Sparse Models in High-Dimensional Dependence Modelling and Index Tracking

Abstract

This thesis is divided into two parts. The first part proposes parsimonious models to the vine copula. The second part is devoted to the index tracking problem.

Vine copulas provide a flexible tool to capture asymmetry in modelling multivariate distributions. Nevertheless, the computational expense of its flexibility increases exponentially as the dimension of the joint distribution grows. To alleviate this issue, the simplifying assumption (SA) is commonly adopted in special applications of vine copula models. In order to relax SA, Chapter 2 proposes generalized linear models (GLMs) to model parameters in conditional bivariate copulas. In the spirit of the principle of parsimony, a regularization methodology is developed to control the number of parameters. This leads to sparse vine copula models. The conventional vine copula with the SA, the proposed GLM-based vine copula and the sparse vine copula are applied to several financial datasets. Empirical results show that the proposed models in this chapter outperform the one with SA significantly in terms of the Bayesian information criterion.

Index tracking is a dominant method among passive investment strategies. It attempts to reproduce the return of stock-market indices. Chapter 3 focuses on selecting stocks to construct tracking portfolios. In order to do that, principal component analysis (PCA) is applied via a two-step procedure. In the first step, the index return is expressed as a function of the principal components (PCs) of stock returns, and a subset of PCs is selected according to Sobol's total sensitivity index. In the second step, a subset of stocks, which is most "similar" to those selected PCs, is detected. This similarity is measured by Yanai's generalized coefficient of determination, the distance correlation, or Heller-Heller-Gorfine test statistics. Given selected stocks, their weights in the tracking portfolio can be determined by minimizing a specific tracking error. Compared with existing methods, constructing tracking portfolios based on stocks selected by this PCA-based method is more computationally efficient and comparably effective at minimizing the tracking error.

When the number of index components is large, it is too computationally demanding to apply methods in Chapter 3 or most of existing methods, such as those relying on mixed-integer quadratic programming. In Chapter 4, factor models are used to describe stock returns. Under this assumption, the tracking error is partitioned into two parts: one depends on common economic factors, and the other depends on idiosyncratic risks. According to this partition, a 2-stage method is introduced to construct tracking portfolios by minimizing the tracking error. Stage 1 relies on a mixed-integer linear program to identify stocks that are able to reduce factors' impacts on the tracking error, and Stage2 determines weights of identified stocks by minimizing the tracking error. This 2-stage method efficiently constructs tracking portfolios benchmarked to indices with thousands of components. It reduces out-of-sample tracking errors significantly.

In Chapter 5, the index tracking problem is solved by repeatedly solving one-period tracking problems. Each one-period tracking strategy is determined by a quadratic optimization with the L-1 regularization on asset weights. This formulation considers transaction costs and other practical constraints. Since the true joint distribution of financial returns is usually unknown, we solve one-period tracking problems under empirical distributions. With the L-1 regularization on asset weights, our one-period tracking strategy enjoys persistent properties in the high-dimensional setting. More specifically, the variable number d=d(n)=O(n^ α), where n is the sample size and α>1. Simulation studies are carried out to support our one-period tracking strategy's performance with finite samples. Applications on real financial data provide evidence that, in dealing with one-period tracking, this tracking strategy outperforms the L-q penalty tracking method in terms of tracking performance and computational efficiency. In terms of multi-period tracking, this proposed method outperforms the full-replication strategy.