Statistical Modeling Using Local Gaussian Approximation

By Dag Tjøstheim, Håkon Otneim and Bård Støve

Statistical Modeling using Local Gaussian Approximation extends powerful characteristics of the Gaussian distribution, perhaps, the most well-known and most used distribution in statistics, to a large class of non-Gaussian and nonlinear situations through local approximation. This extension enables the reader to follow new methods in assessing dependence and conditional dependence, in estimating probability and spectral density functions, and in discrimination. Chapters in this release cover Parametric, nonparametric, locally parametric, Dependence, Local Gaussian correlation and dependence, Local Gaussian correlation and the copula, Applications in finance, and more.

Additional chapters explores Measuring dependence and testing for independence, Time series dependence and spectral analysis, Multivariate density estimation, Conditional density estimation, The local Gaussian partial correlation, Regression and conditional regression quantiles, and a A local Gaussian Fisher discriminant.

• Reviews local dependence modeling with applications to time series and finance markets
• Introduces new techniques for density estimation, conditional density estimation, and tests of conditional independence with applications in economics
• Evaluates local spectral analysis, discovering hidden frequencies in extremes and hidden phase differences
• Integrates textual content with three useful R packages

• Language: English
• Publisher: Academic Press
• Release date: Oct 5, 2021
• ISBN: 9780128154458

Dag Tjøstheim

Dag Tjøstheim is Emeritus Professor, Department of Mathematics, University of Bergen. He has a PhD in applied mathematics from Princeton University (1974). He has authored more than 120 papers in international journals. He is a member of the Norwegian Academy of Sciences and has received several prizes for his scientific work. His main interests are in econometrics, nonlinear time series, nonparametric methods, modeling of dependence, spatial variables, and fishery statistics.

Book preview

Preface

Dag Tjøstheim     

Håkon Otneim     

Bård Støve     

The central idea of this book is the approximation of a general multivariate density f by a family of Gaussian distributions. Locally around a point x in the support of f, f is approximated by a multivariate Gaussian distribution. This makes it possible to define a local mean, a local variance, and a local correlation matrix.

This idea is powerful and can be applied to a number of tasks and problems for continuous stochastic variables. This has been done in several recent papers. These papers, thirteen altogether, form the basis of this book. In particular, following two introductory chapters, each of the main Chapters 3–11 and 13 is composed from one or more of these papers.

In Chapter 4 the emphasis is on the local correlation, its properties, and its use as a measure of dependence. It can be defined on the original x-scale, but also on a normalized z-scale obtained by transforming the marginals of f to the standard normal. This is in a way analogous to the transformation to uniform variables in a copula construction, and the relationship between the copula concept and the Gaussian approximation concept is explored in Chapter 5.

It has long been realized that a multivariate Gaussian distribution fitted to data in finance or econometrics may not be a good idea. In fact, data in finance and econometrics have thick tails, and a global Gaussian fit may lead to disastrous results with a large underestimation of economic risk. In Chapter 6, we apply local Gaussian approximation to financial data, including financial contagion and a preliminary attempt of portfolio construction.

With the establishment of local Gaussian correlation as a measure of local dependence, an obvious next step is employing this measure to testing of independence. This is done in Chapter 7 in three stages: testing of independence between two sequences, each consisting of independent identically distributed variables, testing of serial independence in a time series, and testing of independence between two stationary time series. This implies the introduction of a local autocorrelation and cross-correlation concept.

The locally Gaussian autocorrelation introduced in Chapter 7 is used in Chapter 8 to construct a locally Gaussian spectral density. It coincides with the ordinary power spectral density in the Gaussian time series case. For non-Gaussian data, the new local spectral concept can be used to pick up spectral peaks that may be hidden in a conventional spectral estimation.

Chapters 9 and 10 are devoted to estimation of multivariate density functions and to estimation of multivariate conditional densities. This is done by merging the Gaussian densities of the local approximating families. In the conditional case the unique properties of the conditional (global) Gaussian distribution is of crucial importance. Further, the curse of dimensionality is sought circumvented by a simplified Gaussian approximation, in a sense similar to the use of the additive simplification in nonparametric regression.

The local Gaussian approximation also makes it possible to introduce a local Gaussian partial correlation. In Chapter 11, it is shown how this can be used to construct a local measure of conditional dependence and to test for conditional independence. We believe that this idea has potential for network theory and causality.

