Generalized Autoregressive Score Trees and Forests*

ABSTRACTConditional risk measuring plays an important role in financial regulation and depends on volatility estimation. A new class of parameter models called Generalized Autoregressive Score (GAS) model has been successfully applied for different error's densities and for different problems of time series prediction in particular for volatility modeling and VaR estimation. To improve the estimating accuracy of the GAS model, this study proposed a semi-parametric method, LS-SVR and FS-LS-SVR applied to the GAS model to estimate the conditional VaR. In particular, we fit the GAS(1,1) model to the return series using three different distributions. Then, LS-SVR and FS-LS-SVR approximate the GAS(1,1) model. An empirical research was performed to illustrate the effectiveness of the proposed method. More precisely, the experimental results from four stock indexes returns suggest that using hybrid models, GAS-LS-SVR and GAS-FS-LS-SVR provides improved performances in the VaR estimation.

This paper compares Generalized Autoregressive Score (GAS) models and GARCH‐type models on their forecasting abilities for crude oil and natural gas spot and futures returns from developing and developed markets over multiple horizons. The out‐of‐sample forecasting results based on two loss functions and the Diebold–Mariano predictive accuracy test for multiple models show that the GAS framework outperforms GARCH and EGARCH models, particularly for crude oil assets. For natural gas, no specific model retains an advantage over the other two models as the predictive accuracy changes over forecasting horizons and varies across markets. Meanwhile, the GAS model performs well in both developed and developing markets. The cumulated sum of squared forecast error differential (CSSFED) graphically monitors the evolution of the relative forecasting performance of different models and shows that the superiority of GARCH is vulnerable to extraordinary event shocks. Over the short‐term forecasting (less than or equal to 1 month ahead), the GAS framework shows a prominent advantage over GARCH and EGARCH models for crude oil assets.

PurposeThis paper examines and forecasts correlations between cryptocurrencies and major fiat currencies using Generalized Autoregressive Score (GAS) time-varying copulas. The authors examine to which extent the multivariate GAS method captures the volatility persistence and the nonlinear interaction effects between cryptocurrencies and major fiat currencies.Design/methodology/approachThe authors model tail dependence between conventional currencies and Bitcoin utilizing a Glosten-Jagannathan-Runkle Generalized Autoregressive Conditional Heteroscedastic model (GJR-GARCH)-GAS copula specification, which allows detecting the leptokurtic feature and clustering effects of currency returns distribution.FindingsThe authors' results show evidence of multiple tail dependence regimes, implying the unsuitability of applying static models to entirely describe the extreme dependence between Bitcoin and fiat currencies. Compared to the most common constant copulas, the authors find that the multivariate GAS copulas better forecast the volatility and dependency between cryptocurrencies and foreign exchange markets. Furthermore, based on the value-at-risk (VaR) and expected shortfall (ES) analyses, the authors show that the multivariate GAS models produce accurate risk measures by adding cryptocurrencies to a portfolio of fiat currencies.Originality/valueThis paper has two main contributions to the existing literature on cryptocurrencies. First, the authors empirically examine the tail dependence structure between common conventional currencies and bitcoin using GJR-GARCH GAS copulas which consider the leptokurtic feature and clustering effects of currency returns distribution. Second, by modeling VaR and ES, the authors test the implication of using time-varying models on the performance of currency portfolios, including cryptocurrencies.

This study compares the performance of hedged equity index portfolios constructed using either a generalized autoregressive score (GAS) or a realized GAS (GRAS) model. GAS models encompass popular models, and studies indicate that high‐frequency data improve a model's forecasting ability. The in‐sample estimation results demonstrate that the GRAS model has better explanatory power and more robust time‐varying variance and dependence parameters when fat‐tailed distributions are accounted for. The out‐of‐sample comparison confirms its superiority in reducing hedged portfolio variance and accruing economic benefits to highly risk‐averse hedgers.

A new framework for the joint estimation and forecasting of dynamic Value-at-Risk (VaR) and Expected Shortfall (ES) is proposed by incorporating intraday information into a generalized autoregressive score (GAS) model, introduced by Patton, Ziegel, and Chen (2019) to estimate risk measures in a quantile regression setup. We consider four intraday measures: the realized variance at 5-min and 10-min sampling frequencies, and the overnight return incorporated into these two realized variances. In a forecasting study, the set of newly proposed semiparametric models is applied to 4 international stock market indices: the S&P 500, the Dow Jones Industrial Average, the NIKKEI 225 and the FTSE 100, and is compared with a range of parametric, nonparametric and semiparametric models including historical simulations, GARCH and the original GAS models. VaR and ES forecasts are backtested individually, and the joint loss function is used for comparisons. Our results show that GAS models, enhanced with the realized variance measures (especially at 5 minutes frequency), outperform the benchmark models consistently across all indices and various probability levels.

