Score-driven Models Research Articles

Generalized Autoregressive Conditional Betas: A New Multivariate Score-Driven Filter

Abstract In this paper, we extend the recent Gaussian autoregressive conditional beta (Gaussian-ACB) model from the literature on score-driven models. In the new asset pricing model, named the t generalized ACB (t-GACB) model, a multivariate score-driven filter for the t-distribution drives dynamic interaction effects among the beta coefficients. We present the econometric formulation and statistical inference for the t-GACB model, which we apply to 15 stocks from the United States (US) from 1999 to 2022. In our empirical application, we use the three Fama–French factors as asset pricing factors, and we also use the Volatility Index, TED Spread, and ICE BofA US High Yield Index Option-Adjusted Spread as exogenous explanatory variables that influence the beta coefficients. We compare the statistical and realized tracking error performances of the Gaussian-ACB, t-ACB, and t-GACB models, and we find that the t-GACB model improves the Gaussian-ACB model.

On the optimality of score-driven models

Summary Score-driven models have recently been introduced as a general framework to specify time-varying parameters of conditional densities. The score enjoys stochastic properties that make these models easy to implement and convenient to apply in several contexts, ranging from biostatistics to finance. Score-driven parameter updates have been shown to be optimal in terms of locally reducing a local version of the Kullback–Leibler divergence between the true conditional density and the postulated density of the model. A key limitation of such an optimality property is that it holds only locally both in the parameter space and sample space, yielding to a definition of local Kullback–Leibler divergence that is in fact not a divergence measure. The current paper shows that score-driven updates satisfy stronger optimality properties that are based on a global definition of Kullback–Leibler divergence. In particular, it is shown that score-driven updates reduce the distance between the expected updated parameter and the pseudo-true parameter. Furthermore, depending on the conditional density and the scaling of the score, the optimality result can hold globally over the parameter space, which can be viewed as a generalization of the monotonicity property of the stochastic gradient descent scheme. Several examples illustrate how the results derived in the paper apply to specific models under different easy-to-check assumptions, and provide a formal method to select the link function and the scaling of the score.

Comparison of Score-Driven Equity-Gold Portfolios During the COVID-19 Pandemic Using Model Confidence Sets

Abstract Gold may have a hedge, safe haven, or diversifier property when added to stock portfolios. Motivated by the favorable statistical properties and out-of-sample performance of score-driven models, we investigate for equity-gold portfolios whether score-driven mean, volatility, and copula models can improve the performances of DCC (dynamic conditional correlation) portfolios, the naïve portfolio strategy, and the Standard & Poor’s 500 (S&P 500) index. We consider 2880 score-driven portfolio strategies. We use score-driven Clayton, rotated Clayton, Frank, Gaussian, Gumbel, rotated Gumbel, Plackett, and Student’s t copulas. We use several classical and score-driven models of marginal distribution. We use weekly, monthly, quarterly, semi-annual, and annual updates of portfolio weights. We use minimum-variance, maximum Sharpe ratio, and maximum utility function strategies. We use rolling data-windows for portfolio optimization for the COVID-19 investment period of February 2020 to September 2021. We classify competing portfolios by using a new robust multi-step model confidence set (MCS) test approach and provide evidence of the superiority of score-driven portfolios.

Score-driven models for realized volatility

This paper sets up a statistical framework for modeling realized volatility (RV) using a Dynamic Conditional Score (DCS) model. It first shows how, for a dataset on stock indices, a preliminary analysis of RV, based on fitting a linear Gaussian model to its logarithm, suggests the use of a two component dynamic specification. It also indicates a departure from normality, a weekly pattern in the data and the presence of heteroscedasticity. Fitting the two component DCS specification with leverage and a day of the week effect is then carried out directly on RV with a Generalized Beta of the Second Kind (GB2) conditional distribution or, equivalently, on the logarithm of RV with an Exponential Generalized Beta of the Second Kind (EGB2) distribution. The forecasting performance of this model, with and without heteroscedasticity, is compared with that of the Heterogeneous Autoregression (HAR), some extensions of it and some other models. Overall there is a clear gain from using the GB2-DCS model, even when the HAR model uses additional information, such as realized semi-variance. When the aim is to forecast tail behavior, the fat-tailed GB2 model performs much better than models with thin-tailed distributions. A further exercise uses a dataset on exchange rates to compare GB2-DCS models with models that use realized quarticity. Again the additional information offers no forecasting advantage. Overall the GB2-DCS models are transparent, provide a comprehensive description of the properties of RV, and are difficult to beat for forecasting.

