José Casals

An Exact Multivariate Model-based Structural Decomposition

We propose a simple and structured procedure for decomposing a vector of time series into trend, cycle, seasonal, and irregular components. Contrary to common practice, we do not assume these components to be orthogonal conditional on their past. However, the state–space representation employed ensures that their estimates converge to values with null variances and covariances. Null variances are very important, as they ensure that the components do not change when the sample increases. This lack of “revisions” is the most important feature of our method, in comparison with most alternative procedures. On the other hand, null covariances provide a solid statistical foundation for the decomposition, as it ensures that a given component can be analyzed and interpreted independently of any other component(s). Other convenient properties of our method derive from the use of a state–space approach. First, defining the problem in state–space avoids dependence on particular model specifications, so the same procedure can be applied to a wide class of data representations, including ARIMA, VARMAX, univariate transfer functions, and structural time series models. Also, state–space methods deal easily with nonstandard situations, such as samples with missing values or constraints upon the structural components. Practical application of the procedure is illustrated with both simulated and real data. A MATLAB toolbox for time series modeling and decomposition is available via the Internet.

A fast and stable method to compute the likelihood of time invariant state-space models

Abstract We propose two fast and stable methods to compute the likelihood of time invariant state-space models. The first one exploits the properties of the Kalman filter when applied to steady-state innovations models. The second procedure allows for more general structures.

Exact smoothing for stationary and non-stationary time series

Abstract In this work we derive an analytical relationship between exact fixed-interval smoothed moments and those obtained from an arbitrarily initialized smoother. Combining this result with a conventional smoother we obtain an exact algorithm that can be applied to stationary, non-stationary or partially non-stationary systems. Other advantages of our method are its computational efficiency and numerical stability. Its extension to forecasting, filtering, fixed-point and fixed-lag smoothing is immediate, as it only requires modification of a conditioning information set. Three examples illustrate the adverse effect of an inadequate initialization on smoothed estimates.

Fast estimation methods for time series models in state-space form

We propose two new, fast and stable methods to estimate time-series models written in their equivalent state–space form. They are useful both to obtain adequate initial conditions for a maximum likelihood iteration and to provide final estimates when maximum likelihood is considered inadequate or computationally expensive. The state–space foundation of these procedures provides flexibility, as they can be applied to any linear fixed-coefficients model, such as ARIMA, VARMAX or structural time-series models. A simulation exercise shows that their computational costs and finite-sample performance are very good.

Exact initial conditions for maximum likelihood estimation of state space models with stochastic inputs

Abstract We derive exact expressions for the conditional mean and variance of the initial state of a State Space system with stochastic inputs, under stationarity or non-stationarity. These results provide an initialization method to obtain maximum likelihood estimates of the parameters.

Original article: From general state-space to VARMAX models

Fixed coecients State-Space and VARMAX models are equivalent, meaning that they are able to represent the same linear dynamics, being indistinguishable in terms of overall fit. However, each representation can be specically adequate for certain uses, so it is relevant to be able to choose between them. To this end, we propose two algorithms to go from general State-Space models to VARMAX forms. The rst one computes the coecients of a standard VARMAX model under some assumptions while the second, which is more general, returns the coecients of a VARMAX echelon. These procedures supplement the results already available in the literature allowing one to obtain the State-Space model matrices corresponding to any VARMAX. The paper also discusses some applications of these procedures by solving several theoretical and practical problems.

The exact likelihood for a state space model with stochastic inputs

Abstract In this work, we derive exact and approximate expressions for the conditional mean and variance of the initial state of a state space model, allowing for unit roots and stochastic inputs. These results provide adequate initial conditions to compute the exact likelihood using the Kalman filter. The exact conditional moments are the best choice when the stochastic structure of the inputs is known. If this is not the case, the approximate expressions are a good alternative, as some simulation results illustrate.

Unit roots and cointegration modelling through a family of flexible information criteria

We propose a fast and consistent procedure to detect unit roots based on subspace methods. It has three distinctive features. First, the same method can be applied to single or multiple time series. Second, it employs a flexible family of information criteria, whose loss functions can be adapted to the statistical properties of the data. Last, it does not require the specification of a model for the analysed series. In addition, we provide a subspace-based consistent estimator for the cointegrating rank and the cointegrating matrix. Simulation exercises show that these procedures have good finite sample properties.

A New Approach to the Unconditional Measurement of Default Risk

This paper analyzes the unconditional measurement of default risk and proposes an alternative modeling approach. We begin the analysis by showing that when conducted under non-stationarity, the objective of the unconditional measurement changes and that some relevant problems appear as a consequence of the sample dependence. Based on this result, we introduce our approach and discuss its consistency, practical advantages, and the main differences from the conventional static framework. An empirical analysis is also conducted. Under non-stationarity, the regulatory model for the unconditional probability of default distribution performs badly when compared to our approach. Results also show that the capital figure presents a determinant and nontrivial dependence on the homogeneity and severity of the economic scenario represented in the sample.

Decomposition of a state-space model with inputs

This article shows how to compute the in-sample effect of exogenous inputs on the endogenous variables in any linear model written in a state–space form. Estimating this component may be either interesting by itself, or a previous step before decomposing a time series into trend, cycle, seasonal and error components. The practical application and usefulness of this method is illustrated by estimating the effect of advertising on the monthly sales of Lydia Pinkhams vegetable compound.