Modelling multivariate volatilies via conditionally uncorrelated components

ArticleinJournal of the Royal Statistical Society Series B (Statistical Methodology) 70(4):679-702 · February 2008with 52 Reads 
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Abstract
We propose to model multivariate volatility processes on the basis of the newly defined conditionally uncorrelated components (CUCs). This model represents a parsimonious representation for matrix-valued processes. It is flexible in the sense that each CUC may be fitted separately with any appropriate univariate volatility model. Computationally it splits one high dimensional optimization problem into several lower dimensional subproblems. Consistency for the estimated CUCs has been established. A bootstrap method is proposed for testing the existence of CUCs. The methodology proposed is illustrated with both simulated and real data sets. Copyright (c) 2008 Royal Statistical Society.

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    This paper presents theoretical results in the formulation and estimation of multivariate generalized ARCH models within simultaneous equations systems. A new parameterization of the multivariate ARCH process is proposed and equivalence relations are discussed for the various ARCH parameterizations. Constraints sufficient to guarantee the positive definiteness of the conditional covariance matrices are developed, and necessary and sufficient conditions for covariance stationarity are presented. Identification and maximum likelihood estimation of the parameters in the simultaneous equations context are also covered. * This paper began as a synthesis of at least three UCSD Ph.D. dissertations on various aspects of multivariate ARCH modelling, byYoshi Baba, Dennis Kraft and Ken Kroner. In fact, an early version of this paper was written by Baba, Engle, Kraft and Kroner, which led to the acronym (BEKK) used in this paper for the new parameterization presented. In the interests of continui...
  • Chapter
    In the preceding chapters, the authors introduced several different estimation principles and algorithms for independent component analysis (ICA). In this chapter, they provide an overview of these methods. First, they show that all these estimation principles are intimately connected, and the main choices are between cumulant-based vs. negentropy/likelihood-based estimation methods, and between one-unit vs. multi-unit methods. They compare the algorithms experimentally, and show that the main choice here is between on-line (adaptive) gradient algorithms vs. fast batch fixed-point algorithms. At the end of the chapter, they provide a short summary of basic ICA estimation.
  • Book
    Introduction.- Stationary Time Series.- Smoothing in Time Series.- ARMA Modeling and Forecasting.- Parametric Nonlinear Time Series Models.- Nonparametric Models.- Hypothesis Testing.- Continuous Time Models in Finance.- Nonlinear Prediction.
  • Article
    We propose a new method to predict time series using the technique of Independent Component Analysis (ICA) as a preprocessing tool. If certain assumptions hold, we show that ICA can be used to transform a set of time series into another set that is easier to predict. These assumptions are not unrealistic for many real-world time series, including financial time series. We have tested this approach on two sets of data: artificial toy data and financial time series. Simulations with a set of foreign exchange rate time series suggest that these can be predicted more accurately using the ICA preprocessing.
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    Squared-residual autocorrelations have been found useful in detecting non-linear types of statistical dependence in the residuals of fitted autoregressive-moving average (ARMA) models [cf. C. W. J. Granger and A. P. Andersen, An introduction to bilinear time series models. (1978; Zbl 0379.62074)]. In this note it is shown that the normalized squared-residual autocorrelations are asymptotically unit multivariate normal. The results of a simulation experiment confirming the small- sample validity of the proposed tests is reported.
  • Book
    A comprehensive introduction to ICA for students and practitionersIndependent Component Analysis (ICA) is one of the most exciting new topics in fields such as neural networks, advanced statistics, and signal processing. This is the first book to provide a comprehensive introduction to this new technique complete with the fundamental mathematical background needed to understand and utilize it. It offers a general overview of the basics of ICA, important solutions and algorithms, and in-depth coverage of new applications in image processing, telecommunications, audio signal processing, and more.Independent Component Analysis is divided into four sections that cover:* General mathematical concepts utilized in the book* The basic ICA model and its solution* Various extensions of the basic ICA model* Real-world applications for ICA modelsAuthors Hyvarinen, Karhunen, and Oja are well known for their contributions to the development of ICA and here cover all the relevant theory, new algorithms, and applications in various fields. Researchers, students, and practitioners from a variety of disciplines will find this accessible volume both helpful and informative.
