable in a market-making environment.</p></abstract>

Comparing stochastic volatility specifications for large Bayesian VARs

Large Bayesian vector autoregressions with various forms of stochastic volatility have become increasingly popular in empirical macroeconomics. One main difficulty for practitioners is to choose the most suitable stochastic volatility specification for their particular application. We develop Bayesian model comparison methods–based on marginal likelihood estimators that combine conditional Monte Carlo and adaptive importance sampling–to choose among a variety of stochastic volatility specifications. The proposed methods can also be used to select an appropriate shrinkage prior on the VAR coefficients, which is a critical component for avoiding over-fitting in high-dimensional settings. Using US quarterly data of different dimensions, we find that both the Cholesky stochastic volatility and factor stochastic volatility outperform the common stochastic volatility specification. Their superior performance, however, can mostly be attributed to the more flexible priors that accommodate cross-variable shrinkage.

On the evolution of income elasticities of imports and Thirlwall’s Law: A Bayesian analysis for Latin America (1961-2018)

Abstract This paper analyzes the evolution of income elasticities of imports and verifies the validity original Thirlwall’s Law for eight Latin American countries overs the period 1961-2018. Given the econometric issues inherent to classical time-varying parameter regressions, a Bayesian estimation procedure is implemented in order to provide more robust parameter estimates. A stochastic volatility specification is also included to take into account the potential presence of conditional heteroskedasticity. While some Latin American economies showed a rather stable behavior, the income elasticities of imports for Brazil, Colombia and Mexico presented an upward trend from 1961 to 2018. Regarding the validity of original Thirlwall’s Law, there were evidences of the balance-of-payments (BOP) posing a restriction to the growth performance in all of the eight selected Latin American countries.

The horseshoe prior for time-varying parameter VARs and Monetary Policy

Time-varying parameter VARs have become the workhorse models in empirical macroeconomics. These models are usually equipped with tightly parametrized prior distributions which favor a small and gradual change in parameters. Do such prior distributions suppress some degree of time variation in the VAR coefficients? We address this question by proposing a flexible global-local prior, namely the horseshoe prior. It turns out that conventional priors may suppress economically relevant patterns of time variation. Using the global-local prior, we observe that parameter change and changes in systematic monetary policy can be abrupt rather than smooth. Furthermore, we provide a comparison of the horseshoe prior with a range of plausible alternatives. Finally, we find that a VAR with a stochastic volatility specification using the horseshoe prior is well suited to modelling the extreme observations due to Covid-19. In contrast, the conventional prior (spuriously) picks up an increase of volatilities even before the Covid crisis.

Global Impacts of Us Monetary Policy Uncertainty Shocks

We build a new empirical model to estimate the global impact of an increase in the volatility of US monetary policy shocks. Specifically, we admit time-varying variances of local structural shocks from a stochastic volatility specification. By allowing for rich dynamic interaction between the endogenous variables and time-varying volatility in the global setting, we find that US interest rate uncertainty not only drives local output and inflation volatility, but also causes declines in output, inflation, and the interest rate. Moreover, we document strong global impacts, making the world move in a very synchronous way. Crucially, spillback effects are found to be significant even for the US economy. JEL Classification: C32, C54, E52, E58, F44

Time-varying monetary policy reaction function under asymmetric preferences: revisiting the Brazilian inflation targeting experience

