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Time-Varying Parameter Models for Discrete Valued Time Series Isbn 978 90 5170 755 7 TIME-VARYING PARAMETER MODELS FOR DISCRETE VALUED TIME SERIES ISBN 978 90 5170 755 7 Cover design: Crasborn Graphic Designers bno, Valkenburg a.d. Geul Image: Wiki Commons (template) and Marcin Zamojski (design) This book is no. 642 of the Tinbergen Institute Research Series, established through coopera- tion between Rozenberg Publishers and the Tinbergen Institute. A list of books which already appeared in the series can be found in the back. VRIJE UNIVERSITEIT TIME-VARYING PARAMETER MODELS FOR DISCRETE VALUED TIME SERIES ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus prof.dr. V. Subramaniam, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de Faculteit der Economische Wetenschappen en Bedrijfskunde op dinsdag 9 februari 2016 om 11.45 uur in de aula van de universiteit, De Boelelaan 1105 door Rutger Lit geboren te Alkmaar promotoren: prof.dr. A. Lucas prof.dr. S.J. Koopman Acknowledgements The last two and a half years went by at a fast pace. It was a period of time that I greatly enjoyed and the end product, this dissertation, is something that I am proud of. I could not have done this research alone. I am especially grateful to my supervisors Siem Jan Koopman and Andr´e Lucas who were essential for the realisation of this dissertation. Their expert advice on numerous topics in combination with frequent meetings, despite their busy schedules, were very important and much appreciated. I look forward to continuing our collaboration. From all my colleagues at the finance and econometrics department at the VU Am- sterdam, a special thanks goes out to Istv´an Barra. The originality and quality of our research discussions were very helpful to me and very much appreciated. Another thanks goes out to Marcin Zamojski for his advice on several econometric and finance topics as well as making my digital life easier and more secure. I would also like to thank Marco Bazzi, Falk Br¨auning, Artem Duplinskiy, Wenqian Huang (Ä)), Anne Opschoor, Kristian Støre, Andries van Vlodrop, and Marius Zoican for the discussions and laughs we shared. I am grateful to the Dutch National Science Foundation (NWO) and the Tinbergen Institute for the financial support. Last but not least I thank my parents for never doubting the research path that I have chosen. Amsterdam, November 2015 Contents Acknowledgements v 1 Introduction 1 1.1 Introduction for a general audience ......................... 1 1.2Econometricmethodologies............................. 2 1.2.1 Non-Gaussianstatespacemodels...................... 2 1.2.2 Score-drivenmodels.............................. 4 1.3Contributions..................................... 4 1.3.1 Chapter2................................... 4 1.3.2 Chapter3................................... 5 1.3.3 Chapter4................................... 6 1.3.4 Chapter5................................... 7 2 A Dynamic Bivariate Poisson Model for Analysing Football Match Results 9 2.1Introduction...................................... 9 2.2Thestatisticalmodellingframework......................... 12 2.2.1 BivariatePoissonmodel........................... 12 2.2.2 Dynamic specification for goal scoring intensities .............. 13 2.2.3 Someextensionsofthebasicmodel..................... 14 2.2.4 General state space representation of the model .............. 16 2.2.5 Evaluation of likelihood function and estimation .............. 17 2.3 Empirical application ................................. 19 2.3.1 Data description ............................... 19 2.3.2 Detailsofthebasicmodel.......................... 20 2.3.3 Parameterestimates............................. 21 2.3.4 Signal estimates of strengths of attack and defence ............ 22 2.3.5 Model evaluations: in-sample and out-of-sample .............. 24 2.4 Out-of-sample performance in a betting strategy .................. 28 2.5Conclusions...................................... 32 Appendices......................................... 33 A Likelihoodevaluation............................. 33 B Constructionofapproximatingmodel.................... 34 C Thederivativesforthemodelobservationdensity............. 36 D Computational details ............................ 37 E Tablesandfigures............................... 40 3 Intraday Stochastic Volatility in Discrete Price Changes 49 3.1Introduction...................................... 49 3.2ThedynamicSkellammodel............................. 52 3.2.1 The Skellam distribution ........................... 52 3.2.2 The modified Skellam distribution ...................... 52 3.2.3 TheSkellammodelwithdynamicmeanandvariance........... 55 3.3Analysisofhigh-frequencySkellampricechanges................. 