Perhaps the most important use of nonparametric estimation methods is currently in nonparametric regression. The local Gaussian approximation concept is developed for multivariate statistical analysis, where all of the variables are, so to speak, on the same basis, in contradistinction to regression analysis, where one variable or a group of variables are dependent variables expressed as a function of another group of explanatory variables. Nevertheless, in Chapter 12, we make an attempt to apply local Gaussian approximation techniques to regression estimation and quantile regression estimation. More work is required to determine in what way local methods can complement nonparametric techniques like, for instance, additive modeling.

The local Gaussian approach can be applied to other fields of statistics as well. As an example of such an application, in Chapter 13, we look at applications to classification and discrimination, involving among other things a local Fisher discriminant.

To put local Gaussian approximation analysis into context with other methods, the book also contains three introductory chapters. Chapter 1 contains a general introduction explaining the overall features and concepts of our approach. Chapter 2 gives a brief but at the same time quite broad overview over parametric, nonparametric, and locally parametric approaches to statistics. Chapter 3, based on a very recent survey paper to appear in Statistical Science, contains an overview of the statistical concept of dependence, how it can be measured, and how we can test for independence. The survey concentrates on methods developed in the last two decades, going beyond the most used measure of dependence, the Pearson correlation. The local Gaussian correlation is put directly into this context in Chapter 4.

There is some overlap between the various chapters in the book. This has been done intentionally, so that a reader can single out the chapters of primary interest to her/him. Most chapters can be read independently of each other as the basic material from Chapter 4 is included briefly as an introductory material in each of the following chapters. The mathematical and technical level of each chapter is quite modest. For readers with more interest in technical details, we give references, often to supplementary material to the papers that the book is composed from.

There are three R-packages that have been developed for various types of analysis in the book. We do not present details of use of these packages in this book, but references to the packages are given in Chapter 1.

The local Gaussian approach is a recently developed methodology. Some of the chapters are based on papers that have just appeared or are in the process of appearing in journals. Putting this in a book, the emphasis is on presenting the fundamental concepts inherent in a local Gaussian approximation and in demonstrating their usefulness in several areas in statistics. At the same time, we hope that the book may serve as a starting point and inspiration for further research and applications in the subject matters taken up in each of the chapters of the book, as well as in new subject areas.

The chapters of the book have been primarily based on papers by the three authors of the book, but some chapters have also benefited from joint work and joint papers with others, namely Karl Ove Hufthammer (Chapters 4 and 6), Geir Berentsen (Chapters 5 and 7), Viginia Lacal (Chapter 7), Lars Arne Jordanger (Chapter 8), Martin Jullum (Chapter 13), and Anders Sleire (parts of Chapter 6). Without their contributions the book had not been possible in its present form, and we are very grateful to them for their good work and cooperation on these subjects.

Bergen, May 2021

Chapter 1: Introduction

Abstract

This book introduces the local Gaussian approximation as a modeling philosophy for continuous stochastic variables. In the introductory chapter, this approach to modeling is explained and motivated, the main idea being that the unique properties of the global Gaussian distribution are extended to a much more general modeling situation by using the unique local properties of the Gaussian for given density functions. In the introduction, we explain that this procedure has advantages in different situations such as measuring local dependence, describing copula dependence, measuring conditional dependence, testing for independence, serial independence and conditional independence, performing nonlinear and local spectral analysis, estimating multivariate and conditional density functions, undertaking local Gaussian regression estimation and local Gaussian discrimination.

Keywords

Local Gaussian approximation; measuring local dependence; describing copula dependence; measuring conditional dependence; testing for independence; serial independence and conditional independence; estimating multivariate and conditional density functions; nonlinear and local spectral estimation; local Gaussian regression estimation and local Gaussian discrimination; R

Chapter Outline

1.1  Computer code

References

The most important distribution in statistics is the Gaussian distribution. It has a number of very useful and special properties, particularly in the multivariate case. Just think about a normally distributed vector of dimension p, where T denotes transposed. Its distribution is given by the density function

where and , , are the mean vector and covariance matrix of X, respectively. Looking at this familiar expression, it is easy to forget its simplicity and elegance. Here we have a distribution whose location is completely determined by its means , the scale by the variances , and whose dependence relations have the amazing property that they are completely determined by the pairwise covariances .

