| Title: | A General Framework for Observation Driven Time-Varying Parameter Models |
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| Authors: | Drew Creal, Siem Jan Koopman and Andre Lucas |
| Summary: | We propose a new class of observation driven time series models referred to as Generalized Autoregressive Score (GAS) models. The driving mechanism of the GAS model is the scaled score of the likelihood function. This approach provides a unified and consistent framework for introducing time-varying parameters in a wide class of non-linear models. The GAS model encompasses other well-known models such as the generalized autoregressive conditional heteroskedasticity, the autoregressive conditional duration, the autoregressive conditional intensity, and the single source of error models. In addition, the GAS specification provides a wide range of new observation driven models. Examples include non-linear regression models with time-varying parameters, observation driven analogues of unobserved components time series models, multivariate point process models with time-varying parameters and pooling restrictions, new models for time-varying copula functions, and models for time-varying higher order moments. We study the properties of GAS models and provide several non-trivial examples of their application. |
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| Title: | Spline Smoothing over difficult regions: a state space approach |
| Authors: | Siem Jan Koopman and Soon Yip Wong |
| Summary: | We consider the problem of smoothing data on two-dimensional grids with holes or gaps. Such grids are often referred to as difficult regions. Since the data is not observed on these locations, the gap is not part of the domain. We cannot apply standard smoothing methods since they smooth over and across difficult regions. More unfavorable properties of standard smoothers become visible when the data is observed on an irregular grid in a non-rectangular domain. In this paper, we adopt smoothing spline methods within a state space framework to smooth data on one- or two-dimensional grids with difficult regions. We make a distinction between two types of missing observations to handle the irregularity of the grid and to ensure that no smoothing takes place over and across the difficult region. For smoothing on two-dimensional grids, we introduce a two-step spline smoothing method. The proposed solution applies to all smoothing methods that can be represented in a state space framework. We illustrate our methods for three different cases of interest. |
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| Title: | The effect of the great moderation on the U.S. business cycle in a time-varying multivariate trend-cycle model |
| Authors: | Siem Jan Koopman, Drew Creal and Eric Zivot |
| Summary: | In this paper we investigate whether the dynamic properties of the U.S. business cycle have changed in the last fifty years. For this purpose we develop a flexible business cycle indicator that is constructed from a moderate set of macroeconomic time series. The coincident economic indicator is based on a multivariate trend-cycle decomposition model that accounts for time variation in macroeconomic volatility, known as the great moderation. In particular, we consider an unobserved components time series model with a common cycle that is shared across different time series but adjusted for phase shift and amplitude. The extracted cycle can be interpreted as the result of a model-based bandpass filter and is designed to emphasize the business cycle frequencies that are of interest to applied researchers and policymakers. Stochastic volatility processes and mixture distributions for the irregular components and the common cycle disturbances enable us to account for all the heteroskedasticity present in the data. The empirical results are based on a Bayesian analysis and show that time-varying volatility is only present in the a selection of idiosyncratic components while the coefficients driving the dynamic properties of the business cycle indicator have been stable over time in the last fifty years. |
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| Title: | Likelihood-based Analysis for Dynamic Factor Models |
| Authors: | B. Jungbacker and Siem Jan Koopman |
| Summary: | We present new results for the likelihood-based analysis of the dynamic factor model that possibly includes intercepts and explanatory variables. The latent factors are modelled by stochastic processes. The idiosyncratic disturbances are specified as autoregressive processes with mutually correlated innovations. The new results lead to computationally efficient procedures for the estimation of the factors and parameter estimation by maximum likelihood and Bayesian methods. An illustration is provided for the analysis of a large panel of macroeconomic time series. |
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| Title: | Unobserved Components, Time-Varying Spectra and the U.S. Business Cycle |
| Authors: | Siem Jan Koopman and Soon Yip Wong |
