Search

International audience; 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.

This article deals with the problem of estimating the volatility of a financial security in a market with frictions. We propose a microstructural model with time-varying fundamental price volatility in which the trading price varies only if the value of the information signal is large enough to guarantee a profit in excess of transaction costs. Using transaction data only, the proposed approach allows to recover: (i) the conditional volatility of the information signal, which is thus cleaned out by market frictions and (ii) an estimate of transaction costs. Our analysis reveals that, after correcting for frictions, the risk of illiquid securities is substantially different from what is predicted by traditional volatility models. Furthermore, using a big dataset of intraday returns, we show that our transaction cost estimate is highly correlated with the main illiquidity measures and that such correlations are significant under different volatility regimes.

The occurrence of extreme observations in a time series depends on the heaviness of the tails of its distribution. The paper proposes a dynamic conditional score model (DCS) for modelling dynamic shape parameters that govern the tail index. The model is based on the Generalised t family of conditional distributions, allowing for the presence of asymmetric tails and therefore the possibility of specifying different dynamics for the left and right tail indices. The paper examines through simulations both the convergence properties of the model and the implications of the link functions used. In addition the paper introduces and studies the size and power properties of a new Lagrange Multiplier (LM) test based on fitted scores to detect the presence of dynamics in the tail index parameter. The paper also shows that the novel LM test is more effective than existing tests based on fitted scores. The model is fitted to Equity Indices and Credit Default Swaps returns. It is found that the tail index for equities has dynamics driven mainly by either the upper or lower tail depending if leverage is taken or not into account. In the case of Credit Default Swaps the test identifies very persistent dynamics for both the tails. Finally the implications of dynamic tail indices for the estimated conditional distribution are assessed in terms of conditional distribution forecasting showing that the novel model predicts more accurately expected shortfalls and value-at-risk than existing models.

We provide general conditions under which a class of discrete-time volatility models driven by the score of the conditional density converges in distribution to a stochastic differential equation as the interval between observations goes to zero. We show that the form of the diffusion limit depends on: (i) the link function, (ii) the conditional second moment of the score, (iii) the normalization of the score. Interestingly, the properties of the stochastic differential equation are strictly entangled with those of the discrete-time counterpart. Score-driven models with fat-tailed densities lead to continuous-time processes with finite volatility of volatility, as opposed to fat-tailed models with a GARCH update, for which the volatility of volatility is explosive. We examine in simulations the implications of such results on approximate estimation and filtering of diffusion processes. An extension to models with a time-varying conditional mean and to conditional covariance models is also developed. We provide general conditions under which a class of discrete-time volatility models driven by the score of the conditional density converges in distribution to a stochastic differential equation as the interval between observations goes to zero. We show that the form of the diffusion limit depends on: (i) the link function, (ii) the conditional second moment of the score, (iii) the normalization of the score. Interestingly, the properties of the stochastic differential equation are strictly entangled with those of the discrete-time counterpart. Score-driven models with fat-tailed densities lead to continuous-time processes with finite volatility of volatility, as opposed to fat-tailed models with a GARCH update, for which the volatility of volatility is explosive. We examine in simulations the implications of such results on approximate estimation and filtering of diffusion processes. An extension to models with a time-varying conditional mean and to conditional covariance models is also developed.

We propose a class of score-driven realized covariance models where volatilities and correlations are separately estimated. We can thus combine univariate realized volatility models with a recently introduced class of score-driven realized covariance models based on Wishart and matrix- F distributions. Compared to the latter, the proposed models remain computationally simple at high dimensions and allow for higher flexibility in parameter estimation. Through a Monte-Carlo study, we show that the two-step maximum likelihood procedure provides accurate parameter estimates in small samples. Empirically, we find that the proposed models outperform those based on joint estimation, with forecasting gains that become more significant as the cross-section dimension increases.

In this paper we develop a general framework to analyze state space models with timevarying system matrices where time variation is driven by the score of the conditional likelihood. We derive a new filter that allows for the simultaneous estimation of the state vector and of the time-varying parameters. We use this method to study the timevarying relationship between the price dividend ratio, expected stock returns and expected dividend growth in the US since 1880. We find a significant increase in the long-run equilibrium value of the price dividend ratio over time, associated with a fall in the longrun expected rate of return on stocks. The latter can be attributed mainly to a decrease in the natural rate of interest, as the long-run risk premium has only slightly fallen.

This paper proposes a novel approach to introduce time-variation in the variances of the structural shocks of DSGE models. The variances are allowed to evolve over time via an observation-driven updating equation. The estimation of the resulting DSGE model can be easily performed by maximum likelihood without the need of time-consuming simulation-based methods. An empirical application to a DSGE model with time-varying volatility for structural shocks shows a significant improvement in the accuracy of density forecasts.

Despite their effectiveness, linear models for realized variance neglect measurement errors on integrated variance and exhibit several forms of misspecification due to the inherent nonlinear dynamics of volatility. We propose new extensions of the popular approximate long-memory heterogeneous autoregressive (HAR) model apt to disentangle these effects and quantify their separate impact on volatility forecasts. By combining the asymptotic theory of the realized variance estimator with the Kalman filter and by introducing time-varying HAR parameters, we build new models that account for: (i) measurement errors (HARK), (ii) nonlinear dependencies (SHAR) and (iii) both measurement errors and nonlinearities (SHARK). The proposed models are simply estimated through standard maximum likelihood methods and are shown, both on simulated and real data, to provide better out-of-sample forecasts compared to standard HAR specifications and other competing approaches.

