Alessandra Luati

Filtering With Heavy Tails

An unobserved components model in which the signal is buried in noise that is non-Gaussian may throw up observations that, when judged by the Gaussian yardstick, are outliers. We describe an observation-driven model, based on a conditional Student’s t-distribution, which is tractable and retains some of the desirable features of the linear Gaussian model. Letting the dynamics be driven by the score of the conditional distribution leads to a specification that is not only easy to implement, but which also facilitates the development of a comprehensive and relatively straightforward theory for the asymptotic distribution of the maximum likelihood estimator. The methods are illustrated with an application to rail travel in the United Kingdom. The final part of the article shows how the model may be extended to include explanatory variables.

Maximum Fisher information in mixed state quantum systems

We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101-102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439-3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481-4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states.

Generalised Partial Autocorrelations and the Mutual Information between Past and Future

The paper introduces the generalised partial autocorrelation (GPAC) coefficients of a stationary stochastic process. The latter are related to the generalised autocovariances, the inverse Fourier transform coefficients of a power transformation of the spectral density function. By interpreting the generalized partial autocorrelations as the partial autocorrelation coefficients of an auxiliary process, we derive their properties and relate them to essential features of the original process. Based on a parameterisation suggested by Barndorff-Nielsen and Schou (1973) and on Whittle likelihood, we develop an estimation strategy for the GPAC coefficients. We further prove that the GPAC coefficients can be used to estimate the mutual information between the past and the future of a time series.

Real time estimation in local polynomial regression, with application to trend-cycle analysis

The paper focuses on the adaptation of local polynomial filters at the end of the sample period. We show that for real time estimation of signals (i.e., exactly at the boundary of the time support) we cannot rely on the automatic adaptation of the local polynomial smoothers, since the direct real time filter turns out to be strongly localized, and thereby yields extremely volatile estimates. As an alternative, we evaluate a general family of asymmetric filters that minimizes the mean square revision error subject to polynomial reproduction constraints; in the case of the Henderson filter it nests the well-known Musgraves surrogate filters. The class of filters depends on unknown features of the series such as the slope and the curvature of the underlying signal, which can be estimated from the data. Several empirical examples illustrate the effectiveness of our proposal.

Hyper-Spherical and Elliptical Stochastic Cycles

A univariate first-order stochastic cycle can be represented as an element of a bivariate first-order vector autoregressive process, or VAR(1), where the transition matrix is associated with a rotation along a circle in the plane, and the reduced form is ARMA(2,1). This paper generalizes this representation in two directions. According to the first, the cyclical dynamics originate from the motion of a point along an ellipse. The reduced form is also ARMA(2,1), but the model can account for certain types of asymmetries. The second deals with the multivariate case: the cyclical dynamics result from the projection along one of the coordinate axis of a point moving in along an hyper-sphere. This is described by a VAR(1) process whose transition matrix is obtained by a sequence of n-dimensional Givens rotations. The reduced form of an element of the system is shown to be ARMA(n, n - 1). The properties of the resulting models are analysed in the frequency domain, and we show that this generalization can account for a multimodal spectral density. The illustrations show that the proposed generalizations can be fitted successfully to some well-known case studies of the time series literature. Copyright Copyright 2010 Blackwell Publishing Ltd

A linear transformation and its properties with special applications in time series filtering

Abstract The main purpose of this paper is to introduce a linear transformation, called t , and to derive its algebraic properties by means of permutation matrices that represent it. To demonstrate the importance of the t -transformation for the estimation of latent variables in time series decomposition, we obtain a general expression for smoothing matrices characterized by symmetric and asymmetric weighting systems. We show that the submatrix of the symmetric weights (to be applied to central observations) is t -invariant whereas the submatrices of the asymmetric weights (to be applied to initial and final observations) are the t -transform of each other. By virtue of this relation, the properties of the t -transformation provide useful information on the smoothing of time series data. Finally, we illustrate the role of the t -transformation on the weighting systems of several smoothers often applied for trend-cycle estimation, such as the locally weighted regression smoother (loess), the cubic smoothing spline, the Gaussian kernel and the 13-term trend-cycle Henderson filter.

