Dependence Modeling: Vine Copula Handbook

✍ Scribed by Dorota Kurowicka, Harry Joe, Editors

• Publisher: World Scientific Publishing Company
• Year: 2010
• Tongue: English
• Leaves: 370
• Category: Library

✦ Synopsis

This book is a collaborative effort from three workshops held over the last three years, all involving principal contributors to the vine-copula methodology. Research and applications in vines have been growing rapidly and there is now a growing need to collate basic results, and standardize terminology and methods. Specifically, this handbook will (1) trace historical developments, standardizing notation and terminology, (2) summarize results on bivariate copulae, (3) summarize results for regular vines, and (4) give an overview of its applications. In addition, many of these results are new and not readily available in any existing journals. New research directions are also discussed.

✦ Table of Contents

Contents......Page 8
Preface......Page 6
1.1 Introduction......Page 11
1.2 Investment Example......Page 12
1.3 Vines......Page 16
1.3.1 Graphical representation......Page 17
1.3.2 Vine density......Page 19
1.3.3 Estimation......Page 20
1.3.4 Properties and applications......Page 21
1.4 Outline......Page 23
1.5 Glossary and Notation......Page 25
References......Page 26
2. Multivariate Copulae M. Fischer......Page 29
2.1 Copulae......Page 30
2.2.1 Elliptical copulae......Page 32
2.2.2 Generalized t-copulae......Page 35
2.3.2 Non-exchangeable Archimedean copulae......Page 37
2.3.3 Generalized multiplicative Archimedean copulae......Page 39
2.3.4 Koehler–Symanowski copulae......Page 41
2.4 Combinations of Arbitrary Copulae into a New One......Page 42
References......Page 44
3. Vines Arise R. M. Cooke, H. Joe and K. Aas......Page 47
3.1 Introduction......Page 48
3.2 Regular Vines......Page 49
3.3.1 Vine copula or pair-copula construction......Page 53
3.3.2.1 Partial correlation......Page 56
3.3.2.2 Partial correlation vine......Page 57
3.3.2.3 Applications......Page 58
3.4 Historical Origins......Page 60
3.5 Compatibility of Marginal Distributions......Page 62
3.6.1 Sampling a D-vine......Page 65
3.6.2 Sampling an arbitrary regular vine......Page 66
3.6.3 Density approach sampling......Page 67
3.7 Parametric Inference for a Specific Pair-Copula Construction......Page 68
3.7.1 Inference for a C-vine......Page 69
3.7.2 Inference for a D-vine......Page 71
3.8 Model Inference......Page 72
3.8.1 Sequential selection......Page 73
3.8.2.1 Definitions and theorems......Page 74
3.9.1 Multivariate data analysis......Page 77
3.9.2 Non-parametric Bayesian belief nets......Page 78
References......Page 79
4. Sampling Count Variables with Specified Pearson Correlation: A Comparison between a Naive and a C-Vine Sampling Approach V. Erhardt and C. Czado......Page 83
4.1 Introduction......Page 84
4.2 Copulae and Multivariate Distributions......Page 85
4.3 Naive Sampling with Illustration to GP Count Data......Page 88
4.4 Simulation Study......Page 90
4.5 Summary and Discussion......Page 95
References......Page 96
5. Micro Correlations and Tail Dependence R. M. Cooke, C. Kousky and H. Joe......Page 99
5.2 Micro Correlations......Page 100
5.3 Tail Dependence and Aggregation......Page 102
5.3.1 Latent variable models for tail dependence......Page 104
5.3.2 Sum of damages over extreme events......Page 107
5.3.3 L1-symmetric measures......Page 111
5.3.4 Tail dependence for sums of L1 measures......Page 113
5.3.5 Lower tail dependence......Page 115
Appendices......Page 116
References......Page 121
6. The Copula Information Criterion and Its Implications for the Maximum Pseudo-Likelihood Estimator S. Grønneberg......Page 123
6.1 Introduction......Page 124
6.2 The Developments Leading to the CIC......Page 126
6.2.1 The fully parametric MLE......Page 127
6.2.2 Kullback–Leibler divergence and model selection......Page 129
6.2.3 The MPLE, the empirical copula and invariance considerations......Page 132
6.2.4 What about semiparametric efficiency?......Page 134
6.2.5 Large-sample theory for the MPLE......Page 135
6.3 Model Selection with the MPLE......Page 136
6.3.1 Non-existence of bias-correction terms and implications for the MPLE......Page 140
6.4 Illustrations......Page 142
6.5 Concluding Remarks......Page 146
References......Page 147
7.1 Introduction......Page 149
7.2 Equivalence Classes of Regular Vines......Page 150
7.3 Simulation from Vine Copulae......Page 155
7.4 Comparing Dependence of Vine Copulae......Page 164
7.5 Gaussian Vines and Generalized Toeplitz Matrices......Page 166
7.6 More Comparisons of Dependence for Different Vines......Page 169
