Nonparametric Estimation of Copulas for Time Series

We consider a nonparametric method to estimate copulas, i.e. functions linking joint distributions to their univariate margins. We derive the asymptotic properties of kernel estimators of copulas and their derivatives in the context of a multivariate stationary process satisfactory strong mixing conditions. Monte Carlo results are reported for a stationary vector autoregressive process of order one with Gaussian innovations. An empirical illustration containing a comparison with the independent, comotonic and Gaussian copulas is given for European and US stock index returns.Nonparametric, Kernel; Time Series; Copulas; Dependence Measures; Risk Management; Loss Severity Distribution

Robust estimation of the vector autoregressive model by a least trimmed squares procedure.

The vector autoregressive model is very popular for modeling multiple time series. Estimation of its parameters is typically done by a least squares procedure. However, this estimation method is unreliable when outliers are present in the data, and therefore we propose to estimate the vector autoregressive model by using a multivariate least trimmed squares estimator. We also show how the order of the autoregressive model can be determined in a robust way. The robust procedure is illustrated on a real data set.Robustness; Multivariate time series; Outliers; Trimming; Vector autoregressive models;

Robust estimation of the vector autoregressive model by a trimmed least squares procedure.

The vector autoregressive model is very popular for modeling multiple time series. Estimation of its parameters is done by a least squares procedure. However, this estimation method is unreliable when outliers are present in the data, and there is a need for robust alternatives. In this paper we propose to estimate the vector autoregressive model by using a trimmed least squares estimator. We show how the order of the autoregressive model can be determined in a robust way, and how confidence bounds around the robustly estimated impulse response functions can be constructed. The resistance of the estimators to outliers is studied on real and simulated data.Advantages; Calibration; Data; Estimator; Least-squares; M-estimators; Methods; Model; Optimal; Outliers; Partial least squares; Precision; Prediction; Regression; Research; Robust regression; Robustness; Squares; Variables; Yield; Robust estimation; Time; Time series; Order; Functions;

A Scalable MCEM Estimator for Spatio-Temporal Autoregressive Models

Very large spatio-temporal lattice data are becoming increasingly common across a variety of disciplines. However, estimating interdependence across space and time in large areal datasets remains challenging, as existing approaches are often (i) not scalable, (ii) designed for conditionally Gaussian outcome data, or (iii) are limited to cross-sectional and univariate outcomes. This paper proposes an MCEM estimation strategy for a family of latent-Gaussian multivariate spatio-temporal models that addresses these issues. The proposed estimator is applicable to a wide range of non-Gaussian outcomes, and implementations for binary and count outcomes are discussed explicitly. The methodology is illustrated on simulated data, as well as on weekly data of IS-related events in Syrian districts.Comment: 29 pages, 8 figure

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

On the prediction of stationary functional time series

This paper addresses the prediction of stationary functional time series. Existing contributions to this problem have largely focused on the special case of first-order functional autoregressive processes because of their technical tractability and the current lack of advanced functional time series methodology. It is shown here how standard multivariate prediction techniques can be utilized in this context. The connection between functional and multivariate predictions is made precise for the important case of vector and functional autoregressions. The proposed method is easy to implement, making use of existing statistical software packages, and may therefore be attractive to a broader, possibly non-academic, audience. Its practical applicability is enhanced through the introduction of a novel functional final prediction error model selection criterion that allows for an automatic determination of the lag structure and the dimensionality of the model. The usefulness of the proposed methodology is demonstrated in a simulation study and an application to environmental data, namely the prediction of daily pollution curves describing the concentration of particulate matter in ambient air. It is found that the proposed prediction method often significantly outperforms existing methods