| Title: | Seasonality with Trend and Cycle Interactions in Unobserved Components Models |
|---|---|
| Authors: | Siem Jan Koopman and K.M. Lee |
| Summary: | Unobserved components time series models decompose a time series into a trend, a season, a cycle, an irregular disturbance, and possibly other components. These models have been successfully applied to many economic time series. The standard assumption of a linear model, often appropriate after a logarithmic transformation of the data, facilitates estimation, testing, forecasting and interpretation. However, in some settings the linear-additive framework may be too restrictive. In this paper, we formulate a non-linear unobserved components time series model which allows interactions between the trend-cycle component and the seasonal component. The resulting model is cast into a non-linear state space form and estimated by the extended Kalman filter, adapted for models with diffuse initial conditions. We apply our model to UK travel data and US unemployment and production series, and show that it can capture increasing seasonal variation and cycle dependent seasonal fluctuations. |
| Published: | Journal of the Royal Statistical Society Series C, (forthcoming) |
| Links: | PDF document Working paper version |
| Title: | Analyzing the Term Structure of Interest Rates using the Dynamic Nelson-Siegel Model with Time-Varying Parameters |
| Authors: | Siem Jan Koopman, M.I.P. Mallee and M. van der Wel |
| Summary: | In this paper we introduce time-varying parameters in the dynamic Nelson-Siegel yield curve model for the simultaneous analysis and forecasting of interest rates of different maturities, known as the term structure. The Nelson-Siegel model has been recently reformulated as a dynamic factor model where the latent factors are interpreted as the level, slope and curvature of the term structure. The factors are modelled by a vector autoregressive process. We propose to extend this framework in two directions. First, the factor loadings are made time-varying through a simple single step function and we show that the model fit increases significantly as a result. The step function can be replaced by a spline function to allow for more smoothness and flexibility. Second, we investigate empirically whether the volatility in interest rates across different time periods is constant. For this purpose, we introduce a common volatility component that is specified as a spline function of time and scaled appropriately for each series. Based on a data-set that is analysed by others, we present empirical evidence where time-varying loadings and volatilities in the dynamic Nelson-Siegel framework lead to significant increases in model fit. Improvements in the forecasting of the term structure are also reported. Finally, we provide an illustration where the model is applied to an unbalanced dataset. It shows that missing data entries can be estimated accurately. |
| Published: | Journal of Business and Economic Statistics, (forthcoming) |
| Links: | PDF document Working paper version |
| Title: | Testing the assumptions behind importance sampling |
| Authors: | S.J. Koopman, Neil Shephard and Drew Creal |
| Summary: | Importance sampling is used in many areas of modern econometrics to approximate un- solvable integrals. Its reliable use requires the sampler to possess a variance, for this guar- antees a square root speed of convergence and asymptotic normality of the estimator of the integral. However, this assumption is seldom checked. In this paper we use extreme value theory to empirically assess the appropriateness of this assumption. Our main application is the stochastic volatility model, where importance sampling is commonly used for maximum likelihood estimation of the parameters of the model. |
| Published: | Journal of Econometrics, 2009, Volume 149, Pages 2-11. |
| Links: | Abstract + paper |
| Title: | Credit Cycles and Macro Fundamentals |
| Authors: | Siem Jan Koopman, Roman Kraussl, Andre Lucas and Andre Monteiro |
| Summary: | We study the relation between the credit cycle and macro economic fundamentals in an intensity based framework. Using rating transition and default data of U.S. corporates from Standard and Poor's over the period 1980-2005 we directly estimate the credit cycle from the micro rating data. We relate this cycle to the business cycle, bank lending conditions, and financial market variables. In line with earlier studies, these variables appear to explain part of the credit cycle. As our main contribution, we test for the correct dynamic specification of these models. In all cases, the hypothesis of correct dynamic specification is strongly rejected. Moreover, if we account for the dynamic mis-specification, many of the variables thought to explain the credit cycle, turn out to be insignificant. The main exceptions are GDP growth, and to some extent stock returns and stock return volatilities. Their economic significance appears low, however. This raises the puzzle of what macro-economic fundamentals explain default and rating dynamics. |
| Published: | Journal of Empirical Finance, 2009, Volume 16, Pages 42–54 |
| Links: | Abstract + paper |
| Title: | An Hourly Periodic State Space Model for Modelling French National Electricity Load |
| Authors: | V. Dordonnat, Siem Jan Koopman, M. Ooms, A. Dessertaine and J. Collet |