  • Book
    This book provides an account of weak convergence theory and empirical processes and their applications to a wide variety of applications in statistics. The first part of the book presents a thorough account of stocastic convergence in its various forms. Part 2 brings together the theory of empirical processes in a form accessible to statisticians and probabilists. In Part 3, the authors cover a range of topics which demonstrate the applicability of the theory to important questions such as: limit theorems in asymptotic statistics; measures of goodness of fit; the bootstrap; and semiparametric estimation. Most of the sections conclude with "problems and complements". Some of these are exercises to help the reader's understanding of the material whereas others are intended to supplement the text.
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    In the present paper we examine the strict stationarity and the existence of higher-order moments for the GARCH(p,q) model under general and tractable assumptions.
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    Volatility plays an important role in controlling and forecasting risks in various �nancial operations. For a univariate return series, volatility is often represented in terms of conditional variances or conditional standard deviations. Many statistical models have been developed for modelling univariate conditional variance processes. While univariate descriptions are useful and important, problems of risk assessment, asset allocation, hedging in futures markets and options pricing require a multivariate framework, since high volatilities are often observed in the same time periods across di�erent assets. Statistically this boils down to model time-varying conditional variance and covariance matrices of a vector-valued time series. Section 2 below lists some existing statistical models for multivariate volatility processes. We refer to Bauwens, Laurent and Rombouts (2005) for a more detailed survey on this topic. We propose a new and ad hoc method with numerical illustration in section 3. We concludes in section 4 with a brief summary.
  • Article
    A new representation of the diagonal Vech model is given using the Hadamard product. Sufficient conditions on parameter matrices are provided to ensure the positive definiteness of covariance matrices from the new representation. Based on this, some new and simple models are discussed. A set of diagnostic tests for multivariate ARCH models is proposed. The tests are able to detect various model misspecifications by examing the orthogonality of the squared normalized residuals. A small Monte-Carlo study is carried out to check the small sample performance of the test. An empirical example is also given as guidance for model estimation and selection in the multivariate framework. For the specific data set considered, it is found that the simple one and two parameter models and the constant conditional correlation model perform fairly well.
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    In this paper we consider several tests for model misspecification after a multivariate conditional heteroscedasticity model has been fitted. We examine the performance of the recent test due to Ling and Li (J. Time Ser. Anal. 18 (1997), 447–64), the Box–Pierce test and the residual-based F test using Monte Carlo methods. We find that there are situations in which the Ling–Li test has very weak power. The residual-based diagnostics demonstrate significant under-rejection under the null. In contrast, the Box–Pierce test based on the cross-products of the standardized residuals often provides a useful diagnostic that has reliable empirical size as well as good power against the alternatives considered.
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    This paper gives sufficient conditions for the weak convergence to Gaussian processes of empirical processes andU-processes from stationary kp/(p - 2) (logk)2(p - 1)/(p - 2) bk ® 0 as k ® ¥k^{p/(p - 2)} (\log k)^{2(p - 1)/(p - 2)} \beta _k \to 0 as k \to \infty In the case that the functions in theV-C subgraph class are uniformly bounded, we obtain uniform central limit theorems for the empirical process and theU-process, provided that the decay rate of the mixing coefficient satisfies k =O(k –r ) for somer>1. These conditions are almost minimal.
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    Bahadur-Kiefer approximations for generalized quantile processes as defined in Einmahl and Mason (1992) are given which generalize results for the classical one-dimensional quantile processes. An as application we consider the special case of the volume process of minimum volume sets in classes of subsets of the d-dimensional Euclidean space. Minimum volume sets can be used as estimators of level sets of a density and might be useful in cluster analysis. The volume of minimum volume sets itself can be used for robust estimation of scale. Consistency results and rates of convergence for minimum volume sets are given. Rates of convergence of minimum volume sets can be used to obtain Bahadur-Kiefer approximations for the corresponding volume process and vice versa. A generalization of the minimum volume approach to non-i.i.d. problems like regression and spectral analysis of time series is discussed.