PurposeThis paper analyzes the potential presence of time-varying asymmetries in the preference parameters of the Central Bank of Brazil during the inflation targeting regime.Design/methodology/approachGiven the econometric issues inherent to classical time-varying parameter (TVP) regressions, a Bayesian estimation procedure is implemented in order to provide more robust parameter estimates. A stochastic volatility specification is also included to take into account the potential presence of conditional heteroskedasticity.FindingsThe obtained results show that the reduced form and structural parameters were not constant during the period considered. Moreover, the subsequent analysis of the preference parameters provided evidences of short periods in which asymmetry was an important feature to the conduction of monetary policy in Brazil. Yet, during most of the sample period, the loss function was considered to be symmetrical.Originality/valueThis paper aims to contribute to the rather scarce monetary debate on time-varying central bank preferences. The study of Lopes and Aragón (2014) is, to the best of the authors’ knowledge, the only study for Brazil considering specifically TVPs. The authors applied Kalman filter estimation to data from 2000:M1 to 2011:M12. Despite the similar structure of TVPs, the present paper extends the latter study by controlling for stochastic volatility. Ignoring conditional heteroskedasticity might lead to spurious movements in time-varying variables and inaccurate inference (Hamilton, 2010). Thus, the stochastic volatility specification is included to take this issue into account. The authors follow the theoretical scheme put forward by Surico (2007) and Aragón and Portugal (2010), in which the economy is modeled from a New Keynesian perspective and the central bank loss function is assumed to be asymmetric regarding the responses to inflation and output deviations from their targets. On the empirical side, the authors propose a TVP univariate regression with stochastic volatility for the Brazilian reduced-form reaction function, following closely the Bayesian econometric procedure developed by Nakajima (2011). Given the nonlinear non-Gaussian nature of the TVP regression with stochastic volatility, the choice of a nonlinear Bayesian approach using the Markov chain Monte Carlo (MCMC) method is justified due to the intractability of the associated likelihood function (Primiceri, 2005). Finally, based on the theoretical model specification, the authors intend to recover the central bank preference parameters as to further evaluate the degree of asymmetry and its potential time-variation under the inflation targeting regime.

A multi-country dynamic factor model with stochastic volatility for euro area business cycle analysis.

This paper develops a dynamic factor model that uses euro area country‐specific information on output and inflation to estimate an area‐wide measure of the output gap. Our model assumes that output and inflation can be decomposed into country‐specific stochastic trends and a common cyclical component. Comovement in the trends is introduced by imposing a factor structure on the shocks to the latent states. We moreover introduce flexible stochastic volatility specifications to control for heteroscedasticity in the measurement errors and innovations to the latent states. Carefully specified shrinkage priors allow for pushing the model towards a homoscedastic specification, if supported by the data. Our measure of the output gap closely tracks other commonly adopted measures, with small differences in magnitudes and timing. To assess whether the model‐based output gap helps in forecasting inflation, we perform an out‐of‐sample forecasting exercise. The findings indicate that our approach yields superior inflation forecasts, both in terms of point and density predictions.

Capturing Macroeconomic Tail Risks with Bayesian Vector Autoregressions

A rapidly growing body of research has examined tail risks in macroeconomic outcomes. Most of this work has focused on the risks of significant declines in GDP, and it has relied on quantile regression methods to estimate tail risks. Although much of this work discusses asymmetries in conditional predictive distributions, the analysis often focuses on evidence of downside risk varying more than upside risk. We note that this pattern in risk estimates over time could obtain with conditional distributions that are symmetric but subject to simultaneous shifts in conditional means (down) and variances (up). Building on that insight, we examine the ability of Bayesian VARs with stochastic volatility to capture tail risks in macroeconomic forecast distributions and outcomes. We consider both a conventional stochastic volatility specification and a specification with a common factor in volatility that enters the VAR’s conditional mean. Even though the one-step-ahead conditional predictive distributions from the conventional stochastic volatility specification are symmetric, the model estimates yield more time variation in downside risk as compared to upside risk. Results from the model that includes a volatility factor in the conditional mean and thereby allows for asymmetries in conditional distributions are very similar. Our paper also extends the recent literature by formally evaluating the accuracy of tail risk forecasts and assessing the performance of Bayesian quantile regression, as well as our Bayesian VARs, in this context. Overall, the BVAR models perform comparably to quantile regression for estimating and forecasting tail risks, complementing BVARs’ established performance for forecasting and structural analysis.