| Summary: | We present a model for hourly electricity load forecasting based on stochastically time-varying processes that are designed to account for changes in customer be- haviour and in utility production efficiencies. The model is periodic: it consists of different equations and different parameters for each hour of the day. Dependence between the equations is introduced by covariances between disturbances that drive the time-varying processes. The equations are estimated simultaneously. Our model consists of components that represent trends, seasons at different levels (yearly, weekly, daily, special days and holidays), short-term dynamics and weather regres- sion effects including nonlinear functions for heating effects. The implementation of our forecasting procedure relies on the multivariate linear Gaussian state space framework and is applied to national French hourly electricity load. The analy- sis focuses on two hours, 9 AM and 12 AM, but forecasting results are presented for all twenty-four hours. Given the time series length of nine years of hourly ob- servations, many features of our model can be readily estimated including yearly patterns and their time-varying nature. The empirical analysis involves an out-of- sample forecasting assessment up to seven days ahead. The one-day ahead forecasts from fourty-eight bivariate models are compared with twenty-four univariate mod- els for all hours of the day. We find that the implied forecasting function strongly depends on the hour of the day. |
| Published: | International Journal of Forecasting, 2008, Volume 24, Number 4, Pages 566-587. |
| Links: | Abstract + paper |
| Title: | A Non-Gaussian Panel Time Series Model for Estimating and Decomposing Default Risk |
| Authors: | S.J. Koopman and A. Lucas |
| Summary: | We model 1981–2002 annual default frequencies for a panel of US firms in different rating and age classes from the Standard and Poor’s database. The data is decomposed into a systematic and firm-specific risk component, where the systematic component reflects the general economic conditions and default climate. We have to cope with (i) the shared exposure of each age cohort and rating class to the same systematic risk factor; (ii) strongly non-Gaussian features of the individual time series; (iii) possible dynamics of an unobserved common risk factor; (iv) changing default probabilities over the age of the rating, and (v) missing observations. We propose a non-Gaussian multivariate state space model that deals with all of these issues simultaneously. The model is estimated using importance sampling techniques that have been modified to a multivariate setting. We show in a simulation study that such a multivariate approach improves the performance of the importance sampler. |
| Published: | Journal of Business and Economic Statistics, 2008, Volume 26, Number 4, Pages 510-525. |
| Links: | Abstract + paper |
| Title: | Estimating Systematic Continuous-time Trends in Recidivism using a Non-Gaussian Panel Data Model |
| Authors: | Siem Jan Koopman, André Lucas, Marius Ooms, Kees van Montfort and Victor van der Geest |
| Summary: | We model panel data of crime careers of juveniles from a Dutch Judicial Juvenile Institution. The data are decomposed into a systematic and an individual-specific component, of which the systematic component reflects the general time-varying conditions including the criminological climate. Within a model-based analysis, we treat (1) shared effects of each group with the same systematic conditions, (2) strongly non-Gaussian features of the individual time series, (3) unobserved common systematic conditions, (4) changing recidivism probabilities in continuous time, (5) missing observations. We adopt a non-Gaussian multivariate state space model that deals with all of these issues simultaneously. The parameters of the model are estimated by Monte Carlo maximum likelihood methods. This paper illustrates the methods empirically. We compare continuous-time trends and standard discrete-time stochastic trend specifications. We find interesting common time-variation in the recidivism behavior of the juveniles during a period of 13 years, while taking account of significant heterogeneity determined by personality characteristics and initial crime records. |
| Published: | Statistica Neerlandica, 2008, Volume 62, Issue 1, Pages 104-130. |
| Links: | Abstract + paper |
| Title: | Measuring Synchronisation and Convergence of Business Cycles |
| Authors: | S.J. Koopman and Joao Valle e Azevedo |
| Summary: | This paper investigates business cycle relations among different economies in the Euro area. Cyclical dynamics are explicitly modelled as part of a time series model. We introduce mechanisms that allow for increasing or diminishing phase shifts and for time-varying association patterns in different cycles. Standard Kalman filter techniques are used to estimate the parameters simultaneously by maximum likelihood. The empirical illustrations are based on gross domestic product (GDP) series of seven European countries which are compared with the GDP series of the Euro Area and that of the United States. The original integrated time series are band-pass filtered. We find that there is an increasing resemblance between the business cycle fluctuations of the European countries analysed and those of the Euro area, although with varying patterns. |
| Published: | Oxford Bulletin of Economics and Statistics, 2008, Volume 70, Issue 1, Pages 23–51. |
| Links: | Abstract + paper |
| Title: | Model-based measurement of latent risk in time series with applications |
| Authors: | Frits Bijleveld, Jacques Commandeur, Phillip Gould and S.J. Koopman |
| Summary: | Risk is at the center of many policy decisions in companies, governments and other institutions. The risk of road fatalities concerns local governments in planning counter- measures, the risk and severity of counterparty default concerns bank risk managers on a daily basis and the risk of infection has actuarial and epidemiological consequences. However, risk can not be observed directly and it usually varies over time. Measuring risk is therefore an important exercise. In this paper we introduce a general multivariate framework for the time series analysis of risk that is modelled as a latent process. The latent risk time series model extends existing approaches by the simultaneous modelling of (i) the exposure to an event, (ii) the risk of that event occurring and (iii) the severity of the event. First, we discuss existing time series approaches for the analysis of risk which have been applied to road safety, actuarial and epidemiological problems. Second, we present a general model for the analysis of risk and discuss its statistical treatment based on linear state space methods. Third, we apply the methodology to time series of insurance claims, credit card purchases and road safety. It is shown that the general methodology can be effectively used in the assessment of risk. |
