(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 Jo˜ao Cabral Morais

Examination Committee Chairperson: Prof. Ant´onio Manuel Pacheco Pires Supervisor: Prof. Manuel Jo˜ao 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 suggestion 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 Y i.i.d. N(0, σ2)...... 43 t ∼ 0 3.2 Estimated ARL, SDR, CVRL and MdRL when Y : t N ARCH(1), with α = 1.0 { t ∈ } ∼ 0 and α1 =0.0(i.i.d.case)...... 44 3.3 Statistics and control limits — individual modiﬁed EWMA charts for µ and for σ. .... 59

3.4 Pairs of smoothing parameters (λ1,λ2) — simultaneous modiﬁed EWMA schemes. . . . . 60

3.5 Estimated ARL and the correspondent pair (λ1, λ2) — simultaneous modiﬁed EWMA schemes(ProcessI)...... 61

3.6 Estimated ARL and the correspondent pair (λ1, λ2) — simultaneous modiﬁed EWMA schemes(ProcessII)...... 62

4.1 Types of misleading and unambiguous signals...... 66

4.2 Behavior of the estimated PMSIII (θ)...... 69

4.3 Behavior of the estimated PMSIV (δ)...... 69

4.4 Behavior of the estimated PUNSIII (θ)...... 71

4.5 Behavior of the estimated PUNSIV (δ)...... 74

4.6 Minimum ARL, minimum PMSIII (θ) and maximum PUNSIII (θ)(ProcessI)...... 75

ix 4.7 Minimum ARL, minimum PMSIII (θ) and maximum PUNSIII (θ)(ProcessII)...... 75

4.8 Minimum ARL, minimum PMSIV (δ) and maximum PUNSIV (δ)(ProcessI)...... 76

4.9 Minimum ARL, minimum PMSIV (δ) and maximum PUNSIV (δ)(ProcessII)...... 76

(I) (I) A.1 Critical values c1, c2 and c3 — simultaneous modified EWMA scheme with chart for σ basedonthesquaredobservations...... 89 (II) (II) A.2 Critical values c1, c2 and c3 — simultaneous modified EWMA scheme with chart for σ basedontheconditionalvariance...... 89 (III) (III) A.3 Critical values c1, c2 and c3 — simultaneous modified EWMA scheme with chart for σ based on the exponentially weighted variance...... 89 (IV) (IV) A.4 Critical values c1, c2 and c3 — simultaneous modified EWMA scheme with chart for σ basedonthelogarithmofsquaredobservations...... 89

A.5 Estimated PMSIII (θ)(ProcessI)...... 90

A.6 Estimated PMSIII (θ)(ProcessII)...... 91

A.7 Estimated PMSIV (δ)(ProcessI)...... 92

A.8 Estimated PMSIV (δ)(ProcessII)...... 92

A.9 Estimated PUNSIII (θ)(ProcessI)...... 93

A.10 Estimated PUNSIII (θ)(ProcessII)...... 94

A.11 Estimated PUNSIV (δ)(ProcessI)...... 95

A.12 Estimated PUNSIV (δ)(ProcessII)...... 95

x List of Figures

1.1 Earlytimegraphs...... 2 1.2 Quarterly U.S. gross national product (GNP) in billions of chained 1996 dollars, from 1947(1)to2002(3)...... 4 1.3 Quarterly growth rate of U.S. GNP, from 1947(1) to 2002(3)...... 8

2.1 Logarithm of quarterly consumption of non-durables in the UK, from 1955(1) to 1988(4). 10 2.2 Canadian index of real effective exchange rates, from 1980(1) to2008(12)...... 10 2.3 Stock price of the Siemens AG share, from November 28, 1997 to October12,2015.. . . . 11 2.4 Daily returns of the Siemens AG share, from December 1, 1997 to October12,2015. . . . 11 2.5 Historical deviation of the Siemens AG share, from December 1, 1997 to October 12, 2015. 11 2.6 Model building and forecasting scheme...... 22 2.7 Monthly log returns of the Intel stock, from 1973(1) to 2008(12)...... 23 2.8 Sample ACF and PACF of various functions of the monthly log returns of the Intel stock, from1973(1)to2008(12)...... 24 2.9 Diagnostics for the estimated ARCH(1) fit of monthly log returns of the Intel stock, from 1973(1)to2008(12)...... 25 2.10 Monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12)...... 27 2.11 Sample ACF of the monthly excess returns of the S&P 500 index and PACF of the squared monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12)...... 27 2.12 Estimated volatility and standardized residuals of the GARCH(1, 1) model fitted to the monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12)...... 29 2.13 Sample ACF of the standardized residuals and of the squared standardized residuals of the GARCH(1, 1) fit to the monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12)...... 29 2.14 QQ-plot of the standardized residuals 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

3.1 Estimates of ARL and SDRL, in the presence of a sustained shift in scale — ARCH(1) model...... 45 3.2 Estimates of CVRL and MdRL, in the presence of a sustained shift in scale — ARCH(1) model...... 46

xi 3.3 Estimates of ARL and SDRL, in the presence of a short-lived shift in location — ARCH(1) model...... 47 3.4 Estimates of CVRL and MdRL, in the presence of a short-lived shift in location — ARCH(1)model...... 48 3.5 Estimates of ARL and SDRL, in the presence of a short-lived shift in location and a sustained shift in scale — ARCH(1) model...... 49 3.6 Estimates of CVRL and MdRL, in the presence of a short-lived shift in location and a sustained shift in scale — ARCH(1) model...... 50 3.7 Estimates of ARL and SDRL, in the presence of a sustained shift in scale — GARCH(1,1) model...... 52 3.8 Estimates of CVRL and MdRL, in the presence of a sustained shift in scale — GARCH(1,1) model...... 53 3.9 Estimates of ARL and SDRL, in the presence of a short-lived shift in location — GARCH(1,1) model...... 54 3.10 Estimates of CVRL and MdRL, in the presence of a short-lived shift in location — GARCH(1,1)model...... 55 3.11 Estimates of ARL and SDRL, in the presence of a short-lived shift in location and a sustained shift in scale — GARCH(1,1) model...... 56 3.12 Estimates of CVRL and MdRL, in the presence of a short-lived shift in location and a sustained shift in scale — GARCH(1,1) model...... 57 3.13 Stock price of the Deutsche Bank share, from May 21, 1997 to September 22, 1997. . . . . 63 3.14 Returns of the Deutsche Bank share, from May 21, 1997 to September22,1997...... 63 3.15 Squared returns of the Deutsche Bank share, from May 21, 1997 to September 22, 1997. . 64

4.1 Estimated PMSIII (θ)...... 68

4.2 Estimated PMSIV (δ)...... 70

4.3 Estimated PUNSIII (θ)...... 72

4.4 Estimated PUNSIV (δ)...... 73

A.1 Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a sustained shift in scale —ARCH(1)model...... 86 A.2 Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a short-lived shift in location—ARCH(1)model...... 86 A.3 Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a short-lived shift in location and a sustained shift in scale — ARCH(1) model...... 87 A.4 Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a sustained shift in scale —GARCH(1,1)model...... 87 A.5 Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a short-lived shift in location—GARCH(1,1)model...... 88

xii A.6 Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a short-lived shift in location and a sustained shift in scale — GARCH(1,1) model...... 88

xiii xiv Glossary

0 vector of zeros. a vector. a⊤ transpose of a.

I identity matrix.

1S indicator function of the set S. r.v. random variable. i.i.d. independent and identically distributed.

xv Acronyms

ARL average run length.

ARL∗ pre-speciﬁed in-control average run length.

ARCH autoregressive conditionally heteroscedastic.

CVRL coeﬃcient of variation of the run length.

EWMA exponentially weighted moving average.

GARCH generalized autoregressive conditionally heteroscedastic.

LCL lower control limit.

MdRL median of the run length.

MS misleading signal.

PMS probability of a misleading signal.

PUNS probability of an unambiguous signal.

RL run length.

SDRL standard deviation of the run length.

SPC statistical process control.

UCL upper control limit.

UNS unambiguous signal.

xvi Chapter 1

Time series

From the earliest times man has measured the passage of time with candles or clepsydras or clocks, has constructed calendars, sometimes with remarkable accuracy. . . Sir Maurice Kendall

A time series is a sequence of observations (usually) ordered in time (Anderson, 1971, p. 1); examples arise in several fields, ranging from economics to engineering (Chatfield, 1975, p. 1). The study of sequences whose development and variation with the passing of time is of interest, by explicitly recognizing the importance of the order in which the observations are made (Anderson, 1971, p. 1), as a science in itself is relatively recent, even though the recording of events on a chart whose horizontal axis is marked with equal spaces of time (Kendall and Ord, 1990, p. 1) is a quite old practise. According to Kendall and Ord (1990, p. 1), the earliest time graph known in the Western world dates from the 10th, possibly 11th century, and it is part of a manuscript of Macrobius Ambrosius Theodosius1 on Cicero’s In Somnium Scipionis (Dream of Scipio).2 Many early medieval manuscripts of Macrobius include maps of the Earth, zonal maps showing the Ptolemaic climates derived from the concept of a spherical Earth and a diagram showing the Earth (Wikipedia, n.d.-a). This particular time graph was apparently meant to represent a plot of the inclinations of the planetary orbits as a function of time, and the time abscissa and the variable ordinate were certainly used albeit in a crude and limited way (Kendall and Ord, 1990, p. 1), as shown by Figure 1.1(a) taken from Epanechnikov (2011) but also found in Kendall and Ord (1990, Fig. 1.1, p. 1). It is believed that one of the earliest one (possibly the first) time series diagram displayed in the modern way is due to William Playfair (1759–1823)3 and dates from 1821 (Kendall and Ord, 1990, p. 2). However, this chart combining a graph and a histogram from Playfair’s A letter on our Agricultural Distress (Kendall and Ord, 1990, Fig. 1.2, p. 2) is predated by Playfair’s trade-balance time-series chart, published in his Commercial and Political Atlas in 1786 and reproduced in Figure 1.1(b), taken from Wikipedia (n.d.-b).

1Cameron (1966) argues that Macrobius, known to his contemporaries as Theodosius, was a praetorian prefect of Italy in 430 AD. 2Macrobius is the author of a commentary in two books on the Dream of Scipio narrated by Cicero at the end of his Republic (Wikipedia, n.d.-a). 3William Playfair was a Scottish engineer and political economist, considered to be the founder of graphical methods of statistics (FitzPatrick, 1960), namely for having invented the line graph and bar chart of economic data, both in 1786, and in 1801 the pie chart (Wikipedia, n.d.-b).

1 (a) Macrobius’ inclinations of the planetary orbits plot. (b) Playfair’s trade-balance plot.

Figure 1.1: Early time graphs.

Until 1925 or thereabouts, the behavior of any time series was considered completely described by a set of deterministic laws, and consequently the deviations from trends, cycles and other patterns were considered errors similar to errors of observation (Kendall and Ord, 1990, p. 3). This approach to the study of time series was discarded in 1927 by George Udny Yule (1871–1951), who broke new ground while studying the sun-spot numbers (Kendall and Ord, 1990, p. 3). This statistician was struck by the irregularities of this time series and instead of attributing any diﬀerence between theory and observation to an ephemeral error (Kendall and Ord, 1990, pp. 3–4), but to true disturbances responsible for variations in the amplitude and phase in a continual manner (Yule, 1927, p. 268). This seminal idea underlies much of the subsequent work in time series analysis and led to the theory of stochastic processes (Kendall and Ord, 1990, pp. 3–4). What distinguishes time series analysis from other statistical analysis is that the temporal order of the observations plays an important role, in particular it leads to the study of the relationship between observations that are one after other, i.e., to the investigation of serial correlations (Kendall and Ord, 1990, p. 4) that plays an important role in the statistical modeling of the time series.

The following sections of this chapter comprise selections of Morais (2012, chapters 1–2) that remain deliberately unaltered most of the time. A few fundamental concepts in time series are recalled, an example of a time series is given and the simplest real-valued time series models, such as autoregressive (AR), moving average (MA) and mixed autoregressive moving average (ARMA) processes, are brieﬂy described.

1.1 Fundamental concepts

Informally, a time series is an ordered sequence of observations and the ordering should be taken into account while describing and modeling the time series and forecasting future values of the time series (Morais, 2012, p. 1). As referred by Brockwell and Davis (1991, p. 8) and Morais (2012, p. 1), to allow for the unpredictable nature of future observations, it is supposed that each observation at time t is a realization of a random variable Xt.

2 Definition 1.1. — Stochastic process (Karr, 1993, p. 45; Brockwell and Davis, 1991, p. 8; Morais, 2012, p. 1) A stochastic process with index set T is a collection X : t T of random variables (or random vectors) { t ∈ } defined on a probability space (Ω, , ). F P • Definition 1.2. — Time series (Wei, 1990, p. 6; Brockwell and Davis, 1991, p. 8; Morais, 2012, p. 1) The collection of values x : t T is said to be a time series if it is a realization (or part of a realization) { t ∈ 0} of the stochastic process X : t T , where T T . { t ∈ } ⊇ 0 • Remark 1.3. — Time series in discrete (resp. continuous) time (Brockwell and Davis, 1991, pp. 8–9; Wei, 1990, p. 1; Morais, 2012, p. 2) Z N R+ R If T = or T = then the time series is said to be in discrete time; if T = 0 or T = then the time series is said to be in continuous time. • In this thesis, the attention is restricted to time series in discrete time, with the observations made at fixed time intervals and which can take values on a continuous set.

Examples of time series abound in such fields as economics, business, the natural and the social sciences (Box et al., 1994, p. 1; Morais, 2012, p. 2). The following time series illustrates data observed at different time epochs, as well some of the statistical questions that might be asked about such data (Shumway and Stoffer, 2006, p. 4; Morais, 2012, p. 2). The starting point of the study of this or any other time series is its graphical representation (Murteira et al., 1993, p. 2; Morais, 2012, p. 2) or time series plot.

Definition 1.4. — Time series plot (Shumway and Stoffer, 2006, p. 12; Pires, 2001, p. 5; Morais, 2012, p. 2) It is conventional to display a time series x : t T graphically by plotting the observed values of X { t ∈ 0} t on the vertical axis (or ordinate) and the time points as the abscissae; the values at adjacent times points are often connected. The resulting graph is usually termed as time series plot. • Example 1.5. — U.S. GNP data (Shumway and Stoffer, 2006, p. 144–145; Morais, 2012, pp. 2–4) The data set refers to the quarterly real U.S. gross national product (GNP)4 in billions of chained 1996 dollars5 from the first quarter of 1947, 1947(1), to the third quarter of 2002, 2002(3). The total of n = 223 observations were obtained from Shumway (2000). To plot the time series using the R statistical software,6 the following commands are typed and ran in order to read the data, define its initial year and frequency, and then plot the time series in Figure 1.2. quatgnp = c(1488.9, 1496.9,[...], 9376.7, 9477.9) quartgnp = ts(quartgnp, start=1947, frequency=4) plot(quartgnp, xlab="Year", ylab="Quarterly U.S. GNP")

4 GNP is the total value of all final goods and services produced within a nation in a particular year, plus income earned by its citizens (including income of those located abroad), minus income of non-residents located in that country. GNP measures the value of goods and services that the country’s citizens produced regardless of their location. GNP is one measure of the economic condition of a country, under the assumption that a higher GNP leads to a higher quality of living, all other things being equal (Wikipedia, n.d.-c; Morais, 2012, pp. 2–3). 5 Chained dollars is a method of adjusting real dollar amounts for inflation over time, so as to allow comparison of figures from different years. The U.S. Department of Commerce introduced the chained-dollar measure in 1996 (Wikipedia, n.d.-d; Morais, 2012, p. 3). 6All programs used to obtain the results presented in this and in the following chapters were implemented in R. They will be made available to those who request them from the author.

3 Quarterly U.S. GNP 2000 4000 6000 8000

1950 1960 1970 1980 1990 2000 Year

Figure 1.2: Quarterly U.S. gross national product (GNP) in billions of chained 1996 dollars, from 1947(1) to 2002(3).

Figure 1.2 shows a plot of a long times series x : t = 1,..., 223 with a strong increasing trend that { t } probably hides some other eﬀect, as pointed out by Shumway and Stoﬀer (2006, p. 145) and Morais (2012, p. 3). Moreover, it is not clear if the variance also increases with time. •

When dealing with a stochastic process X : t Z , it is necessary to extend the concept of { t ∈ } mean vector, covariance or correlation matrices — the mean, the autocovariance and the autocorrelation functions provide that extension (Brockwell and Davis, 1991, p. 11; Morais, 2012, p. 8).

Deﬁnition 1.6. — Mean, variance, autocovariance and autocorrelation functions (Wei, 1990, p. 7; Morais, 2012, p. 8) A stochastic process in discrete time X : t Z can be partially described by the following functions: { t ∈ }

µ = E(X ), t Z; t t ∈ σ2 = V(X ), t Z; t t ∈ γ(t ,t ) = Cov(X ,X ), t ,t Z; 1 2 t1 t2 1 2 ∈ γ(t1,t2) Z ρ(t1,t2) = Corr(Xt1 ,Xt2 )= , t1,t2 . σ2 σ2 ∈ t1 × t2 q They represent the mean, variance, autocovariance and autocorrelation functions, respectively. The autocorrelation function is usually referred to as ACF. •

In addition to the ACF, ρ = Corr(X ,X ), k N, the correlation between X and X can be de- k t t+k ∈ t t+k termined, after their mutual linear dependency on the intervening random variables Xt+1,Xt+2,...,Xt+k−1 has been removed (Wei, 1990, p. 12; Morais, 2012, p. 11), leading to the notion of partial autocorrelation function.

Deﬁnition 1.7. — Partial autocorrelation function (Wei, 1990, pp. 12–16; Pires, 2001, pp. 14–15; Morais, 2012, p. 11) Let X : t Z be a stationary stochastic process with zero mean. In time series analysis, { t ∈ }

φ = Corr(X ,X X ,X ,...,X ), k N , (1.1) k,k t t+k| t+1 t+2 t+k−1 ∈ 0

4 is referred to as the partial autocorrelation function (PACF) and is deﬁned by

1 ρ1 ρ2 ρk−2 ρ1 1 ρ1 ρ2 ρk−2 ρk−1 ρ 1 ρ ··· ρ ρ ρ 1 ρ ··· ρ ρ 1 1 ··· k−3 2 1 1 ··· k−3 k−2 ρ2 ρ1 1 ρk−4 ρ3 ρ2 ρ1 1 ρk−4 ρk−3 ··· ··· φk,k = ...... , . . . . . •

ρ ρ ρ 1 ρ ρ ρ ρ 1 ρ k−2 k−3 k−4 ··· k−1 k−2 k−3 k−4 ··· 1 ρ ρ ρ ρ ρ ρ ρ ρ ρ 1 k−1 k−2 k−3 ··· 1 k k−1 k−2 k−3 ··· 1

A crucial issue in the development of time series models is an assumption of some form of statistical equilibrium (Box et al., 1994, p. 21; Morais, 2012, p. 8) or stationarity. Since it is very diﬃcult or virtually impossible to verify stationarity in distribution or strict stationarity in time series analysis7 weaker notions of stationarity are often used and deﬁned in terms of the moments of the stochastic process (Wei, 1990, p. 8; Pires, 2001, p. 11; Morais, 2012, p. 10).

Deﬁnition 1.8. — First and second order weak stationarity (Wei, 1990, p. 8; Pires, 2001, p. 11; Morais, 2012, p. 10)

A ﬁrst order weakly stationary process X : t Z has constant mean function µ = µ, t Z. • { t ∈ } t ∈ A second order weakly stationary process X : t Z — or simply stationary (or covariance • { t ∈ } 2 stationary) from now on — has constant mean and variance functions (i.e., µt = µ and σt = σ , for t Z), and and its autocovariance and autocorrelation are functions of the time lag alone: ∈ γ(t,t + k)= γ and ρ(t,t + k)= γ /γ = ρ for any t, k Z. k k 0 k ∈ •

2 A stationary time series is (partially) characterized by its mean µ, variance σ , ACF (ρk) and PACF

(φkk). The exact values of these parameters can be only calculated if the ensemble of all possible realizations is known (Wei, 1990, p. 17; Morais, 2012, p. 12); otherwise, they can be estimated if multiple or just one realization of the stochastic process are available (Morais, 2012, p. 12).

1.2 Some simple time series models

In this section, a few well known time series models are deﬁned, such as autoregressive (AR), moving average (MA) and mixed autoregressive moving average (ARMA). Even though the following stochastic process rarely occurs in applied time series, it plays a major role as a basic building block in the construction of time series models (Wei, 1990, p. 17; Morais, 2012, p. 14).

Definition 1.9. — White noise process (Wei, 1990, p. 16; Box et al., 1994, p. 47; Pires, 2001, p. 12; Morais, 2012, pp. 13–14) X : t Z is called a white noise (WN) process with expected value µ (usually assumed to be 0) and { t ∈ } variance σ2 if, for t, k Z: µ = µ, t Z and γ(t,t + k) = γ with γ equal to σ2, if k = 0, and 0, ∈ t ∈ k k otherwise. When it is written X : t Z WN µ, σ2 , the sequence random variables is uncorrelated, but { t ∈ } ∼ not necessarily independent. In particular, they come from a fixed distribution with constant mean and variance. • 7For the definitions of stationarity in distribution or strict stationarity, the reader is referred to Morais (2012, pp. 8–10).

5 By deﬁnition, it immediately follows that a WN process is stationary.

Deﬁnition 1.10. — Gaussian white noise process (Wei, 1990, p. 17; Brockwell and Davis, 1991, p. 13; Morais, 2012, p. 14) X : t Z is called a Gaussian white noise (GWN) process with expected value µ (usually assumed to { t ∈ } be 0) and variance σ2 — in short X : t Z GWN µ, σ2 — if the joint distributions of all (ﬁnite) { t ∈ }∼ subsets of random variables Xt are multivariate normal. •

According to Box et al. (1994, pp. 47–48), a fundamental result in the development of stationary processes is due to Wold (1938), who established that any zero-mean purely nondeterministic stationary process possesses a linear representation (Morais, 2012, p.14).