Large Hybrid Time-Varying Parameter VARs

Time-varying parameter VARs with stochastic volatility are routinely used for structural analysis and forecasting in settings involving a few macroeconomic variables. Applying these models to high-dimensional datasets has proved to be challenging due to intensive computations and over-parameterization concerns. We develop an efficient Bayesian sparsification method for a class of models we call hybrid TVP-VARs - VARs with time-varying parameters in some equations but constant coefficients in others. Specifically, for each equation, the new method automatically decides (i) whether the VAR coefficients are constant or time-varying, and (ii) whether the error variance is constant or has a stochastic volatility specification. Using US datasets of various dimensions, we find evidence that the VAR coefficients and error variances in some, but not all, equations are time varying. These large hybrid TVP-VARs also forecast better than standard benchmarks.

Real‐time inflation forecast combination for time‐varying coefficient models

AbstractWe use real‐time macroeconomic variables and combination forecasts with both time‐varying weights and equal weights to forecast inflation in the USA. The combination forecasts compare three sets of commonly used time‐varying coefficient autoregressive models: Gaussian distributed errors, errors with stochastic volatility, and errors with moving average stochastic volatility. Both point forecasts and density forecasts suggest that models combined by equal weights do not produce worse forecasts than those with time‐varying weights. We also find that variable selection, the allowance of time‐varying lag length choice, and the stochastic volatility specification significantly improve forecast performance over standard benchmarks. Finally, when compared with the Survey of Professional Forecasters, the results of the best combination model are found to be highly competitive during the 2007/08 financial crisis.

RCAR and CHARMA Models Versus Stochastic Volatility Models: A Forecasting Analysis Using Macrofinancial Data

Using inflation and return time series, we first evaluate the forecasting performance of two classes of conditional heteroscedastic models: the random coefficient autoregressive (RCAR) models and the conditional heteroscedastic autoregressive moving average (CHARMA) models. Markov Chain Monte Carlo schemes are developed for the estimation of the model parameters. Furthermore, the forecasting ability of these two models is compared against that of several stochastic volatility models that control for time-varying parameters, in mean effects, leverage effects and moving average errors. We found that our proposed models can produce better point and density forecasts than various stochastic volatility specifications, with the CHARMA models outperforming the RCAR models.

A Hybrid SLV Model with Multifactor Stochastic Volatility

The need for more satisfactory pricing models has brought the attention of researchers and practitioners on Stochastic Local Volatility models. Despite the growing interest on the topic, however, it seems that no particular attention has been paid to the use of multifactor specifications for the stochastic volatility part. The additional flexibility given by this kind of approach in describing the skew dynamics becomes extremely important when we deal with the pricing of forward volatility sensitive contracts like forward start or cliquet options.This paper tries to fill the gap: we introduce the Wishart Stochastic Local Volatility model that describes the stochastic volatility by means of a Wishart process as proposed. The goal of the new framework is twofold: adding a realistic volatility dynamics to Local Volatility models without the limitations of standard 1-factor Stochastic Volatility approaches, and improving European claims pricing performance of pure Wishart Stochastic Volatility model. The latter is quite relevant since we need to introduce stringent parameters restrictions to satisfy existence and uniqueness conditions for the solution of Wishart SDE. These conditions are required when we want to properly simulate the variance process in order to price more exotic derivatives. However such conditions are not usually met when market calibration is performed and so the resulting constrained parameters set is not able to reproduce accurately the market implied volatility surface. The additional Local Volatility component acts then as compensator.We further present an innovative simulation scheme for the asset path in the pure stochastic volatility specification based on the approach. The simulation scheme is also used to tackle the calibration of the so-called leverage function and price derivatives in the new hybrid framework.

