Contents Essays on Modelling and Forecasting Financial Time Series

Universit¶e Libre de Bruxelles

Facult¶e des Sciences Sociales et Politiques

Solvay Brussels School of Economics and Management

Essays on Modelling and Forecasting Financial Time Series

Dissertation pr¶esent¶ee en vue de l'obtention du grade de Docteur en Sciences ¶Economiques

par

Laura Coroneo

Promoteur: Professeur David Veredas Co-promoteur: Professeur Catherine Dehon Membres du Jury:

Professeur Luc Bauwens Professeur Valentina Corradi Professeur Christine De Mol

Professeur Marc Hallin Ann¶ee acad¶emique 2008-2009

Contents

Acknowlegments vii Introduction 1

1 A Quantile Regression Approach to Intraday Seasonality 6

1.1 Introduction . . . 7

1.2 Market and Data . . . 11

1.3 Quantile Regression as Density Regression . . . 17

1.4 Estimation Results . . . 22

1.4.1 Speci¯cation Tests . . . 27

1.4.2 Conditional Quantile-based Shape Measures . . . 29

1.5 Intraday Value at Risk . . . 34

1.5.1 Comparison with GARCH Models . . . 39

1.6 Conclusion . . . 43

1.7 Appendix . . . 45

2 Forecasting the yield curve using large macroeconomic information 47 2.1 Introduction . . . 48

2.2 Model . . . 52

2.2.1 Alternative Models . . . 56

2.3 Data . . . 58

2.4 Estimation Procedure . . . 64

2.4.1 Model Selection . . . 65

i 2.5 Estimation Results . . . 71

2.6 Out-of-sample Forecast . . . 75

2.7 Conclusion . . . 87

2.8 Appendix . . . 89

3 How Arbitrage-Free is the Nelson-Siegel Model? 96 3.1 Introduction . . . 97

3.2 Modelling Framework . . . 101

3.2.1 The Nelson-Siegel Model . . . 102

3.2.2 Gaussian A±ne Arbitrage-free Model . . . 104

3.2.3 Testing Framework . . . 106

3.3 Data . . . 109

3.4 Estimation Procedure . . . 112

3.4.1 Resampling Procedure . . . 113

3.5 Results . . . 118

3.5.1 Testing Results . . . 121

3.5.2 In-sample Comparison . . . 133

3.5.3 Out-of-sample Comparison . . . 135

3.6 Conclusion . . . 139

References 140 ii

List of Figures

1.2 The pdf and the quantile function . . . 18

1.3 Fourier Series . . . 21

1.4 Estimated parameters for TEF . . . 23

1.5 Seasonality . . . 26

1.6 Shape measures . . . 33

1.7 Value at Risk decomposition . . . 36

1.8 Out-of-sample one-step ahead VaR . . . 38

2.1 Yield data . . . 60

2.2 Macro-yields model in-sample ¯t: yields . . . 67

2.3 Macro-yields model in-sample ¯t: key macro variables/1 . . . 68

2.4 Macro-yields model in-sample ¯t: key macro variables/2 . . . 69

2.5 Estimated macro-yields factors . . . 72

2.6 Smoothed square forecast errors (6 months ahead) . . . 83

2.7 Smoothed square forecast errors (12 months ahead) . . . 84

2.8 Smoothed absolute forecast errors (6 months ahead) . . . 85

2.9 Smoothed absolute forecast errors (12 months ahead) . . . 86

3.1 Nelson-Siegel factor loadings . . . 103

3.2 No-arbitrage latent factors and Nelson and Siegel factors . . . 107

3.3 Zero-coupon yields data . . . 110

3.4 No-Arbitrage loadings of the Nelson and Siegel factors . . . 126

iii 3.5 Empirical distribution of ba^NA . . . 129

3.6 Empirical distribution of b^NA(1) . . . 130

3.7 Empirical distribution of bNA(2) . . . 131

3.8 Empirical distribution of b^NA(3) . . . 132

iv

List of Tables

1.2 Sample moments at di®erent hours of the day . . . 16

1.3 Quantile crossings . . . 28

1.4 Likelihood ratio tests for the ¯tted conditional quantiles . . . 30

1.5 Christo®ersen's likelihood ratio tests on the VaR forecasts . . . 40

1.6 Kupiec test on the VaR forecasts . . . 42

2.1 Summary statistics of the US zero-coupon data . . . 59

2.2 Macroeconomic series . . . 62

2.3 Model selection . . . 66

2.4 Summary statistics of the estimated factors . . . 73

2.5 In-sample ¯t . . . 74

2.6 Summary statistics of the 1-step ahead forecast errors . . . 77

2.7 Summary statistics of the 6-steps ahead forecast errors . . . 78

2.8 Summary statistics of the 12-steps ahead forecast errors . . . 79

2.9 Out-of-sample performance . . . 81

3.1 Summary statistics of the US zero-coupon data . . . 111

3.2 Autocorrelations . . . 115

3.3 Parameter estimates . . . 119

3.4 Parameter estimates (continued) . . . 120

3.5 Estimation results for a^NA . . . 122

3.6 Estimation results for bNA(1) . . . 123

v 3.7 Estimation results for b^NA(2) . . . 124

3.8 Estimation results for bNA(3) . . . 125

3.9 Empirical distributions of the estimated loadings . . . 128

3.10 Measures of ¯t . . . 134

3.11 Out-of-sample performance . . . 138 vi