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    We adapt the Lagrange multiplier (LM) principle to test for noncausality in variance of financial returns. The new test is compared with a Portmanteau statistic [Cheung, Y.W., Ng, L.K., 1996. A causality in variance test and its application to financial market prices. Journal of Econometrics 72, 33–48.]. A Monte Carlo study reveals superior power of the LM test.
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    We propose a new model for the variance between multiple time series, the regime switching dynamic correlation. We decompose the covariances into correlations and standard deviations and the correlation matrix follows a regime switching model; it is constant within a regime but different across regimes. The transitions between the regimes are governed by a Markov chain. This model does not suffer from a curse of dimensionality and it allows analytic computation of multi-step ahead conditional expectations of the variance matrix when combined with the ARMACH model (Taylor (Modelling Financial Time Series. Wiley, New York) and Schwert (J. Finance 44(5) (1989) 1115)) for the standard deviations. We also present an empirical application which illustrates that our model can have a better fit of the data than the dynamic conditional correlation model proposed by Engle (J. Business Econ. Statist. 20(3) (2002) 339).
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    This paper develops a test for causality in variance. The test is based on the residual cross-correlation function (CCF) and is robust to distributional assumptions. Asymptotic normal and asymptotic χ2 statistics are derived under the null hypothesis of no causality in variance. Monte Carlo results indicate that the proposed CCF test has good empirical size and power properties. Two empirical examples illustrate that the causality test yields useful information on the temporal dynamics and the interaction between two time series.
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    This paper extends the work by Ding, Granger, and Engle (1993) and further examines the long memory property for various speculative returns. The long memory property found for S&P 500 returns is also found to exist for four other different speculative returns. One significant difference is that for foreign exchange rate returns, this property is strongest when instead of at d = 1 for stock returns. The theoretical autocorrelation functions for various GARCH(1, 1) models are also derived and found to be exponential decreasing, which is rather different from the sample autocorrelation function for the real data. A general class of long memory models that has no memory in returns themselves but long memory in absolute returns and their power transformations is proposed. The issue of estimation and simulation for this class of model is discussed. The Monte Carlo simulation shows that the theoretical model can mimic the stylized empirical facts strikingly well.
  • Article
    This paper discusses the application of a modern signal processing technique known as independentcomponent analysis (ICA) or blind source separation to multivariate financial time series such as aportfolio of stocks. The key idea of ICA is to linearly map the observed multivariate time series into a newspace of statistically independent components (ICs). This can be viewed as a factorization of the portfoliosince joint probabilities become simple products in the coordinate system of the ICs.We apply ICA to three years of daily returns of the 28 largest Japanese stocks and compare the results withthose obtained using principal component analysis. The results indicate that the estimated ICs fall into twocategories, (i) infrequent but large shocks (responsible for the major changes in the stock prices), and (ii)frequent smaller fluctuations (contributing little to the overall level of the stocks). We show that the overallstock price can be reconstructed surprisingly well by using a small number of thresholded weighted ICs.In contrast, when using shocks derived from principal components instead of independent components, thereconstructed price is less similar to the original one. Independent component analysis is a potentially powerfulmethod of analyzing and understanding driving mechanisms in financial markets. There are furtherpromising applications to risk management since ICA focuses on higher-order statistics.
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    We present a dynamic stochastic general equilibrium (DSGE) New Keynesian model with indivisible labor and a dual labor market: a Walrasian one where wages are fully flexible and a unionized one characterized by real wage rigidity. We show that the negative effect of a productivity shock on inflation and the positive effect of a cost-push shock are crucially determined by the proportion of firms that belong to the unionized sector. The larger this number, the larger are these effects. Consequently, the larger the union coverage, the larger should be the optimal response of the nominal interest rate to exogenous productivity and cost-push shocks. The optimal inflation and output gap volatility increases as the number of the unionized firms in the economy increases.
