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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.

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.