## 1 INTRODUCTION

In many settings of empirical interest, time variation in a selection of model parameters is important for capturing the dynamic behavior of univariate and multivariate time series processes. Time series models with time-varying parameters have been categorized by Cox (1981) into two classes of models: observation-driven models and parameter-driven models. In the observation-driven approach, time variation of the parameters is introduced by letting parameters be functions of lagged dependent variables as well as contemporaneous and lagged exogenous variables. Although the parameters are stochastic, they are perfectly predictable given the past information. This approach simplifies likelihood evaluation and explains why observation-driven models have become popular in the applied statistics and econometrics literature. Typical examples of these models are the generalized autoregressive conditional heteroskedasticity (GARCH) models of Engle (1982), Bollerslev (1986) and Engle and Bollerslev (1986), the autoregressive conditional duration and intensity (ACD and ACI, respectively) models of Engle and Russell (1998) and Russell (2001), the dynamic conditional correlation (DCC) model of Engle (2002a), the Poisson count models discussed by Davis *et al*. (2003), the dynamic copula models of Patton (2006), and the time-varying quantile model of Engle and Manganelli (2004). In our modeling framework for time-varying parameters, many of the existing observation-driven models are encompassed as mentioned above. In addition, new models can be formulated and investigated.

In parameter-driven models, the parameters are stochastic processes with their own source of error. Given past and concurrent observations, the parameters are not perfectly predictable. Typical examples of parameter-driven models are the stochastic volatility (SV) model (see Shephard, 2005, for a detailed discussion) and the stochastic intensity models of Bauwens and Hautsch (2006) and Koopman *et al*. (2008). Estimation is usually more involved for these models because the associated likelihood functions are not available in closed form. Exceptions include linear Gaussian state space models and discrete-state hidden Markov models (see Harvey, 1989, and Hamilton, 1989, respectively). In most other cases, computing the likelihood function requires the evaluation of a high-dimensional integral based on simulation methods such as importance sampling and Markov chain Monte Carlo (MCMC) (see, for example, Shephard and Pitt (1997).

The main contribution of this paper is the development of a framework for time-varying parameters which is based on the score function of the predictive model density at time *t*. We will argue that the score function is an effective choice for introducing a driving mechanism for time-varying parameters. In particular, by scaling the score function appropriately, standard observation-driven models such as the GARCH, ACD, and ACI models can be recovered. Application of this framework to other nonlinear, non-Gaussian, possibly multivariate, models will lead to the formulation of new observation driven models.

We refer to our observation-driven model based on the score function as the generalized autoregressive score (GAS) model. The GAS model has the advantages of other observation-driven models. Likelihood evaluation is straightforward. Extensions to asymmetric, long memory, and other more complicated dynamics can be considered without introducing further complexities. Since the GAS model is based on the score, it exploits the complete density structure rather than means and higher moments only. It differentiates the GAS model from other observation-driven models in the literature, such as the generalized autoregressive moving average models of Shephard (1995) and Benjamin *et al*. (2003) and the vector multiplicative error models of Cipollini *et al*. (2006).

In our first illustration, we develop new models for time-varying copulas. The copula function provides an important tool for the econometrics of financial risk measurement. Patton (2006) introduced the notion of time-varying copulas and provided the main properties of dynamic copula functions. Other models for time-varying copulas include Giacomini *et al*. (2007), who developed locally constant copula models, and the stochastic copula model of Hafner and Manner (2011). Another interesting copula-based model is developed by Lee and Long (2009) where the multivariate GARCH model is extended with copula functions to capture any remaining dependence in the volatility of the time series. An extended review of the recent developments of copula functions in time series models is given by Patton (2009).

In our second illustration, we create a new class of multivariate point-process models for credit risk. Models for counterparty default and rating transition risk are an important element in the current regulatory system for financial institutions. Many of the new models are based on marked point processes with time-varying intensities for different levels of risk. Parameter estimation relies on computationally demanding methods (see for example, Duffie *et al*. (2009). One of the main challenges when modeling credit events is the sparse number of transitions for each individual company. We show how a multi-state model for pooled marked point processes follows naturally within our framework. We analyze an extensive data set of Moody's rating histories of more than 8000 US corporates over a time span of almost 30 years. We compare the results of the GAS model with those of its parameter-driven counterpart. The parameters in the benchmark model need to be estimated using a MCMC method, which is computationally more demanding compared to our maximum likelihood procedure. Despite the substantial differences in computing time, the GAS model produces almost identical estimates of time-varying defaults and rating transition probabilities when compared with those of the parameter-driven model.

The remainder of the paper is organized as follows. In Section 2 we provide the basic GAS specification together with a set of motivating examples. Section 3 describes several new copula models with time-varying parameters. Section 4 presents the model for marked point processes with time-varying parameters. Section 5 concludes.