Moreover, if X is subdivided into two components , then any linear combination of and is again Gaussian, and the conditional distribution is Gaussian. The dependence properties of these derived distributions are again determined by the pairwise covariances, in the latter case, through the partial covariances. These properties make the Gaussian especially suitable for linear statistical modeling. Further, the properties of the conditional distribution imply that in a Gaussian system the optimal least squares predictor, given by the conditional mean, is linear and equals the optimal linear predictor. Finally, uncorrelatedness is equivalent to independence in the Gaussian distribution, that is, and are independent if and only if they are uncorrelated. In this case, we can test for independence by computing covariances.

Unfortunately, data are not always well described by a Gaussian distribution and a linear model. In particular, for data in economics and finance the data are usually governed by distributions having thicker tails, and the dependence properties are not well described by pairwise covariances only, as is inherent in the Gaussian distribution. In fact, assuming a linear Gaussian model can lead to disastrous results with drastic underestimation of the risk involved in economic and financial transactions; see, for example, Taleb (2007).

To some degree, these problems can be avoided, or at least lessened, by trying to fit other parametric families to the data or by using a semiparametric or nonparametric approach. An important concept in a nonparametric methodology is the concept of a local approximation. In nonparametric density estimation, we may take as the starting point a locally smoothed version of a histogram of the available data. In a regression, the regression relationship may be approximated by locally fitted polynomials (the particular case of the locally constant case being the regression kernel estimator). What is local is determined by a bandwidth parameter, which for a given point x, selects the neighboring points close to x, close being determined by the bandwidth acting as a distance measure. A density, a conditional density, or a regression can be estimated nonparametrically in this manner. As the dimension p increases, the curse of dimensionality emerges, and simplifying assumptions, such as the additive model for regression, have to be introduced.

We are now ready to formulate the main idea of this book and of the papers it consists of. The idea is simply to approximate an arbitrary p-dimensional density function f locally by a family of Gaussians distributions. This can be viewed as an example of a semiparametric or a locally parametric approach. In principle, another family of parametric distributions could be used as a local approximant (as has been done by Hjort and Jones (1996), who considered the locally parametric density estimator), but we believe that the well-known simple and elegant properties of Gaussians makes this family of distributions the optimal choice. So, for a point x, in a neighborhood of x, we fit a Gaussian distribution, the neighborhood being determined by a bandwidth parameter. The parameters of this Gaussian distribution will be functions of the coordinates of the point x. Moving to another point y and fitting another Gaussian in the neighborhood of y will in general result in another set of parameters depending on y. The exception is when f itself is Gaussian. In that case, as the number of available observations tends to infinity at the same time as the bandwidth tends to zero, the estimated parameters at x and y will ultimately coincide and be equal to the parameters of the Gaussian f. The advantage of using the Gaussian distribution as an approximating family is that the unique properties of the Gaussian can be locally used for a general, possibly non-Gaussian, density f. For instance, for a thick-tailed distribution, it can be locally approximated in the tail by a Gaussian with large variance. In the multivariate case, we have the potential of approximating multivariate tail behavior locally by an appropriate multivariate Gaussian. This turns out to be useful for multivariate financial market data.

Using this idea as a statistical modeling philosophy has many ramifications and applications as we try to illustrate throughout the book. For example, we can define local covariances and correlations, and even local partial covariances. Local dependence and conditional dependence can be measured by these quantities, and local independence and conditional independence can be tested. Dependence properties may be analyzed by aggregation, and independence may be tested over larger regions, ultimately over the entire range of the data. Further, by using the same principle in approximating locally the joint distribution of time series variables, we can introduce concepts of local autocorrelation and cross-correlation. A local spectral density can be constructed, which makes it possible to derive one local spectral density describing the oscillatory properties at one level (e.g., close to extremes) and another local one describing the frequency distribution close to the center of the data.

Seen from this perspective, the unique properties of the Gaussian can be utilized in a non-Gaussian environment but, again, locally. In this book, we discuss many such examples, but there are other avenues that have not been explored so far. Local principal component analysis is one of them. The ordinary principal components are found by solving an eigenvalue problem involving the covariance matrix. Local principal components can be found by replacing the ordinary global covariance matrix by a matrix of local covariances. Another area where research has been initiated, is multiple spectral analysis, where a local amplitude spectrum and phase spectrum can be introduced. There are other possibilities as well, and we believe local Gaussian approximation to be a very comprehensive tool.

