Contagion Phenomena with Applications in Finance

 
 
Elsevier Science Ltd (Verlag)
  • 1. Auflage
  • |
  • erschienen am 17. August 2015
  • |
  • 166 Seiten
 
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978-0-08-100478-4 (ISBN)
 
Much research into financial contagion and systematic risks has been motivated by the finding that cross-market correlations (resp. coexceedances) between asset returns increase significantly during crisis periods. Is this increase due to an exogenous shock common to all markets (interdependence) or due to certain types of transmission of shocks between markets (contagion)? Darolles and Gourieroux explain that an attempt to convey contagion and causality in a static framework can be flawed due to identification problems; they provide a more precise definition of the notion of shock to strengthen the solution within a dynamic framework. This book covers the standard pracitce for defining shocks in SVAR models, impulse response functions, identitification issues, static and dynamic models, leading to the challenges of measurement of systematic risk and contagion, with interpretations of hedge fund survival and market liquidity risks



- Features the standard practice of defining shocks to models to help you to define impulse response and dynamic consequences
- Shows that identification of shocks can be solved in a dynamic framework, even within a linear perspective
- Helps you to apply the models to portfolio management, risk monitoring, and the analysis of financial stability
  • Englisch
  • London
Elsevier Science & Technology
  • Für Beruf und Forschung
  • |
  • Upper-division undergraduates, graduate students, and researchers working on market linkages, pricing and risk management in financial markets and industries.
  • Höhe: 229 mm
  • |
  • Breite: 152 mm
  • 2,08 MB
978-0-08-100478-4 (9780081004784)
0081004788 (0081004788)
weitere Ausgaben werden ermittelt
1. Contagion and Causality in Static Models
2. Contagion in Structural VARMA Models
3. Common Frailty versus Contagion in Linear Dynamic Models
4. Applications of Linear Dynamic Models
5. Common Frailty and Contagion in Nonlinear Dynamic Models
6. An Application of Nonlinear Dynamic Models: The Hedge Fund Survival
1

Contagion and Causality in Static Models


Abstract


A significant part of the academic and applied literature analyzes and measures contagion and causality in a static framework. The aim of this chapter is to review and discuss approaches and to show that such attempts are (almost) hopeless due to identification problems. However, this chapter does not send only a negative message, since it helps to define the notion of shock more precisely: does a shock have to be related to an equation, or to a variable to be systematically aware of identification issues and, last but not least, in noting that such contagion or causality issues can only be considered in a dynamic framework?

Keywords

Arbitrage pricing theory (APT)

Exogenous switching regime model

Factor model

Gaussian model

Linear dependencies

Nonlinear transformations

Principal component analysis (PCA)

A significant part of the academic and applied literature analyzes and measures contagion and causality in a static framework. The aim of this chapter is to review and discuss approaches and to show that such attempts are (almost) hopeless due to identification problems. However, this chapter does not send only a negative message, since it helps to define the notion of shock more precisely: does a shock have to be related to an equation, or to a variable to be systematically aware of identification issues and, last but not least, in noting that such contagion or causality issues can only be considered in a dynamic framework?

1.1 Linear dependence in a static model


The linear dependencies, also called weak dependencies, between variables are the links, which can be detected through the second-order moments, that are the variances and covariances. Equivalently, these are the links, which can be revealed in a Gaussian static framework. For expository purposes, we consider the following Gaussian framework and discuss equivalent specifications of the model, such as its simultaneous equations specification, its recursive forms, but also the specifications underlying exploratory factor analysis and principal component analysis.

1.1.1 The basic model


We consider n (random) variables Yi, i = 1, ., n. They can be stacked into a random vector Y of dimension n. This vector is assumed to follow a Gaussian distribution with zero-mean and a variance-covariance matrix S:

~N(0,S).

  [1.1]

This matrix S admits the variances i2=V(Yi), i = 1, ., n, as diagonal elements, and the covariances si,j = co?(Yi, Yj), i ? j, out of the diagonal.

Example 1.1

Bidimensional case

For instance, in the bidimensional case n = 2, we obtain:

Y1Y2)~N(0,(s12s12s12s22)).

1.1.2 Alternative specifications of the basic model


There exist alternative ways of writing the static Gaussian model. These alternative specifications make the values of variables Y depend (linearly) on the values of n independently underlying Gaussian variables. Then, these underlying variables are assumed to be the variables on which exogenous shocks will be applied. Moreover, these shocks can be applied separately by the independence property.