Deﬁnition 1.11. — Moving average representation (Wei, 1990, p. 23; Morais, 2012, pp. 14–15) Let X : t Z be a stochastic process. Then its moving average representation is given by { t ∈ } ∞ Xt = µ + ψj ǫt−j , (1.2) j=0 X where ǫ : t Z WN(0, σ2), ψ = 1 and ∞ ψ2 < . { t ∈ }∼ ǫ 0 j=0 j ∞ • P Remark 1.12. — Backshift operator, moving average representation and stationarity (Wei, 1990, p. 24; Box et al., 1994, pp. 47, 49; Morais, 2012, p. 15) By introducing the backshift operator, Bj X = X , j N , (1.2) can be rewritten in a compact form: t t−j ∈ 0

X˙ t = ψ(B) ǫt, (1.3) where X˙ = X µ is the deviation of the original process from its mean, ψ(B) = ∞ ψ Bj , with t t − j=0 j ∞ 2 ψ0 = 1, and ψ < . Moreover, it can be easily proven that a stochastic processPXt : t Z that j=0 j ∞ { ∈ } admits the movingP average representation (1.3) is indeed a stationary process (with ﬁnite variance). •

Although the moving average representation is useful, it is not the most convenient model to work with in time series for it is associated to an inﬁnite number of parameters that are impossible to estimate from a ﬁnite data set. Hence, the following time series model.

Deﬁnition 1.13. — qth order moving average process (Wei, 1990, p. 46; Morais, 2012, p. 20) X : t Z is said to be a qth moving average process (q N) — denoted X : t Z MA(q) — if { t ∈ } ∈ { t ∈ }∼

X˙ t = θ(B)ǫt (1.4) where θ(B)=1 q θ Bj . − j=1 j • P The moving average processes are useful in describing phenomena in which events produce an imme- diate eﬀect that only lasts for shorts periods of time (Wei, 1990, p. 46; Morais, 2012, p. 20).

An autoregressive representation is another form to express a time series process, in which we regress

X˙ t on the past and add a random shock (Morais, 2012, p. 15).

6 Definition 1.14. — Autoregressive representation (Wei, 1990, p. 25; Morais, 2012, pp. 15–16) The autoregressive representation of the stochastic process X : t Z is given by { t ∈ } ∞ X˙ t = πj X˙ t−j + ǫt, (1.5) j=1 X or, equivalently, π(B)X˙ = ǫ , where π(B)=1 ∞ π Bj , and the coefficients π are absolutely t t − j=1 j j ∞ summable, i.e., πj < . P j=1 | | ∞ • P A stochastic process is called invertible if it admits an autoregressive representation (Wei, 1990, pp. 25–26; Morais, 2012, p. 16). Unsurprisingly, the autoregressive representation is not useful in practice because it also contains an infinite number of parameters. Thus, the description of another well known time series model arising from the autoregressive representation when only a finite number of πj are nonzero (Morais, 2012, p. 17).

Deﬁnition 1.15. — pth order autoregressive process (Wei, 1990, p. 32; Morais, 2012, p. 17) A stochastic process X : t Z is said to be a pth order autoregressive process (p N) — denoted { t ∈ } ∈ Xt : t Z AR(p) — if { ∈ }∼ p X˙ = φ X˙ + ǫ , t Z, (1.6) t j t−j t ∈ j=1 X where X˙ = X µ, ǫ : t Z WN(0, σ2), ∞ φ < . Applying once again the backshift t t − { t ∈ } ∼ ǫ j=0 | j | ∞ ˙ p j operator, the AR(p) process can be written as φ(BP)Xt = ǫt, where φ(B)=1 φj B . − j=1 • P ˙ 1 To be stationary the AR(p) process has to admit the moving average representation Xt = φ(B) ǫt = ψ(B) ǫ , such that ∞ ψ2 < holds, i.e., such that all the roots of φ(B) = 0 must lie outside the unit t j=0 j ∞ circle (Wei, 1990, p.P 32; Morais, 2012, p. 17).

The rationale behind autoregressive models is the idea that the current value Xt of the time series can be explained as a function of p past values, Xt−1,...,Xt−p, where p determines the number of steps into the past needed to forecast Xt (Shumway and Stoﬀer, 2006, p. 85; Morais, 2012, p. 18).

In model building it may be necessary to include a ﬁnite number of both autoregressive and moving average terms in the model, leading to the following mixed process (Wei, 1990, p. 56).

Deﬁnition 1.16. — Mixed autoregressive-moving average process (Wei, 1990, p. 57; Box et al., 1994, p. 77; Morais, 2012, p. 23) A stochastic process X : t Z a mixed autoregressive moving average process of order p and q { t ∈ } (p, q N) — in short X : t Z ARMA(p, q) —, if ∈ { t ∈ }∼

φ(B) X˙ t = θ(B) ǫt, (1.7) where: ǫ : t Z WN(0, σ2), φ(B)=1 p φ Bj and θ(B)=1 q θ Bj are polynomials of { t ∈ }∼ ǫ − j=1 j − j=1 j degree p and q in B which have no roots in common.P P • For the ARMA(p, q) to be invertible (resp. stationary), we require that the roots of θ(B) = 0 (resp. φ(B) = 0) lie outside the unit circle.

7 This section ends illustrating the fact that it is possible to obtain a stationary series by transforming a non-stationary original time series, as mentioned by Pires (2001, p. 37).

Example 1.17. — U.S. GNP data (bis) (Shumway and Stoffer, 2006, p. 144–151, 154; Morais, 2012, pp. 40–47) The quarterly U.S. GNP data from 1947(1) to 2002(3) is considered once again. The non stationary character of this time series is all too apparent in Figure 1.2. However, the growth rate, y = ln(x ), t ∇ t recommended by Shumway and Stoffer (2006, p. 145) and that can be interpreted as the percentage quarterly growth of the U.S. GNP, has a different behavior. In fact, its corresponding time series plot, obtained by running gnpgrowthrate = diff(log(quartgnp)) plot(gnpgrowthrate, xlab="Year", ylab="Quarterly growth rate U.S. GNP") and in Figure 1.3, suggests that this particular transformation of the original process is a stable process (Morais, 2012, Example 2.34, p. 41). Quarterly U.S. GNP rate growth −0.02 0.00 0.02 0.04

1950 1960 1970 1980 1990 2000 Year

Figure 1.3: Quarterly growth rate of U.S. GNP, from 1947(1) to 2002(3).

Furthermore,8 Morais (2012, pp. 43, 47) adds that the estimated AR(1) and MA(2) models for the quarterly growth rate (Yt) are

Y = 0.0083+ 0.3467 (Y 0.0083) + ǫ (1.8) t × t−1 − t Y = 0.0083 + ǫ +0.3028 ǫ +0.2035 ǫ . (1.9) t t × t−1 × t−2 and seem to adequately ﬁt that data set. • Interestingly enough, Shumway and Stoﬀer (2006, Example 5.3, pp. 283–285) mention that it has been claimed that the U.S. GNP data, such as many economic series, follows a more sophisticated time series model and investigates this claim. The next chapter is devoted to such models.

8For all the details about the model identiﬁcation, parameter estimation, diagnostic checking and model selection regarding this data set, the reader is referred to (Morais, 2012, Example 2.34, pp. 40–47).

8 Chapter 2

(Generalized) autoregressive conditionally heteroscedastic models

Each day I would get a little further on this new model and would talk with Sargan or Durbin or Hendry or Harvey about its properties and my proofs. David Hendry eventually named it AutoRe- gressive Conditional Heteroskedasticity and oﬀered to have Frank Srba program it. We applied it to UK inﬂation data and the ARCH model was launched. Robert F. Engle III

Time series models were initially introduced for descriptive purposes like prediction and seasonal correction or dynamic control (Gouriéroux, 1997, p. 1). C. Gouriéroux aptly adds that in the 1970s, the research in time series models essentially focused on ARMA processes, which were easy to implement back then. Financial time series is among the field of applications for which standard ARMA fit is poor (Gouriéroux, 1997, p. 1). The chief cause of the inadequacy of ARMA models is closely related to several features of economic and financial time series.

Many of them contain a clear trend (Franses, 1998, p. 9; Enders, 2010, p. 121), as the upward trend • exhibited by the time series plot of the quarterly U.S. gross national product in Figure 1.2. When economic and financial time series are observed each month or quarter, such series often • display a seasonal pattern (Franses, 1998, pp. 13–14) possibly associated with a trend,1 as illustrated by the time series plot of the quarterly consumption of non-durables in the UK in Figure 2.1.2 Some economic and financial time series seem to meander in the sense that they show no particular • tendency to increase or decrease (Enders, 2010, pp. 123–124), as evidenced by the time series plot of the Canadian index of real effective exchange rates in Figure 2.2.3 It seems to have a sort of random walk behavior typical of nonstationary series (Enders, 2010, pp. 123–124).4 Distorting and aberrant observations can occur and have a major influence on financial time series • 1It is important to add that if it is possible identify and eliminate from the data the trend and seasonal parts, the remaining is a fluctuating series that usually is between a purely random state and a smooth oscillatory movement (Kendall and Ord, 1990, pp. 14-15). 2It is identical to Figure 2.7 of (Franses, 1998, p. 17) and was produced with the aid of Kleiber and Zeileis (2015, pp. 167–168). 3Note that this time series plot was obtained using the data set found in BIS (2015) and differs slightly from the one found in Enders (2010, Figure 3.5, p. 124). 4The decrease in the exchange rate in 2008 is most likely due to the effect of the subprime mortgage crisis (begun on January 21, 2008) and the effect of the financial crisis (started on the Black Monday, October 6, 2008).

9 10.2 10.4 10.6 10.8 11.0 70 80 90 100 Quarterly in UK consumption of non−durables Canadian index of real effective exchange rate exchange of real effective Canadian index 1955 1965 1975 1985 1980 1985 1990 1995 2000 2005 2010 Year Year

Figure 2.1: Logarithm of quarterly consumption of non- Figure 2.2: Canadian index of real eﬀective exchange durables in the UK, from 1955(1) to 1988(4). rates, from 1980(1) to 2008(12).

modeling and forecasting (Franses, 1998, p. 20); moreover, these observations tend to appear in clusters and in some cases they can be interpreted as resulting, for instance, from the impact of certain news on a stock market (Franses, 1998, p. 24). This phenomenon is called volatility5 clustering or, otherwise, conditional heteroscedasticity6 (Franses, 1998, p. 24).

Many economic and ﬁnancial time series feature indeed this outstanding form of nonlinear dynamics, the strong dependence of the instantaneous variability of the series on its own past (Gouri´eroux, 1997, p. 1). That is the case of returns of assets or stocks.

Deﬁnition 2.1. — Return (Tsay, 2010, p. 3) If P is the price of an asset at time index t then holding the asset for one period from date t 1 to date t − t results in a percentage return (or one-period simple net return) given by

Pt Pt−1 Rt = − 100%. (2.1) Pt−1 ×

• Most financial studies involve returns, instead of prices, of assets7 (Tsay, 2010, p. 2). Figure 2.3 comprises the time series plot of the stock price of Siemens AG share (on trading days) from November 28, 1997 to October 12, 2015 (Siemens, 2015), and suggests a typical financial time series that meanders; the stock price appears to go through sustained periods of appreciation and then depreciation with no tendency to revert a long-run mean. However, the time series plot of the associated daily returns in Figure 2.4 reveals high volatility, especially in 2008 and 2009, coinciding with the start of a dramatic world financial crisis. Furthermore, the historical deviation of the daily returns, for a window width of 20 (trading) days,8 in Figure 2.5, leads

5Recall that the word volatile means liable to change rapidly and unpredictably (Stevenson and Lindberg, 2010). 6It should be noted that the term heteroscedastic (or heteroskedastic) ﬁnd its origin in two Greek words (Wikipedia, n.d.-e): heteros diﬀerent (or other Stevenson and Lindberg, 2010); and skedasis dispersion. 7For investors, return of an asset is a complete and scale-free summary of the investment opportunity, and return series are easier to handle than price series because the former have more attractive statistical properties (Tsay, 2010, p. 2). 8 k 1 k−1 k 2 The historical deviation for a window width of k days is calculated as HD = (R R ) , where Rt t k−1 i=0 t−i − t k 1 k−1 q denotes the daily return at time t and Rt = k i=0 Rt−i (Grimes, 2011). The historicalP deviation can be regarded as a window that moves over the (daily returns) data (Schipper, 2001, p. 5); moreover, it gives an idea of how the variance behaves over time. P

10 125

100 10

75 0 Stock price

50 Daily return (in %) −10

25 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 Year Year

Figure 2.3: Stock price of the Siemens AG share, from Figure 2.4: Daily returns of the Siemens AG share, from November 28, 1997 to October 12, 2015. December 1, 1997 to October 12, 2015. to belief that the associated (short-run) variance is not constant over time, a phenomenon econometricians call heteroscedasticity.

7.5

5.0

2.5 Historical deviation (in %) Historical deviation

0.0 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 Year

Figure 2.5: Historical deviation (window width = 20 days) of the daily returns of the Siemens AG share, from December 1, 1997 to October 12, 2015.

2.1 The birth of the autoregressive conditionally heteroscedastic model

While on sabbatical leave at the London School of Economics in 1979, Robert F. Engle III was looking for a model that could assess the validity of Milton Friedman’s conjecture (Friedman, 1977), that the unpredictability of inﬂation was a primary cause of business cycles not the level of inﬂation, and realized that this could only be plausible if the uncertainty was changing over time (Engle, 2004). Coincidentally, Clive W.J. Granger had recently developed a test for bilinear time series models,9 based on the dependence over time of squared residuals even though the residuals themselves were not dependent, and Engle suspected that it was detecting something besides bilinearity (Engle, 2004). The rationale behind of what David Forbes Hendry (b. 1944) coined autoregressive conditional heteroskedasticity (ARCH) is found in Engle’s seminal paper, Engle (1982), and can be described as follows.

9One of the most popular functional forms for representing between a dependent variable Y and two independent variables U and V is the bilinear form Y = a + bU + c V + d U V , where bU and c V can be interpreted as the main eﬀects and d U V as the interaction or cross-impact eﬀect (Granger and Andersen, 1978, p. 14). Realistic situations can frequently lead to bilinear-type models thoroughly discussed by Granger and Andersen (1978). Take, for instance, the return rate for a price process Pt : t N , Rt = (Pt Pt−1)/Pt−1, and suppose this rate is generated by a MA(1) model; then { ∈ } − Pt = Pt−1 + Pt−1 (εt + θεt−1) is already in a bilinear form (Granger and Andersen, 1978, pp. 14–15). ×

11 The forecast of today’s value X based upon the past information set available at time t 1, ψ , is t − t−1 E(X ψ ), under standard assumptions. t | t−1 The variance of this one-period forecast is given by V(X ψ ) and its expression suggests that past t | t−1 information plays a vital role in this conditional variance, which may therefore be a random variable. Yet traditional econometric models assume that the V(X ψ ) is constant. Consider, for example, the t | t−1 AR(1) model Xt = φXt−1 + εt, where εt is the usual WN process. Its conditional mean and variance E(X ψ ) E(X x ) and V(X x ) are equal to φx and σ2, whereas the unconditional t | t−1 ≡ t | t−1 t | t−1 t−1 mean and variance are given by E(X ) = 0 and V(X )= σ2/(1 φ2), respectively. t t − Engle (1982) expertly enunciates that for real processes one might expect to improve forecast intervals (in other words, improve the conditional variance) if past information is allowed to aﬀect the (one-period) forecast variance, hence the need for another class of time series models. This author continues: the standard approach of heteroscedasticity is/was to introduce an exogenous variable and the model might be Xt = εt Yt−1; moreover, the forecast interval depends on how the exogenous variable evolves. However, due to the fact that this model does not recognize that both conditional means and variances may jointly evolve over time, this sort of heteroscedasticity correction is rarely considered in time series analysis (Engle, 1982). According to Engle (1982), another approach is to allow the conditional variance to depend on past realizations of the time series suchlike in the bilinear models described by Granger and Andersen (1978). A particular and simple bilinear model is

Xt = εt Xt−1, (2.2) for which the conditional variance is equal to V(X x ) = σ2 x2 and the unconditional variance is t | t−1 t−1 either zero or inﬁnity,10 hence, an unattractive formulation and the suggestion of a preferable model. It reads as follows:

1/2 Xt = εt ht , (2.3)

2 ht = α0 + α1 Xt−1, (2.4) where ε : t Z (G)WN(0, 1) (Engle, 1982). { t ∈ }∼ This is the simplest example of the autoregressive conditional heteroscedasticity (ARCH) models introduced by Engle (1982).11 It is not exactly a bilinear model but according to Engle (1982) very close to one. While conventional time series operate under an assumption of constant variance, the ARCH model allows the conditional variance to change over time as function of past errors leaving the unconditional variance constant (Bollerslev, 1986). Moreover: the ARCH processes have zero mean and consist of uncorrelated r.v., as stated in the next section; they may be considered as a speciﬁc nonlinear time series models from the point view of statistical theory (Gouri´eroux, 1997, p. 2).

10Note, however, that the bilinear models constitute a class shown to contain (non-explosive and) invertible models potentially useful in forecasting (Granger and Andersen, 1978, p. 13); Engle (1982) also recognizes that there are slight generalizations of the bilinear model that avoid the curse of the zero or inﬁnite unconditional variance. 11Other authors favour terms such as: autoregressive conditionally heteroscedastic models (Gouri´eroux, 1997, p. 17); autoregressive conditional heteroscedastic models (Tsay, 2010, p. 64).

12 The ARCH models are perhaps the most signiﬁcant of Engle’s ﬁndings and cannot be dissociated from Granger’s bilinear models. Unsurprisingly, Robert F. Engle III shared the Nobel Prize for Economics in 2003 with C.W.J. Granger for methods of analyzing economic time series with time varying volatility (Francq and Zako¨ıan, 2010, p. xi).

2.2 Autoregressive conditionally heteroscedastic processes

The purpose of this section is to familiarize the reader with ARCH processes, their general properties and limitations. The ARCH model is deﬁned following closely Engle (1982), namely by adding the assumption of normality and expressing it in terms of ψt, the information set available at time t (Engle, 1982).

Deﬁnition 2.2. — ARCH(q) model (Engle, 1982) X : t Z is said to be a q th–order ARCH — or a ARCH(q) — process if it is characterized by the { t ∈ } following set of equations:

1/2 Xt = εt ht (2.5)

ht = h(Xt−1,Xt−2,...,Xt−q, α) (2.6)

X ψ N(0,h ), (2.7) t | t−1 ∼ t where: ε : t Z GWN(0, 1); ε is independent of h , for all t; α = (α , α , α ,...,α ) is a vector of { t ∈ }∼ t t 0 1 2 q (q + 1) (unknown) parameters with α > 0 and α 0, for i =1,...,q. 0 i ≥ • Interestingly enough, we can capitalize on the fact that E(ε ) = E(ε ε )=0(n N) and on the t t t−n ∈ independence between εt and ht to conclude that

1/2 E(Xt) = E(εt ht ) = E(ε ) E(h1/2) t × t = 0 (2.8)

E(X X ) = E(ε ε ) E(h1/2 h1/2 ) t t−n t t−n × t t−n = 0, n N, (2.9) ∈ that is, the ARCH(q) process has indeed zero mean and consists of uncorrelated r.v., as pointed out by Enders (2010, p. 127).

There are several formulations of the ARCH model that are appropriate for particular applications, such as the exponential and absolute value forms. However, the linear formulation is the simplest and consequently the most popular in practice.

Deﬁnition 2.3. — Linear ARCH process (Engle, 1982) A ARCH(q) model is said to be linear if

q 2 h(Xt−1,Xt−2,...,Xt−q, α)= α0 + αi Xt−i. (2.10) i=1 X •

13 According to Engle (1982), the linear ARCH(q) process, with α > 0 and α ,...,α 0, is covariance 0 1 q ≥ stationary (i.e., second order weakly stationary) if and only if the associated characteristic equation, formed from the α’s, has all roots outside the unit circle. In that case, the stationary variance is given by α E(X2)= 0 . (2.11) t 1 q α − i=1 i Some properties of the simplest and often very usefulP ARCH model (Engle, 1982) — the linear ARCH(1)

1/2 Xt = εt ht

2 ht = α0 + α1 Xt−1 X ψ N(0, h ) t | t−1 ∼ t

—, are stated next. Firstly, note that: if α = 0 then X : t Z GWN(0, α ); if α > 0 the expression of h suggests 1 { t ∈ }∼ 0 1 t that a large observation of Xt−1 will lead to a large variance in the next period, but the memory is restricted to one time period. 1/2 2 Secondly, the odd moments of Xt = εt ht , where ht = α0 + α1 Xt−1, are immediately seen to be zero (Engle, 1982). This follows from the fact that these innovations are normally distributed with zero mean, hence, with null odd moments (Johnson and Kotz, 1970, p. 47) and the independence between εt and ht:

E(X2r−1) = E(ε2r−1) E(hr−1/2) t t × t = 0 E(hr−1/2) × t = 0, r N. ∈

2r Thirdly, a condition that ensures the ﬁniteness of the unconditional moment E(Xt ) can be found in the following proposition.

Proposition 2.4. — Finiteness of the 2rth moment of the linear ARCH(1) process (Engle, 1982) For a positive integer r, the 2rth moment of the linear ARCH(1) process, with α > 0 and α 0, is 0 1 ≥ ﬁnite12 if and only if r αr (2i 1) < 1. (2.12) 1 − i=1 Y • Moreover, when 0 α2 < 1/3, the unconditional variance and 4th moment of X are given by ≤ 1 t α V(X ) = 0 (2.13) t 1 α − 1 3α2(1 + α ) E(X4) = 0 1 , (2.14) t (1 α )(1 3α2) − 1 − 1 12Engle (1982) wrote exists instead of is ﬁnite. Recall that the expectation of a non negative r.v. exists in the set of + extended non negative real numbers IR0 (Karr, 1993, p. 104).

14 respectively. Suﬃce to say that the variance of the process will be inﬁnite when α 1. 1 ≥ Finally, it can be shown that the unconditional kurtosis of Xt is equal to 4 E [Xt E(Xt)] κ(Xt) = { − 2 } [V(Xt)] 6α2 = 1 +3 (2.15) 1 3α2 − 1 2 and since κ(Xt) > 3 for 0 < α1 < 1/3, the linear ARCH(1) process generates data, as put by Engle (1982), with fatter tails than the normal distribution.