Compound option pricing under stochastic volatility

The paper proposes a flexible and computationally efficient lattice-based approximation for evaluating European and American compound options under stochastic volatility models. In comparison with the existing evaluation procedures, the method is more flexible because it may accommodate several stochastic volatility specifications of the asset price process, and more efficient because it is computationally faster in computing accurate compound option prices. The method is obtained as an extension of Costabile et al. (2012) discretisation, which consists in approximating the stochastic volatility process by a recombining binomial lattice, and considers the asset value as an auxiliary variable whose dynamics is captured by generating subsets of representative realisations to cover the range of possible asset prices at each time slice. The backward induction scheme based on a linear interpolation technique is adapted to compute both the underlying daughter option and the compound option prices. Numerical experiments confirm the method efficiency and accuracy.

Compound option pricing under stochastic volatility

The paper proposes a flexible and computationally efficient lattice-based approximation for evaluating European and American compound options under stochastic volatility models. In comparison with the existing evaluation procedures, the method is more flexible because it may accommodate several stochastic volatility specifications of the asset price process, and more efficient because it is computationally faster in computing accurate compound option prices. The method is obtained as an extension of Costabile et al. (2012) discretisation, which consists in approximating the stochastic volatility process by a recombining binomial lattice, and considers the asset value as an auxiliary variable whose dynamics is captured by generating subsets of representative realisations to cover the range of possible asset prices at each time slice. The backward induction scheme based on a linear interpolation technique is adapted to compute both the underlying daughter option and the compound option prices. Numerical experiments confirm the method efficiency and accuracy.

Empirical Performance of Commodity Pricing Models: When is it Worthwhile to Use a Stochastic Volatility Specification?

We compare the empirical pricing performance of three models: a constant volatility model, a two‐factor stochastic volatility model, and a one‐factor stochastic volatility model with a model‐free implied variance specification. Results of applying these models to oil, copper, and gold derivatives are consistent for all commodities and highlight the relative benefits of the different models implying that in choosing the best model to implement in a real situation, careful consideration must be given to the tradeoffs between effort and precision. We believe our results are not only new, but also relevant for practitioners. © 2015 Wiley Periodicals, Inc. Jrl Fut Mark 36:457–487, 2016

Particle Efficient Importance Sampling

The efficient importance sampling (EIS) method is a general principle for the numerical evaluation of high-dimensional integrals that uses the sequential structure of target integrands to build variance minimising importance samplers. Despite a number of successful applications in high dimensions, it is well known that importance sampling strategies are subject to an exponential growth in variance as the dimension of the integration increases. We solve this problem by recognising that the EIS framework has an offline sequential Monte Carlo interpretation. The particle EIS method is based on non-standard resampling weights that take into account the look-ahead construction of the importance sampler. We apply the method for a range of univariate and bivariate stochastic volatility specifications. We also develop a new application of the EIS approach to state space models with Student's t state innovations. Our results show that the particle EIS method strongly outperforms both the standard EIS method and particle filters for likelihood evaluation in high dimensions. Moreover, the ratio between the variances of the particle EIS and particle filter methods remains stable as the time series dimension increases. We illustrate the efficiency of the method for Bayesian inference using the particle marginal Metropolis-Hastings and importance sampling squared algorithms.

Particle efficient importance sampling

The efficient importance sampling (EIS) method is a general principle for the numerical evaluation of high-dimensional integrals that uses the sequential structure of target integrands to build variance minimising importance samplers. Despite a number of successful applications in high dimensions, it is well known that importance sampling strategies are subject to an exponential growth in variance as the dimension of the integration increases. We solve this problem by recognising that the EIS framework has an offline sequential Monte Carlo interpretation. The particle EIS method is based on non-standard resampling weights that take into account the construction of the importance sampler as a sequential approximation to the state smoothing density. We apply the method for a range of univariate and bivariate stochastic volatility specifications. We also develop a new application of the EIS approach to state space models with Student’s t state innovations. Our results show that the particle EIS method strongly outperforms both the standard EIS method and particle filters for likelihood evaluation in high dimensions. We illustrate the efficiency of the method for Bayesian inference using the particle marginal Metropolis–Hastings and importance sampling squared algorithms.