From the existing literature, it is important to note that multivariate GARCH models are widely used to forecast correlation and hedging among different kinds of assets, but without providing a comparison with other types of modelling. This paper compares the forecasting performances of classical DCC-GARCH model with that of the novel multivariate Generalized Autoregressive Score (GAS) model in analyzing the volatilities, correlations and hedging effectiveness of Artificial Intelligence (AI) stock ETF with both Clean and Dirty energies. The estimation results show that the degree of connectedness for the Clean-AI pair is more pronounced than that of Dirty-AI. Furthermore, AI-based assets provide significantly better hedging potential and stronger diversification gains for Dirty energy than for Clean energy. DCC-GARCH model effectively captures the persistent hedging structure in Clean energy portfolios, whereas the GAS framework proves more suitable for Dirty energy due to its ability to track abrupt market adjustments and geopolitical shocks. Overall, our empirical results reveal important practical implications

Five different distributions for the Lee–Carter model of mortality forecasting: A comparison using GAS models

Forecasting the dynamic relationship between crude oil and stock prices since the 19th century

In this article, we develop a two-step method for conditional Value at Risk (VaR) estimation in the context of the Generalized Autoregressive Score (GAS) models. The first step consists of estimating the volatility parameter by the generalized Quasi Maximum Likelihood Estimator (gQMLE) and in the second step we estimate the theoretical quantile of the innovations by the empirical quantile of the residuals. When the instrumental density q of the gQMLE is not the Gaussian density used in the standard QMLE, or is not the true distribution of the innovations, both the estimations of the volatility and of the quantile are asymptotically biased. In spite of that the two errors counterbalance each other, and we finally get a consistent estimator of the conditional VaR. We establish the asymptotic properties of the gQMLE for GAS models as well as the conditional VaR two-step estimator. Moreover, we discuss how to apply the gQMLE by giving examples of densities distribution and we get worthwhile results.

A multivariate econometric model can be used to forecast volatilities and dynamic correlations between various assets. Volatilities and dynamic correlations forecasting are important applied in risk management, hedging, asset allocation, and asset pricing options. This study uses a multivariate Generalized Autoregressive Score (GAS) model to analyze and forecast volatilities and dynamic correlations of the weekly prices of Crude Palm Oil, coconut oil, soybean oil, and crude oil. The GAS model is a new framework based on a score-driven time series model for updating time-varying parameters. Another multivariate econometric model is the Dynamic Conditional Correlation (DCC) GARCH model as a comparison. The empirical experiments regarding the distribution of return data show that the data can be approximated by the multivariate t-student distribution. The likelihood ratio test in the multivariate GAS model shows that the time-varying parameters in the GAS model are volatility, correlation, and location parameters. The evaluation performance model based on RMSE and MAE show that the multivariate GAS model has better performance in estimating and forecasting volatilities than the DCC GARCH model. Multivariate GAS models have better performance in estimating and forecasting the dynamic correlation on returns of CPO and soybean oil.

Forecasting volatility and correlation between oil and gold prices using a novel multivariate GAS model

SUMMARYWe propose a class of observation‐driven time series models referred to as generalized autoregressive score (GAS) models. The mechanism to update the parameters over time is the scaled score of the likelihood function. This new approach provides a unified and consistent framework for introducing time‐varying parameters in a wide class of nonlinear models. The GAS model encompasses other well‐known models such as the generalized autoregressive conditional heteroskedasticity, autoregressive conditional duration, autoregressive conditional intensity, and Poisson count models with time‐varying mean. In addition, our approach can lead to new formulations of observation‐driven models. We illustrate our framework by introducing new model specifications for time‐varying copula functions and for multivariate point processes with time‐varying parameters. We study the models in detail and provide simulation and empirical evidence. Copyright © 2012 John Wiley & Sons, Ltd.

An extension of the Generalized Autoregressive Score (GAS) model is presented for time series with excess null observations to include explanatory variables. An extension of the GAS model proposed by Harvey and Ito is suggested, and it is applied to precipitation data from a city in Chile. It is concluded that the model provides adequate prediction, and furthermore, an analysis of the relationship between the precipitation variable and the explanatory variables is shown. This relationship is compared with the meteorology literature, demonstrating concurrence.

Generalized autoregressive score models based on sinh-arcsinh distributions for time series analysis

Forecasting mortality rates is crucial for evaluating life insurance company solvency, especially amid disruptions caused by phenomena like COVID-19. The Lee–Carter model is commonly employed in mortality modelling; however, extensions that can encompass count data with diverse distributions, such as the Generalized Autoregressive Score (GAS) model utilizing the COM–Poisson distribution, exhibit potential for enhancing time-to-event forecasting accuracy. Using mortality data from 29 countries, this research evaluates various distributions and determines that the COM–Poisson model surpasses the Poisson, binomial, and negative binomial distributions in forecasting mortality rates. The one-step forecasting capability of the GAS model offers distinct advantages, while the COM–Poisson distribution demonstrates enhanced flexibility and versatility by accommodating various distributions, including Poisson and negative binomial. Ultimately, the study determines that the COM–Poisson GAS model is an effective instrument for examining time series data on mortality rates, particularly when facing time-varying parameters and non-conventional data distributions.