| Published: | Journal of the Royal Statistical Society Series A, 2008, Volume 171, Issue 1, Pages 265-277. |
| Links: | Abstract + paper |
| Title: | The Multi-State Latent Factor Intensity Model for Credit Rating Transitions |
| Authors: | S.J. Koopman, A. Lucas and A. Monteiro |
| Summary: | A new empirical reduced-form model for credit rating transitions is introduced. It is a parametric intensity-based duration model with multiple states and driven by exogenous covariates and latent dynamic factors. The model has a generalized semi-Markov structure designed to accommodate many of the stylized facts of credit rating migrations. Parameter estimation is based on Monte Carlo maximum likelihood methods for which the details are discussed in this paper. A simulation experiment is carried out to show the effectiveness of the estimation procedure. An empirical application is presented for transitions between investment grade, subinvestment grade, and default ratings for U.S. corporates. The model strongly suggests the presence of a common dynamic component that can be interpreted as the credit cycle. We also show that the impact of this credit cycle is asymmetric with respect to downgrade and upgrade probabilities. |
| Published: | Journal of Econometrics, 2008, Volume 142, Issue 1, Pages 399-424 |
| Links: | Abstract + paper |
| Title: | Monte Carlo estimation for nonlinear non-Gaussian state space models |
| Authors: | Borus Jungbacker Siem Jan Koopman |
| Summary: | We consider likelihood inference and state estimation by means of importance sampling for state space models with a nonlinear non-Gaussian observation y ~ p(y|alpha) and a linear Gaussian state alpha ~ p(alpha). The importance density is chosen to be the Laplace approximation of the smoothing density p(alpha|y). We show that computationally efficient state space methods can be used to perform all necessary computations in all situations. It requires new derivations of the Kalman filter and smoother and the simulation smoother which do not rely on a linear Gaussian observation equation. Furthermore, results are presented that lead to a more effective implementation of importance sampling for state space models. An illustration is given for the stochastic volatility model with leverage. |
| Published: | Biometrika, 2007, Volume 94, Pages 827-839 |
| Links: | Abstract + paper and Appendix |
| Title: | Modelling Round-the-Clock Price Discovery for Cross-Listed Stocks using State Space Methods |
| Authors: | Albert J. Menkveld, S.J. Koopman and Andre Lucas |
| Summary: | U.S. trading in non-U.S. stocks has grown dramatically. Round-the-clock, these stocks trade in the home market, in the U.S. market and, potentially, in both markets simultaneously. We develop a general methodology based on a state space model to study 24-hour price discovery in a multiple markets setting. As opposed to the standard variance ratio approach, this model deals naturally with (i) simultaneous quotes in an overlap, (ii) missing observations in a non-overlap, (iii) noise due to transitory microstructure effects, and (iv) contemporaneous correlation in returns due to market-wide factors. We provide an application of our model to Dutch-U.S. stocks. Our findings suggest a minor role for the NYSE in price discovery for Dutch shares, in spite of its non-trivial and growing market share. The results differ significantly from the variance ratio approach. |
| Published: | Journal of Business and Economic Statistics, 2007, Volume 25, Number 2, April 2007, pp. 213-225. |
| Links: | Abstract + paper |
| Title: | Periodic seasonal Reg-ARFIMA-GARCH models for daily electricity spot prices |
| Authors: | S.J. Koopman, Marius Ooms and M. Angeles Carnero |
| Summary: | Novel periodic extensions of dynamic long memory regression models with autoregressive conditional heteroskedastic errors are considered for the analysis of daily electricity spot prices. The parameters of the model with mean and variance specifications are estimated simultaneously by the method of approximate maximum likelihood. The methods are implemented for time series of 1, 200 to 4, 400 daily price observations in four European power markets. Apart from persistence, heteroskedasticity and extreme observations in prices, a novel empirical finding is the importance of day-of-the-week periodicity in the autocovariance function of electricity spot prices. In particular the very persistent daily log prices from the Nord Pool power exchange of Norway are modeled effectively by our framework, which is also extended with explanatory variables to capture demand- and supply effects. The daily log prices of the other three electricity markets, EEX in Germany, Powernext in France, and APX in The Netherlands, are less persistent, but periodicity is also highly significant. The dynamic behaviour differs from market to market, and depends primarily on the method of power generation: hydro power, power generated from fossil fuels, or nuclear power. The paper improves upon existing models in capturing the memory characteristics, which are important in derivative pricing and real option analysis. |
| Published: | Journal of the American Statistical Association, 2007, Volume 102, Number 477, March 2007, pp. 16-27. |
| Links: | Abstract + paper |