The ARCH model has proven useful in modeling several economic phenomena such as the inﬂation rate, as mentioned by Bollerslev (1986); ARMA models with ARCH errors have been also found to be successful by Weiss (1984) in modeling thirteen diﬀerent US macroeconomic time series (Bollerslev,

1986). However, the ARCH(q) model relies on the conditional variance ht written in terms of a moving 2 2 average of q previous shocks, Xt−1,...,Xt−q, hence, it is bound to have some limitations (and natural extensions). The most interesting and revealing ones were expertly listed by Tsay (2010, p. 119).

Although it is well know that in practice the prices of financial assets react differently to positive • and negative shocks, and yet the ARCH model considers that those shocks have the same effect on the conditional variance. The ARCH model is very restrictive parameter wise. For example, the linear ARCH(1) process • has a finite fourth moment iff 0 α2 < 1/3. The associated constraints are rather complicated ≤ 1 for larger values of q, thus, restricting the ability of ARCH models with Gaussian innovations to capture excess kurtosis. These models only provide a mechanical way to describe the behavior of the conditional variance; • they give no explanation(s) for such behavior.

The interested reader is referred to Gouri´eroux (1997, p. 3) for a few other (practical) limitations to the assumptions on the underlying ARCH models.

2.3 Generalized autoregressive conditionally heteroscedastic processes

The extension of the ARCH model tacked in this section bears, as Bollerslev (1986) pointed out, much resemblance to the extension of the standard AR process to the general ARMA process. Engle’s ARCH(q) model explicitly recognizes the diﬀerence between unconditional and conditional variance allowing the latter to be time-varying (Bollerslev, 1986) and dependent of past squared obser- 2 2 vations (Xt−1,...,Xt−q).

It seems natural and of practical interest to consider that the conditional variance ht depends not only on those MA terms, but also on AR terms, the lagged conditional variances (ht−1,...,ht−p), thus, corresponding to what Bollerslev (1986) considers a sort of adaptive learning mechanism. This leads to the generalized autoregressive conditionally heteroscedastic (GARCH)13 processes introduced by Bollerslev (1986) and in Deﬁnition 2.5.

13Bollerslev (1986) favours the term generalized autoregressive conditional heteroskedastic.

15 Deﬁnition 2.5. — GARCH(p, q) process (Bollerslev, 1986) X : t Z is said to be a GARCH(p, q) process if it satisﬁes the following set of equations: { t ∈ } 1/2 Xt = εt ht (2.16) q p 2 ht = α0 + αi Xt−i + βi ht−i, (2.17) i=1 i=1 X X where: ε : t Z GWN(0, 1); ε is independent of h , for all t; q 0; α > 0, α 0, for i =1,...,q; { t ∈ }∼ t t ≥ 0 i ≥ p 0; β 0, for i =1,...,p. Once again X ψ N(0,h ). ≥ i ≥ t | t−1 ∼ t • The ARCH and GWN processes are particular cases of the GARCH processes. When p = 0 one is dealing with a ARCH(q) process, whereas p = q = 0 leads to X : t Z GWN(0, α ). { t ∈ }∼ 0 One ought to note that the conditional variance of the GARCH(p, q) can be rewritten in terms of the backshift operator and also as a distributed lag of the past Xt’s:

2 ht = α0 + α(B) Xt + β(B) ht (2.18) = α [1 β(1)]−1 + α(B) [1 β(B)]−1 X2, (2.19) 0 − − t

q i p i p where: α(B)= i=1 αi B ; β(B)= i=1 βi B ; β(1) = i=1 βi. In additionP to this, Bollerslev (1986)P states in TheoremP 1 that the GARCH(p, q) is stationary iﬀ (0 )α(1) + β(1) = q α + p β < 1, and in this case ≤ i=1 i i=1 i P P E(Xt) = 0 (2.20) α E(X2) = 0 (2.21) t 1 q α p β − i=1 i − i=1 i E(X X ) = 0, n N. (2.22) t t−n P∈ P Once more one is dealing with a stochastic process with zero mean and consisting of uncorrelated r.v. The simplest but frequently useful GARCH process is the GARCH(1, 1) (Bollerslev, 1986):

1/2 Xt = εt ht (2.23)

2 ht = α0 + α1 Xt−1 + β1 ht−1 (2.24) X ψ N(0,h ). (2.25) t | t−1 ∼ t It is stationary if the coefficients satisfy 0 α +β < 1; its odd moments are null, as in the ARCH(q) ≤ 1 1 process, for the same reasons pointed out previously. 2m A condition that guarantees that the unconditional moment E(Xt ) is finite for a fixed positive integer m is set down in the next proposition.

Proposition 2.6. — Finiteness of the 2mth moment of the GARCH(1, 1) process (Bollerslev, 1986) The 2mth (m N) moment of the GARCH(1, 1) process is finite iff ∈ m m µ(α ,β ,m)= a αj βm−j < 1, (2.26) 1 1 j j 1 1 j=0 X where a = 1 and a = j (2i 1), j =1,...,m. 0 j i=1 − • Q 16 th The 2m moment of Xt can be obtained recursively: m−1 −1 2n m−n m am a E(X ) α µ(α1,β1,n) E(X2m)= n=0 n t 0 m−n . (2.27) t 1 µ(α ,β ,m) P − 1 1 In particular, if 0 3α2 +2α β + β2 < 1 then the fourth-order moment exists and ≤ 1 1 1 1 α E(X2) = 0 (2.28) t 1 α β − 1 − 1 3α2(1 + α + β ) E(X4) = 0 1 1 (2.29) t (1 α β )(1 3α2 2α β β2) − 1 − 1 − 1 − 1 1 − 1 (Bollerslev, 1986). Therefore the coefficient of kurtosis is given by 6α2 κ = 1 + 3 (2.30) 1 3α2 2α β β2 − 1 − 1 1 − 1 2 2 which is larger than 3 by assumption (3α1 +2α1β1 +β1 < 1) (Bollerslev, 1986) and by considering α1 > 0, leading to the conclusion that the Xt is a leptokurtic (or heavily tailed) r.v., as in the ARCH(1) case.

2.4 Model building and forecasting for GARCH processes

Finding an appropriate model for the available time series involves an iterative procedure based on model identiﬁcation, parameter estimation, diagnostic checking and model selection (Morais, 2012, p. 27), brieﬂy discussed in this section.

2.4.1 Model identiﬁcation

It is well known that the best tool to identify a time series available is the time series plot. Generally, when the time series is generated according to a GARCH model, sporadic periods of increased variation are observed spread through the series. th th 2 The plots of the sample n autocorrelation (ρn) and the k partial autocorrelation (φkk) of Xt obtained by solving the GARCH analogue to the Yule-Walker equations in (2.32) and (2.33) are also b b useful in the identiﬁcation of a GARCH model (Aradyula and Holt, 1988).

Remark 2.7. — Autocovariance, autocorrelation and partial autocorrelation functions of a squared GARCH(p, q) process (Bollerslev, 1986) If the GARCH(p, q) process has ﬁnite fourth-order moment, then the autocovariance function of X2 : { t t Z is given by ∈ } q p m γ = γ = Cov(X2,X2 )= α γ + β γ = ϕ γ , n p +1, (2.31) n −n t t−n i n−i i n−i i n−i ≥ i=1 i=1 i=1 X X X where: m = max p, q ; ϕ = α + β , i =1,...,m; α := 0, for i > q; and β := 0, for i>p. { } i i i i i The analogue of the Yule-Walker equations is given by m γn ρn = = ϕiρn−i, n p +1, (2.32) γ0 ≥ i=1 X th 2 and the k partial autocorrelation for Xt , φkk, can be found by solving the following set of k equations in terms of φk1,...,φkk: k ρn = φkiρn−i, n =1, . . . , k. (2.33) i=i X •

17 For an ARCH(q) process X : t Z , the PACF of X2 : t Z cuts oﬀ after the qth lag (Aradyula { t ∈ } { t ∈ } and Holt, 1988; Bollerslev, 1986).14 Consequently, the plot of the sample PACF of the squared process can be used to determine the ARCH order (Tsay, 2010, pp. 119–120). Moreover, the sample PACF of a squared GARCH process is in general not null and it dampens slowly (Aradyula and Holt, 1988).

2.4.2 Parameter estimation

After identifying a tentative model, the next step of model building comprises the estimation of the parameters of the GARCH model: α0, α1,...,αq and β1,...,βp. Bollerslev (1986) described the derivation of the maximum likelihood estimates of the parameters of a slightly more complex model: the GARCH regression model, obtained by letting the Xt’s be innovations in a linear regression (Bollerslev, 1986).

Deﬁnition 2.8. — GARCH regression model (Bollerslev, 1986) ⊤ 2 2 ⊤ Let zt = (1,Xt−1,...,Xt−q,ht−1,...,ht−p) and ω = (α0, α1,...,αq,β1,...,βp), then the GARCH regression model can be written as

X = ε h1/2 = Z y⊤b (2.34) t t t t − t q p 2 ⊤ ht = α0 + αi Xt−i + βi ht−i = zt ω (2.35) i=1 i=1 X X X ψ N(0,h ), (2.36) t| t−1 ∼ t where Zt is the dependent variable, yt a vector of explanatory variables and b a vector of unknown parameters. • Please note that the GARCH(p, q) process is a particular case of the GARCH(p, q) regression model when b = 0. According to Bollerslev (1986), the log likelihood function for a sample of T observations is apart from some constant, T 1 L (θ)= l (θ), (2.37) T T T t=1 X 2 ⊤ ⊤ 1 1 Xt where θ = (b , ω ) and lT (θ)= log(ht) . − 2 − 2 ht If lT (θ) is diﬀerentiated with respect to ω one gets

∂l 1 ∂h X2 t = t t 1 , (2.38) ∂ω 2h ∂ω h − t t ∂2l X2 ∂ 1 ∂h 1 ∂h ∂h X2 t = t 1 t t t t , (2.39) ∂ω∂ω⊤ h − ∂ω⊤ 2h ∂ω − 2h2 ∂ω ∂ω⊤ h t t t t

∂ht ⊤ p ∂ht−i where ∂ω = zt + i=1 βi ∂ω (Bollerslev, 1986). Since the conditionalP expectation of the ﬁrst term in (2.39) is zero, the Fisher’s information matrix of ω is consistently estimated by the sample analogue of the last term in (2.39) which only involves ﬁrst derivatives (Bollerslev, 1986).

14As the PACF for an AR(q) process (Bollerslev, 1986).

18 Bollerslev (1986) also mentioned an iterative procedure that can be used to obtain the maximum likelihood estimates: the Berndt, Hall, Hall and Hausman algorithm (Berndt et al., 1974), described in the following remark.

Remark 2.9. — Berndt, Hall, Hall and Hausman algorithm (Bollerslev, 1986) Let θ(i) denote the estimate of θ after the ith iteration of the algorithm. Then

T −1 T ∂l ∂l ∂l θ(i+1) = θ(i) + λ t t t , (2.40) i ∂θ ⊤ ∂θ t=1 ∂θ ! t=1 X X (i) where ∂lt/∂θ is evaluated at θ and λi is a variable step length chosen to maximize the likelihood function in the given direction. The direction vector is calculated from a least squares regression of a T 1 vector of ones on ∂l /∂θ. × t • 2.4.3 Diagnostic checking

Once the parameters have been estimated, it is necessary to assess the model adequacy by checking whether the model assumptions are satisfied (Wei, 1990, p. 149; Morais, 2012, p. 38). The adequacy of the fitted GARCH model can be verified by examining the standardized residual series X : t =1,...,T , where X = X /h1/2 and h1/2 is the estimated conditional volatility. { t } t t t t For a properly specified GARCH model, X and X2 should not exhibit serial correlation (Zivot, 2012), e e b t bt after all, X /h1/2 = ε , t Z, are i.i.d. r.v., and so are X2/h = ε2, t Z. t t t ∈ e e t t t ∈ In addition to the visual inspection of the sample ACF of the standardized residuals, one can perform the Ljung-Box(-Pierce) test that takes into consideration the magnitudes of the squared sample autocorrelations as a group and checks whether any of the first K process autocorrelations is different from zero (Shumway and Stoffer, 2006, p. 149; Morais, 2012, p. 39).15

Deﬁnition 2.10. — Ljung-Box(-Pierce) test (McLeod and Li, 1983) The Ljung-Box(-Pierce) lack of ﬁt test has an observed statistic equal to

K ρ 2 (k) Q (K)= T (T + 2) W , (2.41) W T k k=1 − X b where ρ (k) = T w w / T w 2. Under the null hypothesis H : ρ (1) = = ρ (K) = 0, w t=k+1 t t−k t=1 t 0 W ··· W 2 16 it follows approximatelyP the χKP−m distribution, where m is the number of parameters estimated. b b b b One would reject H at the signiﬁcance level α if Q (K) exceeds the (1 α) quantile of the χ2 0 W − − K−m −1 distribution, Fχ2 (1 α). K−m − • Furthermore, one can also use the Lagrange multiplier test statistic to check the conditional het- 2 eroscedasticity of the square series Xt (Tsay, 2010, p. 114; Bollerslev, 1986).

The model assumption that thee innovations ε : t Z are normally distributed r.v. should also { t ∈ } be checked using the standardized residuals (Zivot, 2012); to do so, one can use the Jarque-Bera and

15Please note that this lack-of-fit test can also be performed during the model identification step, to check for the presence of a GARCH effect. 16And the value K is chosen somewhat arbitrarily, for instance, K = 10.

19 Shapiro-Wilk tests (Shumway and Stoﬀer, 2006, p. 285) described below. Additionally, a QQplot of the standardized residuals can help in identifying departures from normality (Zivot, 2012). The Jarque-Bera statistic tests the residuals of a ﬁtted model for normality based on the observed skewness and kurtosis, whereas the Shapiro-Wilk test does it based on the empirical order statistics.

Deﬁnition 2.11. — Jarque-Bera test (Jarque and Bera, 1987)

Let (W1,...,WT ) be a set of T independent observations on a random variable W , denote the unknown population mean of Wi by µ = E(Wi) and, for convenience, write Wi = µ + ui. Moreover, assume that 17 the probability density function of ui, f(ui) is a member of the Pearson family, it means that df(u ) (c u )f(u ) i = 1 − i i , u R. (2.42) du c c u + c u2 i ∈ i 0 − 1 i 2 i It follows that the logarithm of the likelihood function of the T observations is given by +∞ +∞ c u l(µ,c ,c ,c )= T log exp 1 − i du du 0 1 2 − c c u + c u2 i i Z−∞ Z−∞ 0 − 1 i 2 i T +∞ c1 ui + − 2 dui . (2.43) c0 c1u + c2u i=1 −∞ i i X Z − To test the hypothesis of normality means testing H0 : c1 = c2 = 0 and the Jarque-Bera test statistic is given by T (b 3)2 JB = b + 2 − (2.44) 6 1 4 b b j µ3 µ4 T (Wi−W ) T Wi 18 where √b1 = 3/2 , b2 = 2 , µj = and W = . (µb2) µb2 i=1 T i=1 T The hypothesis of normality is rejected,P for large samples,P if the computed value of JB is larger than b the appropriate signiﬁcance point of a chi-square distribution with two degrees of freedom. • Deﬁnition 2.12. — Shapiro-Wilk test (Shapiro and Wilk, 1965) ⊤ Let m = (m1,...,mT ) denote a vector of expected values of standard normal order statistics, and let V = v be the corresponding T T covariance matrix. That is, if (U ,U ,...,U ) denotes an ordered { ij } × 1 2 T random sample from a Normal distribution with mean 0 and variance 1, then

E(Ui)= mi, i =1,...,T

Cov(Ui,Uj )= vij , i,j =1,...,T.

⊤ Let W = (W1, W2,...,WT ) denote a vector of ordered random variables, where Wi = µ + σUi. The null hypothesis is that W is an ordered sample from a Normal distribution with unknown mean µ and unknown variance σ2. The Shapiro-Wilk test statistic for normality is given by 2 T i=1 aiWi SW = (2.45) PT (W W)2 i=1 i − ⊤ −1 ⊤ m V −1 −1 P where a = (a1,...,aT )= (m⊤V V m)1/2 . The minimum and maximum values of SW are Ta2/(T 1) and 1 (respectively); and small values of 1 − SW are significant, i.e., indicate non-normality. Thus, one would reject H0 at the significance level α if the observed value of SW exceeds the (1 α) quantile of its null distribution. − − • 17The Pearson family includes a wide range of distributions, namely, the Normal, Beta, Gamma, Student’s t and F distributions. 18 Note that √b1 and b2 are, respectively, the skewness and kurtosis sample coefficients.

20 2.4.4 Model selection

Choosing the most suitable model is the essence of data analysis since it has as outcome good forecasting results with less prediction errors. The amount of parameters of a model plays an important role in either analysis and forecasting, since the addition of unnecessary lags reduces the sum of squares of the estimated residuals and the forecasting performance as well (Javed and Mantalos, 2013). There are several information criteria to select the model that should be used to forecast. The Akaike’s, Bayesian (also known as Schwarz’s information criterion), Shibata’s and Hannan-Quinn’s information criteria are provided by the output of the function garchFit implemented in the fGarch package of the statistical software R. They combine the log likelihood value corresponding to the parameters estimated \ (log(L)), the number of parameters estimated (m) and the number of observations that are used to estimate the model (T ). These four information criteria are defined assuming that an ARMA(a,b) GARCH(p, q) model, with − m = a + b + p + q + 2 parameters, is fitted to the time series x : t =1,...,T .19 { t } Definition 2.13. — Akaike’s, Bayesian, Shibata’s and Hannan-Quinn’s information criteria (Ghalanos, 2014, p. 23) The Akaike’s (AIC), Bayesian (BIC), Shibata’s (SIC) and Hannan-Quinn’s (HQIC) information criteria are given by: \ log(L) 2m AIC(m) = 2 + (2.46) − T T \ log(L) m log(T ) BIC(m) = 2 + (2.47) − T T \ log(L) 2m SIC(m) = 2 + log 1+ (2.48) − T T \ log(L) 2m log(log(T )) HQIC(m) = 2 + . (2.49) − T T

• The optimal order m of the model is such that those criteria are minimum.

2.4.5 Forecasting

In this subsection one provides an expression for the multistep-ahead volatility forecasts of a GARCH(1, 1) model, considering the forecast origin s. The 1-step-ahead forecast is given by

2 hs(1) = α0 + α1Xs + β1hs, (2.50) where Xs and hs are assumed to be known at time s (Tsay, 2010, p. 133). 2 2 Since Xt = εt ht, the volatility equation can be written as

h = α + (α + β )h + α h (ε2 1). (2.51) t+1 0 1 1 t 1 t t − 19A standard practise is to use simultaneously a GARCH(p,q) model to account for the conditional variance and an ARMA(a, b) model for the conditional mean. The resulting model is termed an ARMA(a, b) GARCH(p,q) model. −

21 When t = s + 1 then h = α + (α + β )h + α h (ε2 1) (2.52) s+2 0 1 1 s+1 1 s+1 s+1 − and since E(ε2 1 ψ ) = 0, the 2-step-ahead volatility forecast is s+1 − | s

hs(2) = α0 + (α1 + β1)hs(1). (2.53)

As a consequence, the l-step-ahead forecast (l> 1) satisﬁes the equation

h (l)= α + (α + β )h (l 1) (2.54) s 0 1 1 s − that can be rewritten as

α [1 (α + β )l−1] h (l)= 0 − 1 1 + (α + β )l−1h (1). (2.55) s 1 α β 1 1 s − 1 − 1 Therefore the multistep-ahead volatility forecasts of a GARCH(1, 1) model converges to the unconditional variance of X , α /(1 α β ), as the forecast horizon l increases to inﬁnity provided that α + β < 1 t 0 − 1 − 1 1 1 (Tsay, 2010, p. 133).

The ﬂowchart displayed in Figure 2.6 summarizes how the GARCH model building and forecasting should be done, as suggested in Akkeren et al. (2005).

Observed data Estimation of Class of models Model Identification parameters Model validation (i) A class of models is (ii) Based on data analysis, (iii) The model (iv) Model is /isn't considered assuming a subset of models is parameters validated using a certain hypothesis identified are estimated hypothesis testing

Yes Obtain the forecast (v) The model parameters are defined and the out- of-sample forecasting can be initiated

Figure 2.6: Model building and forecasting scheme.

2.5 Illustrations

This section comprises two instructive illustrations of the model building procedure, by making use of the R statistical software, in particular the fGarch package.20

20Interesting enough, the Mathematica software refers to the GARCH(p,q) process as GARCHProcess[q,p]. This process can be simulated, estimated from data, and used to produce forecasts of future behavior when combined with the functions RandomFunction, EstimatedProcess and TimeSeriesForecast, respectively (Wolfram, 2015).

22 Example 2.14. — Intel stock data: ARCH process ﬁt (Tsay, 2010, pp. 111–112, 123–131)21 The data set refers to the monthly log returns22 of Intel stock from January 1973 to December 2008 and it is available in Tsay (2005).

Model identiﬁcation and parameter estimation • After having typed and run the following commands for the R statistical software, the data set was loaded from a txt ﬁle and one obtained the time series plot of the monthly log returns in Figure 2.7. I7308=read.table("m-intc7308.txt",header=TRUE) LRI7308=log(I7308[,2]+1) LRI7308TS = ts(LRI7308, start=1973, frequency=12) plot(LRI7308TS,type="l",xlab="Time index", ylab="Log Return") Log Return −0.6 −0.2 0.2 1975 1980 1985 1990 1995 2000 2005 2010 Year

Figure 2.7: Monthly log returns of the Intel stock, from 1973(1) to 2008(12).

