Alessio Sancetta

THE BERNSTEIN COPULA AND ITS APPLICATIONS TO MODELING AND APPROXIMATIONS OF MULTIVARIATE DISTRIBUTIONS

We define the Bernstein copula and study its statistical properties in terms of both distributions and densities. We also develop a theory of approximation for multivariate distributions in terms of Bernstein copulas. Rates of consistency when the Bernstein copula density is estimated empirically are given. In order of magnitude, this estimator has variance equal to the square root of the variance of common nonparametric estimators, e.g., kernel smoothers, but it is biased as a histogram estimator.We would thank Mark Salmon for interesting us in the copula function and Peter Phillips, an associate editor, and the referees for many valuable comments. All remaining errors are our sole responsibility.

Universality of Bayesian Predictions

Given the sequential update nature of Bayes rule, Bayesian methods find natural application to prediction problems. Advances in computational methods allow to routinely use Bayesian methods in econometrics. Hence, there is a strong case for feasible predictions in a Bayesian framework. This paper studies the theoretical properties of Bayesian predictions and shows that under minimal conditions we can derive finite sample bounds for the loss incurred using Bayesian predictions under the Kullback-Leibler divergence. In particular, the concept of universality of predictions is discussed and universality is established for Bayesian predictions in a variety of settings. These include predictions under almost arbitrary loss functions, model averaging, predictions in a non stationary environment and under model miss-specification. Given the possibility of regime switches and multiple breaks in economic series, as well as the need to choose among different forecasting models, which may inevitably be miss-specified, the finite sample results derived here are of interest to economic and financial forecasting.

RECURSIVE FORECAST COMBINATION FOR DEPENDENT HETEROGENEOUS DATA

This paper studies a procedure to combine individual forecasts that achieve theoretical optimal performance. The results apply to a wide variety of loss functions and only require a tail condition on the data sequences. The theoretical results show that the bounds are also valid in the case of time varying combination weights.

Changing Correlation and Equity Portfolio Diversification Failure for Linear Factor Models during Market Declines

The paper considers a linear factor model (LFM) to study the behaviour of the correlation coefficient between various stock returns during a downturn. Changing correlation is related to the tail distribution of the driving factors, which is the market for Sharpes one‐factor model. General classes of distribution functions are considered and asymptotic conditions found on the tails of the distribution, which determine whether diversification will succeed or fail during a market decline.

Bernstein Approximations to the Copula Function and Portfolio Optimization

The copula function is considered within the context of financial multivariate data sets that are not normally distributed. The Bernstein polynomial approximation to copulae is given and motivated by its desirable properties. The multivariate convergence properties are analysed. The concept of Bernstein copula is introduced as a generalisation of some bivariate and higher dimensional families of copulae. Statistical properties of the Bernstein copula are studied together with implementation issues related to portfolio theory and expected utility optimisation.

Greedy algorithms for prediction

In many prediction problems, it is not uncommon that the number of variables used to construct a forecast is of the same order of magnitude as the sample size, if not larger. We then face the problem of constructing a prediction in the presence of potentially large estimation error. Control of the estimation error is either achieved by selecting variables or combining all the variables in some special way. This paper considers greedy algorithms to solve this problem. It is shown that the resulting estimators are consistent under weak conditions. In particular, the derived rates of convergence are either minimax or improve on the ones given in the literature allowing for dependence and unbounded regressors. Some versions of the algorithms provide fast solution to problems such as Lasso.

Nearest neighbor conditional estimation for Harris recurrent Markov chains

This paper is concerned with consistent nearest neighbor time series estimation for data generated by a Harris recurrent Markov chain on a general state space. It is shown that nearest neighbor estimation is consistent in this general time series context, using simple and weak conditions. The results proved here, establish consistency, in a unified manner, for a large variety of problems, e.g. autoregression function estimation, and, more generally, extremum estimators as well as sequential forecasting. Finally, under additional conditions, it is also shown that the estimators are asymptotically normal.

Online Forecast Combination for Dependent Heterogeneous Data

This paper studies a procedure to combine individual forecasts that achieve theoretical optimal performance. The results apply to a wide variety of loss functions and no conditions are imposed on the data sequences and the individual forecasts apart from a tail condition. The theoretical results show that the bounds are also valid in the case of time varying combination weights, under specific conditions on the nature of time variation. Some experimental evidence to confirm the results is provided.

Calculating hedge fund risk: the draw down and the maximum draw down

Hedge funds, defined in this context as geared financial entities, frequently use some measure of point loss as a risk measure. This paper considers the statistical properties of an uninterrupted fall in a security price; called a draw down. The distribution of the draw downs in an N‐trading period is derived together with an approximation to the distribution of the maximum. Complementary results are provided which are useful for risk calculations. A brief empirical study of the S&P futures is included in order to highlight some of the limitations in the presence of extreme events.

Conditional estimation for dependent functional data

Suppose we observe a Markov chain taking values in a functional space. We are interested in exploiting the time series dependence in these infinite dimensional data in order to make non-trivial predictions about the future. Making use of the Karhunen-Loeve (KL) representation of functional random variables in terms of the eigenfunctions of the covariance operator, we present a deliberately over-simplified nonparametric model, which allows us to achieve dimensionality reduction by considering one dimensional nearest neighbour (NN) estimators for the transition distribution of the random coefficients of the KL expansion. Under regularity conditions, we show that the NN estimator is consistent even when the coefficients of the KL expansion are estimated from the observations. This also allows us to deduce the consistency of conditional regression function estimators for functional data. We show via simulations and two empirical examples that the proposed NN estimator outperforms the state of the art when data are generated both by the functional autoregressive (FAR) model of Bosq (2000) [8] and by more general data generating mechanisms.