We propose a new class of score-driven time series models that allows for a more flexible weighting of score innovations for the filtering of time varying parameters. The parameter for the score innovation is made time-varying by means of an updating equation that accounts for the autocorrelations of past innovations. We provide the theoretical foundations for this acceleration method by showing optimality in terms of reducing Kullback–Leibler divergence. The empirical relevance of this accelerated score-driven updating method is illustrated in two empirical studies. First, we include acceleration in the generalized autoregressive conditional heteroskedasticity model. We adopt the new model to extract volatility from exchange rates and to analyze daily density forecasts of volatilities from all individual stock return series in the Standard & Poor's 500 index. Second, we consider a score-driven acceleration for the time-varying mean and use this new model in a forecasting study for US inflation.

This paper proposes a novel approach to introduce time-variation in structural parameters of DSGE models. Structural parameters are allowed to evolve over time via an observation-driven updating equation. The estimation of the resulting DSGE model can be easily performed by maximum likelihood without the need of time-consuming simulation-based methods. An application to a DSGE model with time varying volatility for structural shocks is presented. The results indicate a significant improvement in forecasting performance.

We first consider an extension of the generalized autoregressive conditional heteroskedasticity (GARCH) model that allows for a more flexible weighting of financial squared-returns for the filtering of volatility. The parameter for the squared-return in the GARCH model is time- varying with an updating function similar to GARCH but with the squared-return replaced by the product of the volatility innovation and its lagged value. This local estimate of the first order autocorrelation of volatility innovations acts as an indicator of the importance of the squared-return for volatility updating. When recent volatility innovations have the same sign (positive autocorrelation), the current volatility estimate needs to adjust more quickly than in a period where recent volatility innovations have mixed signs (negative autocorrelation). The empirical relevance of the accelerated GARCH updating is illustrated by forecasting daily volatility in return series of all individual stocks present in the Standard & Poor’s 500 index. Major improvements are reported for those stock return series that exhibit high kurtosis. The local adjustment in weighting new observational information is generalised to score-driven time-varying parameter models of which GARCH is a special case. It is within this general framework that we provide the theoretical foundations of accelerated updating. We show that acceleration in updating is more optimal in terms of reducing Kullback-Leibler divergence and in comparison to fixed updating. The robustness of our proposed extension is highlighted in a simulation study within a misspecified modelling framework. The score-driven acceleration is also empirically illustrated with the forecasting of US inflation using a model with time-varying mean and variance; we report significant improvements in the forecasting accuracy at a yearly horizon.

We establish the strong consistency and asymptotic normality of the maximum likelihood estimator for time-varying parameter models driven by the score of the predictive likelihood function. We formulate primitive conditions for global identification, invertibility, strong consistency, and asymptotic normality under both correct specification and mis-specification of the model. A detailed illustration is provided for a conditional volatility model with disturbances from the Student's t distribution.

This paper proposes a novel and flexible framework to estimate autoregressive models with time-varying parameters. Our setup nests various adaptive algorithms that are commonly used in the macroeconometric literature, such as learning-expectations and forgetting-factor algorithms. These are generalized along several directions: specifically, we allow for both Student-t distributed innovations as well as time-varying volatility. Meaningful restrictions are imposed to the model parameters, so as to attain local stationarity and bounded mean values. The model is applied to the analysis of inflation dynamics. Allowing for heavy-tails leads to a significant improvement in terms of fit and forecast. Moreover, it proves to be crucial in order to obtain well-calibrated density forecasts.

This thesis addresses different aspects of observation-driven time series modeling. The main contributions concern the reliability of likelihood-based inference and the specification of dynamic models to capture complex behaviors observed in time series data. As concerns inference, the main focus of the thesis is on invertibility conditions for observation-driven time series models. Invertibility plays a key role in ensuring the consistency of likelihood-based estimators. However, the invertibilty conditions typically employed in the literature are often unfeasible to be checked. Therefore, the reliability of inference fails to be guaranteed in practice. This thesis contributes to the literature by deriving feasible conditions that ensure the consistency of the maximum likelihood estimator for a wide class of models. One of the most appealing features of our consistency results is that they hold for both correctly specified and misspecified models. Several empirical examples covering different observation-driven models are presented. These examples highlight the practical relevance of the theoretical results. As concerns model specification, we cover two lines of research. The first is related to integer-valued time series data. We propose an extension to the class of Integer-valued Autoregressive models that allows the survival probability to vary over time. We show how our model can be easily estimated by maximum likelihood and we prove the consistency of the estimator. The flexibility of the proposed approach is shown through a simulation experiment and an application to a real time series of crime reports. Finally, the second line of research on model specification is an extension of the Generalized Autoregressive Score framework. We propose a class of models that updates time-varying parameters at different speeds in different time periods. The new updating equation can be employed to describe time series where the amount of information contained in the data is changing over time. This peculiarity is highlighted through a simulation study and we provide theoretical foundations for the proposed approach. Furthermore, two empirical applications to S&P 500 stock returns and US inflation illustrate how our method can be useful in practice.