Aggregation of exponential smoothing processes with an application to portfolio risk evaluation

In this paper we propose a unified framework to analyse contemporaneous and temporal aggregation of exponential smoothing (EWMA) models. Focusing on a vector IMA(1,1) model, we obtain a closed form representation for the parameters of the contemporaneously and temporally aggregated process as a function of the parameters of the original one. In the framework of EWMA estimates of volatility, we present an application dealing with Value-at-Risk (VaR) prediction at different sampling frequencies for an equally weighted portfolio composed of multiple indices. We apply the aggregation results by inferring the decay factor in the portfolio volatility equation from the estimated vector IMA(1,1) model of squared returns. Empirical results show that VaR predictions delivered using this suggested approach are at least as accurate as those obtained by applying the standard univariate RiskMetrics TM methodology.contemporaneous and temporal aggregation, EWMA, volatility, Value-at-Risk

Sharp Oracle Inequalities for Aggregation of Affine Estimators

We consider the problem of combining a (possibly uncountably infinite) set of affine estimators in non-parametric regression model with heteroscedastic Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a PAC-Bayesian type inequality that leads to sharp oracle inequalities in discrete but also in continuous settings. The framework is general enough to cover the combinations of various procedures such as least square regression, kernel ridge regression, shrinking estimators and many other estimators used in the literature on statistical inverse problems. As a consequence, we show that the proposed aggregate provides an adaptive estimator in the exact minimax sense without neither discretizing the range of tuning parameters nor splitting the set of observations. We also illustrate numerically the good performance achieved by the exponentially weighted aggregate

Theory and observations of ice particle evolution in cirrus using Doppler radar: evidence for aggregation

Vertically pointing Doppler radar has been used to study the evolution of ice particles as they sediment through a cirrus cloud. The measured Doppler fall speeds, together with radar-derived estimates for the altitude of cloud top, are used to estimate a characteristic fall time tc for the `average' ice particle. The change in radar reflectivity Z is studied as a function of tc, and is found to increase exponentially with fall time. We use the idea of dynamically scaling particle size distributions to show that this behaviour implies exponential growth of the average particle size, and argue that this exponential growth is a signature of ice crystal aggregation.Comment: accepted to Geophysical Research Letter

Information Aggregation in Exponential Family Markets

We consider the design of prediction market mechanisms known as automated market makers. We show that we can design these mechanisms via the mold of \emph{exponential family distributions}, a popular and well-studied probability distribution template used in statistics. We give a full development of this relationship and explore a range of benefits. We draw connections between the information aggregation of market prices and the belief aggregation of learning agents that rely on exponential family distributions. We develop a very natural analysis of the market behavior as well as the price equilibrium under the assumption that the traders exhibit risk aversion according to exponential utility. We also consider similar aspects under alternative models, such as when traders are budget constrained

A note on forecasting demand using the multivariate exponential smoothing framework

Simple exponential smoothing is widely used in forecasting economic time series. This is because it is quick to compute and it generally delivers accurate forecasts. On the other hand, its multivariate version has received little attention due to the complications arising with the estimation. Indeed, standard multivariate maximum likelihood methods are affected by numerical convergence issues and bad complexity, growing with the dimensionality of the model. In this paper, we introduce a new estimation strategy for multivariate exponential smoothing, based on aggregating its observations into scalar models and estimating them. The original high-dimensional maximum likelihood problem is broken down into several univariate ones, which are easier to solve. Contrary to the multivariate maximum likelihood approach, the suggested algorithm does not suffer heavily from the dimensionality of the model. The method can be used for time series forecasting. In addition, simulation results show that our approach performs at least as well as a maximum likelihood estimator on the underlying VMA(1) representation, at least in our test problems