This plot suggests that there are a few periods of high volatility, namely in the middle 1970s, in the late 1980s and in the early 2000s. The first period of high volatility is probably related to the expansion of the Intel’s business in the 1970s since that led to the dominance of this company in several competitive markets. The second one, one believes, is related to the PC industry boom in the late 1980s that beneficiated primary Intel. Finally, the last one should be linked to the slowing of PC demand and to the rise of low cost PC that caused the end of Intel dominance and that might be associated to the instability of the Intel stock price (Wikipedia, n.d.-f). The R statistical software not only calculates the autocorrelation and partial autocorrelation functions of a stochastic process, but also provides estimates of ρ and φ (k N) (Morais, 2012, p. k k,k ∈ 41). To obtain the graphs with the estimates of the ACF and PACF of various functions of the monthly log returns of the Intel stock (Figure 2.8), it suffices to issue the commands: acf(LRI7308,main="",xlim=c(1,25),ylim=c(-0.15,0.15)) acf(LRI7308ˆ2,main="",xlim=c(1,25),ylim=c(-0.05,0.25)) acf(abs(LRI7308),main="",xlim=c(1,25),ylim=c(-0.05,0.20)) pacf(LRI7308ˆ2,main="",xlim=c(1,25)) According to Tsay (2010, p. 111), the sample ACF of the log returns (Figure 2.8(a)) suggests no serial correlation, except for two slight spikes at lags 7 and 14. The sample ACF of the squared log returns (Figure 2.8(b)) and of the absolute log returns (Figure 2.8(c)) indicate that the monthly log returns are not serially independent (Tsay, 2010, p. 111). The combination of those three plots

21Please note that this author analyzed this data set using both S-Plus and the R statistical softwares; the associated estimates are slightly diﬀerent. 22 If Rt is the return of an asset at time index t then the log return (or compounded return) is calculated as rt = ln(1+Rt) (Tsay, 2010, p. 5).

23 ACF ACF

−0.15 0.05 5 10 15 20 25 −0.05 0.15 5 10 15 20 25 Lag Lag

(a) ACF log returns. (b) ACF squared log returns. ACF Partial ACF −0.05 0.10 5 10 15 20 25 −0.10 0.10 5 10 15 20 25 Lag Lag

Figure 2.8: Sample ACF and PACF of various functions of the monthly log returns of the Intel stock, from 1973(1) to 2008(12).

seems to suggest that the log returns are serially uncorrelated but dependent (Tsay, 2010, p. 111) and this behavior can be captured by an ARCH model. The sample PACF of squared log returns (Figure 2.8(b)) decreases suddenly after lag 3 and that indicates that the ARCH(3) model might be adequate.23 Consequently, an ARCH(3) model was ﬁtted by using the commands24 LRI7308ARCH3=garchFit(LRI7308 garch(3,0),data=LRI7308,trace=FALSE) summary(LRI7308ARCH3) ∼ and the estimated model was

rt = 0.011852 + Xt (2.56)

1/2 2 2 2 Xt = εt ht = εt 0.010588+ 0.237149Xt−1 +0.072747Xt−2 +0.053080Xt−3, (2.57) q where ε i.i.d. N(0, 1). t ∼ The p-values associated with the standard error of the estimates of α2 and α3 are 0.1216 and 0.2539 (respectively) and suggest that these coeﬃcients are statistically nonsigniﬁcant at the 5% level. Therefore a simpler model might be considered: the ARCH(1) model. The estimated model, obtained by running LRI7308ARCH1=garchFit(LRI7308 garch(1,0),data=LRI7308,trace=FALSE) summary(LRI7308ARCH3) ∼ was

rt = 0.012637 + Xt (2.58)

1/2 2 Xt = εt ht = εt 0.011195+ 0.379492Xt−1, (2.59) q 23According to Tsay (2010, pp. 119–120) the PACF of the squared series can be used to determine the ARCH order. The terms of the squared series can be written in the form of an AR model (Tsay, 2010, pp. 119–120). It is also known that the PACF of an AR(p) cuts of at lag p (Morais, 2012, Table 2.1, p. 39). Therefore when there is a sudden change in the PACF of the squared series in lag p and the series presents evidences of heteroscedasticity, an ARCH(p) model might be adequate. 24Please note that, in the fGarch package, the garchFit is implemented in such a way that garch(q,p) denotes a GARCH(p,q) model.

24 where ε i.i.d. N(0, 1). t ∼ Then the estimated unconditional variance of Xt is

α 0.011195 σ2 = 0 0.01804167 X 1 α ≈ 1 0.379492 ≈ − 1 − b It is also interesting to addb that α2 =0.3794922 =0.1440142 < 1/3, i.e., the unconditional fourth 1 b moment of the monthly log return of Intel stock seems to exist. b Diagnostic checking • The plots of the sample ACF of various functions of the standardized residuals and of the standardized residuals themselves were obtained by running LRI7308ARCH1RES=residuals(LRI7308ARCH1,standardize=TRUE) acf(LRI7308ARCH1RES,main="",xlim=c(1,25),ylim=c(-0.15,0.15)) acf(LRI7308ARCH1RESˆ2,main="",xlim=c(1,25),ylim=c(-0.05,0.2)) acf(abs(LRI7308ARCH1RES),main="",xlim=c(1,25),ylim=c(-0.15,0.15) LRI7308ARCH1RESTS = ts(LRI7308ARCH1RES, start=1973, frequency=12) plot(LRI7308ARCH1RESTS,type="l",main="",xlab="Year", ylab="Stres") and can be found in Figure 2.9. It is known that a white noise series has all sample ACF close to zero (Tsay, 2010, p. 36), that is the case of the standardized residuals (Figure 2.9(a)) and of the absolute standardized residuals (Figure 2.9(c)). As for the sample ACF of the squared standardized residuals, it has two spikes at lags 12 and 21 that, one believes, does not rule out the hypothesis that this series behaves according to a white noise process. ACF ACF

−0.15 0.05 5 10 15 20 25 −0.05 0.10 5 10 15 20 25 Lag Lag

(a) ACF of the standardized residuals. (b) ACF of the squared standardized residuals. ACF Stres −0.15 0.05 5 10 15 20 25 −4 −2 0 2 4 1975 1980 1985 1990 1995 2000 2005 2010 Lag Year

Figure 2.9: Diagnostics for the estimated ARCH(1) ﬁt of monthly log returns of the Intel stock, from 1973(1) to 2008(12).

The p-values associated with the Ljung-Box-Pierce tests performed for the original and squared standardized residuals can be found in Table 2.1. They were computed for lag 10 and by using the commands: Box.test(LRI7308ARCH1RES,10,type="Ljung") Box.test(LRI7308ARCH1RESˆ2,10,type="Ljung") These p-values lead to the conclusion that the ﬁtted ARCH(1) model is adequate to describe the monthly volatility of Intel stock returns, at the signiﬁcance level of 5%.

25 Table 2.1: Ljung-Box test for the estimated ARCH(1) ﬁt of monthly log returns of the Intel stock, from 1973(1) to 2008(12).

Ljung-Box statistic p-value standardized residuals Q(10) = 12.54002 0.25053820 squared standardized residuals Q(10) = 16.01590 0.09917815

Additionally, the Jarque-Bera and the Shapiro-Wilk tests were applied to the standardized residuals and the respective p-values, available on the summary of the estimated ARCH(1), can be found in Table 2.2. They are close to 0, therefore the standardized residuals do not seem to be normally distributed.25

Table 2.2: Tests to check the normality for the estimated ARCH(1) ﬁt of monthly log returns of the Intel stock, from 1973(1) to 2008(12).

Test Statistic p-value Jarque-Bera 73.0484 0.000000000 Shapiro-Wilk 0.98580 0.000000596

Model selection • It is time to select one of the tentative models, by comparing the corresponding information criterion \ values. Table 2.3 contains the values of the log(L), m, AIC, BIC, SIC and HQIC corresponding to the ARCH(3) and ARCH(1) estimated models for the monthly log returns of Intel stock from January 1973 to December 2008. Those values are provided when the summary of the ﬁtted model is requested in order to obtain the estimates of the model parameters.

Table 2.3: Information criteria for the estimated models for the monthly log returns of the Intel stock, from 1973(1) to 2008(12).

Information criteria \ Processes log(L) m AIC BIC SIC HQIC ARCH(3) 291.8891 5 1.328 1.281 1.328 1.310 ARCH(1) 288.0589 3 −1.320 −1.291 −1.320 −1.309 − − − −

The values of the information criteria obtained for the two models are quite similar, therefore the ARCH(1) model should be chosen since it is the one with fewer parameters.26 • Example 2.15. — Standard & Poor’s (S&P) 500 index data: GARCH process ﬁt and forecasting (Tsay, 2010, Example 3.3, pp. 134–139)27 The data set was found in Tsay (2005) and refers to the monthly excess returns28 of Standard & Poor’s (S&P) 500 index, from January 1926 to December 1991.

Model identiﬁcation and parameter estimation • To identify a model for the data set, one starts by obtaining the time series plot of the 792 observations in Figure 2.10, by issuing the following commands:

25Expectedly, Tsay (2010, pp. 125–126) also fits an ARCH(1) model with Student-t innovations to the monthly log returns of the Intel stock series. One deemed this new model out of the scope of this illustration. 26One ought to add that Tsay (2010, p. 126) mentions that the GARCH(1, 1) fit is a more appropriate conditionally heteroscedastic model for the monthly log returns of Intel stock. 27As in the previous example, this author analyzed this data set using both S-Plus and the R statistical softwares and there are also small differences in the estimates. 28 The (simple) excess return of an asset at time t is equal to Zt = Rt R0t where Rt and R0t are the asset’s return and the return of some reference asset at time t, respectively (Tsay, 2010, p.− 6).

26 ER2691SP500=scan(file="sp500.txt") ER2691SP500TS=ts(ER2691SP500, start=1926, frequency=12) plot(ER2691SP500TS,type="l",xlab="Year", ylab="Excess return") This time series also presents periods of high volatility, specially from the late 1920s until the early 1940s. This period of major instability is most likely related to the Wall Street Crash of 1929 that was followed by the Great Depression, a severe economic depression that took place during the 1930s (Wikipedia, n.d.-g; Wikipedia, n.d.-h). Excess return −0.2 0.0 0.2 0.4

1930 1940 1950 1960 1970 1980 1990 Year

Figure 2.10: Monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12).

Moreover, by looking at the graphs of the sample ACF of the monthly excess returns of the S&P 500 index from 1926 to 1991 and of the sample PACF of the squared monthly excess returns, obtained by running acf(ER2691SP500,main="",xlim=c(1,25),ylim=c(-0.15,0.15)) pacf(ER2691SP500ˆ2,main="",xlim=c(1,25)) ACF Partial ACF −0.1 0.2 −0.15 0.05 5 10 15 20 25 5 10 15 20 25 Lag Lag

(a) ACF excess return. (b) PACF squared excess return.

Figure 2.11: Sample ACF of the monthly excess returns of the S&P 500 index and PACF of the squared monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12).

and shown in Figure 2.11, one can add that the excess return series has some serial correlations at lags 1 and 3 and that the PACF of the squared excess returns shows strong linear dependence (Tsay, 2010, p. 135). Consequently, one chooses a MA(3) model to specify the equation of the time series mean. By running the command ER2691SP500MA3 = arima(ER2691SP500,order=c(0,0,3),method = "ML") one gets the following ML estimated MA(3) model for the excess returns:

Z =0.0062 + X +0.0949X +0.0096X 0.1415X , (2.60) t t t−1 t−2 − t−3

2 with σX = 0.003321. Nonetheless, Tsay (2010, p. 135) recommends the use of a simpler model: the AR(3) model. Under the normality assumption, the ML estimated AR(3) model for the excess b

27 returns was provided by issuing the command ER2691SP500AR3 = arima(ER2691SP500,order=c(3,0,0),method = "ML") and the ﬁtted model was

Z =0.0062+ 0.089Z 0.0238Z 0.1229Z + X . (2.61) t t−1 − t−2 − t−3 t Posteriorly, to model also the conditional heteroscedasticity apparent in this data set, Tsay (2010, p. 135) decided to combine the GARCH(1, 1) and the AR(3) models. Consequently, one proceeded with the joint estimation of a AR(3) GARCH(1, 1) model. The commands − ER2691SP500AR3GARCH11=garchFit( arma(3,0)+garch(1,1),data=ER2691SP500,trace=FALSE) summary(ER2691SP500AR3GARCH11) ∼ were used and the model obtained was

Z = 0.0077+ 0.03197Z 0.03026Z 0.01065Z + X (2.62) t t−1 − t−2 − t−3 t 1/2 2 Xt = εt ht = εt 0.000079746+ 0.12425Xt−1 +0.85302ht−1. (2.63) q Therefore the estimated unconditional variance of Xt is

2 α0 0.000079746 σX = = 0.003508579. 1 α β 1 0.85302 0.12425 ≈ 1 1 − − − b − However, the t ratiosb of the parameters in the mean equation suggest that all three AR coeﬃcients b b are insigniﬁcant at the 5% level and therefore they should be dropped (Tsay, 2010, p. 136). The resulting estimated GARCH(1, 1) model was

Zt = 0.0074497 + Xt (2.64)

1/2 2 Xt = εt ht = εt 0.000080615+ 0.12198Xt−1 +0.85436ht−1, (2.65) q and it was provided by issuing the following commands: ER2691SP500GARCH11=garchFit( garch(1,1),data=ER2691SP500,trace=FALSE) summary(ER2691SP500GARCH11) ∼ The estimated unconditional variance of Xt is

2 α0 0.000080615 σX = = 0.003407227. 1 α β 1 0.85436 0.12198 ≈ 1 1 − − − b − Please note that b b b α + β 0.85436+ 0.12198 0.97634 1. 1 1 ≈ ≈ ≈ As put by Tsay (2010, p. 137),b b this phenomenon is usually observed in practice and it leads to

imposing the constraint α1 + β1 = 1 in a GARCH(1, 1) model, resulting in an integrated GARCH model (for further details see Section 2.6). Diagnostic checking • The plots of the estimated volatility and of the standardized residuals associated with the GARCH (1, 1) model ﬁtted for the S&P 500 index monthly excess returns are shown in Figure 2.12 and were obtained by running: [email protected] ER2691SP500GARCH11CSDTS = ts(ER2691SP500GARCH11CSD, start=1926, frequency=12) plot(ER2691SP500GARCH11CSDTS,type="l",main="",xlab="Year", ylab="h t") ER2691SP500GARCH11RES=residuals(ER2691SP500GARCH11,standardize=TRUE) ER2691SP500GARCH11RESTS = ts(ER2691SP500GARCH11RES, start=1926, frequency=12) plot(ER2691SP500GARCH11RESTS,type="l",main="",xlab="Year", ylab="Stres")

28 h_t Stres −4 −2 0 2 0.05 0.10 0.15 0.20

1930 1940 1950 1960 1970 1980 1990 1930 1940 1950 1960 1970 1980 1990 Year Year

(a) Estimated volatility. (b) Standardized residuals.

Figure 2.12: Estimated volatility and standardized residuals of the GARCH(1, 1) model ﬁtted to the monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12).

The standardized residuals series seems to behave according to a white noise process (Tsay, 2010, p. 137). In order to conﬁrm that, the sample ACF of the standardized residuals (Figure 2.13(a)) and of the squared standardized residuals (Figure 2.13(b)) were plotted by issuing: acf(ER2691SP500GARCH11RES,main="",xlim=c(1,25),ylim=c(-0.15,0.15)) acf(ER2691SP500GARCH11RESˆ2,main="",xlim=c(1,25),ylim=c(-0.15,0.15)) ACF ACF

−0.15 0.05 5 10 15 20 25 −0.15 0.05 5 10 15 20 25 Lag Lag

(a) ACF of the standardized residuals. (b) ACF of the squared standardized residuals.

Figure 2.13: Sample ACF of the standardized residuals and of the squared standardized residuals of the GARCH(1, 1) ﬁt to the monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12).

These plots do not suggest any signiﬁcant serial correlations or conditional heteroscedasticity in the standardized residual series (Tsay, 2010, p. 137). The previous conclusions are also substantiated by the large p-values of Ljung-Box test in Table 2.4 that were computed by running Box.test(ER2691SP500GARCH11RES,12,type="Ljung") Box.test(ER2691SP500GARCH11RES,24,type="Ljung") Box.test(ER2691SP500GARCH11RESˆ2,12,type="Ljung") Box.test(ER2691SP500GARCH11RESˆ2,24,type="Ljung")

Table 2.4: Ljung-Box test for the estimated GARCH(1, 1) ﬁt of the monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12).

Ljung-Box statistic p-value standardized residuals Q(12) = 12.5674 0.4178 Q(24) = 28.2482 0.2482 squared standardized residuals Q(12) = 13.2654 0.3500 Q(24) = 25.6193 0.3728

The summary of the ﬁtted GARCH(1, 1) model also provides the observed values of the Jarque- Bera and the Shapiro-Wilk statiscs applied to the residuals and the respective p-values can be found in Table 2.4. The Jarque-Bera statistic tests the residuals of the ﬁt for normality based on

29 the observed skewness and kurtosis, and it appears that the standardized residuals have some non- normal skewness and kurtosis, since the associated p-value is approximately zero. The Shapiro-Wilk statistic tests the residuals of the ﬁt for normality based on the empirical order statistics and in this case the associated p-value is also rather small, thus, one suspects that the residuals are not normal.29

Table 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).

Test Statistic p-value Jarque-Bera 80.3211 0.000000000 Shapiro-Wilk 0.98505 0.000000314

The QQ-plot of the standardized residuals presented in Figure 2.14 was obtained as follows: qqnorm(ER2691SP500GARCH11RES, main="", xlab="Normal quantile", ylab="Quantile of residual") qqline(ER2691SP500GARCH11RES, distribution = qnorm). This plot also suggests that the standardized residuals are not normally distributed; in particular, the distribution of the standardized residuals seems to have tails heavier than the normal distribution.30 −4 −2 0 2 Quantile of residual

−3 −1 0 1 2 3 Normal quantile

Figure 2.14: QQ-plot of the standardized residuals for the estimated GARCH(1, 1) ﬁt of the monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12).

Model selection • Once more, the values of four information criteria (AIC, BIC, SIC and HQIC) in Table 2.6 were used to select the most suitable model. The information criteria for the ARMA models were computed using equations (2.46), (2.47), (2.48) and (2.49). The ones referring to the GARCH models were directly obtained can be found in the summary of the ﬁtted model. The GARCH(1, 1) model is associated with the smallest values of all four information criteria, therefore it seems the best model to describe the behavior of the monthly excess returns of the S&P 500 index.

29 Note that Tsay (2010, pp. 138–139) reestimated the GARCH(1, 1) model considering εt with a Student-t distribution with 5 degrees of freedom. 30It is worth mentioning that the command garchFit allows for several conditional distributions that have to be speciﬁed in cond.dist.

30 Table 2.6: Information criteria for the estimated models for the monthly excess returns of the S&P 500 index, from 1926(1) to 1991(12).

Information criteria \ Processes log(L) m AIC BIC SIC HQIC MA(3) 1136.300 4 2.859 2.836 2.859 2.850 AR(3) 1135.250 4 −2.857 −2.833 −2.857 −2.848 AR(3) GARCH(1, 1) 1272.179 7 −3.195 −3.154 −3.195 −3.179 GARCH− (1, 1) 1269.455 4 −3.196 −3.172 −3.196 −3.187 − − − −

Forecasting • 2 The volatility equation ht =0.000080615+ 0.12198Xt−1 +0.85436ht−1 can be used to forecast the volatility of monthly excess returns of the S&P 500 index. In fact, at the forecast origin s, the 1-step-ahead forecast is given by

2 hs(1) = 0.000080615+ 0.12198Xs +0.85436hs, (2.66)

where Xs is the residual of the mean equation (Tsay, 2010, pp. 137–138); the l-step-ahead forecast obtained using the recursion h (l) = α + (α + β )h (l 1), for l = 2, 3,... . The starting value s 0 1 1 s − h0 is ﬁxed at either zero or the (estimated) unconditional variance of Xt (Tsay, 2010, p. 138), i.e., 2 h0 =0 or h0 = σX . The mean and volatility forecasts for the monthly excess returns of the S&P 500 index with forecast b origin s = 792, based on the GARCH(1, 1) model deﬁned in equations (2.64) and (2.65), are available in Table 2.7. The l-step-ahead forecasts (l = 1,..., 5) were computed by issuing the following command:

predict(ER2691SP500GARCH11,5)

Table 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).

l 1 2 3 4 5 Monthly excess return 0.00744972 0.00744972 0.00744972 0.00744972 0.00744972 Volatility 0.05377242 0.05388567 0.05399601 0.05410353 0.05420829

•

2.6 Particular cases and extensions

According to Enders (2010, p. 154), financial analysts are very eager to obtain precise estimates of the conditional variance of an asset price. Moreover, since GARCH models can forecast conditional volatility thus enabling to measure the risk of an asset over the holding period, several extensions of the basic GARCH model have been developed (Enders, 2010, p. 154). The particular cases and extensions briefly described in the next subsection tend to be especially suited to to estimating the conditional volatility of univariate financial data.

31 2.6.1 Other univariate models of conditional variance

In ﬁnancial time series, if one estimates a GARCH(1, 1) model using a long time series of stocks returns, one often ﬁnds that α1 + β1 is very close to one (Enders, 2010, p. 154). In fact, when α1 + β1 = 1 q p or more generally i=1 αi + i=1 βi = 1, one is dealing with a non stationary process, the so called integrated GARCHP process —P for short IGARCH(p, q) (Francq and Zako¨ıan, 2010, p. 38; Tsay, 2010, p. 140). Interestingly enough, the conditional variance of a IGARCH(1, 1) process can be written as a geometrically decaying function of realizations:

∞ α0 i 2 ht = + (1 β1) β1 Xt−1−i (2.67) 1 β1 − i=0 − X (Enders, 2010, p. 155). IGARCH processes are commonly used because they provide a reasonable ap- proximation to the true data-generating volatility process, as referred by Ter¨asvirta (2009, p. 29).

Fan and Yao (2003, p. 170) points out that GARCH models, in their classic form, fail to catch some features of financial return series, such as asymmetry 31 and long range dependence.32 Consequently, one ought to introduce more flexible volatility specifications, such as the ones of the models proposed up to 1995, mentioned on Table 2.8 and briefly discussed in the following paragraphs.

Table 2.8: Other models of conditional variance.