Another potential extension is to broaden the Gaussian family to a more general family like the family of elliptic distributions. Then we lose some of the simple properties of the Gaussian (e.g., independence is not equivalent to uncorrelatedness), but, on the other hand, a multivariate t-distribution, belonging to the elliptic family, is much easier to approximate. Other families are considered by Hjort and Jones (1996), but just in the situation of deriving an alternative density estimator to the kernel estimator.

Local quantities like the local correlation is sensitive to the curse of dimensionality as the dimension of X increases. Throughout the book, we discuss ways of bypassing it. Somewhat similarly to the additive approximation in regression analysis, we try to approximate the elements of the covariance matrix by a function of two coordinates . Still we have to be careful at the edges of the data set where there are few observations. Another device that has been much used in this book is transforming the data to a standard normal marginal scale by using the marginal empirical distribution function. A very different approach, which may deserve a closer examination, is trying to fit a parametric model to the local quantities.

Here is a brief overview of the contents of the book:

To put our method into perspective, we summarize briefly traditional parametric, semiparametric, nonparametric, and locally parametric modeling in Chapter 2. Very briefly, properties of the Gaussian and elliptic distribution are also included.

Local correlation represents one way of measuring dependence as a local version of the traditional Pearson correlation. Chapter 3 presents a fairly self-contained review of recent developments in nonlinear dependence analysis, among them, the Brownian distance covariance and reproducing kernel Hilbert space measures, both having received considerable attention lately.

In Chapter 4, we contrast the dependence measures of Chapter 3 with the local Gaussian correlation (LGC). We define this concept as well as the concept of local Gaussian approximation. Two versions are described, one on the original x-scale and one on the z-scale obtained by transforming the marginals to standard normals. We give a number of properties and illustrate these on simulated and financial real data.

The connection to description of dependence by means of copulas is explored in Chapter 5. We compute the local correlation for traditional copulas like the Clayton, Gumbel, and Frank copulas.

Much of what we do is motivated by problems met in the description of financial markets. Chapter 6 contains a number of applications to financial and econometric data. It is shown that key typical dependence properties of such markets are well described by the local Gaussian correlation, and we also look at applications to financial contagion, portfolio analysis, and value at risk.

In Gaussian distributions, independence and uncorrelatedness are equivalent, so it is natural to use the local Gaussian correlation to test for local and nonlinear independence. In Chapter 7, we extend this to tests of serial dependence in a univariate time series and to independence testing between two time series. We compare to other tests like the Brownian distance covariance for both simulated and real data.

In Chapter 8 the time series frame is kept, but here we focus on the local autocorrelation and the local spectrum that can be derived from it. It is shown that frequency behavior that cannot be detected by ordinary spectral analysis can be detected by the local spectrum. The chapter also contains a brief review of alternative nonlinear spectral techniques.

Chapters 9 and 10 are devoted to density estimation and conditional density estimation, respectively. The density estimation is the aspect stressed by Hjort and Jones (1996) in their local parametric analysis. We carry this through for the local Gaussian approximation of a density and compare with other methods as the dimension increases. In the conditional density estimation, we exploit locally the fact that the conditional density in a joint Gaussian density framework is again a Gaussian density, where the local mean vector and covariance matrix can be found by explicit formulas.

In a sense testing for conditional independence is more important than testing for independence. This is due to the applications to causality analysis among other things. For globally Gaussian data, the partial correlation coefficient is an important tool, for example, in path analysis. In Chapter 11, we introduce the local partial correlation and use it both for measuring conditional dependence and for testing of conditional independence. We compare with alternative tests and give applications to testing Granger causality.

Regression and conditional quantile estimation is covered in Chapter 12. We note that the local Gaussian approach is primarily suited to a situation where all the variables are treated on the same basis. It is perhaps less well suited to a situation where there is one dependent variable and one or several explanatory variables. Nevertheless, we show in this chapter that the local Gaussian approximation can be applied and that in particular cases it may offer an alternative to the additive approximation in regression models.

The traditional Fisher discriminant for discriminating between two or more populations is based on a Gaussian assumption. In Chapter 13, we make the parameters of the Gaussian local and derive a local Gaussian Fisher discriminant, which is applied to simulated and real data. It is easy to find examples where the global Fisher discriminant does not work, whereas the local one does.

