(G)ARCH Processes and the Phenomenon of Misleading and Unambiguous Signals

(G)ARCH processes and the phenomenon of misleading and unambiguous signals Ana Beatriz Torres de Sousa Thesis to obtain the Master of Science Degree in Mathematics and Applications Supervisor: Prof. Manuel João Cabral Morais Examination Committee Chairperson: Prof. António Manuel Pacheco Pires Supervisor: Prof. Manuel João Cabral Morais Members of the Committee: Prof. Patr´ıcia Alexandra de Azevedo Carvalho Ferreira e Pereira Ramos December 2015 Acknowledgments During the preparation of this dissertation, I received the help and support of several people to whom I would like to thank. First of all, I want to express my genuine gratitude to my supervisor, Professor Manuel Cabral Morais, for his guidance, constructive criticism, proofreading and specially for his patience during our weekly and stimulating meetings via Skype. My thanks are also extensive to: Professor Yarema Okhrin because without his programming sug- gestion I would have waited much longer for the simulations to end; and Professor Wolfgang Schmid for providing crucial critical values and, thus, saving me from excruciating computations and time consuming simulations. This thesis could not have been written without the support and infinite patience of my family and friends. I am deeply grateful to: my parents, Manuela and Fernando, for always being there pushing me forward; my sister Rita who showed me that even the hardest obstacles can be overcome with persistence; and my grandmother Alice who always believed in me even when I did not. i ii Resumo Uma série temporal éuma sequência de observa¸cões ordenadas no tempo. No estudo destas sequências, énecessário reconhecer explicitamente a importância da ordem pela qual as observa¸cões são recolhidas, logo recorrer àanálise e aos modelos de séries temporais. Os modelos de séries (estacionárias) lineares de valor real são brevemente descritos no Cap. 1, após terem sido revistos alguns conceitos fundamentais. Uma vez que estes modelos mais simples são manifestamente inadequados para descrever séries temporais económicas e financeiras, que tendem a não satisfazer o pressuposto de que a variância condicional éconstante, no Cap. 2 são revistos os modelos com heteroscedasticidade condicional autoregressiva (gen- eralizada) ((G)ARCH), bem como as suas propriedades, identifica¸cão, estima¸cão, previsão e limita¸cões. Em várias áreas, nomeadamente em Finan¸cas, écrucial detetar altera¸cões estruturais o mais rapi- damente poss´ıvel após a sua ocorrência. Como seria de esperar, as cartas de controlo têm sido usadas para detectar desvios devido a altera¸cões pontuais, valores extremos e fases de maior volatilidade. As primeiras cartas de controlo constru´ıdas com o propósito de detectar altera¸cões no valor esperado (resp. na variância) de processos GARCH foram propostas em 1999 (resp. 2001). Mais, o impacto de rumores ou outros eventos no processo (G)ARCH pode ser frequentemente descrito por uma mudan¸ca na variância ou por um valor extremo responsável por uma altera¸cão no valor esperado do processo, sugerindo assim a utiliza¸cão de esquemas conjuntos para o valor esperado e a variância do processo, tais como os propostos em 2001 e descritos no Cap. 3. Uma vez que as altera¸cões no valor esperado e na variância requerem açcões diferentes dos corretores e negociantes bolsistas, o Cap. 4 édedicado às probabilidades de sinais erróneos e não amb´ıguos (PMS e PUNS) desses esquemas conjuntos que permitem compreender um pouco melhor o respectivo desempenho. O Cap. 5 completa esta disserta¸cão com algumas considera¸cões finais e recomenda¸cões de trabalho futuro. Foram utilizados programas para o software estat´ıstico R de forma a obter todos os resultados e adiantar ilustra¸cões instrutivas apresentadas ao longo da tese. Palavras-chave: séries temporais; processos (G)ARCH; controlo estat´ıstico de processos; esquemas conjuntos EWMA; sinais erróneos e não amb´ıguos; software estat´ıstico R. iii iv Abstract A time series is a sequence of observations ordered in time. To study such sequences, we should explicitly recognize the importance of the order in which the observations are made, hence, resort to time series analysis and models. Linear real-valued time series models are briefly described in Chap. 1, after having recalled a few fundamental concepts. Since these simple models prove to be inadequate to describe economic and financial time series that tend not to operate under the assumption of constant conditional variance, we get familiar in Chap. 2 with (generalized) autoregressive conditionally heteroscedastic ((G)ARCH) processes, their general properties, model building, forecasting and limitations. In several domains suchlike Finance it is crucial to detect deviations