Sonia Sotoca

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.

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.

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.

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.

Single and multiple error state-space models for signal extraction

We compare the results obtained by applying the same signal-extraction procedures to two observationally equivalent state-space forms. The first model has different errors affecting the states and the observations, while the second has a single perturbation term which coincides with the one-step-ahead forecast error. The signals extracted from both forms are very similar but their variances are drastically different, because the states for the single-source error representation collapse to exact values while those coming from the multiple-error model remain uncertain. The implications of this result are discussed both with theoretical arguments and practical examples. We find that single error representations have advantages to compute the likelihood or to adjust for seasonality, while multiple error models are better suited to extract a trend indicator. Building on this analysis, it is natural to adopt a ‘best of both worlds’ approach, which applies each representation to the task in which it has comparative advantage.

Conditional Coverage and Its Role in Determining and Assessing Long-Term Capital Requirements

We define the vector of conditional coverage values generated over the business cycle by a constant capital figure. Using a convenient analytical framework, we explore its properties and propose two applications based on it. For the former, we state a result that links the concepts of conditional and unconditional solvency and offers an alternative interpretation of the unconditional capital. For the latter, we propose using the minimum of the conditional coverage vector in the determination of long-term capital requirements, as well as using its minimum and its standard deviation in the long-term assessment of a given capital figure. Both applications are illustrated empirically. The entire analysis can be understood as an attempt to recognize and incorporate capital cyclicality into the measurement and analysis of default risk.

Minimally conditioned likelihood for a nonstationary state space model

Computing the Gaussian likelihood for a nonstationary state-space model is a difficult problem which has been tackled by the literature using two main strategies: data transformation and diffuse likelihood. The data transformation approach is cumbersome, as it requires nonstandard filtering. On the other hand, the diffuse likelihood value depends on the scale of the diffuse states, so two observationally equivalent models may yield different likelihood values in some nontrivial cases. In this paper we present an alternative approach: computing a likelihood function conditional to the minimum subsample required to eliminate the effect of a diffuse initialization. Our procedure has three convenient features: (a) it can be computed with standard Kalman filters, (b) it is scale-free, and (c) its values are coherent with those resulting from differencing, being this the most popular approach to deal with nonstationary data.