CHARACTERIZATION OF EXPONENTIAL DIVERGENCE OF THE KALMAN FILTER FOR TIME-VARYING SYSTEMS

This paper studies semistability of the recursive Kalman filter in the context of linear time-varying (LTV), possibly nondetectable systems with incorrect noise information. Semistability is a key property, as it ensures that the actual estimation error does not diverge exponentially. We explore structural properties of the filter to obtain a necessary and sufficient condition for the filter to be semistable. The condition does not involve limiting gains nor the solution of Riccati equations, as they can be difficult to obtain numerically and may not exist. We also compare semistability with the notions of stability and stability w.r.t. the initial error covariance, and we show that semistability in a sense makes no distinction between persistent and nonpersistent incorrect noise models, as opposed to stability. In the linear time invariant scenario we obtain algebraic, easy to test conditions for semistability and stability, which complement results available in the context of detectable systems. Illustrative examples are included.

Some remarks on static-feedback linearization for time-varying systems

This work summarizes some results about static state feedback linearization for time-varying systems. Three different necessary and sufficient conditions are stated in this paper. The first condition is the one by [Sluis, W. M. (1993). A necessary condition for dynamic feedback linearization. Systems & Control Letters, 21, 277-283]. The second and the third are the generalizations of known results due respectively to [Aranda-Bricaire, E., Moog, C. H., Pomet, J. B. (1995). A linear algebraic framework for dynamic feedback linearization. IEEE Transactions on Automatic Control, 40, 127-132] and to [Jakubczyk, B., Respondek, W. (1980). On linearization of control systems. Bulletin del` Academie Polonaise des Sciences. Serie des Sciences Mathematiques, 28, 517-522]. The proofs of the second and third conditions are established by showing the equivalence between these three conditions. The results are re-stated in the infinite dimensional geometric approach of [Fliess, M., Levine J., Martin, P., Rouchon, P. (1999). A Lie-Backlund approach to equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic Control, 44(5), 922-937]. (C) 2008 Elsevier Ltd. All rights reserved.

On state representations of time-varying nonlinear systems

This work considers a nonlinear time-varying system described by a state representation, with input u and state x. A given set of functions v, which is not necessarily the original input u of the system, is the (new) input candidate. The main result provides necessary and sufficient conditions for the existence of a local classical state space representation with input v. These conditions rely on integrability tests that are based on a derived flag. As a byproduct, one obtains a sufficient condition of differential flatness of nonlinear systems. (C) 2009 Elsevier Ltd. All rights reserved.

Applying linear time-varying constraints to econometric models: With an application to demand systems

When linear equality constraints are invariant through time they can be incorporated into estimation by restricted least squares. If, however, the constraints are time-varying, this standard methodology cannot be applied. In this paper we show how to incorporate linear time-varying constraints into the estimation of econometric models. The method involves the augmentation of the observation equation of a state-space model prior to estimation by the Kalman filter. Numerical optimisation routines are used for the estimation. A simple example drawn from demand analysis is used to illustrate the method and its application.

Estimating economic relationships subject to firm- and time-varying equality and inequality constraints

Applied econometricians often fail to impose economic regularity constraints in the exact form economic theory prescribes. We show how the Singular Value Decomposition (SVD) Theorem and Markov Chain Monte Carlo (MCMC) methods can be used to rigorously impose time- and firm-varying equality and inequality constraints. To illustrate the technique we estimate a system of translog input demand functions subject to all the constraints implied by economic theory, including observation-varying symmetry and concavity constraints. Results are presented in the form of characteristics of the estimated posterior distributions of functions of the parameters. Copyright (C) 2001 John Wiley Sons, Ltd.

An aggregate sales model for consumer durables incorporating a time-varying mean replacement age

Forecasting category or industry sales is a vital component of a company's planning and control activities. Sales for most mature durable product categories are dominated by replacement purchases. Previous sales models which explicitly incorporate a component of sales due to replacement assume there is an age distribution for replacements of existing units which remains constant over time. However, there is evidence that changes in factors such as product reliability/durability, price, repair costs, scrapping values, styling and economic conditions will result in changes in the mean replacement age of units. This paper develops a model for such time-varying replacement behaviour and empirically tests it in the Australian automotive industry. Both longitudinal census data and the empirical analysis of the replacement sales model confirm that there has been a substantial increase in the average aggregate replacement age for motor vehicles over the past 20 years. Further, much of this variation could be explained by real price increases and a linear temporal trend. Consequently, the time-varying model significantly outperformed previous models both in terms of fitting and forecasting the sales data. Copyright (C) 2001 John Wiley & Sons, Ltd.