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    In this paper, we develop the theoretical and empirical properties of a new class of multivariate GARCH models capable of estimating large time-varying covariance matrices, Dynamic Conditional Correlation Multivariate GARCH. We show that the problem of multivariate conditional variance estimation can be simplified by estimating univariate GARCH models for each asset, and then, using transformed residuals resulting from the first stage, estimating a conditional correlation estimator. The standard errors for the first stage parameters remain consistent, and only the standard errors for the correlation parameters need be modified. We use the model to estimate the conditional covariance of up to 100 assets using S&P 500 Sector Indices and Dow Jones Industrial Average stocks, and conduct specification tests of the estimator using an industry standard benchmark for volatility models. This new estimator demonstrates very strong performance especially considering ease of implementation of the estimator.
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    In this paper we introduce a bootstrap procedure to test parameter restrictions in vector autoregressive models which is robust in cases of conditionally heteroskedastic error terms. The adopted wild bootstrap method does not require any parametric specification of the volatility process and takes contemporaneous error correlation implicitly into account. Via a Monte Carlo investigation empirical size and power properties of the new method are illustrated. We compare the bootstrap approach with standard procedures either ignoring heteroskedasticity or adopting a heteroskedasticity consistent estimation of the relevant covariance matrices in the spirit of the White correction. In terms of empirical size the proposed method clearly outperforms competing approaches without paying any price in terms of size adjusted power. We apply the alternative tests to investigate the potential of causal relationships linking daily prices of natural gas and crude oil. Unlike standard inference ignoring time varying error variances, heteroskedasticity consistent test procedures do not deliver any evidence in favor of short run causality between the two series.
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    This paper constructs a two-country (Home and Foreign) general equilibrium model of Schumpeterian growth without scale effects. The scale effects property is removed by introducing two distinct specifications in the knowledge production function: the permanent effect on growth (PEG) specification, which allows policy effects on long-run growth; and the temporary effects on growth (TEG) specification, which generates semi-endogenous long-run economic growth. In the present model, the direction of the effect of the size of innovations on the pattern of trade and Home’s relative wage depends on the way in which the scale effects property is removed. Under the PEG specification, changes in the size of innovations increase Home’s comparative advantage and its relative wage, while under the TEG specification, an increase in the size of innovations increases Home’s relative wage but with an ambiguous effect on its comparative advantage.
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    Classical empirical process theory for Vapnik-Cervonenkis classes deals mainly with sequences of independent variables. This paper extends the theory to stationary sequences of dependent variables. It establishes rates of convergence for $\beta$-mixing and $\phi$-mixing empirical processes indexed by classes of functions. The method of proof depends on a coupling of the dependent sequence with sequences of independent blocks, to which the classical theory can be applied. A uniform $O(n^{-s/(1+s)})$ rate of convergence over V-C classes is established for sequences whose mixing coefficients decay slightly faster than $O(n^{-s})$.
  • Article
    In this paper two bootstrap procedures are considered for the estimation of the distribution of linear contrasts and of F-test statistics in high dimensional linear models. An asymptotic approach will be chosen where the dimension p of the model may increase for sample size $n\rightarrow\infty$. The range of validity will be compared for the normal approximation and for the bootstrap procedures. Furthermore, it will be argued that the rates of convergence are different for the bootstrap procedures in this asymptotic framework. This is in contrast to the usual asymptotic approach where p is fixed.
  • Article
    The capital asset pricing model provides a theoretical structure for the pricing of assets with uncertain returns. The premium to induc e risk-averse investors to bear risk is proportional to the nondivers ifiable risk, which is measured by the covariance of the asset return with the market portfolio return. In this paper, a multivariate, gen eralized-autoregressive, conditional, heteroscedastic process is esti mated for returns to bills, bonds, and stocks where the expected retu rn is proportional to the conditional covariance of each return with that of a fully diversified or market portfolio. It is found that the conditional covariances are quite variable over time and are a signi ficant determinant of the time-varying risk premia. The implied betas are also time varying and forecastable. Copyright 1988 by University of Chicago Press.
  • Article
    A multivariate time series model with time varying conditional variances and covariances, but constant conditional correlations is proposed. In a multivariate regression framework, the model is readily interpreted as an extension of the Seemingly Unrelated Regression (SUR) model allowing for heteroskedasticity. Parameterizing each of the conditional variances as a univariate Generalized Autoregressive Conditional Heteroskedastic (GARCH) process, the descriptive validity of the model is illustrated for a set of five nominal European U.S. dollar exchange rates following the inception of the European Monetary System (EMS). When compared to the pre- EMS free float period, the comovements between the currenciess are found to be significantly higher over the later period. Copyright 1990 by MIT Press.