Score function scaling for QAR plus Beta-t-EGARCH: an empirical application to the S&P 500

ABSTRACT In the literature on score-driven models, alternative choices of the scaling parameters of the conditional score terms are used, but the optimal choice of those parameters is an open question. Although there are score-driven models for which the choice of the scaling parameters is irrelevant, there are important score-driven models for which score-driven scale filters appear in the information matrix, and the choice of the scaling parameters is relevant. We focus on the quasi-autoregressive (QAR) plus Beta--EGARCH (exponential generalized autoregressive conditional heteroscedasticity) model, by using data on the Vanguard Standard & Poor’s 500 (S&P 500) exchange-traded fund (VOO) and all available S&P 500 stocks for the period of 2013–2023. For QAR plus Beta--EGARCH, each updating term is the product of a scaling parameter and a conditional score, and we use specific alternative scaling parameters from the literature. For different scaling parameters in the scale filter (volatility), alternative location and scale filters coincide. For different scaling parameters in the location filter (expected return), alternative location and scale filters differ significantly. For the statistical and volatility forecasting performances of VOO and most of the S&P 500 stocks, the best-performing scaling parameter for the score-driven location is the conditional inverse information matrix.

Time-Varying Parameters in Econometrics: The editor’s foreword

The Themed Issue Time-Varying Parameters in Econometrics consists of eight papers where theoretical, methodological and empirical developments are reported. Particular attention is given to observation-driven time-varying parameter models of which score-driven models is a sub-class. We will introduce the Themed Issue and review its contributions.

Non-Gaussian score-driven conditionally heteroskedastic models with a macroeconomic application

AbstractWe contribute to the literature on empirical macroeconomic models with time-varying conditional moments, by introducing a heteroskedastic score-driven model with Student’s t-distributed innovations, named the heteroskedastic score-driven $t$ -QVAR (quasi-vector autoregressive) model. The $t$ -QVAR model is a robust nonlinear extension of the VARMA (VAR moving average) model. As an illustration, we apply the heteroskedastic $t$ -QVAR model to a dynamic stochastic general equilibrium model, for which we estimate Gaussian-ABCD and $t$ -ABCD representations. We use data on economic output, inflation, interest rate, government spending, aggregate productivity, and consumption of the USA for the period of 1954 Q3 to 2022 Q1. Due to the robustness of the heteroskedastic $t$ -QVAR model, even including the period of the coronavirus disease of 2019 (COVID-19) pandemic and the start of the Russian invasion of Ukraine, we find a superior statistical performance, lower policy-relevant dynamic effects, and a higher estimation precision of the impulse response function for US gross domestic product growth and US inflation rate, for the heteroskedastic score-driven $t$ -ABCD representation rather than for the homoskedastic Gaussian-ABCD representation.

Score-Driven Modeling with Jumps: An Application to S&P500 Returns and Options

Abstract We introduce a novel score-driven model with two sources of shock, allowing for both time-varying volatility and jumps. A theoretical investigation is performed which yields sufficient conditions to ensure stationarity and ergodicity. We extend the model to consider a time-varying jump intensity. Both an in-sample and an out-of-sample analysis based on the S&P500 time series show that the proposed methodology provides excellent agreement with observed returns, outperforming more standard Generalized Autoregressive Conditional Heteroskedasticity (GARCH) specifications with jumps. Finally, we apply our models to option pricing via risk neutralization. Results show this novel approach produces reliable implied volatility surfaces. Supplementary Materials including proofs, the derivation of the conditional Fisher information, and two figures showing additional empirical results are available online.

COVID-19 Active Case Forecasts in Latin American Countries Using Score-Driven Models

With the aim of mitigating the damage caused by the coronavirus disease 2019 (COVID-19) pandemic, it is important to use models that allow forecasting possible new infections accurately in order to face the pandemic in specific sociocultural contexts in the best possible way. Our first contribution is empirical. We use an extensive COVID-19 dataset from nine Latin American countries for the period of 1 April 2020 to 31 December 2021. Our second and third contributions are methodological. We extend relevant (i) state-space models with score-driven dynamics and (ii) nonlinear state-space models with unobserved components, respectively. We use weekly seasonal effects, in addition to the local-level and trend filters of the literature, for (i) and (ii), and the negative binomial distribution for (ii). We find that the statistical and forecasting performances of the novel score-driven specifications are superior to those of the nonlinear state-space models with unobserved components model, providing a potential valid alternative to forecasting the number of possible new COVID-19 infections.