AVGARCH Absolute Value GARCH (1986)

1/2 q p 1/2 h = α0 + α X + β h t i=1 i | t−i| i=1 i t−i log GARCH LogarithmicP GARCH (1986)P − q 2 p ln(ht)= α0 + i=1 αi ln(Xt−i)+ i=1 βi ln(ht−i) EGARCH ExponentialP GARCH (1991) P q 1/2 p ln(ht)= α0 + i=1 αi g(Xt−i/ht−i)+ i=1 βi ln(ht−i) with g(Z )= θZ + ζ [ Z E( Z )] t−i P t−i | t−i|− | Pt−i| APARCH Asymmetric Power ARCH (1993)

δ/2 q δ p δ/2 h = α0 + α ( X ζ X ) + β h t i=1 i | t−i|− i t−i i=1 i t−i GJR-GARCH Threshold GARCHP by Glosten et al. (1993)P q 1 2 p ht = α0 + i=1(αi + γi {Xt−i>0}) Xt−i + i=1 βi ht−i TGARCH ThresholdP GARCH of Zako¨ıan (1994) P 1/2 q p 1/2 h = α0 + [α max 0,X α min 0,X ]+ β h t i=1 i,+ { t−i}− i,− { t−i} i=1 i t−i GQARCH GeneralizedP quadratic ARCH (1995) P q q 2 q p ht = α0 + i=1 ψi Xt−i + i=1 αi Xt−i + 2 i

AVGARCH (1986) The absolute value GARCH (AVGARCH) model was introduced by Taylor (1986) and it parameterizes the conditional standard deviation as a distributed lag of the absolute innovations and the lagged conditional standard deviations: q p h1/2 = α + α X + β h1/2 (2.68) t 0 i | t−i| i t−i i=1 i=1 X X 31There is evidence that the distribution of stock returns is slightly negatively skewed, possibly because traders react more strongly to negative information than positive information (Fan and Yao, 2003, p. 170). 32The returns of all kinds of assets seldom show any serial correlation, which does not mean that they are independent r.v.; in fact, both squared returns and absolute returns often exhibit persistent autocorrelations, indicating possible long-memory dependence (Fan and Yao, 2003, p. 170).

32 (Bollerslev, 2007, p. 30). This formulation mitigates the influence of large (in an absolute sense) observations relative to the traditional GARCH(p, q) model (Bollerslev, 2007, p. 30). log–GARCH (1986) Suggested independently in slightly different forms in 1986 by Geweke (1986) and Pantalu (1986), the logarithmic GARCH (log–GARCH) process parameterizes the logarithmic conditional variance as a function of the lagged logarithmic variances (Bollerslev, 2007, pp. 19–20) and the lagged logarithmic squared innovations: q p 2 ln(ht)= α0 + αi ln(Xt−i)+ βi ln(ht−i). (2.69) i=1 i=1 X X EGARCH (1991) The exponential GARCH (EGARCH) model was proposed by Nelson (1991), as mentioned by Fan and Yao (2003, p. 170), and the logarithm of the conditional variance is written as a function of past standardized innovations (that is, divided by their conditional standard deviation) instead of past innovations: q p 1/2 ln(ht)= α0 + αi g(Xt−i/ht−i)+ βi ln(ht−i), (2.70) i=1 i=1 X X where g(Z )= θZ + ζ [ Z E( Z )] (Francq and Zako¨ıan, 2010, pp. 246–247). t−i t−i | t−i|− | t−i| This process responds asymmetrically to positive and negative lagged values of Xt, thus, it accom- modates the asymmetry property of financial time series. Moreover, the positivity constraints on the coefficients can be avoided, because the logarithm can be of any sign (Francq and Zako¨ıan, 2010, p. 247).

APARCH (1993) According to Bollerslev (2007, p. 3), the asymmetric power ARCH (APARCH) process was introduced by Ding et al. (1993) and is associated with q p hδ/2 = α + α ( X ζ X )δ + β hδ/2. (2.71) t 0 i | t−i|− i t−i i t−i i=1 i=1 X X The novelty of the APARCH model is in the introduction of the parameter δ which increases the ﬂexibility of GARCH-type models, and allows the a priori selection of an arbitrary power to be avoided (Francq and Zako¨ıan, 2010, pp. 256–257). Unsurprisingly, the family of APARCH models comprises the (standard) GARCH, AVGARCH and log–GARCH models.

GJR–GARCH (1993) and TGARCH (1994) A natural way to introduce asymmetry is to specify the conditional variance as a function of the positive and negative parts of the past values of Xt (Francq and Zako¨ıan, 2010, p. 250), hence introducing a threshold eﬀect into the volatility. One possibility is to consider q p 1 2 ht = α0 + (αi + γi {Xt−i>0}) Xt−i + βi ht−i (2.72) i=1 i=1 X X 33 and the resulting model is named GJR–GARCH after Glosten, Jagannathan and Runkle, who proposed it in Glosten et al. (1993), as referred by Francq and Zako¨ıan (2010, p. 250). Another possible equation for the conditional variance is

q p h1/2 = α + [α max 0,X α min 0,X ]+ β h1/2, (2.73) t 0 i,+ { t−i}− i,− { t−i} i t−i i=1 i=1 X X as proposed by Zako¨ıan (1994), leading to the threshold GARCH (TGARCH) models (Bollerslev, 2007, p. 29). The GJR–GARCH model is a variant of the TGARCH, which corresponds to squaring the variables in (2.73) (Francq and Zako¨ıan, 2010, p. 250). Both threshold models are used to handle leverage eﬀects33 and they use zero as their threshold to separate the impacts of past shocks (Tsay, 2010, p. 149).

GQARCH (1995) Sentana (1995) introduced the generalized quadratic ARCH (GQARCH) model

q q q p 2 ht = α0 + ψi Xt−i + αi Xt−i +2 αij Xt−i Xt−j + βi ht−i (2.74) i=1 i=1 i

This model simpliﬁes to the standard GARCH when ψi = αij = 0, for all i, j = 1,...,q (Bollerslev, 2007, p. 16).

It is possible to simulate and ﬁt some of these processes using the appropriate functions in packages for the R statistical software. For instance, the fGarch package includes the APARCH models (Chalabi and Wuertz, 2013, p. 3) and the rugarch package incorporates additionally the IGARCH, EGARCH and APARCH models (Ghalanos, 2014, pp. 7–11).

2.6.2 Multivariate generalizations

Since economic variables are often interdependent, it usually makes sense to estimate the conditional volatilities of the variables simultaneously while dealing with a data set with several variables, as referred by Enders (2010, p. 165). As superbly added by Bera and Higgins (1993) and Enders (2010, p. 165), the multivariate extension of univariate GARCH models is quite natural and certainly needed for a few good reasons:

apart from possible gains in efficiency in parameter estimation, estimation of a number of financial • coefficients requires sample values of covariances between relevant economic variables; many economic variables are influenced by the same information, and hence, have nonzero covari- • ances conditional on the information set, that is to say that contemporaneous shocks to economic variables can be correlated with each other.

33Leverage eﬀect states that a drop in the value of a stock (negative return) increases the ﬁnancial leverage which makes the stock riskier and thus increases the volatility (Awartani and Corradi, 2005).

34 Multivariate GARCH models, such as the ones concisely deﬁned in this subsection, allow for volatility spillovers in the sense that volatility shocks to one variable might aﬀect the volatility of other related variables (Enders, 2010, p. 165).

Deﬁnition 2.16. — Multivariate GARCH model (Bera and Higgins, 1993) Let X = (X ,...,X )⊤ be a N 1 random vector indexed on t. Then the general form of the t 1t Nt × multivariate GARCH model can be expressed as

X ψ Normal (µ , H ), (2.75) t| t−1 ∼ N t t where µ represents the N 1 conditional mean vector and H represents the N N conditional covariance t × t × matrix. •

One main issue is to specify Ht, a far more serious problem than the deﬁnition of the conditional variance of the univariate GARCH model (Bera and Higgins, 1993). A somewhat general form of the matrix Ht can be found below and it may remind the reader of the conditional variance of the univariate GARCH(p, q) model.

Remark 2.17. — Vech representation of Ht (Bera and Higgins, 1993)

In order to deﬁne the conditional covariance matrix Ht, one needs to use the ’vech’ notation which stacks the lower triangular entries of a symmetric matrix in a single column vector,

q p ⊤ vech(Ht) = vech(A0)+ Ai vech(Xt−i Xt−i)+ Bi vech(Ht−i), (2.76) i=1 i=1 X X where: X = (X ,...,X )⊤; A is a N N positive-deﬁnite matrix; A and B are N(N + 1)/2 t 1t Nt 0 × i i × N(N + 1)/2 matrices. • For a glimpse at this representation for N = 2 and p = q = 1, the reader is referred to equation (6.4) of Bera and Higgins (1993) or Enders (2010, p. 165). It should be noted that in this case one is dealing 1 1 with 2 N(N + 1)[1 + 2 N(N + 1)(p + q)] = 21 parameters (Bera and Higgins, 1993) and this number grows rather quickly as more variables are added, thus, resulting in a model diﬃcult to estimate although simple to conceptualize (Enders, 2010, p. 165).

Bera and Higgins (1993) notes that the specification of Ht should be done in such way that one is dealing with a positive-definite matrix, for all possible realizations of the time series; moreover, some restrictions should be imposed so that the number of parameters to be estimated is not very large. One model that became popular in the early literature and circumvents the second of these two issues is the diagonal vech model (Enders, 2010, p. 166) or diagonal representation (Bera and Higgins, 1993) or diagonal vectorization model (Tsay, 2010, p. 510), used by Bollerslev et al. (1988) in the analysis of returns on bills, bonds and stocks (Bera and Higgins, 1993). One simple assumption that leads to the reduction of the number of parameters is to specify that the conditional variance depends only on lagged squared residuals and lagged values, that is, to assume that the matrices Ai and Bi in the general form of Ht are diagonal. This representation appears to be too restrictive and the positive-definite character of the conditional covariance matrix is not, in general, easy to check, and it also difficult to impose at the estimation stage

35 (Bera and Higgins, 1993). Unsurprisingly, Baba et al (1990) proposed a model that ensures that the conditional variances are positive. This model was popularized by Engle and Kroner (1995) and is now called the Baba–Engle–Kraft–Kroner (BEKK) or Baba–Engle–Kroner (BEK) model (Enders, 2010, p. 167) or parametrization.

Deﬁnition 2.18. — Baba–Engle–Kraft–Kroner parametrization (Enders, 2010, p. 167; Tsay, 2010, p. 513) The BEKK parametrization or representation reads as follows:

q p ⋆ ⊤ ⋆ ⋆ ⊤ ⊤ ⋆ ⋆ ⊤ ⋆ Ht = (A0) A0 + (Ai ) Xt−i Xt−i Ai + (Bi ) Ht−i Bi , (2.77) i=1 i=1 X X where: A⋆ is a lower triangular N N matrix; A⋆ and B⋆ are N N matrices. 0 × i i × • The problem is that the BEKK model is diﬃcult to estimate due to the large number of parameters (Enders, 2010, p. 167), which is equal to N 2(p + q)+ N(N + 1)/2 and quickly increases with p and q (Tsay, 2010, p. 513). For another popular multivariate GARCH speciﬁcation, the constant conditional correlation (CCC) model, please refer to Enders (2010, p. 167).

Finally, it is worth to mention that:

according to Tsay (2010, p. 505), multivariate GARCH models prove to be very useful in portfolio • selection and asset allocation, and they allow the computation of the value at risk of a ﬁnancial position consisting of multiple assets; the R software also allows the estimation of some of these processes, for instance the rmgarch • package package allows the estimation of several multivariate GARCH models and the ccgarch package can be used to estimate the constant conditional correlation model (Eddelbuettel, 2015).

36 Chapter 3

Simultaneous control schemes for the mean and variance of GARCH processes

Progress in modifying our concept of control [. . . ] requires the application of statistical methods which up to the present time have been for the most part left undisturbed in the journal in which they appeared. Walter A. Shewhart

Irrespective of the care taken in defining the production process, the manufacturer realizes that he cannot make all units of a given kind of product identical. This is equivalent to assuming the existence of non- assignable causes of variation in quality of product. [... ] The reason to find assignable causes is obvious — it is only through the control of such factors that we are able to improve the product without changing the whole manufacturing process. These sentences can be found on the first page of Shewhart (1926) and probably sum up the main driving force behind the little and yet historic memorandum of May 16, 1924, Walter A. Shewhart wrote to his superiors at Bell Laboratories. According to one of them, George D. Edwards, about a third of that one page memorandum was devoted to a simple diagram which can be recognized today as a schematic quality control chart (ASQ, n.d.). As mentioned by Morais (2012, p. 69), monitoring a production process by means of one of the most important tools of Statistical Process Control (SPC), the quality control chart, requires to:

choose a quality characteristic; • select a relevant parameter (e.g. the process mean µ or variance σ2); • collect data regularly; • sequentially plot the observed value of a control statistic (e.g. the sample mean or variance) together • with appropriate (control) limits; trigger a signal if the observed value of the statistic is, for example, beyond appropriate control • limits.

Control charts have been thoroughly used to detect the presence of assignable causes responsible for changes in parameters such as the process mean or variance that may cause a deterioration in the quality

37 (Morais, 2012, p. 69) of a product or service. Expectedly, such a graphical tool can also be used to detect relevant structural deviations in the observed process X : t N underlying an economic or ﬁnancial time series, from a target process { t ∈ } Y : t N with mean µ and variance σ2.1 { t ∈ } 0 0 As aptly noted by Schipper and Schmid (2001a), additive outliers and changes in the variance of the returns can be frequently observed, therefore it is reasonable to relate the observed and target processes as follows: Yt for t < τ

X =  µ0 + θ (Yt µ0)+ δ σ0 for t = τ (3.1) t −   µ + θ (Y µ ) for t > τ, 0 t − 0  + where τ (τ N), δ (δ R) and θ (θ R ) denote the time epoch of the structural change, the relative ∈ ∈ ∈ size of the short-lived shift (Montgomery, 2009, p. 183) in location due to the additive outlier and the magnitude of the sustained shift in scale, respectively.

Since the mean and variance of Xt are given by

E(X ) = µ + δ σ 1 (t) (3.2) t 0 0 × {τ} 2 2 V(X ) = [1 (1 θ ) 1 N (t)] σ (3.3) t − − × {k∈ :k≥τ} 0

(Schipper and Schmid, 2001a), one can write

E(Xt) µ0 1 δ δ(t)= − {τ}(t) (3.4) ≡ σ0 × V(Xt) 1 1 θ θ(t)= {k∈N:k≥τ}(t)+ {k∈N:k<τ}(t). (3.5) ≡ p σ0 × It is also important to note that the process X : t N is said to be out-of-control if δ =0 or θ = 1, { t ∈ } 6 6 and deemed in-control otherwise.

While developing control charts for real-valued output, two major assumptions are usually made:

the underlying distribution of the collected data is normal; • the output is i.i.d. • However, one or both assumptions are frequently violated in practice (Wieringa, 1999, p. 2; Morais, 2012, p. 70), such as when we are dealing with economic or ﬁnancial data. According to Morais (2012, p. 70), there are essentially three approaches to monitor the mean of an autocorrelated process:

ignore the autocorrelation structure of the output and use a standard control chart that assumes • independent output; plot the original time series in a standard chart, however, with readjusted control limits to account • for the autocorrelation — leading to what is called a modiﬁed chart;

1While dealing with economic and financial data, Schipper and Schmid (2001a) emphasize the distinction between the target and the observed processes and suggest that: the former corresponds to the assumed (or in-control) model and can be obtained namely by fitting a suitable model to a starting (or historical) block of data; the observed process is associated with the time series. Moreover, these authors add that control charts should be used to check whether or not the observed process can be reasonably described by the fitted model.

38 plot the residuals of the time series (instead of the original data) in a standard chart — this sort • of chart is usually termed a residual chart.

The ﬁrst approach is rather na¨ıve and has dramatic consequences in the ability of the chart to detect changes (Knoth et al., 2009; Morais, 2012, p. 70), as illustrated in the next section.

3.1 On the impact of falsely assuming independence

For illustration purposes,

one assumes that the target process is a linear ARCH(1) (resp. GARCH(1, 1)), with zero mean and • 2 α0 2 α0 variance equal to σ0 = 1−α1 resp. σ0 = 1−α1−β1 , and yet a Shewhart control chart for individual measurements (Montgomery, 2009, pp. 260–261) is used • admitting that Y i.i.d. N(0, σ2).2 t ∼ 0

The control statistic of this chart is nothing but Xt. The observed value xt is plotted against time t and compared with a pair of control limits; a point lying outside the control limits indicates the potential presence of assignable causes that should be investigated and eliminated (Ramos, 2013, p. 3), thus, a signal should be triggered. This signal is a false alarm in the absence of assignable causes, and a valid one otherwise. As put by Ramos (2013, p. 4) a control chart should be associated to infrequent false alarms and should trigger a valid signal as quickly as possible when the process is out-of-control. Unsurprisingly, the performance of this Shewhart control chart for individual measurements (or any other chart) is commonly assessed by making use of the run length (RL), which is the number of samples collected before a signal is triggered by the chart (Wetherill and Brown, 1991, p. 100). The control limits are usually set in such way that the in-control average run length (ARL) attains some desired (and reasonably large) value, say ARL∗. Since one assumed that Y i.i.d. N(0, σ2), the lower t ∼ 0 and upper control limits of the Shewhart chart for individual measurements are

LCL = γ σ (3.6) S − S × 0 UCL = +γ σ , (3.7) S S × 0 where the critical value γS is given by

1 γ =Φ−1 1 . (3.8) S − 2ARL∗ Furthermore, one assumes from now on that τ = 1, thus the observed process is given by

θ Yt + δ σ0 for t =1 Xt = × × (3.9) ( θ Yt for t =2, 3,... × Under these conditions one can state a proposition regarding the probabilities of a signal being triggered by this Shewhart chart if the target process is indeed i.i.d.

2One chose not to jointly use a moving range chart (Montgomery, 2009, p. 260) in order to avoid any details on a chart, whose control statistics are not independent according to Morais (2012, p. 90), and to keep the illustrations as simple as possible.

39 i.i.d. 2 Proposition 3.1. — Probabilities of triggering a signal when Yt ∼ N(0,σ0) Let p p(δ, θ) (resp. p′ p′(θ)) be the probability that the ﬁrst (resp. tth, t N 1 ) observation is ≡ ≡ ∈ \{ } responsible for a signal, while using the Shewhart control chart for individual measurements with control limits (3.6) and (3.7) and τ = 1. Then

γ δ γ δ p = 1 Φ S − Φ − S − (3.10) − θ − θ γ p′ = 2 1 Φ S . (3.11) − θ h i

•

The proof of these two results follows immediately because X N(δ σ ,θ2 σ2) and X 1 ∼ × 0 × 0 t ∼ N(0,θ2 σ2), t N 1 , when Y i.i.d. N(0, σ2), t N. × 0 ∈ \{ } t ∼ 0 ∈ In the next proposition one can ﬁnd not only the probability function (p.f.) and the survival function (s.f.) of the RL of the chart considered so far, but also the associated ARL, standard deviation (SDRL), coeﬃcient of variation (CVRL) and median (MdRL).

i.i.d. 2 Proposition 3.2. — RL related performance measures when Yt ∼ N(0,σ0) The p.f. and s.f. of the RL of the Shewhart chart for individual measurements, with control limits (3.6) and (3.7) and τ = 1, are given by

p for t =1 P (RL = t) = (1 p)(1 p′)t−2p′ for t N 1 (3.12)  − − ∈ \{ }  0 otherwise  1 for t< 1 P (RL>t) = (3.13) (1 p)(1 p′)⌊t−1⌋ for t 1, − − ≥ where p and p′ are the probabilities of trigger a signal deﬁned on Proposition 3.1. The associated ARL, SDRL, CVRL and MdRL are equal to

p′ +1 p ARL = − (3.14) p′ 1 p2 1 p SDRL = − − (3.15) s (p′)2 − p′ (1 p)(1 + p p′) CVRL = − − (3.16) p′ +1 p p − ln(1 p′) ln(1 p) + ln(0.5) MdRL = max − − − , 1 . (3.17) ln(1 p′) −

•

Proof. If Y i.i.d. N(0, σ2) then the RL of this control chart has a sort of delayed geometric distribution, t ∼ 0 hence, the p.f. and s.f. above.

40 In addition, since the RL is a positive r.v., its expected value can be obtained as follows:

+∞ ARL = P (RL>t) t=0 X +∞ =1+ (1 p)(1 p′)t−1 − − t=1 X 1 p =1+ − p′ p′ +1 p = − . p′ To derive SDRL and CVRL, one ﬁrst computes the second moment of RL:

+∞ E(RL2)= t2 (1 p)(1 p′)t−1 × − − t=0 X +∞ = p + p′(1 p) t2(1 p′)t−2 − − t=2 X +∞ +∞ +∞ = p + p′(1 p) t(t + 1)(1 p′)t−1 + t(1 p′)t−1 + (1 p′)t−1 − − − − "t=1 t=1 t=1 # X X X 2 1 = p + (1 p) + +1 . − (p′)2 p′ Consequently:

SDRL = E(RL2) E2(RL) − q 2 1 2(1 p) (1 p)2 = p + (1 p) + +1 1+ − + − − (p′)2 p′ − p′ (p′)2 s 2 (1 p) 1 2 = (1 p) − − + − − (p′)2 p′ s 1 p2 1 p = − − ; s (p′)2 − p′ SD(RL) CVRL = E(RL) 1+ p 1 p′ = (1 p) − (p′)2 − p′ × p′ +1 p s − (1 p)(1 + p p′) = − − . p′ +1 p p − Finally since the RL is a positive integer r.v., its MdRL can be derived as follows:

MdRL = inf t N : P (RL>t) 0.5 { ∈ ≤ } = inf t N : (1 p)(1 p′)t−1 0.5 { ∈ − − ≤ } = inf t N : (t 1) ln(1 p′) ln(1 p) + ln(0.5) { ∈ − − ≤− − } ln(1 p′) ln(1 p) + ln(0.5) = inf t N : t − − − ∈ ≥ ln(1 p′) − ln(1 p′) ln(1 p) + ln(0.5) = max − − − , 1 . ln(1 p′) −

41 Deriving analogues of the two previous propositions, when the target process is governed by a GARCH model, is rather diﬃcult.3 Therefore one has to rely on Monte Carlo simulations to estimate ARL, SDRL, CVRL and MdRL and, thus, assess the performance of the Shewhart chart for individual measurements 2 under the false assumption of an i.i.d. N(0, σ0 ) target process, when it is in fact a linear ARCH or GARCH 2 process with zero mean and variance equal to σ0 . In addition to τ = 1 and ε : t N GWN(0, 1), one has taken: { t ∈ }∼ δ =0.0 and θ =0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.5, 2.0, 3.0; • δ =0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 and θ =1.0; • (δ, θ)=(1.5, 0.7), (2.0, 0.8), (0.0, 1.0), (1.0, 1.1), (3.0, 1.1), (1.5, 1.5), (2.0, 2.0), (0.5, 3.0), (3.0, 3.0). • The in-control situation, (δ, θ)=(0.0, 1.0), is comprised by any of these parameter constellations.4 They also refer to three out-of-control scenarios one will treat separately in this section:

a sustained shift in the process standard deviation with magnitude θ; • a short-lived shift in the process mean due to an additive outlier with magnitude δ; • a short-lived shift in the process mean accompanied by a sustained shift in the process standard • deviation with magnitude (δ, θ).