  • Article
    This paper surveys the most important developments in multivariate ARCH-type modelling. It reviews the model speciÞcations, the inference methods, and the main areas of application of these models in Þnancial econometrics.
  • Article
    This paper studies a broad class of nonnegative ARCH( ) models. Sufficient conditions for the existence of a stationary solution are established and an explicit representation of the solution as a Volterra type series is found. Under our assumptions, the covariance function can decay slowly like a power function, falling just short of the long memory structure. A moving average representation in martingale differences is established, and the central limit theorem is proved.
  • Article
    Hall & Yao (2003) showed that, for ARCH/GARCH, i.e. autoregressive conditional heteroscedastic/generalised autoregressive conditional heteroscedastic, models with heavy-tailed errors, the conventional maximum quasilikelihood estimator suffers from complex limit distributions and slow convergence rates. In this paper three types of absolute deviations estimator have been examined, and the one based on logarithmic transformation turns out to be particularly appealing. We have shown that this estimator is asymptotically normal and unbiased. Furthermore it enjoys the standard convergence rate of n-super-1/2 regardless of whether the errors are heavy-tailed or not. Simulation lends further support to our theoretical results. Copyright Biometrika Trust 2003, Oxford University Press.
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    Full-text available
    Matching university places to students is not as clear cut or as straightforward as it ought to be. By investigating the matching algorithm used by the German central clearinghouse for university admissions in medicine and related subjects, we show that a procedure designed to give an advantage to students with excellent school grades actually harms them. The reason is that the three-step process employed by the clearinghouse is a complicated mechanism in which many students fail to grasp the strategic aspects involved. The mechanism is based on quotas and consists of three procedures that are administered sequentially, one for each quota. Using the complete data set of the central clearinghouse, we show that the matching can be improved for around 20% of the excellent students while making a relatively small percentage of all other students worse off.
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    Full-text available
    The second alternative has been proposed by Andersen et al. (2003). In this case, a daily measure of variances and covariances is computed as an aggregate measure from intraday returns. More specifically, a daily realized variance for day t is computed as the sum of the squared intraday equidistant returns for the given trading day and a daily realized covariance is obtained by summing the products of intraday returns. Once such daily measures have been obtained, they can be modelled, e.g. for a prediction purpose. A nice feature of this approach is that unlike MGARCH and multivariate stochastic volatility models, the N(N − 1)/2 covariance components of the conditional variance matrix (or, rather, the components of its Choleski decomposition) can be forecasted independently, using as many univariate models. As shown by Andersen et al. (2003), although the use of the realized covariance matrix facilitates rigorous measurement of conditional volatility in much higher dimensions than is feasible with MGARCH and multivariate SV models, it does not allow the dimensionality to become arbitrarily large. Indeed, to ensure the positive definiteness of the realized covariance matrix, the number of assets (N) cannot exceed the number of intraday returns for each trading day. The main drawback is that intraday data remain relatively costly and are not readily available for all assets. Furthermore, a large amount of data handling and computer programming is usually needed to retrieve the intraday returns from the raw data files supplied by the exchanges or data vendors. On the contrary, working with daily data is relatively simple and the data are broadly available.
  • Article
    Multivariate GARCH specifications are typically determined by means of practical considerations such as the ease of estimation, which often results in a serious loss of generality. A new type of multivariate GARCH model is proposed, in which potentially large covariance matrices can be parameterized with a fairly large degree of freedom while estimation of the parameters remains feasible. The model can be seen as a natural generalization of the O-GARCH model, while it is nested in the more general BEKK model. In order to avoid convergence difficulties of estimation algorithms, we propose to exploit unconditional information first, so that the number of parameters that need to be estimated by means of conditional information is more than halved. Both artificial and empirical examples are included to illustrate the model. Copyright © 2002 John Wiley & Sons, Ltd.