7.7 Discussion and Further Research......Page 172
References......Page 173
8.1 Introduction......Page 175
8.2 Tail Dependence in Different Multivariate Copula Families......Page 176
8.3 Tail Dependence Parameters and Functions......Page 177
8.3.1 Bivariate tail dependence......Page 178
8.3.2 Multivariate tail dependence functions......Page 183
8.3.3 Conditional tail dependence functions......Page 185
8.4 Main Theorem on Tail Dependence for Vine Copulae......Page 186
8.5 Reflection Asymmetry of Vine Copulae......Page 189
8.6 Choice of Tail Asymmetric Bivariate Linking Copulae......Page 190
Acknowledgments......Page 194
Appendix......Page 195
References......Page 196
9. Counting Vines O. Morales-Napoles......Page 199
9.2 Basic Definitions......Page 200
9.3 Regular Vines and Prufer Codes......Page 204
9.4 Regular Vines and Line Graphs......Page 207
9.5 Regular Vines and Regular Vine Arrays......Page 208
9.6 Classifying Regular Vines......Page 213
Appendix......Page 215
References......Page 227
10.1 Introduction......Page 229
10.2 Naming Convention for Vines......Page 231
10.3 Number of Equivalence Classes......Page 232
10.4 Examples......Page 240
Reference......Page 241
11. Optimal Truncation of Vines D. Kurowicka......Page 243
11.1 Introduction......Page 244
11.2 Vines......Page 246
11.3 Vine Distributions......Page 247
11.3.1 Markov trees......Page 248
11.3.2 Vines in trees......Page 249
11.4.1 Generating regular vines......Page 250
11.5 Optimal Truncation: Results......Page 252
11.5.1 Example......Page 253
11.5.2 Comparison......Page 254
References......Page 257
12. Bayesian Inference for D-Vines: Estimation and Model Selection C. Czado and A. Min......Page 259
12.1 Introduction......Page 260
12.2 D-Vine......Page 261
12.3 D-Vine PCC Based on t-Copulae......Page 263
12.4 Bayesian Inference for D-Vine PCC Based on t-Copulae......Page 265
12.5 Application: Australian Electricity Loads......Page 267
12.6 Bayesian Model Selection for Australian Electricity Loads......Page 269
12.7 Summary and Discussion......Page 270
References......Page 272
13. Analysis of Australian Electricity Loads Using Joint Bayesian Inference of D-Vines with Autoregressive Margins C. Czado, F. G¨artner and A. Min......Page 275
13.1 Introduction......Page 276
13.2 Multivariate Time Series with D-Vine Dependency and Marginal Autoregressive Structure......Page 277
13.3 Bayesian Analysis of Multivariate Time Series with D-Vine Dependency and Marginal Autoregressive Structure......Page 279
13.4 Modeling Australian Electricity Loads......Page 280
13.5 Bayesian Model Selection......Page 284
References......Page 289
14. Non-Parametric Bayesian Belief Nets versus Vines A. Hanea......Page 291
14.1 Introduction or: How to Represent Information Burdened by Uncertainty......Page 292
14.2 Non-Parametric Bayesian Belief Nets: Sampling and Conditionalizing......Page 298
14.2.1 Sampling an NPBBN......Page 301
14.2.2 Conditionalizing an NPBBN......Page 303
14.3 Data Mining with NPBBNs......Page 307
14.4 Applications of NPBBNs......Page 309
14.5 Conclusions......Page 310
References......Page 312
15. Modeling Dependence between Financial Returns Using Pair-Copula Constructions K. Aas and D. Berg......Page 315
15.1 Introduction......Page 316
15.2.2 Partially nested Archimedean construction (PNAC)......Page 317
15.2.3 Pair-copula construction (PCC)......Page 319
15.3.2 PNAC......Page 321
15.3.3 PCC......Page 322
15.4.1 Data set......Page 323
15.4.2.1 PNAC......Page 325
15.4.3 Validation......Page 327
15.5.1 Data set......Page 329
15.5.2.1 PCC......Page 330
15.5.2.2 Four-dimensional Student copula......Page 332
15.5.3 Tail dependence......Page 333
15.5.4 Pair-copula decomposition with copulae from di.erent families......Page 334
15.6 Summary and Conclusions......Page 336
References......Page 337
16. Dynamic D-Vine Model A. Heinen and A. Valdesogo......Page 339
16.1 Introduction......Page 340
16.2.1 Copulae......Page 342
16.2.1.1 Copula-based dependence measures......Page 343
16.2.1.2 Asymmetric dependence, exceedance correlation and tail dependence......Page 344
16.2.2 D-vine copula......Page 345
16.2.3 Dynamic D-vine model......Page 347
16.3.1 Marginal model......Page 348
16.3.2.1 Gaussian copula......Page 349
16.3.2.4 Gumbel and rotated Gumbel copula......Page 350
16.4.2 Marginal models......Page 351
16.4.3 Copula structure......Page 355
16.5 Conclusion......Page 361
References......Page 362
17.1 Summary......Page 365
17.2 Future Research Directions......Page 366
Index......Page 369

✦ Subjects

Финансово-экономические дисциплины;Математические методы и моделирование в экономике;