Lumpy and intermittent retail demand forecasts with score-driven models

Irregular demand in retail is characterized by stock keeping units (SKUs) that present high intermittency, and simultaneously, high erraticness (classified as lumpy demand) or low erraticness (classified as intermittent demand), following the classification of Syntetos et al. (2005). These SKUs are basically defined by periods of zero sales interleaved with positive sales producing series with some degree of variability. Many SKUs at the store/daily level can be characterized as presenting such a type of demand. Therefore methods for adequately forecasting irregular time series are necessary for proper inventory management. This article derives models for intermittent and lumpy time series using the framework of score-driven models as developed by Creal et al. (2013) and Harvey (2013). More precisely we derive Poisson, negative binomial, hurdle Poisson, and hurdle negative binomial models and apply them to real data obtained from a large Brazilian retail chain, comparing the performance of the proposed models to adequate competing methods from the ‘slow’/intermittent demand forecasting literature. Forecasting accuracy is evaluated based on point forecasts and the entire predictive distribution. Our results show that the score-driven models perform well compared to intermittent traditional forecasting methods, providing competitive forecasting models for irregular demand in retailing.

Modelling volatility dependence with score copula models

Abstract I study score-driven models for modelling high persistence dependence between financial volatility series. I model this persistence dependence with two components, one for the long memory and the other for the short-term process. The addition of components offers a parsimonious solution for modelling high persistence and also allows for a short-term component for the transient shocks. I apply the model to emerging equities in the Americas. The estimates are robust to the advent of the pandemic. In addition, data resampling and marginal alternatives deliver similar parameter estimates. The proposed two-component model improves the in-sample diagnostics and generates more accurate out-of-sample forecasts.

Score-driven multi-regime Markov-switching EGARCH: empirical evidence using the Meixner distribution

Abstract In this paper, statistical and volatility forecasting performances of the non-path-dependent score-driven multi-regime Markov-switching (MS) exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models are explored. Three contributions to the existing literature are provided. First, we use all relevant score-driven distributions from the literature - namely, the Student’s t-distribution, general error distribution (GED), skewed generalized t-distribution (Skew-Gen-t), exponential generalized beta distribution of the second kind (EGB2), and normal-inverse Gaussian (NIG) distribution. We then introduce the score-driven Meixner (MXN) distribution-based EGARCH model to the literature on score-driven models. Second, proving the sufficient conditions of the asymptotic properties of the maximum likelihood (ML) estimator for non-path-dependent score-driven MS-EGARCH models is an unsolved problem. We provide a partial solution to that problem by proving necessary conditions for the asymptotic theory of the ML estimator. Third, to the best of our knowledge, this work includes the largest number of international stock indices from the G20 countries in the literature, covering the period of 2000–2022. We provide a discussion on the major events which caused common or non-common switching to the high-volatility regime for the G20 countries. The statistical performance and volatility forecasting results support the adoption of score-driven MS-EGARCH for the G20 countries.

Predicting VaR for China's stock market: A score-driven model based on normal inverse Gaussian distribution

Under the framework of dynamic conditional score, we propose a parametric forecasting model for Value-at-Risk based on the normal inverse Gaussian distribution (Hereinafter NIG-DCS-VaR), which creatively incorporates intraday information into daily VaR forecast. NIG specifies an appropriate distribution to return and the semi-additivity of the NIG parameters makes it feasible to improve the estimation of daily return in light of intraday return, and thus the VaR can be explicitly obtained by calculating the quantile of the re-estimated distribution of daily return. We conducted an empirical analysis using two main indexes of the Chinese stock market, and a variety of backtesting approaches as well as the model confidence set approach prove that the VaR forecasts of NIG-DCS model generally gain an advantage over those of realized GARCH (RGARCH) models. Especially when the risk level is relatively high, NIG-DCS-VaR beats RGARCH-VaR in terms of coverage ability and independence.

Conservatorship, quantitative easing, and mortgage spreads: a new multi-equation score-driven model of policy actions

Abstract In this paper, the effects of United States (US) policy actions on mortgage-backed security and mortgage loan spreads are measured, by using data before, during, and after the US subprime mortgage crisis. We study the effects of the following policy actions: (i) the placement of Fannie Mae and Freddie Mac into US Government conservatorship; (ii) the US Federal Reserve quantitative easing (QE) programs. We provide the following contributions to the literature: (i) for a robust measurement of policy effects, a new multi-equation score-driven t-QVARMA (quasi-vector autoregressive moving average) model is used. (ii) In addition to the measurement of the effects of QE, the effects of government conservatorship are also measured in this paper. (iii) Furthermore, the data period of the relevant literature is extended to the period of June 1998 to March 2020.