1.1 Computer code

The package lg, see Otneim (2021), for the R programming language (see R Core Team, 2017) provides implementations of most of the methodological advances on applications of the local Gaussian approximation presented in this book. This includes estimation of the local Gaussian correlation itself, multivariate density estimation, conditional density estimation, various tests for independence, conditional independence and financial contagion (cf. Chapter 6), and a graphical module for creating dependence maps; see Otneim (2019). Note that the use of local Gaussian correlation in spectral analysis of time series, presented in Chapter 8, has its own computational ecosystem in the localgaussSpec-package¹ for R. The R package localgauss (see Berentsen et al., 2014) provided the first publicly available implementation of the LGC and a test for independence. Note that the lg-package depends on the localgauss-package.

We refer to the R documentation of the mentioned packages and Otneim (2021) for the direct use of various available functions, as this will not be covered in the book.

References

Berentsen et al., 2014 G.D. Berentsen, T. Kleppe, D. Tjøstheim, Introducing localgauss, an R package for estimating and visualizing local Gaussian correlation, Journal of Statistical Software 2014;56(12):1–18.

Hjort and Jones, 1996 N. Hjort, M. Jones, Locally parametric nonparametric density estimation, Annals of Statistics 1996;24(4):1619–1647.

Otneim, 2019 H. Otneim, lg: Locally Gaussian distributions: estimation and methods, https://CRAN.R-project.org/package=lg; 2019 R package version 0.4.1.

Otneim, 2021 H. Otneim, Ig: an R package for local Gaussian approximations, To appear The R Journal 2021. URL: https://journal.r-project.org/archive/2021/RJ-2021-079/index.html.

R Core Team, 2017 R Core Team, R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing; 2017.

Taleb, 2007 N.N. Taleb, The Black Swan: The Impact of the Highly Improbable. Random House; 2007.

---

¹  See https://github.com/LAJordanger/localgaussSpec for details.

Chapter 2: Parametric, nonparametric, locally parametric

Abstract

In this chapter, we give a very brief but up-to-date review of parametric, nonparametric, and locally parametric techniques that forms the background for the main topics of the book. Among topics discussed, there are distributional aspects such as Gaussian and elliptic distributions. Next, parametric regression models, linear, and nonlinear are covered. A compressed version of time series models, including ARMA, GARCH, and nonlinear models, is included. The last part of the chapter deals with nonparametric methods in density estimation, regression estimation, bandwidth choice, and the curse of dimensionality. Additive models, quantile regression, and semiparametric models are also mentioned. Finally, locally parametric models are introduced as leading up to local Gaussian approximations, which are the main topic of the book.

Keywords

Gaussian distributions; elliptic distributions; parametric regression; time series models; nonparametric estimation; semiparametric estimation; curse of dimensionality; bandwidth; locally parametric models

Chapter Outline

2.1  Introduction

2.2  Parametric density models

2.2.1  The Gaussian distribution

2.2.2  The elliptical distribution

2.2.3  The exponential family

2.3  Parametric regression models

2.3.1  Linear regression

2.3.2  Nonlinear regression and some further modeling aspects

2.4  Time series

2.5  Nonparametric density estimation

2.5.1  Nonparametric kernel density estimation

2.5.2  Bandwidth selection

2.5.3  Multivariate and conditional density estimation

2.6  Nonparametric regression estimation

2.6.1  Kernel regression estimation

2.6.2  Local polynomial estimation

2.6.3  Choice of bandwidth in regression

2.7  Fighting the curse of dimensionality

2.7.1  Additive models

2.7.2  Regression trees, splines, and MARS

2.8  Quantile regression

2.9  Semiparametric models

2.9.1  Partially linear models

2.9.2  Index models and projection pursuit

2.10  Locally parametric

References

2.1 Introduction

In statistical modeling, we have to choose between a parametric model and the use of nonparametric statistics. A compromise is a semiparametric model, where both aspects of modeling are taken into consideration.

For a parametric model, the mathematical form of the model and relationships between stochastic variables entering the model and their distributions are explicitly stated and generally assumed to be known except for a set of parameters. The parameters may, for instance, appear as coefficients in a linear regression or as parameters of a distribution function from a certain class. Strictly speaking, a parametric model is never true, or in the words of Box and Draper (1987, p. 424), All models are wrong, but some are useful. A parametric model is often quite simple and has a straightforward interpretation. In certain situations, however, parametric models may lead astray. When the model is seriously wrong, the accuracy of parameter estimates does not help. One well-known example is the estimation of value-at-risk for financial markets. Sometimes, financial crises have been blamed on using Gaussian models in the tail of a distribution when financial objects quite clearly have thicker tails. Using the Gaussian distribution may lead to a disastrous underestimation of the risk; see Taleb (2007). In such situations a parametric model works as an extremely inconvenient strait jacket.