57 3.3.1 Data...................................... 57 3.3.2 Dynamic Skellam with Intraday Stochastic Volatility ........... 58 3.3.3 Parameterestimationresults......................... 60 3.3.4 Signal extraction ............................... 62 3.3.5 Goodness-of-fit ................................ 62 3.3.6 Diagnosticchecking.............................. 64 3.3.7 Forecastingstudy............................... 66 3.4Conclusions...................................... 69 Appendices......................................... 70 A Modified Skellam distribution of type I ................... 70 B Moments of the MSKII(i, j, k) distribution ................. 70 C Simulationstudy............................... 71 D Numerically accelerated importance sampling ............... 73 E Intradaily time series of price changes in 2012 ............... 77 4 A Skellam Model for Analysing the Differences in Count Data 79 4.1Introduction...................................... 79 4.2ThedynamicSkellammodel............................. 80 4.2.1 Skellam distribution ............................. 80 4.2.2 Dynamicspecificationofintensities..................... 81 4.3Analysingfootballscores............................... 82 4.3.1 Estimationresults............................... 84 4.3.2 Modelextensions............................... 85 4.3.3 Signal extraction ............................... 88 4.4Conclusions...................................... 90 viii 5 Dynamic Discrete Copula Models for High Frequency Stock Price Changes 93 5.1Introduction...................................... 93 5.2 Score-driven dynamic discrete copula model .................... 96 5.3Simulationstudy...................................100 5.3.1 Estimatingparameterswhenmodeliscorrectlyspecified.........100 5.3.2 Estimatingtime-varyingpathswhenmodelismisspecified........101 5.4Dependencebetweendiscretepricechanges.....................103 5.4.1 Data description ...............................104 5.4.2 Missingvalues.................................105 5.4.3 Copula selection ................................106 5.4.4 Fullyearresults................................107 5.4.5 Comparison with intraday spline ......................110 5.5Conclusions......................................111 Appendices.........................................114 A Derivationofthescorevector........................114 B Tablesandfigures...............................115 Bibliography 119 Summary 127 ix Chapter 1 Introduction The introduction of this dissertation is organised as follows. Section 1.1 explains the title of this dissertation to a general audience and uses a football example as a lively illustration of the use of time-varying parameter models. Here, the use of mathematical notation is avoided as much as possible. Starting from Section 1.2, the focus is on the more experienced reader in the field of econometrics and statistics. 1.1 Introduction for a general audience Suppose in the very unlikely event that a football aficionado with knowledge of econo- metrics, statistics, and finance is interested in predicting the outcome of the next football match. This dissertation, with the title ‘time-varying parameter models for discrete valued time series’, can assist with these predictions. Let us start by getting a good understanding of what a time series is. Here, I consider a time series as a set of observations taken at (possibly unequally spaced) intervals over time. For example, we can observe the number of goals scored by a football team over a period of time (for example each week for a period of five years). It is crucial that the order of the observations is preserved since without this we cannot do any meaningful time series analysis. If we denote the number of goals scored by the football team at time index t by yt for t =1,...,n,theny =(y1,...,yn) is a time series of length n. The part discrete valued refers to the type of data the time series consists of. In this dissertation I focus on integers (whole numbers) and do not consider categorical data which could also be regarded as discrete. An example of discrete data are the number of goals scored by a football team. This number is in this case restricted to a positive integer since a football team cannot score a negative number of goals. Negative integers, however, play a prominent role in this dissertation in Chapter 3–5. Time-varying parameters form the core of this dissertation. To make a forecast about the next football match we could be interested in a measure of strength of the football CHAPTER 1. INTRODUCTION teams. It is an unrealistic assumption that the strength of a team is constant over time since the composition of football teams change over time as well. Also, recent match results probably tell us more about the current strength of a team than match results from the more
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