Robust Estimation and Forecasting of Climate Change Using Score-Driven Ice-Age Models

We use data on the following climate variables for the period of the last 798 thousand years: global ice volume (Icet), atmospheric carbon dioxide level (CO2,t), and Antarctic land surface temperature (Tempt). Those variables are cyclical and are driven by the following strongly exogenous orbital variables: eccentricity of the Earth’s orbit, obliquity, and precession of the equinox. We introduce score-driven ice-age models which use robust filters of the conditional mean and variance, generalizing the updating mechanism and solving the misspecification of a recent climate–econometric model (benchmark ice-age model). The score-driven models control for omitted exogenous variables and extreme events, using more general dynamic structures and heteroskedasticity. We find that the score-driven models improve the performance of the benchmark ice-age model. We provide out-of-sample forecasts of the climate variables for the last 100 thousand years. We show that during the last 10–15 thousand years of the forecasting period, for which humanity influenced the Earth’s climate, (i) the forecasts of Icet are above the observed Icet, (ii) the forecasts of CO2,t level are below the observed CO2,t, and (iii) the forecasts of Tempt are below the observed Tempt. The forecasts for the benchmark ice-age model are reinforced by the score-driven models.

Quasi score-driven models

This paper introduces the class of quasi score-driven (QSD) models. This new class inherits and extends the basic ideas behind the development of score-driven (SD) models and addresses a number of unsolved issues in the score literature. In particular, the new class of models (i) generalizes many existing models, including SD models, (ii) disconnects the updating equation from the log-likelihood implied by the conditional density of the observations, (iii) allows testing of the assumptions behind SD models that link the updating equation of the conditional moment to the conditional density, (iv) allows QML estimation of SD models, (v) and allows explanatory variables to enter the updating equation. We establish the asymptotic properties of the QLE, QMLE and MLE of the proposed QSD model as well as the likelihood ratio and Lagrange multiplier test statistics. The finite sample properties are studied by means of an extensive Monte Carlo study. Finally, we show the empirical relevance of QSD models to estimate the conditional variance of 400 US stocks.

Signal smoothing for score-driven models: a linear approach

In the present article, a linear approach of signal smoothing for nonlinear score-driven models is suggested, by using results from the literature on minimum mean squared error signals. Score-driven location, trend, and seasonality models with constant and score-driven scale parameters are used, for which the parameters are estimated by using the maximum likelihood method. The smoothing procedure is computationally fast, and it uses closed-form formulas for smoothed signals. Applications for monthly data of the seasonally adjusted and the not seasonally adjusted the United States inflation rate variables for the period of 1948–2020 are presented.

Multivariate Score-Driven Models for Count Time Series To Assess Financial Contagion

Multivariate Score-Driven Models for Count Time Series To Assess Financial Contagion

Score-Driven Modeling of Spatio-Temporal Data

A simultaneous autoregressive score-driven model with autoregressive disturbances is developed for spatio-temporal data that may exhibit heavy tails. The model specification rests on a signal plus noise decomposition of a spatially filtered process, where the signal can be approximated by a nonlinear function of the past variables and a set of explanatory variables, while the noise follows a multivariate Student-t distribution. The key feature of the model is that the dynamics of the space-time varying signal are driven by the score of the conditional likelihood function. When the distribution is heavy-tailed, the score provides a robust update of the space-time varying location. Consistency and asymptotic normality of maximum likelihood estimators are derived along with the stochastic properties of the model. The motivating application of the proposed model comes from brain scans recorded through functional magnetic resonance imaging when subjects are at rest and not expected to react to any controlled stimulus. We identify spontaneous activations in brain regions as extreme values of a possibly heavy-tailed distribution, by accounting for spatial and temporal dependence.

Dynamic Tobit models

Score-driven models provide a solution to the problem of modeling time series when the observations are subject to censoring and location and/or scale may change over time. The method applies to generalized t and EGB2 distributions, as well as to the normal distribution. Explanatory variables can be included, making static Tobit models a special case. A set of Monte Carlo experiments show that the score-driven model provides good forecasts even when the true model is parameter-driven. The viability of the new models is illustrated by fitting them to data on Chinese stock returns.