The purpose of a nonparametric approach is letting the data speak for themselves and thereby preventing such situations from occurring. However, this approach also has its disadvantages. First, the convergence rate of nonparametric estimates is slower than the parametric rate. But perhaps the most important obstacle is the curse of dimensionality: when we have a moderate or large number of variables, the nonparametric approach does not work in practice. It may still be possible to state theorems of convergence rates for nonparametric estimates, but these are so slow that we need astronomically large sample sizes to come close to the true values, which we typically do not have. There are various ways of trying to get around the curse. This can be done by assuming further restrictive assumptions such as an additive model in a regression context. We will come back to this on several occasions later in this chapter and in later chapters of the book.

Another way of tackling the curse of dimensionality is using a semiparametric model, that is, a model where some parts of the model are treated parametrically and other parts of the model are treated nonparametrically. The implicit understanding is that the nonparametric part is specified in such a way that we avoid the curse of dimensionality. However, it may not be obvious which parts of the model should be specified parametrically and which parts should be treated in a nonparametric fashion.

The main philosophy of this book is to try to take advantage of the best features of the nonparametric and parametric methodologies. We do this by letting the parameters of a parametric model depend on the variables involved, which is a local parametric approach advocated by, for example, Hjort and Jones (1996) and Loader (1996). These two references treat localization of parameters for a quite general parametric family. On the other hand, we concentrate on localizing the Gaussian parametric family. It turns out that this has some great potential advantages, and we are not only able to use the local Gaussian structure to lessen the curse of dimensionality, but we are also able to generalize and extend models where the correlation or autocorrelation function plays a natural role. A brief introduction to the idea of local parametric estimation is given in Section 2.10 of this chapter, and a much more detailed account of the local Gaussian approach follows in Chapter 4.

In the present chapter we will give a brief overview of the two main approaches, parametric and nonparametric, and mention the semiparametric hybrid. The overview will be rather subjective in its choice of topics and emphasis. Some special topics are given more coverage than in most reviews of this sort. We do this in such a manner because we not only motivate local Gaussian modeling in its present state, but also briefly include topics that may serve as extensions of the local modeling as presented in this book. There are no data illustrations in this chapter as such illustrations can be found in numerous other books on these topics.

2.2 Parametric density models

Motivated by their use later in the book, as possible candidates for extensions of local Gaussian models, we discuss three types of distributions: the Gaussian, elliptic, and exponential families.

2.2.1 The Gaussian distribution

The Gaussian, or normal, distribution is probably the most used distribution in statistics. There are several reasons for this, but the most important one is probably the central limit theorem, which states that averages are (approximately) normally distributed if we have fairly many observations. Moreover, the normal distribution has a number of attractive and simple mathematical properties, not the least that it can be generalized very easily to the multivariate case. This is one reason why many books on multivariate statistics to a large part are based on the multivariate normal distribution; see, for example, Anderson (2003) and Johnson and Wichern (2007).

In applications, however, and perhaps particularly in finance, the normal distribution does not always give a good approximation to the data and may in fact lead to very wrong and even catastrophic results. The main idea of this book is to use the Gaussian distribution as a local approximation. We are seeking to replace a given distribution, univariate or multivariate, by a family of Gaussian distributions, where each member of the family approximates the properties of the given distribution locally in a neighborhood surrounding a given point. The motivation is that we hope to avoid the inaccuracy of using the pure Gaussian distribution and at the same time retain simple and unique properties of the Gaussian distribution locally.

The Gaussian is of course a prime example of a parametric model in statistics, and in this section, we very briefly survey some fundamental facts of this distribution that we will use in later chapters. The multivariate normal density function for a continuous stochastic variable of dimension p is given by

(2.1)

where with , and Σ is the covariance matrix of X given by with for and , the variance of . Moreover, is the determinant of Σ, and T denotes transpose. We write . A fantastic property of this multivariate distribution is that it is composed of pairwise dependencies only. It suffices to compute the means, variances, and pairwise covariances to obtain the whole distribution. This is a fact that we will use later in our local Gaussian analysis.