| Summary: | Recent empirical studies provide evidence that dynamic properties of macroeconomic time series have been changing over time. Model-based procedures for the measurement of business cycles may require time-varying parameters. For this purpose, we introduce a parsimonious and flexible method based on time-dependent sample spectra. Explicit model specifications for the parameters are not required. Parameter estimation is carried out in the frequency domain by optimizing the spectral likelihood function. The time-dependent spectrum is specified as a semi-parametric smoothing spline function that can be formulated in state space form. Since the spectral likelihood function is time-varying, model parameter estimates become time-varying as a result. This new approach to business cycle extraction includes bootstrap procedures for the computation of confidence intervals. We illustrate the methodology by presenting a business cycle analysis for three U.S. macroeconomic time series. |
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| Title: | Likelihood Functions for State Space Models with Diffuse Initial Conditions |
| Authors: | Mark K. Francke, Siem Jan Koopman and Aart de Vos |
| Summary: | State space models with nonstationary processes and fixed regression effects require a state vector with diffuse initial conditions. Different likelihood functions can be adopted for the estimation of parameters in time series models with diffuse initial conditions. In this paper we consider profile, diffuse and marginal likelihood functions. The marginal likelihood is defined as the likelihood function of a transformation of the data vector. The transformation is not unique. The diffuse likelihood is a marginal likelihood for a specific data transformation that may depend on parameters. Therefore, the diffuse likelihood can not be used generally for parameter estimation. Our newly proposed marginal likelihood function is based on an orthonormal transformation that does not depend on parameters. Likelihood functions for state space models are evaluated using the Kalman filter. The diffuse Kalman filter is specifically designed for computing the diffuse likelihood function. We show that a modification of the diffuse Kalman filter is needed for the evaluation of our proposed marginal likelihood function. Diffuse and marginal likelihood functions have better small sample properties compared to the profile likelihood function for the estimation of parameters in linear time series models. The results in our paper confirm the earlier findings and show that the diffuse likelihood function is not appropriate for a range of state space model specifications. |
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| Title: | Forecasting Cross-Sections of Frailty-Correlated Default |
| Authors: | Siem Jan Koopman, A. Lucas and B. Schwaab |
| Summary: | We propose a novel econometric model for estimating and forecasting cross-sections of time-varying conditional default probabilities. The model captures the systematic variation in corporate default counts across e.g. rating and industry groups by using dynamic factors from a large panel of selected macroeconomic and financial data as well as common unobserved risk factors. All factors are statistically and economically significant and together capture a large part of the time-variation in observed default rates. In this framework we improve the out-of-sample forecasting accuracy associated with conditional default probabilities by about 10-35% in terms of Mean Absolute Error, particularly in years of default stress. |
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| Title: | Long Memory Modelling of Inflation with Stochastic Variance and Structural Breaks |
| Authors: | C.S. Bos, Siem Jan Koopman and M. Ooms |
| Summary: | We investigate changes in the time series characteristics of postwar U.S. inflation. In a model-based analysis the conditional mean of inflation is specified by a long memory autoregressive fractionally integrated moving average process and the conditional variance is modelled by a stochastic volatility process. We develop a Monte Carlo maximum likelihood method to obtain efficient estimates of the parameters using a monthly dataset of core inflation for which we consider different subsamples of varying size. Based on the new modelling framework and the associated estimation technique, we find remarkable changes in the variance, in the order of integration, in the short memory characteristics and in the volatility of volatility. |
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| Title: | Periodic unobserved cycles in seasonal time series with an application to US unemployment |
| Authors: | Siem Jan Koopman, Marius Ooms and Irma Hindrayanto |
| Summary: | This paper discusses identification, specification, estimation and forecasting for a general class of periodic unobserved components time series models with stochastic trend, seasonal and cycle components. Convenient state space formulations are introduced for exact maximum likelihood estimation, component estimation and forecasting. Identification issues are considered and a novel periodic version of the stochastic cycle component is presented. In the empirical illustration, the model is applied to postwar monthly US unemployment series and we discover a significantly periodic cycle. Furthermore, a comparison is made between the performance of the periodic unobserved components time series model and a periodic seasonal autoregressive integrated moving average model. Moreover, we introduce a new method to estimate the latter model. |
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| Title: | Multivariate nonlinear time series modelling of exposure and risk in road safety research |
| Authors: | Frits Bijleveld, Jacques Commandeur, Siem Jan Koopman and Kees van Montfort |