On the equivalence of the weighted least squares and the generalised least squares estimators, with applications to kernel smoothing

This paper establishes the conditions under which the generalised least squares estimator of the regression parameters is equivalent to the weighted least squares estimator. The equivalence conditions have interesting applications in local polynomial regression and kernel smoothing. Specifically, they enable to derive the optimal kernel associated with a particular covariance structure of the measurement error, where optimality has to be intended in the Gauss-Markov sense. For local polynomial regression it is shown that there is a class of covariance structures, associated with non-invertible moving average processes of given orders which yield the Epanechnikov and the Henderson kernels as the optimal kernels.

On the Equivalence of the Weighted Least Squares and the Generalised Least Squares Estimators, with Applications to Kernel Smoothing

This paper establishes the conditions under which the generalised least squares estimator of the regression parameters is equivalent to the weighted least squares estimator. The equivalence conditions have interesting applications in local polynomial regression and kernel smoothing. Specifically, they enable to derive the optimal kernel associated with a particular covariance structure of the measurement error, where optimality has to be intended in the Gauss-Markov sense. For local polynomial regression it is shown that there is a class of covariance structures, associated with non-invertible moving average processes of given orders which yield the Epanechnikov and the Henderson kernels as the optimal kernels.

A Cascade Linear Filter to Reduce Revisions and False Turning Points for Real Time Trend-Cycle Estimation

The problem of identifying the direction of the short-term trend (nonstationary mean) of seasonally adjusted series contaminated by high levels of variability has become of relevant interest in recent years. In fact, major financial and economic changes of global character have introduced a large amount of noise in time series data, particularly, in socioeconomic indicators used for real time economic analysis. The aim of this study is to construct a cascade linear filter via the convolution of several noise suppression, trend estimation, and extrapolation linear filters. The cascading approach approximates the steps followed by the nonlinear Dagum (1996) trend-cycle estimator, a modified version of the 13-term Henderson filter. The former consists of first extending the seasonally adjusted series with ARIMA extrapolations, and then applying a very strict replacement of extreme values. The nonlinear Dagum filter has been shown to improve significantly the size of revisions and number of false turning points with respect to H13. We construct a linear approximation of the nonlinear filter because it offers several advantages. For one, its application is direct and hence does not require some knowledge on ARIMA model identification. Furthermore, linear filtering preserves the crucial additive constraint by which the trend of an aggregated variable should be equal to the algebraic addition of its component trends, thus avoiding the selection problem of direct versus indirect adjustments. Finally, the properties of a linear filter concerning signal passing and noise suppression can always be compared to those of other linear filters by means of spectral analysis.

Global and local statistical properties of fixed-length nonparametric smoothers

The main purpose of this study is to analyze the global and local statistical properties of nonparametric smoothers subject to a priori fixed length restriction. In order to do so, we introduce a set of local statistical measures based on their weighting system shapes and weight values. In this way, the local statistical measures of bias, variance and mean square error are intrinsic to the smoothers and independent of the data to which they will be applied on. One major advantage of the statistical measures relative to the classical spectral ones is their easiness of calculation. However, in this paper we use both in a complementary manner. The smoothers studied are based on two broad classes of weighting generating functions, local polynomials and probability distributions. We consider within the first class, the locally weighted regression smoother (loess) of degree 1 and 2 (L1 and L2), the cubic smoothing spline (CSS), and the Henderson smoothing linear filter (H); and in the second class, the Gaussian kernel (GK). The weighting systems of these estimators depend on a smoothing parameter that traditionally, is estimated by means of data dependent optimization criteria. However, by imposing to all of them the condition of an equal number of weights, it will be shown that some of their optimal statistical properties are no longer valid. Without any loss of generality, the analysis is carried out for 13- and 9-term lengths because these are the most often selected for the Henderson filters in the context of monthly time series decomposition.