1.1.2.1 Simultaneous equation model

Gaussian model [1.1] can be rewritten as:

=CY+?,with?~N(0,Id),

  [1.2]

where the (n, n)-matrix C is assumed to have eigenvalues different from 1, and where the error term ? is standard Gaussian. Under specification [1.2], we have the (misleading) impression that a given variable, say Yi, is affected by the other variables Y1, ., Yn, plus a noise, and that these effects are summarized in the C matrix. Therefore this C matrix plays a key role for contagion analysis.

However, simultaneous equation system [1.2] is equivalent to the system:

=(Id-C)-1?,

  [1.3]

where Id denotes the identity matrix of size n. We deduce the expression of the variance-covariance matrix S as:

=(Id-C)-1(Id-C´)-1.

  [1.4]

Equivalently, Id - C is a "square root" of S. But it is well known that a symmetric positive definite matrix has a large number of different square roots. Some square roots are symmetric matrices: there exist 2n such square roots, whenever S has different eigenvalues. There also exist nonsymmetric square roots, for instance square roots, which are lower triangular matrices and are used in the Cholesky decomposition (see e.g. [HIG 01]). Anyway, if we observe independently at several dates the values of the variables Yt, t = 1, ., T, satisfying [1.1], we can expect to accurately approximate matrix S by its sample counterpart. But, due to the multiplicity of square roots, we cannot deduce a nonambiguous C; in other words matrix C is not identifiable.

1.1.2.2 Recursive specification

Recursive specifications of model [1.1] are frequently considered in practice. For expository purposes, let us discuss the bidimensional case, but the discussion is easily extended to any dimension. For instance, we can write:

Y1=u1,Y2=a2|1Y1+?2|1,

  [1.5]

where u1 and ?2|1 are independent, 1~N(0,s12), 2|1~N(0,s22(1-?2)) and 2|1=s21/s12,?2=s122/(s12+s22).

This recursive specification is a direct consequence of the Bayes formula: the first equation provides the marginal (i.e. unconditional) distribution of Y1, whereas the second equation provides the conditional distribution of Y2 given Y1. a2|1Y1 is the best predictor of Y2 given as Y1, which is linear in Y1 in the Gaussian case; 221-?2 is the variance of the associated prediction error. The independence between the two error terms is a consequence of the no correlation between Y1 and the prediction error ?2|1 and of the equivalence between no correlation and independence in the Gaussian framework. Specification [1.5] may induce a misleading causal interpretation: variable Y1 is fixed first (the first equation), then influence Y2 (the second equation). Such an interpretation is clearly misleading, since it is also possible to get the symmetric recursive representation by changing the orders of indices 1 and 2:

Y1=a1|2Y2+?1|2,Y2=u2,,say.

  [1.6]

A naive view of system [1.6] might lead to detect a causality in the reverse direction from 2 to 1.

Nevertheless, such recursive specifications are still often used in practice for causal interpretations, or for defining the basic shocks when constructing the impulse response functions, that are the changes on Y consequences to some shocks.

For example, Dungey et al. [DUN 05] consider the contagion from country 1 to country 2 using the following specification:

y1=?1w+d1u1,y2=?2w+d2u2+?u1,

where ?1, ?2, w are independent standard normal variables. We have seen in the previous section that this model can also be written as:

y1=?1w+a1?1+??2,y2=?2w+a2?2,

which gives the impression of a contagion in the reverse direction.

Similarly, Sims [SIM 77, SIM 80] proposed to shock the error terms using the recursive form, say [1.5]. In this scheme, u1 is interpreted as the shock on variable Y1, whereas v2|1 is interpreted as the shock on the equation defining Y2. As mentioned earlier, there is no argument to privilegiate this recursive form instead of the second recursive form [1.6]. Nevertheless, this practice leads to important questions:

- how can we define a shock on a variable Y1?;

- how can we define a shock on an equation?;

- are shocks on variables Y1, Y2 (respectively, on equations defining Y1 and Y2) independent or linked?;

- if they are linked, how can we represent this link?

1.1.2.3 Principal component analysis (PCA)

Any variance-covariance matrix is a symmetric positive definite matrix. This matrix can be...

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