When it comes to the simulations, it is important to refer that the rugarch package was used instead of fGarch.5 Moreover, one:

simulated a time series y ,...,y for each i (i = 1,...,rep), drawn from the target process, • { i,1 i,N } where the number of replications is equal to rep = 105 for each set of parameter values;6 obtained the associated values of the control statistics x ,...,x for each i (i = 1,...,rep), • { i,1 i,N } compared them with the control limits deﬁned in (3.6) and (3.7) and counted the number of observations (i.e., the RL) until a signal was triggered by the chart for each i and estimated the ARL, SDRL, CVRL and MdRL.7

Finally, one considered Shewhart charts for individual measurements calibrated in such a way that the in-control ARL (ARL∗) is equal to 60, as in Schipper and Schmid (2001a) and Schipper and Schmid (2001b).8 They will be confronted with the corresponding values obtained using Proposition 3.2 and found in Table 3.1. The estimates of the ARL, SDRL, CVRL and MdRL are summarized in the plots found in the next two subsections, for the linear ARCH(1) and the GARCH(1, 1) processes.9

3 This is essentially due to the fact that one is unable to derive joint distributions regarding Yt : t N and, consequently, { ∈ } simplify the corresponding s.f. P (RL >k)= P (Xt [LCLS,UCLS], t = 1,...,k), for k N. 4Note that the values of δ and θ were also considered∈ by Schipper and Schmid (2001a), wherea∈ s the pairs (δ, θ) considered in the simulation study constitute a subset of the 70 pairs taken by these authors. 5Note that the fGarch package can lead to simulation times up to nine times larger than the ones associated with rugarch. 6One was forced to simulate a ﬁxed number of observations N because the software does not allow to sequentially simulate a GARCH process, say until a signal is triggered. To speed up the simulation study, the time series size N varies according to the values of the parameters δ and θ: N = 100000, for (δ, θ) = (0.0, 0.7), (1.5, 0.7); N = 10000, for (δ, θ) = (0.0, 0.8), (0.0, 0.9), (2.0, 0.8); N = 1000, otherwise. 7For estimation purposes, one dismissed the realizations of the observed process for which no observation was beyond the control limits. 8According to Schipper and Schmid (2001b), the value 60 roughly reﬂects three months at the stock exchange. 9These estimates will be made available to those who are interested and request them from the author.

42 i.i.d. 2 Table 3.1: ARL, SDRL, CVRL and MdRL values when Yt N(0, σ ). ∼ 0 RL based performance measures

θ (δ = 0.0) ARLiid SDRLiid CVRLiid MdRLiid 0.7 1596.7355 1596.2355 0.9997 1107 0.8 361.3689 360.8685 0.9986 251 0.9 127.9658 127.4649 0.9961 89 1.0 60.0000 59.4979 0.9916 42 1.1 33.8645 33.3607 0.9851 24 1.2 21.7180 21.2121 0.9767 15 1.3 15.2567 14.7482 0.9667 11 1.5 9.0504 8.5358 0.9431 6 2.0 4.3232 3.7904 0.8767 3 3.0 2.3536 1.7849 0.7584 2

δ (θ = 1.0) ARLiid SDRLiid CVRLiid MdRLiid 0.0 60.0000 59.4979 0.9916 42 0.5 59.1391 59.4844 1.0058 41 1.0 56.0796 59.3356 1.0581 38 1.5 49.8571 58.5404 1.1742 31 2.0 40.1918 55.9270 1.3915 17 2.5 28.4670 50.1411 1.7614 1 3.0 17.3350 40.9516 2.3624 1

(δ, θ) ARLiid SDRLiid CVRLiid MdRLiid (1.5, 0.7) 1436.8149 1588.1540 1.1053 938 (2.0, 0.8) 249.9138 343.0633 1.3727 117 (0.0, 1.0) 60.0000 59.4979 0.9916 42 (1.0, 1.1) 31.3579 33.2287 1.0597 21 (3.0, 1.1) 10.8492 23.6690 2.1816 1 (1.5, 1.5) 7.5135 8.3043 1.1052 5 (2.0, 2.0) 3.4386 3.5637 1.0364 2 (0.5, 3.0) 2.3386 1.7806 0.7614 2 (3.0, 3.0) 1.9035 1.5916 0.8361 1

One ought to note that the values in Table 3.2 are close to the corresponding values of those four RL performance measures in Table 3.1, when one considers the following i.i.d. target processes: Y : t { t ∈ N ARCH(1), with α =1.0 and α = 0. }∼ 0 1 In ﬁgures 3.1–3.12 one broke apart the ARL, SDRL, CRL and MdRL curves for each value of (0,θ),

(δ, 1) and (δ, θ), and added an horizontal line with ARLiid, SDRLiid, CVRLiid and MdRLiid, the corresponding values obtained using results (3.14), (3.15), (3.16) and (3.17), with (0,θ), (δ, 1) and (δ, θ). 2 Since the ARCH(1) target process corresponds to GWN(0, σ0) when α1 = 0, the curves of those estimates intersect the lines corresponding to ARLiid, SDRLiid, CVRLiid and MdRLiid for α1 = 0, as shown by

ﬁgures 3.1–3.6. Curiously, for the GARCH(1, 1) processes, similar intersections occur when β1 = 0.95 and α1 = 0, as illustrated by ﬁgures 3.7–3.12.

3.1.1 ARCH(1) model

The following plots were obtained assuming that the target process is a linear ARCH(1) model with:

α =1.0; • 0 α =0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9. • 1

Sustained shift in the process variance

The plots in Figure A.1 refer to the estimated ARL, SDRL, CVRL and MdRL. These estimates pertain to the in-control situation and to several sustained shifts in the target process variance.

43 Table 3.2: Estimated ARL, SDR, CVRL and MdRL when Yt : t N ARCH(1), with α0 = 1.0 and α1 = 0.0 (i.i.d. case). { ∈ } ∼

RL based performance measures θ (δ = 0.0) ARL SDRL CVRL MdRL 0.7 1602.6133 1604.9714 1.0015 1108 0.8 362.7448 363.3206 1.0016 250 0.9 128.3633 127.7357 0.9951 89 1.0 60.0101 59.6098 0.9933 42 1.1 33.8384 33.4115 0.9874 24 1.2 21.6748 21.0675 0.9720 15 1.3 15.2944 14.8088 0.9683 11 1.5 9.0607 8.5575 0.9445 6 2.0 4.3139 3.7898 0.8785 3 3.0 2.3564 1.7955 0.7620 2 δ (θ = 1.0) ARL SDRL CVRL MdRL 0.0 60.0101 59.6098 0.9933 42 0.5 58.9830 59.2776 1.0050 41 1.0 56.0756 59.5604 1.0621 37 1.5 49.9293 58.8504 1.1787 31 2.0 40.1839 55.9772 1.3930 18 2.5 28.5506 50.3435 1.7633 1 3.0 17.5033 41.1394 2.3504 1 (δ, θ) ARL SDRL CVRL MdRL (1.5, 0.7) 1439.4903 1591.4824 1.1056 935 (2.0, 0.8) 250.1750 342.5769 1.3693 117 (0.0, 1.0) 60.0101 59.6098 0.9933 42 (1.0, 1.1) 31.2884 33.0572 1.0565 21 (3.0, 1.1) 10.8415 23.6702 2.1833 1 (1.5, 1.5) 7.4992 8.2459 1.0996 5 (2.0, 2.0) 3.4404 3.5689 1.0373 2 (0.5, 3.0) 2.3402 1.7917 0.7656 2 (3.0, 3.0) 1.9041 1.5898 0.8350 1

Those plots suggest that the behavior of the estimates of ARL, SDRL and MdRL is similar, for all the values of θ and α1 one considered. For θ < 0.9 they seem to decrease with α1 and then to increase; ∗ the ARL estimate for α1 = 0 is much larger than ARL = 60, i.e., the chart takes longer, in average, to detect some decreases in the process variance than to trigger a false alarm even in the i.i.d. case. For θ 0.9, the estimates of ARL, SDRL and MdRL seem to be not inﬂuenced by the ARCH parameter ≥ when 0 α < 0.7; for 0.7 α < 1 they increase slightly. The CVRL estimates are almost unperturbed ≤ 1 ≤ 1 by α1 and close to the CVRL for an i.i.d. target process (CVRLiid), except for θ =2.0 and for θ =3.0.

In these two cases the estimates of CVRL increase with α1, from CVRLiid to values close to 1.

Figures 3.1 and 3.2 suggest that the behavior of the estimates of ARL, SDRL and MdRL is diﬀerent for θ < 1.0 and θ 1.0. For example, when θ = 0.7, 0.8, the curves of these estimates are below the ≥ lines corresponding to ARLiid(0,θ), SDRLiid(0,θ) and MdRLiid(0,θ). When θ =0.9, those curves cross those lines at diﬀerent points associated with distinct values of α . Finally, for θ 1.0 (and α > 0.0), 1 ≥ 1 the estimates of ARL, SDRL and MdRL seem to be always larger than ARLiid(0,θ), SDRLiid(0,θ) and

MdRLiid(0,θ) (respectively).

It is also important to add that there is no signiﬁcant diﬀerence between the estimates of CVRL and

CVRLiid(0,θ), except for θ =2.0 and for θ =3.0, as depicted in Figure 3.2.

The results obtained so far suggest that if one neglects the ARCH character of the target process and uses a Shewhart chart for individual measurements for i.i.d. output then:

44 the true ARL, SDRL and MdRL can be underestimated (resp. overestimated) by ARL (0,θ), • iid SDRLiid(0,θ) and MdRLiid(0,θ) when θ > 0.9 (resp. θ< 0.9); the true CVRL does not diﬀer substantially from CVRL (0,θ), as Figure 3.2 portrays quite • iid vividly.

(δ,θ)=(0.0,0.7) (δ,θ)=(0.0,0.8) (δ,θ)=(0.0,0.9) (δ,θ)=(0.0,1.0) (δ,θ)=(0.0,1.1) ARL ARL ARL ARL ARL 0 500 1000 1500 0 100 200 300 0 50 100 150 0 20 40 60 80 100 120 140 0 20 40 60 80 100

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,1.2) (δ,θ)=(0.0,1.3) (δ,θ)=(0.0,1.5) (δ,θ)=(0.0,2.0) (δ,θ)=(0.0,3.0) ARL ARL ARL ARL ARL 0 20 40 60 80 0 20 40 60 80 0 10 20 30 40 50 0 5 10 15 20 25 30 0 2 4 6 8 10 12

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,0.7) (δ,θ)=(0.0,0.8) (δ,θ)=(0.0,0.9) (δ,θ)=(0.0,1.0) (δ,θ)=(0.0,1.1) SDRL SDRL SDRL SDRL SDRL 0 500 1000 1500 0 100 200 300 0 50 100 150 0 20 40 60 80 100 120 140 0 20 40 60 80 100

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,1.2) (δ,θ)=(0.0,1.3) (δ,θ)=(0.0,1.5) (δ,θ)=(0.0,2.0) (δ,θ)=(0.0,3.0) SDRL SDRL SDRL SDRL SDRL 0 20 40 60 80 0 20 40 60 0 10 20 30 40 50 0 5 10 15 20 25 0 2 4 6 8 10 12

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

Figure 3.1: Estimates of ARL and SDRL, in the presence of a sustained shift in scale — ARCH(1) model.

45 (δ,θ)=(0.0,0.7) (δ,θ)=(0.0,0.8) (δ,θ)=(0.0,0.9) (δ,θ)=(0.0,1.0) (δ,θ)=(0.0,1.1) CVRL CVRL CVRL CVRL CVRL 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,1.2) (δ,θ)=(0.0,1.3) (δ,θ)=(0.0,1.5) (δ,θ)=(0.0,2.0) (δ,θ)=(0.0,3.0) CVRL CVRL CVRL CVRL CVRL 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,0.7) (δ,θ)=(0.0,0.8) (δ,θ)=(0.0,0.9) (δ,θ)=(0.0,1.0) (δ,θ)=(0.0,1.1) MdRL MdRL MdRL MdRL MdRL 0 200 400 600 800 1000 0 50 100 150 200 250 0 20 40 60 80 100 120 0 20 40 60 80 100 0 20 40 60 80

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,1.2) (δ,θ)=(0.0,1.3) (δ,θ)=(0.0,1.5) (δ,θ)=(0.0,2.0) (δ,θ)=(0.0,3.0) MdRL MdRL MdRL MdRL MdRL 0 10 20 30 40 50 60 0 10 20 30 40 50 0 10 20 30 40 0 5 10 15 20 0 2 4 6 8

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

Figure 3.2: Estimates of CVRL and MdRL, in the presence of a sustained shift in scale — ARCH(1) model.

Short-lived shift in the process mean

The plots of the estimated ARL, SDRL, CVRL and MdRL can be found in: Figure A.2; ﬁgures 3.3 and 3.4, along with horizontal lines corresponding to ARLiid(δ, 1), SDRLiid(δ, 1), CVRLiid(δ, 1) and

MdRLiid(δ, 1). These estimates refer now to several short-lived shifts in the target process mean and once more to the in-control scenario. The estimates of ARL, SDRL and MdRL (resp. CVRL) seem to decrease (resp. increase) with δ 2, ≤

46 (δ,θ)=(0.0,1.0) (δ,θ)=(0.5,1.0) (δ,θ)=(1.0,1.0) (δ,θ)=(1.5,1.0) (δ,θ)=(2.0,1.0) ARL ARL ARL ARL ARL 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(2.5,1.0) (δ,θ)=(3.0,1.0) ARL ARL 0 10 20 30 40 50 0 5 10 15

0.0 0.4 0.8 0.0 0.4 0.8

α1 α1

(δ,θ)=(0.0,1.0) (δ,θ)=(0.5,1.0) (δ,θ)=(1.0,1.0) (δ,θ)=(1.5,1.0) (δ,θ)=(2.0,1.0) SDRL SDRL SDRL SDRL SDRL 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(2.5,1.0) (δ,θ)=(3.0,1.0) SDRL SDRL 0 20 40 60 80 100 0 10 20 30 40 50

0.0 0.4 0.8 0.0 0.4 0.8

α1 α1

Figure 3.3: Estimates of ARL and SDRL, in the presence of a short-lived shift in location — ARCH(1) model.

as portrayed for example by Figure A.2. Furthermore, the estimates of ARL tend: to be larger than

ARLiid(δ, 1); and to increase with α1 except for large values of δ, suggesting that the larger the ARCH parameter the more one underestimates the true ARL when one falsely assumes independent output and adopt a chart accordingly. For instance, ARLiid(1, 1) = 56.0796, whereas the corresponding estimate for

α1 =0.8 is equal to 86.7673. This behavior means a smaller ability of detecting short-lived shifts in the process mean by the Shewhart chart for individual measurements, even if its control limits account for

47 (δ,θ)=(0.0,1.0) (δ,θ)=(0.5,1.0) (δ,θ)=(1.0,1.0) (δ,θ)=(1.5,1.0) (δ,θ)=(2.0,1.0) CVRL CVRL CVRL CVRL CVRL 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(2.5,1.0) (δ,θ)=(3.0,1.0) CVRL CVRL 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5

0.0 0.4 0.8 0.0 0.4 0.8

α1 α1

(δ,θ)=(0.0,1.0) (δ,θ)=(0.5,1.0) (δ,θ)=(1.0,1.0) (δ,θ)=(1.5,1.0) (δ,θ)=(2.0,1.0) MdRL MdRL MdRL MdRL MdRL 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(2.5,1.0) (δ,θ)=(3.0,1.0) MdRL MdRL 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8

α1 α1

Figure 3.4: Estimates of CVRL and MdRL, in the presence of a short-lived shift in location — ARCH(1) model. the standard deviation of the target ARCH(1) process. A closer look at the curves of the other three RL based performance measures also leads to the conclusion that ignoring the ARCH character of the target process is not beneﬁcial in the presence of this type of shift.10

10 Interestingly, for δ 2.5, the estimate of MdRL is equal to 1 regardless of the the values of α1, as shown in Figure 3.4, that is, for very large short-lived≥ shifts in the process mean falsely assuming independence seems to have no inﬂuence in the median RL.

48 Short-lived shift in the process mean and sustained shift in the process variance

The estimates of ARL, SDRL, CVRL and MdRL in ﬁgures 3.5 and 3.6 and in Figure A.3 refer to the in-control situation and to eight short-lived shifts in the process mean combined with sustained shifts in the process variance.

(δ,θ)=(1.5,0.7) (δ,θ)=(2.0,0.8) (δ,θ)=(0.0,1.0) (δ,θ)=(1.0,1.1) (δ,θ)=(3.0,1.1) ARL ARL ARL ARL ARL 0 200 400 600 800 1200 0 50 100 150 200 250 0 20 40 60 80 100 120 140 0 20 40 60 80 100 0 2 4 6 8 10 12

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(1.5,1.5) (δ,θ)=(2.0,2.0) (δ,θ)=(0.5,3.0) (δ,θ)=(3.0,3.0) ARL ARL ARL ARL 0 10 20 30 40 50 0 5 10 15 20 0 2 4 6 8 10 12 0 1 2 3 4

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1

(δ,θ)=(1.5,0.7) (δ,θ)=(2.0,0.8) (δ,θ)=(0.0,1.0) (δ,θ)=(1.0,1.1) (δ,θ)=(3.0,1.1) SDRL SDRL SDRL SDRL SDRL 0 500 1000 1500 0 50 100 150 200 250 300 350 0 20 40 60 80 100 120 140 0 20 40 60 80 100 0 10 20 30 40

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(1.5,1.5) (δ,θ)=(2.0,2.0) (δ,θ)=(0.5,3.0) (δ,θ)=(3.0,3.0) SDRL SDRL SDRL SDRL 0 10 20 30 40 50 0 5 10 15 20 25 0 2 4 6 8 10 12 0 2 4 6 8

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1

Figure 3.5: Estimates of ARL and SDRL, in the presence of a short-lived shift in location and a sustained shift in scale — ARCH(1) model.

In ﬁgures 3.5 and 3.6, one can verify that:

49 (δ,θ)=(1.5,0.7) (δ,θ)=(2.0,0.8) (δ,θ)=(0.0,1.0) (δ,θ)=(1.0,1.1) (δ,θ)=(3.0,1.1) CVRL CVRL CVRL CVRL CVRL 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(1.5,1.5) (δ,θ)=(2.0,2.0) (δ,θ)=(0.5,3.0) (δ,θ)=(3.0,3.0) CVRL CVRL CVRL CVRL 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1

(δ,θ)=(1.5,0.7) (δ,θ)=(2.0,0.8) (δ,θ)=(0.0,1.0) (δ,θ)=(1.0,1.1) (δ,θ)=(3.0,1.1) MdRL MdRL MdRL MdRL MdRL 0 200 400 600 800 0 20 40 60 80 100 120 140 0 20 40 60 80 100 0 20 40 60 80 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(1.5,1.5) (δ,θ)=(2.0,2.0) (δ,θ)=(0.5,3.0) (δ,θ)=(3.0,3.0) MdRL MdRL MdRL MdRL 0 10 20 30 0 2 4 6 8 10 0 2 4 6 8 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1

Figure 3.6: Estimates of CVRL and MdRL, in the presence of a short-lived shift in location and a sustained shift in scale — ARCH(1) model.

when θ 0.8 (resp. θ 1.0) the ARL (δ, θ) and SDRL (δ, θ) underestimates (resp. overesti- • ≤ ≥ iid iid mates) the true ARL and SDRL, as in the case corresponding to just a sustained shift in the process variance. the estimates of of CVRL seem to diﬀer from CVRL (δ, θ) more than in the two previous cases • iid of isolated shifts.

50 Misapplying a Shewhart chart for individual measurements proves once more to have a great impact in all four RL based performance measures. Furthermore, it is interesting to compare the inﬂuence of a short-lived shift in the process mean and of a sustained shift in the process variance. For instance, one gets for δ = 0.5 or θ = 3.0:

ARLiid(0.5, 1) = 59.1391, ARLiid(0, 3) = 2.3536 and ARLiid(0.5, 3.0) = 2.3386 and the corresponding estimates, when α1 = 0.8, are 87.9767, 6.4502 and 6.3347. Then the associated relative errors are (59.1391 87.9767)/87.9767 100% = 32.78%, 63.51% and 63.08%, respectively. Hence, in this case − × − − − the relative diﬀerence between the true value and the ARLiid(δ, θ) is larger for the shift in the variance than for the shift in the mean or for the combination of the two shifts.

3.1.2 GARCH(1,1) model

To produce the following plots, one assumed that the target process is a GARCH(1, 1) model with:

α =1.0; • 0 α =0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9; • 1 β =0.95 α .11 • 1 − 1 Suﬃce to say that the presentation of the results in this subsection follows along the same lines as the ones referring to the ARCH(1) model.

Sustained shift in the process variance

The plots of the estimated ARL, SDRL, CVRL and MdRL can be found in ﬁgures 3.7 and 3.8 and in Figure A.4. These estimates refer to the in-control scenario and to several sustained shifts in the process variance. Note that the behavior of the curves of the estimated ARL, SDRL and MdRL obtained for the GARCH(1, 1) seems to be similar to the ones obtained for the ARCH(1). There are some diﬀerences though, for instance:

the true out-of-control ARL, SDRL and MdRL are underestimated by the ARL (0, 0.9), SDRL • iid iid (0, 0.9) and MdRLiid(0, 0.9); the true CVRL diﬀers a bit more from CVRL (0,θ), for θ 1, if one compares ﬁgures 3.8 and • iid ≥ 3.2.

Short-lived shift in the process mean

The estimates of the ARL, SDRL, CVRL and MdRL in figures 3.9 and 3.10 and in Figure A.5 pertain to the in-control situation and to six short-lived shifts in the process mean. As in the case where there is a sustained shift in the process variance, one can detect some similarities and a few differences in the behavior of the estimated RL based performance measures obtained for the ARCH(1) and GARCH(1, 1) processes. For instance the estimates of the ARL, SDRL and MdRL also seem to increase with α when δ 2. However, the differences between those estimates and the 1 ≤ correspondent values computed according to Proposition 3.2 are not negligible for small values of α1, as

11These choice of parameters was suggested by Schipper (2001, p. 39) and Schipper and Schmid (2001a).

51 in the ARCH(1) case, and they experience a sort of logarithmic growth with α1 unlike in that simpler process, as depicted by ﬁgures 3.9 and 3.10.

(δ,θ)=(0.0,0.7) (δ,θ)=(0.0,0.8) (δ,θ)=(0.0,0.9) (δ,θ)=(0.0,1.0) (δ,θ)=(0.0,1.1) ARL ARL ARL ARL ARL 0 500 1000 1500 0 100 200 300 400 0 50 100 150 200 250 300 0 50 100 150 200 0 50 100 150

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,1.2) (δ,θ)=(0.0,1.3) (δ,θ)=(0.0,1.5) (δ,θ)=(0.0,2.0) (δ,θ)=(0.0,3.0) ARL ARL ARL ARL ARL 0 50 100 150 0 20 40 60 80 100 120 140 0 20 40 60 80 100 0 10 20 30 40 50 0 5 10 15 20

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,0.7) (δ,θ)=(0.0,0.8) (δ,θ)=(0.0,0.9) (δ,θ)=(0.0,1.0) (δ,θ)=(0.0,1.1) SDRL SDRL SDRL SDRL SDRL 0 500 1000 1500 0 100 200 300 400 0 50 100 150 200 250 300 0 50 100 150 200 0 50 100 150

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,1.2) (δ,θ)=(0.0,1.3) (δ,θ)=(0.0,1.5) (δ,θ)=(0.0,2.0) (δ,θ)=(0.0,3.0) SDRL SDRL SDRL SDRL SDRL 0 50 100 150 0 20 40 60 80 100 120 0 20 40 60 80 100 0 10 20 30 40 50 0 5 10 15 20

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

Figure 3.7: Estimates of ARL and SDRL, in the presence of a sustained shift in scale — GARCH(1,1) model.

52 (δ,θ)=(0.0,0.7) (δ,θ)=(0.0,0.8) (δ,θ)=(0.0,0.9) (δ,θ)=(0.0,1.0) (δ,θ)=(0.0,1.1) CVRL CVRL CVRL CVRL CVRL 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,0.7) (δ,θ)=(0.0,0.8) (δ,θ)=(0.0,0.9) (δ,θ)=(0.0,1.0) (δ,θ)=(0.0,1.1) MdRL MdRL MdRL MdRL MdRL 0 200 400 600 800 1000 0 50 100 150 200 250 0 50 100 150 200 0 50 100 150 0 20 40 60 80 100 120 140

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(0.0,1.2) (δ,θ)=(0.0,1.3) (δ,θ)=(0.0,1.5) (δ,θ)=(0.0,2.0) (δ,θ)=(0.0,3.0) MdRL MdRL MdRL MdRL MdRL 0 20 40 60 80 100 0 20 40 60 80 0 10 20 30 40 50 60 70 0 10 20 30 0 5 10 15

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

Figure 3.8: Estimates of CVRL and MdRL, in the presence of a sustained shift in scale — GARCH(1,1) model.

53 (δ,θ)=(0.0,1.0) (δ,θ)=(0.5,1.0) (δ,θ)=(1.0,1.0) (δ,θ)=(1.5,1.0) (δ,θ)=(2.0,1.0) ARL ARL ARL ARL ARL 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(2.5,1.0) (δ,θ)=(3.0,1.0) ARL ARL 0 20 40 60 80 0 5 10 15 20 25 30

0.0 0.4 0.8 0.0 0.4 0.8

α1 α1

(δ,θ)=(0.0,1.0) (δ,θ)=(0.5,1.0) (δ,θ)=(1.0,1.0) (δ,θ)=(1.5,1.0) (δ,θ)=(2.0,1.0) SDRL SDRL SDRL SDRL SDRL 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(2.5,1.0) (δ,θ)=(3.0,1.0) SDRL SDRL 0 50 100 150 0 20 40 60 80

0.0 0.4 0.8 0.0 0.4 0.8

α1 α1

Figure 3.9: Estimates of ARL and SDRL, in the presence of a short-lived shift in location — GARCH(1,1) model.

54 (δ,θ)=(0.0,1.0) (δ,θ)=(0.5,1.0) (δ,θ)=(1.0,1.0) (δ,θ)=(1.5,1.0) (δ,θ)=(2.0,1.0) CVRL CVRL CVRL CVRL CVRL 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(2.5,1.0) (δ,θ)=(3.0,1.0) CVRL CVRL 0 1 2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5

0.0 0.4 0.8 0.0 0.4 0.8

α1 α1

(δ,θ)=(0.0,1.0) (δ,θ)=(0.5,1.0) (δ,θ)=(1.0,1.0) (δ,θ)=(1.5,1.0) (δ,θ)=(2.0,1.0) MdRL MdRL MdRL MdRL MdRL 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 0 20 40 60 80 100 120 140

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(2.5,1.0) (δ,θ)=(3.0,1.0) MdRL MdRL 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8

α1 α1

Figure 3.10: Estimates of CVRL and MdRL, in the presence of a short-lived shift in location — GARCH(1,1) model.

55 Short-lived shift in the process mean and sustained shift in the process variance

The plots in ﬁgures 3.11, 3.12 and Figure A.6 refer to estimates of ARL, SDRL, CVRL and MdRL. These estimates pertain to eight short-lived shifts in the process mean combined with sustained shifts in the process variance and to the in-control scenario.

(δ,θ)=(1.5,0.7) (δ,θ)=(2.0,0.8) (δ,θ)=(0.0,1.0) (δ,θ)=(1.0,1.1) (δ,θ)=(3.0,1.1) ARL ARL ARL ARL ARL 0 200 400 600 800 1200 0 100 200 300 0 50 100 150 200 0 50 100 150 0 5 10 15 20 25

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(1.5,1.5) (δ,θ)=(2.0,2.0) (δ,θ)=(0.5,3.0) (δ,θ)=(3.0,3.0) ARL ARL ARL ARL 0 20 40 60 80 0 10 20 30 40 0 5 10 15 20 0 1 2 3 4 5 6

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1

(δ,θ)=(1.5,0.7) (δ,θ)=(2.0,0.8) (δ,θ)=(0.0,1.0) (δ,θ)=(1.0,1.1) (δ,θ)=(3.0,1.1) SDRL SDRL SDRL SDRL SDRL 0 500 1000 1500 0 100 200 300 400 0 50 100 150 200 0 50 100 150 0 20 40 60 80

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(1.5,1.5) (δ,θ)=(2.0,2.0) (δ,θ)=(0.5,3.0) (δ,θ)=(3.0,3.0) SDRL SDRL SDRL SDRL 0 20 40 60 80 100 0 10 20 30 40 50 0 5 10 15 20 0 2 4 6 8 10 12 14

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1

Figure 3.11: Estimates of ARL and SDRL, in the presence of a short-lived shift in location and a sustained shift in scale — GARCH(1,1) model.

Once more, one can confront the impact of the shift in the location and of the shift in scale, considering again α1 =0.8 and compare the relative errors for δ =0.5 and θ = 3, in this case the estimated ARL values

56 corresponding to ARLiid(0.5, 1), ARLiid(0, 3) and ARLiid(0.5, 3.0) are 212.0692, 18.8503 and 6.1934 then the associated relative errors are 72.11%, 87.51% and 62.24%, respectively. Once again the relative − − −

(δ,θ)=(1.5,0.7) (δ,θ)=(2.0,0.8) (δ,θ)=(0.0,1.0) (δ,θ)=(1.0,1.1) (δ,θ)=(3.0,1.1) CVRL CVRL CVRL CVRL CVRL 0 1 2 3 4 5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(1.5,1.5) (δ,θ)=(2.0,2.0) (δ,θ)=(0.5,3.0) (δ,θ)=(3.0,3.0) CVRL CVRL CVRL CVRL 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1

(δ,θ)=(1.5,0.7) (δ,θ)=(2.0,0.8) (δ,θ)=(0.0,1.0) (δ,θ)=(1.0,1.1) (δ,θ)=(3.0,1.1) MdRL MdRL MdRL MdRL MdRL 0 200 400 600 800 0 50 100 150 200 250 0 50 100 150 0 20 40 60 80 100 120 140 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1 α1

(δ,θ)=(1.5,1.5) (δ,θ)=(2.0,2.0) (δ,θ)=(0.5,3.0) (δ,θ)=(3.0,3.0) MdRL MdRL MdRL MdRL 0 10 20 30 40 50 60 0 5 10 15 20 0 5 10 15 0.0 0.5 1.0 1.5 2.0

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

α1 α1 α1 α1

Figure 3.12: Estimates of CVRL and MdRL, in the presence of a short-lived shift in location and a sustained shift in scale — GARCH(1,1) model.

discrepancy between the true value and the ARLiid(δ, θ) is larger when a sustained shift in the process variance occurs.

57 In light of all the results reported so far, one concludes that ignoring the ARCH or GARCH character of the target process while designing a chart can have dramatic consequences in its performance. This calls for alternative charts to control the process mean and variance, suchlike the simultaneous modiﬁed EWMA schemes introduced by Schipper (2001) and described in the next section.

3.2 Simultaneous schemes for the process mean and variance

In Finance it is quite usual to assume that a process behaves according to a previously specified GARCH model due to the effectiveness of GARCH modeling (Schipper, 2001, p. 7) in that field. Moreover, according to Schipper (2001, p. 7), the traders need to promptly track structural deviations from that model since they can make a profit when those changes occur. When the control chart triggers a signal, a deviation is found and that information should be considered in the trading strategy, since it can be used to determine if the trader keeps a security in his portfolio (Schipper, 2001, p. 9). That is why it is so important to construct control charts adjusted to take into account the GARCH effect. The first charts specifically designed to control the mean and the variance of a GARCH process were proposed by Severin and Schmid (1999); these control charts were used to monitor the process mean. Schipper and Schmid (2001b) studied control charts to detect solely changes in the process variance; the reader should bear in mind that in Finance the variance measures the risk of an asset, thus, it is extremely important to detect any changes in this process parameter. However, in order to efficiently control a process, one needs to jointly monitor its location and scale and to do that joint (simultaneous or combined) control schemes are used (Ramos, 2013, p. 5). The standard practice is to use simultaneously two charts, one for the mean value (µ) and other for the variance (σ2). That is the case of the simultaneous modified EWMA schemes proposed by Schipper (2001). In Table 3.3, one finds the statistics and the control limits of one individual modified EWMA chart for µ and of four individual modified EWMA charts for σ based on the squared observations (I), the conditional variance (II), the exponentially weighted variance (III) and the logarithm of the squared observations (IV) (Schipper, 2001, pp. 34, 35), with smoothing parameters λ , λ (0, 1]. Please note 1 2 ∈ that according to Schipper (2001, pp. 12–14, 35):

2 σ0 for t =1 2 σt =  β (3.18)  2 2 2 1 2 2 N  σ0 + (α1 + β1)[(Xt−1 µ0) σ0 ]+ [(Xt−1 µ0) σt−1] for t 1 , − − rt−1 − − ∈ \{ } b  2 2 1−2β1α1−β1 2 β1 b where rt = 2 for t = 1 and rt =1+ β for t 2; and 1−(α1+β1) 1 − rt−1 ≥ 2 σ0 for t =0 (σ∗)2 = (3.19) t  0.94 (σ∗ )2 +0.06 (X µ )2 for t N.  t−1 t − 0 ∈ The simultaneous scheme gives a signal when at least one of its individual charts does, that is, its run length is deﬁned as

RL = inf t N : Z(k) µ > UCL(k) Z(k) < LCL(k) Z(k) > UCL(k) , (3.20) { ∈ | 1,t − 0| E−µ ∨ 2,t E−σ ∨ 2,t E−σ}

58 Table 3.3: Statistics and control limits — individual modiﬁed EWMA charts for µ and for σ.

EWMA control chart for µ

(k) µ0 for t = 0 Statistic Z1,t = (k) (1 λ1)Z + λ1Xt for t N ( − 1,t−1 ∈ (k) (k) LCL LCLE−µ = µ0 c1 (k) − (k) UCL UCLE−µ = µ0 + c1 2 EWMA control chart for σ based on Xt 2 (I) σ0 for t = 0 Statistic Z2,t = (I) 2 (1 λ2)Z + λ2(Xt µ0) for t N ( − 2,t−1 − ∈ (I) (I) LCL LCLE−σ = c2 α0 (I) (I) UCL UCLE−σ = c3 α0 2 EWMA control chart for σ based on σt 2 (II) σ0 for t = 0 Statistic Z2,t = (II) 2 b (1 λ2)Z + λ2σ for t N ( − 2,t−1 t ∈ (II) (II) LCL LCLE−σ = c2 α0 (II) (II) b UCL UCLE−σ = c3 α0 ∗ 2 EWMA control chart for σ based on (σt ) 2 (III) σ0 for t = 0 Z2,t = (III) ∗ 2 Statistic (1 λ2)Z + λ2(σ ) for t N ( − 2,t−1 t ∈ (III) (III) LCL LCLE−σ = c2 α0 (III) (III) UCL UCLE−σ = c3 α0 2 EWMA control chart for σ based on ln(Xt ) 2 (IV) E(ln[(Yt µ0) ]) for t = 0 Statistic Z2,t = −(IV) 2 (1 λ2)Z + λ2(ln[(Xt µ0) ]) for t N ( − 2,t−1 − ∈ (IV) (IV) LCL LCLE−σ = c2 (IV) (IV) UCL UCLE−σ = c3 where k I,II,III,IV . ∈{ } In order to compute c(k), c(k) and c(k) for k I,II,III,IV , Schipper and Schmid (2001a) deﬁned a 1 2 3 ∈{ } system of 3 nonlinear equations to obtain an unique solution solution satisfying the following conditions:

the ARL∗ is equal to a pre-speciﬁed value; • E(RLupper) = E(RLlower) = E(RLupper) = E(RLlower), where E(RLupper) (resp. E(RLlower)) • 1 1 2 2 1 1 represents the in-control ARL of an individual upper (resp. lower) one-sided chart for µ, and upper lower E(RL2 ) (resp. E(RL2 )) denotes the in-control ARL of an individual upper (resp. lower) one-sided chart for σ.

That system can be solved using a 3-dimensional secant rule, having as starting values for the iteration the critical values of the individual one-sided charts (Schipper and Schmid, 2001a).

3.3 Estimating the ARL via Monte Carlo simulation

In this subsection, one compares the in-control and out-of-control ARL of the four simultaneous modiﬁed EWMA schemes for the process mean and variance of GARCH(1, 1) models previously studied in Schipper (2001, Section 2.3) and brieﬂy described in Section 3.2. One considered the two following target processes in Schipper (2001, p. 39):

Process I:I α =0.1 α =0.05 β =0.9; • 0 1 1

59 Process II: α =1.0 α =0.25 β =0.7. • 0 1 1 For all pairs (δ, θ) with

δ =0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 • θ =0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.5, 2.0, 3.0, • one:

simulated the time series y ,...,y for each i (i =1,...,rep), drawn from the target process, • { i,1 i,N } where the number of replications is equal to rep = 105 and the number of observations is equal to N = 1000; obtained the correspondent realization of the observed process x ,...,x for each i (i = • { i,1 i,N } 1,...,rep); computed the associated values of the control statistics for the charts for the mean and for the • variance, compared them with the associated control limits, counted the number of observations until a signal was triggered by the simultaneous scheme for each i and estimated the corresponding ARL.12

The four simultaneous schemes were calibrated in such a way that ARL∗ is equal to 60. The sets of values of smoothing parameters λ1 for the chart for µ and λ2 for the chart for σ considered are in Table 3.4 and coincide with the ones considered by Schipper (2001, p. 48) and Schipper and Schmid (2001a).

Table 3.4: Pairs of smoothing parameters: λ1 (chart for µ) and λ2 (chart for σ) — simultaneous modiﬁed EWMA schemes.

λ values (1) (2) (3) (4) (5) (6) (7) (8) (9)

λ1 0.10 0.25 0.50 0.75 1.00 0.10 1.00 1.00 1.00 λ2 0.10 0.25 0.50 0.75 1.00 1.00 0.10 0.25 0.50

Moreover, the three critical values for each simultaneous scheme and for (λ1, λ2) can be found in tables A.1-A.4 on Section A.2. Table 3.5 (resp. 3.6), whose layout was inspired in Schipper and Schmid (2001a, Table 1), contains the simulated ARL for Process I (resp. II). For Process I (resp. II), the ARL were computed for the pair

(λ1, λ2) that corresponds to the minimum ARL value in Schipper (2001, tables A.7, A.9, A.11, A.13, A.15, A.17, A.19 and A.21, pp. 111–126) (resp. Schipper, 2001, tables A.8, A.10, A.12, A.14, A.16, A.18, A.20 and A.22, pp. 111–126).

Each ARL estimate is followed by the pair (λ1, λ2) in Table 3.4 that was used for its computation. Additionally, for each value of (θ,δ), the ARL of the simultaneous modified EWMA scheme with chart for σ are listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV). Bear in mind that there are some differences between the ARL estimates in tables 3.5 and 3.6 and the estimates in Schipper (2001, tables A.7–A.22, pp. 111–126) and Schipper and Schmid (2001a, Table 1). Those differences are more obvious when one adopts the simultaneous modified EWMA scheme with chart for σ based on the conditional variance (scheme II). The major differences were detected for Process

12As before, one dismissed the realizations of the observed process for which no observation of the control statistics was beyond the control limits.

60 Table 3.5: Estimated ARL and the correspondent pair (λ1, λ2) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process I.

θ δ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 29.64 (7) 29.96 (7) 30.70 (7) 30.31 (7) 26.74 (7) 19.05 (7) 10.07 (7) 27.05 (5) 27.47 (5) 28.92 (5) 29.47 (5) 26.58 (5) 19.15 (5) 10.24 (5) 0.7 30.15 (5) 30.50 (5) 31.24 (5) 31.03 (5) 27.61 (5) 19.70 (5) 10.57 (9) 46.68 (7) 47.20 (7) 47.80 (7) 46.15 (7) 39.80 (7) 27.69 (7) 14.35 (7) 44.41 (7) 44.64 (7) 44.94 (7) 42.96 (7) 36.89 (7) 26.62 (7) 14.98 (7) 42.47 (5) 43.30 (7) 43.31 (5) 42.51 (5) 37.07 (5) 27.03 (5) 15.86 (7) 0.8 43.83 (5) 44.09 (5) 44.41 (9) 42.81 (9) 37.02 (5) 26.81 (5) 15.35 (5) 62.33 (1) 62.83 (1) 63.21 (1) 60.99 (7) 51.91 (7) 36.85 (7) 20.65 (7) 58.81 (7) 58.97 (7) 57.95 (7) 54.25 (7) 46.17 (7) 33.30 (7) 20.17 (7) 56.90 (1) 57.19 (1) 57.92 (1) 54.55 (7) 46.74 (7) 34.50 (7) 20.74 (7) 0.9 58.15 (7) 58.31 (8) 57.31 (5) 53.64 (7) 45.67 (7) 33.92 (7) 20.26 (7) 67.97 (1) 68.19 (1) 68.48 (1) 64.70 (7) 54.42 (7) 39.35 (7) 23.37 (7) 61.17 (1) 60.66 (3) 58.59 (4) 71.34 (5) 58.41 (5) 42.04 (5) 25.51 (5) 60.67 (1) 60.84 (5) 58.79 (5) 54.61 (5) 45.74 (5) 33.82 (5) 21.07 (5) 1.0 60.98 (5) 60.34 (7) 58.62 (7) 53.77 (7) 45.55 (7) 33.64 (7) 21.34 (7) 60.86 (7) 60.51 (7) 57.65 (5) 52.69 (5) 43.21 (5) 30.94 (5) 18.94 (5) 51.66 (5) 51.35 (5) 48.93 (5) 43.87 (5) 36.16 (5) 26.67 (5) 17.14 (5) 49.21 (5) 48.81 (5) 46.68 (5) 42.42 (5) 35.44 (5) 26.40 (5) 17.11 (5) 1.1 49.38 (7) 48.79 (7) 46.56 (7) 42.76 (7) 35.55 (7) 26.69 (7) 17.79 (5) 42.27 (9) 41.74 (9) 40.03 (5) 36.26 (5) 29.96 (5) 22.02 (5) 14.20 (5) 34.14 (5) 33.85 (5) 32.21 (5) 28.89 (5) 23.69 (5) 17.95 (5) 12.21 (5) 34.99 (5) 34.44 (5) 32.79 (5) 29.67 (5) 24.67 (5) 18.49 (5) 12.58 (5) 1.2 35.18 (7) 34.84 (5) 33.40 (5) 29.65 (7) 24.87 (7) 18.86 (7) 12.74 (7) 29.55 (9) 29.26 (9) 27.78 (9) 25.03 (9) 21.17 (9) 16.05 (5) 10.88 (5) 24.06 (5) 23.54 (5) 22.26 (5) 19.95 (5) 16.78 (5) 12.99 (5) 9.05 (5) 24.51 (5) 24.05 (5) 22.50 (5) 20.33 (5) 17.11 (5) 13.02 (5) 9.11 (5) 1.3 24.70 (7) 24.37 (8) 22.94 (7) 20.62 (7) 17.35 (7) 13.48 (7) 9.46 (7) 21.28 (9) 21.03 (9) 20.01 (9) 18.04 (9) 15.14 (9) 11.73 (9) 8.35 (9) 13.45 (5) 13.27 (5) 12.43 (5) 11.24 (5) 9.56 (5) 7.72 (5) 5.79 (5) 13.28 (5) 12.93 (5) 12.19 (5) 10.90 (5) 9.19 (5) 7.45 (5) 5.65 (5) 1.5 13.68 (5) 13.33 (9) 12.61 (5) 11.29 (9) 9.65 (5) 7.82 (7) 5.89 (5) 12.35 (9) 12.08 (9) 11.44 (9) 10.35 (9) 8.84 (9) 7.07 (9) 5.44 (9) 5.52 (5) 5.42 (5) 5.19 (5) 4.76 (5) 4.26 (5) 3.71 (5) 3.12 (5) 5.33 (5) 5.20 (5) 5.02 (5) 4.60 (5) 4.11 (5) 3.59 (5) 2.99 (5) 2.0 5.56 (5) 5.47 (5) 5.20 (5) 4.83 (5) 4.33 (5) 3.76 (5) 3.17 (5) 5.26 (9) 5.18 (9) 4.92 (9) 4.60 (9) 4.10 (9) 3.59 (9) 3.04 (9) 2.62 (5) 2.60 (5) 2.54 (5) 2.45 (5) 2.34 (5) 2.19 (5) 2.06 (5) 2.61 (9) 2.59 (5) 2.52 (5) 2.43 (5) 2.32 (5) 2.17 (5) 2.05 (5) 3.0 2.68 (5) 2.66 (5) 2.61 (5) 2.51 (5) 2.39 (5) 2.26 (5) 2.10 (5) 2.60 (9) 2.58 (9) 2.53 (5) 2.46 (5) 2.33 (9) 2.20 (9) 2.05 (5)

I, for certain values of δ when θ 0.7, 0.8, 0.9, 1.1, 1.2, 1.3 , and for Process II, for some of the δ values ∈{ } when θ 0.7, 0.8, 0.9, 1.1, 1.2, 1.3, 1.5 . Note that the in-control ARL values simulated for Process II ∈ { } do not coincide with ARL∗ = 60, unlike Process I. On top of that, a few significant differences were also detected when the simultaneous modified EWMA scheme with chart for σ based on the squared observations (scheme I) was applied to Process I and θ 1.0, 1.1, 1.2 for some values of δ. Those ∈ { } discrepancies are possibly due to: differences between the implementations of the control schemes; the fact that one used a distinct software in the Monte Carlo simulations.

Note that for a ﬁxed δ and for the four simultaneous schemes, the out-of-control ARL increases (resp. decreases) with θ, for θ< 1.0 (resp. θ> 1.0), as expected.

Regretfully, the out-of-control ARL can increase with δ for a ﬁxed θ, namely when the simultaneous

61 Table 3.6: Estimated ARL and the correspondent pair (λ1, λ2) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process II.

θ δ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 40.47 (8) 40.99 (8) 41.09 (8) 38.98 (8) 31.40 (8) 18.78 (8) 7.27 (8) 39.42 (5) 41.76 (7) 40.74 (5) 37.92 (5) 27.86 (5) 14.29 (5) 5.14 (5) 0.7 44.83 (9) 45.02 (9) 45.38 (5) 43.77 (9) 36.00 (9) 21.50 (9) 8.68 (5) 46.13 (7) 46.84 (7) 46.91 (7) 44.88 (7) 36.53 (7) 22.16 (7) 8.97 (7) 50.68 (8) 50.32 (8) 49.97 (8) 46.82 (8) 37.21 (8) 22.81 (8) 10.34 (9) 50.03 (7) 50.12 (7) 50.43 (7) 47.80 (7) 33.68 (5) 23.51 (8) 10.47 (9) 0.8 51.88 (9) 51.88 (9) 51.80 (9) 49.00 (9) 39.46 (9) 24.83 (9) 10.93 (9) 55.48 (7) 55.79 (7) 55.49 (7) 52.21 (7) 42.16 (7) 26.30 (7) 11.94 (7) 57.89 (7) 57.46 (7) 56.55 (7) 52.41 (7) 42.04 (9) 26.68 (9) 12.82 (9) 58.39 (1) 58.01 (7) 57.03 (7) 53.00 (7) 37.87 (5) 27.37 (8) 13.46 (7) 0.9 58.14 (9) 58.06 (9) 57.21 (9) 52.67 (9) 42.58 (9) 27.60 (9) 13.61 (5) 61.52 (7) 61.47 (7) 60.91 (7) 55.90 (7) 44.97 (7) 29.67 (9) 14.02 (9) 61.62 (1) 60.04 (6) 59.15 (9) 53.43 (9) 41.14 (5) 26.38 (5) 13.36 (5) 62.76 (9) 61.20 (2) 60.92 (9) 55.38 (9) 39.07 (5) 29.23 (9) 11.22 (5) 1.0 61.94 (5) 61.01 (7) 59.25 (9) 54.65 (8) 43.45 (9) 29.47 (8) 15.39 (7) 61.59 (4) 61.75 (7) 59.84 (8) 53.38 (9) 41.47 (5) 26.74 (5) 13.39 (5) 51.08 (6) 50.89 (6) 49.89 (6) 44.46 (5) 34.67 (5) 22.76 (5) 12.33 (5) 60.29 (5) 59.23 (5) 56.35 (5) 48.98 (5) 36.77 (5) 23.42 (5) 12.10 (5) 1.1 58.83 (1) 58.60 (1) 56.54 (7) 51.09 (7) 41.21 (7) 27.96 (7) 15.59 (9) 51.58 (6) 51.55 (6) 49.59 (5) 44.20 (5) 34.83 (5) 23.10 (5) 12.42 (5) 42.95 (6) 42.33 (6) 41.61 (6) 36.00 (5) 28.60 (5) 19.30 (5) 11.09 (5) 53.00 (5) 51.74 (5) 49.00 (5) 42.15 (5) 32.39 (5) 20.91 (5) 11.42 (5) 1.2 52.33 (1) 52.31 (1) 49.95 (7) 44.68 (7) 36.15 (7) 24.88 (7) 14.43 (7) 43.64 (4) 43.26 (6) 41.13 (5) 36.10 (5) 28.61 (5) 19.56 (5) 11.35 (5) 35.99 (6) 35.26 (5) 33.20 (5) 29.27 (5) 23.26 (5) 16.22 (5) 9.84 (5) 44.35 (5) 43.34 (5) 40.50 (5) 35.06 (5) 26.97 (5) 18.00 (5) 10.49 (5) 1.3 44.89 (1) 43.96 (7) 41.84 (7) 36.92 (7) 29.64 (7) 21.26 (7) 12.97 (7) 36.25 (5) 35.57 (5) 33.50 (5) 29.53 (5) 23.52 (5) 16.47 (5) 9.93 (5) 24.27 (5) 23.79 (5) 22.32 (5) 19.65 (5) 15.74 (5) 11.58 (5) 7.69 (5) 29.30 (5) 28.71 (5) 26.59 (5) 22.88 (5) 18.02 (5) 12.65 (5) 7.98 (5) 1.5 29.02 (7) 28.49 (7) 26.63 (7) 23.54 (7) 19.33 (7) 13.95 (7) 9.21 (7) 24.58 (5) 24.02 (5) 22.45 (5) 19.97 (5) 16.06 (5) 11.93 (5) 7.72 (5) 10.38 (5) 10.08 (5) 9.44 (5) 8.48 (5) 7.12 (5) 5.71 (5) 4.26 (5) 11.32 (5) 11.04 (5) 10.28 (5) 9.01 (5) 7.58 (5) 5.87 (5) 4.33 (5) 2.0 11.05 (7) 10.95 (7) 10.19 (7) 9.15 (7) 7.71 (7) 6.15 (7) 4.58 (7) 10.37 (5) 10.16 (5) 9.59 (5) 8.51 (5) 7.22 (5) 5.80 (5) 4.39 (5) 3.69 (5) 3.63 (5) 3.48 (5) 3.27 (5) 2.98 (5) 2.70 (5) 2.36 (5) 3.69 (5) 3.64 (5) 3.50 (5) 3.25 (5) 2.99 (5) 2.65 (5) 2.34 (5) 3.0 3.82 (9) 3.80 (9) 3.67 (5) 3.42 (7) 3.12 (9) 2.81 (7) 2.47 (5) 3.71 (5) 3.64 (5) 3.53 (5) 3.28 (5) 3.02 (5) 2.70 (5) 2.39 (5) modiﬁed EWMA scheme with a chart for σ based on the squared observations (scheme I) is applied to the two processes and θ =0.7. This means that the control scheme I becomes less sensitive to some shifts in the process variance, as the short-lived shift in the process mean becomes more severe. Moreover, in Section 4.2, the estimates of the minimum ARL are used to compare the performance of the four simultaneous schemes, in the presence of a sustained shift in the process variance and of a short-lived shift in the process mean.

62 3.4 Illustration

The purpose of this section is to provide an instructive example of the use of simultaneous control schemes in ﬁnance. With that in mind, the daily returns of the Deutsche Bank share are analyzed.

Example 3.3. — Application to the Deutsche Bank share (Schipper, 2001, pp. 49–56) The target process was determined from the data of a starting block13 which consists of the daily returns of the Deutsche Bank shares dated from May 21, 1996 to May 20, 1997. In order to ﬁt a GARCH model to the starting block data, Schipper (2001) used the SAS routine AUTOREG and obtained the following model

1/2 2 Rt = εt ht = εt 0.0629+ 0.228Rt−1 +0.757ht−1, (3.21) q with ε i.i.d. t . Based on this target process, the daily returns of the shares of Deutsche Bank from May t ∼ 6

50 DBK

21/05/97 21/06/97 21/07/97 21/08/97 21/09/97 Date

Figure 3.13: Stock price of the Deutsche Bank share, from May 21, 1997 to September 22, 1997.

21, 1997 to September 22, 199714 were investigated. The stock prices and the daily returns on this time period can be found in figures 3.13 and 3.14, respectively. The data was analyzed sequentially. The aim now is to detect deviations from the previously fitted model and therefore control charts were considered. Consequently, a simultaneous modified Shewhart scheme was applied to the returns and to the squared

RDBK 0

−5

21/05/97 21/06/97 21/07/97 21/08/97 21/09/97 Date

Figure 3.14: Returns of the Deutsche Bank share, from May 21, 1997 to September 22, 1997. returns of the Deutsche Bank share, from May 21, 1997 to September 22, 1997, as shown by figures 3.14 and 3.15. The control limits are represented by dotted lines.15 This control scheme triggers a signal on: 13In practise, this starting block should be chosen based on the experience of an analyst and it should reflect the typical behavior of the share. In this case, it was chosen arbitrarily (Schipper, 2001, p. 49). 14It is worth mentioning that the plots in this section were obtained using the data set found in DB (2015) which is slightly different from the one used by Schipper (2001, pp. 49–56). The major differences are in the data from September 22, 1997 to October 10, 1997 omitted in this example. Moreover, the values of stock prices from May 21, 1997 to September 22, 1997 in DB (2015) are in a different scale; fortunately, this fact does not influence the values of the returns. 15The control limits of the simultaneous modified Shewhart scheme can be found in Schipper (2001, figures 2.14 and 2.15, pp. 52, 54).

63 June 25, 1997; July, 22–24, 1997; August 22, 1997; September 22, 1997. It is also important to refer that when the mean chart prompts an alarm, the chart for the variance also triggers a signal.

40 RQDBK

21/05/97 21/06/97 21/07/97 21/08/97 21/09/97 Date

Figure 3.15: Squared returns of the Deutsche Bank share, from May 21, 1997 to September 22, 1997.

The ﬁrst signal given by the control scheme refers to June 25, 1997 and, according to Schipper (2001, p. 53), it reﬂects the fact that after good Wall-Street results the DAX increased to record high. On July 21, 1997, after the stock exchange had closed, the Vereinsbank and the Hypo-Bank announced their merger. This merger caused an increase on the stock price and, consequently a signal was triggered on July 22, 1997 (Schipper, 2001, p. 53). For more details on these and other signals, the reader is referred to Schipper (2001, p. 53). In light of all these signals, this author recommends the use of a new starting block to estimate a target process with higher volatility (see Schipper, 2001, pp. 53–54). It is worth mentioning that Schipper (2001) continues to explore this example in Section 3.5, where this author describes an objective trading strategy that combines the newest developments in SPC with trading techniques for options (Schipper, 2001, p. 57). •

64 Chapter 4

On the phenomenon of misleading and unambiguous signals

In many applications the diagnostic procedures followed when an out-of-control signal is observed may diﬀer depending on whether that signal emanated from the X-bar chart or the R chart [. . . ] Ralph C. St. John and Daniel J. Bragg

The concept of misleading signal (MS) was ﬁrstly introduced by St. John and Bragg (1991) that identiﬁed the three following types of MS:

I. the process mean increases but the signal is triggered by the chart for the variance or on the negative side of the chart for the mean;

II. the process mean decreases but the signal is triggered by the chart for the variance or on the positive side of the chart for the mean;

III. the process variance has shift up but the signal is triggered by the chart for the mean.

Posteriorly, a fourth type was deﬁned by Morais and Pacheco (2000):

IV. the process mean shifts but the signal is triggered by the chart for the variance.

More generally, a misleading signal (MS) is a valid alarm1 that corresponds to the misinterpretation of a shift on the process mean (resp. variance) as a shift in the process variance (resp. mean) when it is considered a simultaneous scheme for these two parameters (Morais et al., 2014). Additionally, Ramos (2013, p. 112) identiﬁed another class of valid signals – the unambiguous signals (UNS) that were divided into two types by Ralha (2014):

III. the process variance is out-of-control and the ﬁrst chart that triggers a signal is the chart for variance;

IV. the process mean is out-of-control and the correspondent chart is the ﬁrst to signal.

In this thesis, the attention is restricted to the misleading and unambiguous signals of types III and IV that are systematized in Table 4.1 (Ralha et al., 2015).

1An alarm is valid if the signal is triggered in the presence of assignable causes, i.e., when the process is indeed out-of- control (Ramos, 2013, p. 3).

65 Table 4.1: Types of misleading and unambiguous signals.

Valid signals µ σ First chart to signal

MSIII in-control out-of-control chart for µ MSIV out-of-control in-control chart for σ UNSIII in-control out-of-control chart for σ UNSIV out-of-control in-control chart for µ

Regarding the MS and UNS, the main question should be how frequent they are and that can be measured by the probabilities of the MS and UNS (PMS and PUNS). Moreover, a simultaneous control scheme should trigger MS (resp. UNS) as infrequent (resp. often) as possible, i.e., the PMS (resp. PUNS) should be small (resp. large). These probabilities have been used to assess the performance of several simultaneous schemes. For instance, the PMS have been computed and analyzed by several authors for: univariate normal i.i.d. output (Reynolds and Stoumbos, 2001; Morais, 2002; Ralha, 2014; Ralha et al., 2015); univariate stationary Gaussian processes (Knoth et al., 2009; Ramos et al., 2012; Morais et al., 2014); bivariate normal i.i.d. output (Ramos, 2013; Ramos et al., 2013a; Ramos et al., 2013b; Ralha, 2014); and multivariate normal i.i.d. processes (Ramos et al., 2015). Since the assignable causes on charts for µ can be different of those on charts for σ, the appropriate trading strategy that follows a signal can differ depending on whether the signal is triggered by the chart for µ or by the one for σ. Therefore a MS can lead to a reduction in the profit made by the traders/brokers if it led them to use an inappropriate trading strategy. Consequently, PMS and PUNS should be used to evaluate the performance of simultaneous schemes that are used to monitor changes in the mean and variance of financial time series, such as the four simultaneous modified EWMA schemes introduced by Schipper (2001) and previously described in Section 3.2.

4.1 Estimating PMS and PUNS via Monte Carlo simulation

The PMS and PUNS of types III and IV are deﬁned as follows:

PMS (θ)= P [RL (0,θ) > RL (0,θ)], θ = 1 (4.1) III σ µ 6 PMS (δ)= P [RL (δ, 1) > RL (δ, 1)], δ = 0 (4.2) IV µ σ 6 PUNS (θ)= P [RL (0,θ) > RL (0,θ)], θ = 1 (4.3) III µ σ 6 PUNS (δ)= P [RL (δ, 1) > RL (δ, 1)], δ =0, (4.4) IV σ µ 6 where RLµ and RLσ are the RL of the individual charts for µ and for σ, respectively (Ralha et al., 2015). The joint/marginal distribution function of the RL of the individual control charts is necessary to derive the exact expressions of the PMS and PUNS of types III and IV. As mentioned before, when the target process is governed by a GARCH model, one is unable to derive the joint distributions regarding Y : t N that are needed to determine the distribution of the RL of the individual control charts. { t ∈ } Furthermore, RLµ and RLσ are surely dependent random variables. Thus, when one assumes a GARCH target process, those expressions can not be simpliﬁed and one has to rely on Monte Carlo simulations to estimate the PMS and PUNS of types III and IV.

66 Once again, to monitor simultaneously the process mean and variance one used the four EWMA simultaneous control schemes described in the previous chapter. As far as the shifts in the parameters are concerned, two diﬀerent out-of-control scenarios have been considered in order to obtain the estimates of the PMS and PUNS of types III and IV:

in the first case, the process standard deviation suffers a sustained shift of magnitude θ =0.7, 0.8, 0.9, • 1.1, 1.2, 1.3, 1.5, 2.0, 3.0 and the process mean stays on-target (δ = 0); in the second situation, the process mean suffers a short-lived shift of magnitude δ =0.5, 1.0, 1.5, 2.0, • 2.5, 3.0 and the process variance stays on-target (θ = 1).

For each constellation of parameter values and for all the pairs of smoothing parameters λ1 and λ2 in Table 3.4, one:

simulated the time series y ,...,y for each i (i =1,...,rep), drawn from the target process, • { i,1 i,N } where the number of replications is equal to rep = 105 values and the number of observations simulated is equal to N = 1000; obtained the correspondent realization of the observed process x ,...,x for each i (i = • { i,1 i,N } 1,...,rep); calculated for each i the associated values of the control statistics and compared them with the • corresponding control limits; counted the number of misleading signals, the number of unambiguous signals and the number of • signals triggered by the simultaneous schemes, and estimated the corresponding PMS and PUNS.

Finally, one considered once again that the control schemes were calibrated in such a way that the in-control ARL is equal to ARL∗ = 60. The simulations refer to processes I and II. The plots in the next four subsections prove to be essential to devise the behavior of the estimated PMS and PUNS of types III and IV. The associated values can be found in tables A.5–A.12.

4.1.1 PMS of Type III

The plots of the estimated PMS of Type III as a function of θ can be found in Figure 4.1, and the ones on the left (resp. right) refer to Process I (resp. II). If one compares the several plots in Figure 4.1, it seems that the simultaneous modiﬁed EWMA 2 scheme with a chart for σ based in Xt (scheme I), is the one with the smallest PMSIII values, for both processes. (For more related remarks, please refer to Section 4.2.)

The general behavior of the estimated PMSIII is systematized in Table 4.2, for processes I and II, each pair (λ1, λ2) and the four simultaneous modiﬁed EWMA schemes.

One can verify that the PMSIII values are usually not monotonous in θ; on top of that, they tend to increase with the magnitude of the shift in the process variance (an undesirable behavior) and then ∗ 2 decrease. Additionally, for the control simultaneous schemes with charts for σ based on (σt ) and on 2 ln(Xt ), the behavior of the PMSIII values can vary not only with the pair (λ1, λ2), but also with the target process. The reader should also be reminded that Knoth et al. (2009) (resp. Ramos, 2013, p. 27) have previ-

67 (1) (2) (3)

III III (4) (5) (6)

PMS PMS (7) (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 θ θ

2 2 (a) Joint scheme based on Xt and Xt (Process I). (b) Joint scheme based on Xt and Xt (Process II).

(1) (2) (3)

III III (4) (5) (6)

PMS PMS (7) (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 θ θ

2 2 (c) Joint scheme based on Xt and σt (Process I). (d) Joint scheme based on Xt and σt (Process II).

b b

(1) (2) (3)

III III (4) (5) (6)

PMS PMS (7) (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 θ θ

∗ 2 ∗ 2 (e) Joint scheme based on Xt and (σt ) (Process I). (f) Joint scheme based on Xt and (σt ) (Process II).

(1) (2) (3)

III III (4) (5) (6)

PMS PMS (7) (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 θ θ

2 2 (g) Joint scheme based on Xt and ln(Xt ) (Process I). (h) Joint scheme based on Xt and ln(Xt ) (Process II).

Figure 4.1: Estimated PMSIII (θ) for the pairs (λ1, λ2): (1) (0.10,0.10); (2) (0.25,0.25); (3) (0.50,0.50); (4) (0.75,0.75); (5) (1.00,1.00); (6) (0.10,1.00); (7) (1.00,0.10); (8) (1.00,0.25) and (9) (1.00,0.50).

68 Table 4.2: Behavior of the estimated PMSIII (θ).

(λ1, λ2) Proc. (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10)(0.25,0.25)(0.50,0.50)(0.75,0.75)(1.00,1.00)(0.10,1.00)(1.00,0.10)(1.00,0.25)(1.00,0.50) Joint scheme based I 2 ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ on Xt and X II t ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ Joint scheme based I 2 ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ on Xt and σ II t ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ Joint scheme based I ∗ 2 ↑↓↑ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↑↓ ↑↓ on Xt and (σb ) II t ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↑↓ ↑↓ Joint scheme based I 2 ↑↓↑ ↑↓↑ ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↑ ↑↓ on Xt and ln(X ) II t ↑ ↑ ↑ ↑↓ ↑↓ ↑↓ ↑ ↑ ↑

ously established that the PMSIII does not depend on the parameters that translate the autocorrelation structure of the target process, when a simultaneous residual scheme is used to control the process mean and variance of an AR(1) (resp. stationary Gaussian) process. