In this study, we propose the use of mixture copulas in d-dimensional Pair-Copula- Constructions as a new strategy to circumvent the otherwise necessary (and error- prone) selection of parametric forms for the d(d − 1)/2 bivariate pair-copulas. While previous studies in the literature have tried different approaches to select optimally fitting pair-copulas from parametric copula families characterized by different tail de- pendence (e. g., goodness-of-fit tests, graphical tools, etc.), we propose to use convex combinations of these bivariate copulas for each pair-copula in a vine model. Each mixture pair-copula is then estimated using the well-known EM-algorithm yielding a fully specified vine model in which no parametric copula needs to be selected as all parametric candidate copulas can be included in the mixture pair-copulas. After
156Complementing the test of conditional coverage, the results of the Weibull backtest show that with
outlining our proposed Mixture-PCC, we test the performance of our new model in comparison with a benchmark PCC in which each pair-copula is chosen by computing the Akaike’s Information Criterion for each candidate parametric copula and selecting the copula with the optimal AIC value. We perform both a simulation study on the in-sample fit of both models as well as an empirical study in which we assess both models’ out-of-sample forecasting accuracy.
Our main result can be summarized as follows: in our simulations and in the empiri- cal study, both models yield acceptable Value-at-Risk forecasts. However, we show that our proposed Mixture-PCC yields better results in backtesting for at least the 2.5% significance level while the benchmark overestimates portfolio risk. Especially for higher dimensions, our Mixture-PCC model seems to approximate portfolio losses better than the benchmark which is far too conservative in many cases. Consequently, our model can help risk managers to save on regulatory risk capital while at the same time satisfactorily bounding possible portfolio losses.
For future research, one could think of an analysis of extended Mixture-PCCs in which the number of mixture pair-copulas is not fixed as it is done in this study. Fur- thermore, Mixture-PCCs could be combined with pruning strategies to truncate or sim- plify some of the mixture pair-copulas in lower trees to limit the computational cost. We expect all these extensions to lead to further improvements on the forecasting ac- curacy of our Mixture-PCC in comparison to models from the related literature and intend to address them in a future study.
Chapter 5
Extreme Dependence in Investor
Attention to Bank stocks
5.1
Introduction
The effect of investor attention on financial markets has been of long-standing in- terest to economists. Traditionally, measures of attention were restricted to indirect proxies like trading volume, media news, and abnormal returns. Recent research in financial econometrics literature illustrates the usefulness of Google search data as a direct measure for retail investors’ attention. Starting with the work by Da et al. (2011), most applications of Google search data in financial applications focus on asset pric- ing implications and predicting dynamics of stock market volatility (see, e. g., Da et al., 2011, Hamid and Heiden, 2014, Mondria and Wu, 2011, Vozlyublennaia, 2014). In the course of these studies, the co-movement between investor attention and stock prices (volatility) has been investigated in detail.
In this study, we provide a statistical modeling framework for specifying, estimat- ing, and testing time series of investor attention measured by internet search queries. More precisely, our paper is the first to present both a univariate as well as a multi- variate econometric model for Google search data. We find that the dependence struc- ture of high-dimensional Google search data is significantly non-linear and asymmet-
ric. Furthermore, we document the existence of extreme dependence in Google search data pairs and, particularly noteworthy, between stock returns and the corresponding Google search data. Finally, our main contribution is to show a striking similarity in the joint distributions of a multivariate bank stock portfolio and the corresponding portfo- lio of Google search queries, respectively. Following our results, we hypothesize that investor attention as measured by internet search data and stock returns reflect almost the same information.
Starting point of our paper is the proper modeling of internet search queries. While modeling of time series such as stock returns, foreign exchange rates, and CDS spreads has become common practice in financial econometrics, the application of statistical techniques to investor attention measured by internet search data is widely unexplored. To this end, we provide a comprehensive time series analysis and extract meaningful statistics and other characteristics of Google search data. We find both autoregressive dynamics and conditional heteroskedasticity in the underlying data set. Additionally, there is statistical evidence for specific distributional characteristics like the presence of distinct levels of skewness and kurtosis within Google search data. Our analysis shows that the first- and second-moment dependence are well captured by asymmetric ARMA-CS-GARCH models. Finally, we also find that the skewed t as well as the skewed generalized error distribution (sged) provide good fits to the Google search data residuals.
In our multivariate econometric framework, we aim to model the joint distribution of high-dimensional search query data in a flexible way. In this regard, we propose the implementation of a vine copula approach, which is especially appropriate for two reasons: First, it allows to capture both linear dependence as well as potential non- linearities in the dependence structure. In fact, we document the existence of strong non-linear and asymmetric dependence in the Google search data. Second, due to their hierarchical construction, vine copulas allow to model different dependence structures between pairs of variables. As a result, our study provides the first empirical evidence of significant tail dependence in Google search data.
Beside their usefulness in capturing inherent dependency patterns of high dimen- sional data sets, vine copulas provide a powerful tool to detect similarities in the joint distribution of different data sets. To be precise, we find a striking similarity in the joint distributions of a multivariate bank stock portfolio and the corresponding portfolio of Google search queries, respectively. The remarkable similarities necessitates a detailed investigation of the co-movement between investor attention and stock returns. In our analysis, we provide first empirical evidence for the existence of tail dependence be- tween stock returns and the respective search query pairs. Furthermore, we document that stock returns and Google search data evolve concurrently in real time. Our re- sults suggest that investor attention measured by internet search data and stock returns reflect almost the same information.
Our paper makes several major contributions. Firstly, our findings push forward sig- nificantly the knowledge of search data and its characteristics. The study of Dimpfl and Jank (forthcoming) indicates that Google search data collected within one country are characterized by specific properties that are usually common to financial time series, like non-normality and autocorrelation. In contrast to their work, however, we take the discussion further and provide a comprehensive time series analysis of worldwide Google search data. Based on our results, our paper is the first to present an economet- ric model that is well-suited for capturing first- and second-moment dependencies in univariate search queries.
Furthermore, we propose to model the joint distribution of Google search data by using regular vine (R-vine) copulas. In this regard, our paper complements several previous studies in the field of using vine copulas in financial econometrics and quan- titative risk management (see, e. g., Aas et al., 2009, Min and Czado, 2010, Dissmann et al., 2013, Christoffersen et al., 2012, Oh and Patton, 2013). But, to the best of our knowledge, this article provides the first application of the vine copula concept to internet search data.
In our empirical application, we provide first empirical evidence of significant tail dependence in Google search data. Referring to this, we contribute to the current
state of research in documenting extreme dependencies in time series. While non- linear dependence has been shown to exist in financial time series like stock returns (see, e. g., Poon et al., 2004, Bollerslev and Todorov, 2011) and credit risk (see, e. g., Christoffersen et al., 2013), this paper is the first to confirm that investor attention measured by Google search data is characterized by strong tail dependence as well.
Finally, our multivariate econometric framework is also beneficial in the context of modeling co-dependencies between investor attention and stock returns. The idea to examine possible causal relations between investor attention and stock returns is related to several studies in the literature (see, e. g., Da et al., 2011, Dimpfl and Jank, forthcoming, Hamid and Heiden, 2014, Mondria and Wu, 2011, Vozlyublennaia, 2014). However, in the course of these works the co-movement between attention and stock prices (volatility) is being investigated by the classical tools of Vector Au- toregressive models and Granger Causality tests, despite their limitations to capture non-linear and asymmetric dependencies in time and between the data series. In con- trast, our vine copula modeling approach is especially appropriate for non-linear and asymmetric modeling between time series dependencies. In fact, our study is the first to document significant tail dependence between investor attention and stock returns.
The remainder of the paper is structured as follows. In Section 5.2, we specify our data sample and present descriptive statistics of the data sets. The marginal and multivariate models we employ to the data are presented and discussed in Section 5.3. Section 5.4 contains the empirical application and a comprehensive discussion of the economic importance of the empirical findings. Section 5.5 concludes.