Donker van Heel, S., R.-J. Lange, D. van Dijk, and B. van Os, 2025, Stability and performance guarantees for misspecified multivariate score-driven filters, Tinbergen Institute Discussion Paper 2025-006/III.
Abstract: We consider the problem of tracking latent time-varying parameter vectors under model misspecification. We analyze implicit and explicit score-driven (ISD and ESD) filters, which update a prediction of the parameters using the gradient of the logarithmic observation density (i.e., the score). In the ESD filter, the score is computed using the predicted parameter values, whereas in the ISD filter, the score is evaluated using the new, updated parameter values. For both filter types, we derive novel sufficient conditions for the exponential stability (i.e., invertibility) of the filtered parameter path and existence of a finite mean squared error (MSE) bound with respect to the pseudo-true parameter path. In addition, we present expressions for finite-sample and asymptotic MSE bounds. Our performance guarantees rely on mild moment conditions on the data-generating process, while our stability result is entirely agnostic about the true process. As a result, our primary conditions depend only on the characteristics of the filter; hence, they are verifiable in practice. Concavity of the postulated log density combined with simple parameter restrictions is sufficient (but not necessary) for ISD-filter stability, whereas ESD-filter stability additionally requires the score to be Lipschitz continuous. Extensive simulation studies validate our theoretical findings and demonstrate the enhanced stability and improved performance of ISD over ESD filters. An empirical application to U.S. Treasury-bill rates confirms the practical relevance of our contribution.
de Punder, R., C.G.H. Diks, R.J.A. Laeven and D. van Dijk, 2023, Localizing strictly proper scoring rules, Tinbergen Institute Discussion Paper 2023-084/III.
Abstract: When comparing predictive distributions, forecasters are typically not equally interested in all regions of the outcome space. To address the demand for focused forecast evaluation, we propose a procedure to transform strictly proper scoring rules into their localized counterparts while preserving strict propriety. This is accomplished by applying the original scoring rule to a censored distribution, acknowledging that censoring emerges as a natural localization device due to its ability to retain precisely all relevant information of the original distribution. Our procedure nests the censored likelihood score as a special case. Among a multitude of others, it also implies a class of censored kernel scores that offers a multivariate alternative to the threshold weighted Continuously Ranked Probability Score (twCRPS), extending its local propriety to more general weight functions than single tail indicators. Within this localized framework, we obtain a generalization of the Neyman Pearson lemma, establishing the censored likelihood ratio test as uniformly most powerful. For other tests of localized equal predictive performance, results of Monte Carlo simulations and empirical applications to risk management, inflation and climate data consistently emphasize the superior power properties of censoring.
Lange, R.-J., B. van Os and D. van Dijk, 2022, Implicit score-driven filters for time-varying parameter models, Tinbergen Institute Discussion Paper 2022-066/III. (revised June 2024)
Abstract: We propose an observation-driven modeling framework that permits time variation in the model's parameters using an implicit score-driven (ISD) update. The ISD update maximizes the observation log density with respect to the parameter vector, while penalizing the weighted ℓ2 norm relative to the one-step-ahead prediction. This yields an implicit stochastic-gradient update; taking instead the explicit version produces the popular class of score-driven models. Specifically, we show that the explicit score-driven (ESD) update arises as a linear approximation to the ISD update. By preserving the full density, the ISD update globalizes favorable local properties of the ESD update. Namely, for log-concave observation densities (even when misspecified), ISD models are stable for any learning rate and globally contractive to a pseudo-truth. We demonstrate the usefulness of ISD models in both simulations and empirical illustrations for finance and macroeconomics.
Gong, X., M. van der Wel and D. van Dijk, 2019, Improved forecasting of the implied volatility surface.
Abstract: The (implied) volatility surface is the collection of option-implied volatilities for different strike prices and maturities. Existing literature documents that the volatility surface can be modelled by a limited number of factors using simple regression techniques, and that these factors are persistent. However, regression techniques leave substantial serial correlation in the residuals. We propose an autoregressive model and an `equilibrium correction' style model which uses the information of the deviation from put call parity to directly exploit the serial correlation. We apply the models to S&P500 index options and options of 95 stocks, and show that the new models improve the existing models with a 40% decrease of both in-sample RMSE and out-of-sample RMSFE. The economic significance evaluation shows that the new models can generate higher Sharpe ratios than existing models.
Gong, X., M. van der Wel and D. van Dijk, 2019, Using index options to enhance the description of the implied volatility surface of equity options.
Abstract: We put forward a novel factor-based approach for modeling and forecasting the implied volatility surface (IVS) for a cross-section of equity options. Inspired by the findings of Christoffersen et al. (2017) that equity volatility levels, smiles and term structures show strong co-movement, we postulate the presence of common factors in these characteristics. We argue that liquid index options can be used to obtain estimates of these common factors, which renders a feasible modeling approach even in high-dimensional settings. We implement the model for a cross-section of 97 equity options combined with options on the S&P 500 index. We document strong explanatory power of index options factors for the equity IVSs, and find that incorporating the index option information can improve the forecasting performance, particularly for illiquid stocks.