Comparative analysis of time-frequency methods estimating the time-varying microstructure of sleep EEG spindles

Proceedings of the Information Technology Applications in Biomedicine, Ioannina - Epirus, Greece, October 26-28, 2006

Potential dementia biomarkers based on the time-varying microstructure of sleep EEG spindles

Proceedings of the 29th Annual International Conference of the IEEE EMBS Cité Internationale, Lyon, France August 23-26, 2007

Specification and testing of multiplicative time-varying GARCH models with applications

In this article, we develop a specification technique for building multiplicative time-varying GARCH models of Amado and Teräsvirta (2008, 2013). The variance is decomposed into an unconditional and a conditional component such that the unconditional variance component is allowed to evolve smoothly over time. This nonstationary component is defined as a linear combination of logistic transition functions with time as the transition variable. The appropriate number of transition functions is determined by a sequence of specification tests. For that purpose, a coherent modelling strategy based on statistical inference is presented. It is heavily dependent on Lagrange multiplier type misspecification tests. The tests are easily implemented as they are entirely based on auxiliary regressions. Finite-sample properties of the strategy and tests are examined by simulation. The modelling strategy is illustrated in practice with two real examples: an empirical application to daily exchange rate returns and another one to daily coffee futures returns.

Bayesian Inference in the Time Varying Cointegration Model

There are both theoretical and empirical reasons for believing that the parameters of macroeconomic models may vary over time. However, work with time-varying parameter models has largely involved Vector autoregressions (VARs), ignoring cointegration. This is despite the fact that cointegration plays an important role in informing macroeconomists on a range of issues. In this paper we develop time varying parameter models which permit cointegration. Time-varying parameter VARs (TVP-VARs) typically use state space representations to model the evolution of parameters. In this paper, we show that it is not sensible to use straightforward extensions of TVP-VARs when allowing for cointegration. Instead we develop a specification which allows for the cointegrating space to evolve over time in a manner comparable to the random walk variation used with TVP-VARs. The properties of our approach are investigated before developing a method of posterior simulation. We use our methods in an empirical investigation involving a permanent/transitory variance decomposition for inflation.

Large Time-Varying Parameter VARs

In this paper we develop methods for estimation and forecasting in large timevarying parameter vector autoregressive models (TVP-VARs). To overcome computational constraints with likelihood-based estimation of large systems, we rely on Kalman filter estimation with forgetting factors. We also draw on ideas from the dynamic model averaging literature and extend the TVP-VAR so that its dimension can change over time. A final extension lies in the development of a new method for estimating, in a time-varying manner, the parameter(s) of the shrinkage priors commonly-used with large VARs. These extensions are operationalized through the use of forgetting factor methods and are, thus, computationally simple. An empirical application involving forecasting inflation, real output, and interest rates demonstrates the feasibility and usefulness of our approach.

Model Switching and Model Averaging in Time- Varying Parameter Regression Models

This paper investigates the usefulness of switching Gaussian state space models as a tool for implementing dynamic model selecting (DMS) or averaging (DMA) in time-varying parameter regression models. DMS methods allow for model switching, where a different model can be chosen at each point in time. Thus, they allow for the explanatory variables in the time-varying parameter regression model to change over time. DMA will carry out model averaging in a time-varying manner. We compare our exact approach to DMA/DMS to a popular existing procedure which relies on the use of forgetting factor approximations. In an application, we use DMS to select different predictors in an in ation forecasting application. We also compare different ways of implementing DMA/DMS and investigate whether they lead to similar results.

On the time-varying relationship between EMU sovereign spreads and their determinants

We use a dynamic multipath general-to-specific algorithm to capture structural instability in the link between euro area sovereign bond yield spreads against Germany and their underlying determinants over the period January 1999 – August 2011. We offer new evidence suggesting a significant heterogeneity across countries, both in terms of the risk factors determining spreads over time as well as in terms of the magnitude of their impact on spreads. Our findings suggest that the relationship between euro area sovereign risk and the underlying fundamentals is strongly timevarying, turning from inactive to active since the onset of the global financial crisis and further intensifying during the sovereign debt crisis. As a general rule, the set of financial and macro spreads’ determinants in the euro area is rather unstable but generally becomes richer and stronger in significance as the crisis evolves.