Covariance estimation for multivariate conditionally Gaussian dynamic linear models

In multivariate time series, the estimation of the covariance matrix of the observation innovations plays an important role in forecasting as it enables the computation of the standardized forecast error vectors as well as it enables the computation of confidence bounds of the forecasts. We develop an on-line, non-iterative Bayesian algorithm for estimation and forecasting. It is empirically found that, for a range of simulated time series, the proposed covariance estimator has good performance converging to the true values of the unknown observation covariance matrix. Over a simulated time series, the new method approximates the correct estimates, produced by a non-sequential Monte Carlo simulation procedure, which is used here as the gold standard. The special, but important, vector autoregressive (VAR) and time-varying VAR models are illustrated by considering London metal exchange data consisting of spot prices of aluminium, copper, lead and zinc.Comment: 21 pages, 2 figures, 6 table

Comparisons of Hyv\"arinen and pairwise estimators in two simple linear time series models

The aim of this paper is to compare numerically the performance of two estimators based on Hyv\"arinen's local homogeneous scoring rule with that of the full and the pairwise maximum likelihood estimators. In particular, two different model settings, for which both full and pairwise maximum likelihood estimators can be obtained, have been considered: the first order autoregressive model (AR(1)) and the moving average model (MA(1)). Simulation studies highlight very different behaviours for the Hyv\"arinen scoring rule estimators relative to the pairwise likelihood estimators in these two settings.Comment: 14 pages, 2 figure

Dynamic Covariance Models for Multivariate Financial Time Series

The accurate prediction of time-changing covariances is an important problem in the modeling of multivariate financial data. However, some of the most popular models suffer from a) overfitting problems and multiple local optima, b) failure to capture shifts in market conditions and c) large computational costs. To address these problems we introduce a novel dynamic model for time-changing covariances. Over-fitting and local optima are avoided by following a Bayesian approach instead of computing point estimates. Changes in market conditions are captured by assuming a diffusion process in parameter values, and finally computationally efficient and scalable inference is performed using particle filters. Experiments with financial data show excellent performance of the proposed method with respect to current standard models

Aggregation and long memory: recent developments

It is well-known that the aggregated time series might have very different properties from those of the individual series, in particular, long memory. At the present time, aggregation has become one of the main tools for modelling of long memory processes. We review recent work on contemporaneous aggregation of random-coefficient AR(1) and related models, with particular focus on various long memory properties of the aggregated process

Joint Covariance Estimation with Mutual Linear Structure

We consider the problem of joint estimation of structured covariance matrices. Assuming the structure is unknown, estimation is achieved using heterogeneous training sets. Namely, given groups of measurements coming from centered populations with different covariances, our aim is to determine the mutual structure of these covariance matrices and estimate them. Supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices, we develop a new efficient algorithm discovering the structure and using it to improve the estimation. Our technique is based on the application of principal component analysis in the matrix space. We also derive an upper performance bound of the proposed algorithm in the Gaussian scenario and compare it with the Cramer-Rao lower bound. Numerical simulations are presented to illustrate the performance benefits of the proposed method

On Locally Dyadic Stationary Processes

We introduce the concept of local dyadic stationarity, to account for non-stationary time series, within the framework of Walsh-Fourier analysis. We define and study the time varying dyadic ARMA models (tvDARMA). It is proven that the general tvDARMA process can be approximated locally by either a tvDMA and a tvDAR process.Comment: 27 pages, 2 figure