| Summary: | In this paper we consider a multivariate nonlinear time series model for the analysis of traffic volumes and road casualties inside and outside urban areas. The model consists of dynamic unobserved factors for exposure and risk that are related in a nonlinear way. The multivariate dimension of the model is due to the inclusion of different time series for inside and outside urban areas. The analysis is based on the extended Kalman filter. Quasimaximum likelihood methods are utilised for the estimation of unknown parameters. The latent factors are estimated by extended smoothing methods. We present a case study of yearly time series of numbers of fatal accidents (inside and outside urban areas) and numbers of driven kilometers by motor vehicles in the Netherlands between 1961 and 2000. The analysis accounts for missing entries in the disaggregated numbers of driven kilometres although the aggregated numbers are observed throughout. It is concluded that the salient features of the observed time series are captured by the model in a satisfactory way. |
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| Title: | Measuring asymmetric stochastic cycle components in U.S. macroeconomic time series |
| Authors: | S.J. Koopman and K. M. Lee |
| Summary: | To gain insights in the current status of the economy, macroeconomic time series are often decomposed into trend, cycle and irregular components. This can be done by nonparametric band-pass filtering methods in the frequency domain or by model-based decompositions based on autoregressive moving average models or unobserved components time series models. In this paper we consider the latter and extend the model to allow for asymmetric cycles. In theoretical and empirical studies, the asymmetry of cyclical behavior is often discussed and considered for series such as unemployment and gross domestic product (GDP). The number of attempts to model asymmetric cycles is limited and it is regarded as intricate and nonstandard. In this paper we show that a limited modification of the standard cycle component leads to a flexible device for asymmetric cycles. The presence of asymmetry can be tested using classical likelihood based test statistics. The trend-cycle decomposition model is applied to three key U.S. macroeconomic time series. It is found that cyclical asymmetry is a prominent salient feature in the U.S. economy. |
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| Title: | Intervention Time Series Analysis of Crime Rates |
| Authors: | Sanjeev Sridharan, Suncica Vujic and S.J. Koopman |
| Summary: | The Commonwealth of Virginia abolished parole and reformed sentencing for all felony offenders committed on or after January 1, 1995. We examine the impact of this legislation on reported crime rates using different time series approaches. In particular, structural time series models are considered as an alternative to the Box-Jenkins ARIMA models that form the standard time series approach to intervention analysis. Limited support for the deterrent impact of parole abolition and sentence reform is obtained using univariate modelling devices, even after including unemployment as an explanatory variabIe. Finally, the flexibility of structural time series models is illustrated by presenting a multivariate analysis that provides some additional evidence of the deterrent impact of the new legislation. |
| Download: | Abstract or PDF file (408 kB) |
| Title: | Forecasting the Variability of Stock Index Returns with Stochastic Volatility Models and Implied Volatility |
| Authors: | E. Hol and S.J. Koopman |
| Summary: | We compare the predictive ability of Stochastic Volatility (SV) models to that of volatility forecasts implied by option prices. An SV model is proposed with implied volatility as an explanatory variable in the variance equation which allows the use of statistical testing; we refer to this model as the SVX model. Next we obtain a Stochastic Implied Volatility (SIV) model by restricting the volatility persistence parameter in the SVX model to equal zero. All SV models are estimated by exact maximum likelihood using Monte Carlo importance sampling methods. The performance of the models is evaluated both within-sample and out-of-sample for daily returns on the Standard & Poor's 100 index. Our in-sample results confirm the information content of implied volatility measures as the SVX and SIV models produce more effective estimates of the underlying volatility process than the standard SV model based solely on historical returns. The out-of-sample volatility forecasts are evaluated against daily squared returns and intraday volatility measures for forecasting horizons ranging from 1 to 20 days. For both the squared daily returns and the cumulative intraday squared 10-minute returns we find that the SIV model outperforms both the SV and the SVX model on several evaluation criteria but that the SV model produces volatility forecasts with the smallest bias. All models underestimate the volatility process on average which in our opinion is closely related to the fact that the average level of volatility in the estimation samples is lower than in the evaluation sample. |
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Time Series Analysis by State Space Methods.
2001, with J. Durbin, Oxford University Press.
For workpage of the book, please click here.
State Space and Unobserved Component Models: Theory and Applications. Proceedings of a Conference in Honour of James Durbin. 2004, with A. Harvey and N. Shephard, pp. 393, Cambridge University Press.
An Introduction to State Space Time Series Analysis. 2007, with Jacques J.F. Commandeur pp. 192, Oxford University Press.
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