ESTIMATING THE CONDITIONAL COVARIANCE MATRIX OF THE SYSTEM

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INTRODUCTION

Understanding how different variables react to one another has long been at the core of economics. Variables react to one another not only through the mean, but also through the second moments. This implies that the change in one variable might result not only in a change in the level of another variable, but also affect the volatility of other variables. Depending on the purpose of the research one will be interested in the mean effect or the higher moments, or perhaps both. Many techniques have been developed to obtain a consistent, efficient and unbiased estimate for these relationships that allows for the most accurate analysis. These analyses differ in objective – it might be for forecasting purposes, or understanding the structure of the relationships for policy analysis. Whatever the objective, the best estimate (i.e. in terms of bias, efficiency and consistency) under the given circumstances is always important.

OBJECTIVE AND RESEARCH METHODOLOGY

The primary objective of this study is to decompose the conditional covariance matrix of a system of variables. Therefore, a structural GARCH model is proposed which makes use of existing multivariate GARCH (MGARCH) models to decompose the covariance matrix. This type of analysis allows for the structural analysis of the volatility generated within a system of variables, as well as the volatility generated from factors outside the system. In most multivariate GARCH models the structural relationships between the variables are ignored, thereby leaving the investor without any idea of how the volatility is generated and what drives it. However, this type of analysis is important, for depending on the source of the innovation, the volatility of variables will differ in periods following the innovation.

OUTLINE OF THE STUDY

The outline of the study is as follows. In chapter 2 the problems associated with the estimation of simultaneous equations are discussed. The problem of identification is explained as well as solutions proposed in the literature. This problem is very important, for identifying structural parameters can be extremely problematic. Wrong applications of solutions can result in spurious relationships.

REDUCED-FORM VS. STRUCTURAL PARAMETERS

The question of when it is necessary to use reduced-form parameters and when it is necessary to use structural-form parameters depends on the purpose of the estimation. If the purpose of estimation is to forecast variables, to describe various characteristics of the data, or to search for hypotheses of interest to test a theory, the reduced-form parameters are sufficient. However, using the reduced-form of a system is not sufficient if the aim is to evaluate structural innovation and economic policy. Also, related impulse response functions are less useful if not done using structural equations (Cooley and LeRoy, 1985). Since the aim of this paper is to analyse, amongst others, the effects of structural innovations on a portfolio of assets, the reduced-form is not sufficient for using in the research.

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1. INTODUCTION AND BACKGROUND
1.1. INTRODUCTION
1.2. OBJECTIVE AND RESEARCH METHODOLOGY
1.3. CONTRIBUTION OF THE STUDY
1.4. OUTLINE OF THE STUDY
2. THE PROBLEM OF IDENTIFICATION
2.1. INTRODUCTION
2.2. IDENTIFICATION
2.3. REDUCED-FORM VS. STRUCTURAL PARAMETERS
2.4.OTHER METHODS OF ESTIMATING CONSISTENT PARAMETERS IN A SYSTEM OF EQUATIONS
2.4.1. Instrumental Variables (Two-stage least squares)
2.4.2. Three-stage least squares
2.4.3. Full information maximum likelihood
2.4. CONCLUSION
3. IDENTIFICATION THROUGH HETEROSCEDASTICITY
3.1. INTRODUCTION
3.2. IDENTIFICATION THROUGH HETEROSCEDASTICITY
3.3.EMPIRICAL STUDIES USING IDENTIFICATION THROUGH HETEROSCEDASTICITY
3.4. CONCLUSION
4. MULTIVARIATE GARCH MODELS
4.1. INTRODUCTION
4.2. OVERVIEW OF MGARCH MODELS
4.2.1. VEC and BEKK models
4.2.2. Factor and Orthogonal Models
4.2.3. Conditional Correlation Models
4.3. CONCLUSION
5. A STRUCTURAL GARCH MODEL
5.1. INTRODUCTION
5.2. STEP 1: ESTIMATING THE EXOGENOUS CONDITIONAL COVARIANCE MATRIX DRIVEN BY THE STRUCTURAL INNOVATIONS IN A SYSTEM
5.3. STEP 2: ESTIMATING THE ENDOGENOUS CONDITIONAL COVARIANCE MATRIX OF THE VARIABLES IN THE SYSTEM
5.4. CONCLUSION
6. LITERATURE REVIEW ON EMPIRICAL RESEARCH
6.1. INTRODUCTION
6.2. STOCK PRICES AND THE EXCHANGE RATE
6.3. STOCK PRICES AND THE INTEREST RATE
6.4. THE EXCHANGE RATE AND THE INTEREST RATE
6.5. CONCLUSION
7. ESTIMATING A STRUCTURAL GARCH MODEL
7.1. INTRODUCTION
7.2. THE DATA
7.3. ESTIMATING THE CONDITIONAL COVARIANCE MATRIX OF THE SYSTEM
7.3.1. Step 1: Estimating the exogenous conditional covariance matrix driven by the structural innovations in the system
7.3.2. Step 2: Estimating the endogenous conditional covariance matrix of variables in the system
7.4. CONCLUSION
8. IMPULSE RESPONSES AND AN APPLICATION TO PORTFOLIO RISK MANAGEMENTS
8.1. INTRODUCTION
8.2. IMPULSE RESPONSES
8.3. AN APPLICATION TO PORTFOLIO RISK MANAGEMENT
8.4. CONCLUSION
9. SUMMARY AND CONCLUSION
9.1. INTRODUCTION
9.2. METHODOLOGY
9.3. EMPIRICAL RESULTS
9.4. CONCLUDING REMARKS
BIBLIOGRAPHY