In the bivariate case , we write

where is the correlation between and . It is well known that uncorrelatedness does not imply independence in general, but for the Gaussian, as is easy to check from the form of the distribution function, uncorrelatedness and independence are equivalent. In a local Gaussian approximation, this property is used to asses dependence by means of the local correlation, and tests of independence are constructed by accumulating the local Gaussian correlation, see Chapter 7.

Another very important property of the Gaussian distribution is that marginal distributions and conditional distributions are again Gaussian. Let be a p-dimensional column vector, and let Y be a q-dimensional column vector with mean and covariance matrix . Assume that the covariance matrix is nonsingular, and assume that is jointly -variate normally distributed. Then the conditional distribution of Y given X is Gaussian with mean

(2.2)

with , and with covariance matrix

(2.3)

In the bivariate case , this reduces to

and

These formulas are starting points for defining the partial correlation function of two vectors X and Y given a third vector Z. In Chapter 11, we will introduce local versions of these quantities by straight analogy and use them in a description of conditional density functions and tests for conditional independence. The local partial correlation function derived from the formulas for the global Gaussian will play an essential role in these derivations.

Other useful properties of a multivariate Gaussian are its simple transformation rules. If X is a multivariate Gaussian of dimension p, c is a vector of scalars having dimension q, B is a matrix of scalars, and if ), then . In particular, any linear combination of the components of X is again normally distributed. We will make use of this in Section 6.4 on nonlinear local portfolio construction.

The estimation of the parameters μ and Σ in (2.1) can be done by maximizing the log likelihood function, which becomes very simple for the density function (2.1). The analogue for estimating a local mean and a local covariance is a local log likelihood function as explained in some detail in Chapter 4.

2.2.2 The elliptical distribution

The normal distribution plays a very central role in this book, since it forms the basis for a local Gaussian representation. For a distribution that is not normal but belongs to a specific class of distributions, say the t-distribution, it may of course be a disadvantage to try to approximate it with a family of Gaussian distributions. Then the question arises whether we should try to use a family of more general distributions. In particular, we might be interested in distributions that retain as many as possible of the very simple properties of Gaussian distributions and that are easily extendable to the multivariate case. A natural class for this purpose is the elliptical distributions. This class of distributions is also of interest in its own right when it comes to applications in finance, since it contains several heavy-tailed distributions. A classic paper in this context is by Owen and Rabinovitch (1983). A more recent contribution is by Landsman and Valdez (2003). There is also a separate chapter on elliptical distributions in McNeil et al. (2005).

There are several ways to define an elliptic distribution, which can be defined for both discrete and continuous variables. Perhaps the simplest definition is taking as a starting point the class of spherical distributions. The random q-dimensional variable is said to have a spherical distribution if its characteristic function can be written as

for some function ψ, which is called the characteristic generator. We write . A particular case of spherical multivariate distributions is obtained by taking , which is the multivariate normal with being the q-dimensional identity matrix.

Then the general p-dimensional distribution is defined by the distribution of a stochastic variable X of dimension p given by

where μ is a p-dimensional vector of real numbers, A is a matrix, and .

Hence we obtain elliptical distributions by multivariate affine transformations of spherical distributions. The characteristic function is

(2.4)

where . Denote the elliptic distribution by , where μ is the location vector, Σ is the dispersion matrix, and ψ is the characteristic generator of the distribution.

Note that the representation in (2.4) is not unique. Even though the location vector μ is uniquely determined, the dispersion matrix Σ and the characteristic generator ψ are only determined up to a multiplicative constant c. If , then we also have that . This implies that if , then Σ may not necessarily be directly identified with the covariance matrix of X. Provided that all variances are finite, however, it is always possible to find a representation where Σ can be identified with the covariance matrix, but this may not be the standard representation of the distribution; see McNeil et al. (2005, p. 93).

It is possible to find a representation of X in terms of the density function, if it exists, by taking the inverse Fourier transform. Let g be a non-negative function on such that

where . Further, let , and let Σ be a positive definite matrix. Then an elliptic density function parameterized by , μ, and Σ is given by

(2.5)

where is a normalizing factor given by

There is exactly the same type of scaling non-uniqueness in this representation as for the representation in terms of a characteristic function, and Σ cannot necessarily be identified with the covariance matrix of X. One case in which Σ can be identified with the covariance matrix is the multivariate t-distribution. The density of this distribution is given