I derive an oracle inequality for a family of possibly misspecified multivariate conditional autoregressive quantile models. The family includes standard specifications for (nonlinear) quantile prediction proposed in the literature. This inequality is used to establish that the predictor that minimizes the in-sample average check loss achieves the best out-of-sample performance within its class at a near optimal rate, even when the model is fully misspecified. An empirical application to backtesting global Growth-at-Risk shows that a combination of the generalized autoregressive conditionally heteroscedastic model and the vector autoregression for Value-at-Risk performs best out-of-sample in terms of the check loss.

Empirical risk minimization is a standard principle for choosing algorithms in learning theory. In this paper we study the properties of empirical risk minimization for time series. The analysis is carried out in a general framework that covers different types of forecasting applications encountered in the literature. We are concerned with 1-step-ahead prediction of a univariate time series belonging to a class of location-scale parameter-driven processes. A class of recursive algorithms is available to forecast the time series. The algorithms are recursive in the sense that the forecast produced in a given period is a function of the lagged values of the forecast and of the time series. The relationship between the generating mechanism of the time series and the class of algorithms is not specified. Our main result establishes that the algorithm chosen by empirical risk minimization achieves asymptotically the optimal predictive performance that is attainable within the class of algorithms.

We propose a novel specification of the Dynamic Conditional Correlation (DCC) model based on an alternative normalization of the pseudo-correlation matrix called Projected DCC (Pro-DCC). Our modification consists in projecting, rather than rescaling, the pseudo-correlation matrix onto the set of correlation matrices in order to obtain a well defined conditional correlation matrix. A simulation study shows that projecting performs better than rescaling when the dimensionality of the correlation matrix is large. An empirical application to the constituents of the S&P 100 shows that the proposed methodology performs favorably to the standard DCC in an out-of-sample asset allocation exercise. 

Este capítulo trata el problema de selección de carteras de inversión con un gran número de activos financieros. En particular, se repasa la literatura en modelización de correlaciones condicionales dinámicas de elevadas dimensiones (DCC, por sus siglas en inglés). Consideramos diferentes tipos de especificaciones, en particular, la versión estándar del modelo DCC, el DCC con estructura de factores, y el DCC con regularización. Introducimos métodos de estimación específicamente diseñados para modelos de elevada dimensionalidad. Evaluamos su capacidad de predicción a través de una aplicación en selección de carteras de inversión con los constituyentes del índice S&P 500.

We introduce a monitoring tool based on the time-varying probability of joint exceedence of two or more components of a multivariate time series conditional on covariates. A general location-scale specification is used to model the dynamics of the multivariate time series. A semiparametric estimation approach is proposed and its statistical properties are established under mild assumptions. In order to choose among different candidate specifications and covariates, we introduce an out-of-sample conditional coverage error test and derive the asymptotic distribution of the test statistic under the null of equal predictive ability. We apply our framework to study stagflation in the US.