from a target process as soon as possible after their occurrence. Expectedly, control charts have been used to detect such deviations due to change points, outliers or phases of higher volatility. The first control charts with the purpose of detecting changes in the mean (resp. variance) of a GARCH process can be traced back to 1999 (resp. 2001). Furthermore, the impact of rumors or other events on the target (G)ARCH process can be frequently described by a change in the variance or an outlier responsible for a shift in the process mean, thus calling for the use of joint schemes for the process mean and variance, as the ones proposed in 2001 and described in Chap. 3. Since changes in the mean and in the variance require different actions from the traders/brokers, Chap. 4 provides an account on the probabilities of misleading and unambiguous signals (PMS and PUNS) of those joint schemes, thus providing further insights on their out-of-control performance. Chap. 5 wraps up the dissertation with some final thoughts, namely a few recommendations for future work. Programs for the R statistical software were written to produce all the results and to provide striking illustrations throughout the thesis. Keywords: time series; (G)ARCH models; statistical process control; simultaneous EWMA schemes; misleading and unambiguous signals; R statistical software. v vi Contents Acknowledgments i Resumo iii Abstract v List of Tables ix List of Figures xi Glossary xv Acronyms xvi 1 Time series 1 1.1 Fundamentalconcepts .............................. ........ 2 1.2 Some simple time series models . ..... 5 2 (Generalized) autoregressive conditionally heteroscedasticmodels 9 2.1 The birth of the autoregressive conditionally heteroscedastic model............. 11 2.2 Autoregressive conditionally heteroscedastic processes . ................ 13 2.3 Generalized autoregressive conditionally heteroscedastic processes.............. 15 2.4 Model building and forecasting for GARCH processes . ........... 17 2.4.1 Model identification . .. 17 2.4.2 Parameterestimation ............................... ... 18 2.4.3 Diagnostic checking . .. 19 2.4.4 Modelselection.................................... .. 21 2.4.5 Forecasting ...................................... .. 21 2.5 Illustrations....................................... ..... 22 2.6 Particularcasesandextensions . ......... 31 2.6.1 Other univariate models of conditional variance . ........ 32 2.6.2 Multivariate generalizations . .... 34 vii 3 Simultaneous control schemes for the mean and variance of GARCHprocesses 37 3.1 On the impact of falsely assuming independence . ......... 39 3.1.1 ARCH(1)model .................................... 43 3.1.2 GARCH(1,1)model .................................. 51 3.2 Simultaneous schemes for the process mean and variance . .............. 58 3.3 Estimating the ARL via Monte Carlo simulation . ...... 59 3.4 Illustration ........................................ .... 63 4 Onthephenomenonofmisleadingandunambiguoussignals 65 4.1 Estimating PMS and PUNS via Monte Carlo simulation . ...... 66 4.1.1 PMSofTypeIII .................................... 67 4.1.2 PMSofTypeIV ..................................... 69 4.1.3 PUNSofTypeIII ................................... 71 4.1.4 PUNSofTypeIV .................................... 71 4.2 Summaryoffindings .................................. ..... 74 5 Final thoughts 77 References 79 A Additional plots and tables 85 A.1 EstimatesofRLbasedperformancemeasures . ............ 86 A.2 Critical values of the simultaneous modified EWMA schemes . .......... 89 A.3 Estimates of PMS and PUNS via Monte Carlo simulation . ....... 90 viii List of Tables 2.1 Ljung-Box test for the estimated ARCH(1) fit of monthly log returns of the Intel stock, from1973(1)to2008(12). ....... 26 2.2 Tests to check the normality for the estimated ARCH(1) fit of monthly log returns of the Intelstock,from1973(1)to2008(12). ........... 26 2.3 Information criteria for the estimated models for the monthly log returns of the Intel stock, from1973(1)to2008(12). ....... 26 2.4 Ljung-Box test for the estimated GARCH(1, 1) fit of the monthly excess returns of the S&P500index,from1926(1)to1991(12). ......... 29 2.5 Tests to check the normality for the estimated GARCH(1, 1) fit of the monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12). ............. 30 2.6 Information criteria for the estimated models for the monthly excess returns of the S&P 500index,from1926(1)to1991(12). ......... 31 2.7 Monthly excess return and volatility forecasts for the estimated GARCH(1, 1) fit of the monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12). .......... 31 2.8 Other models of conditional variance. ......... 32 3.1 ARL, SDRL, CVRL and MdRL values when

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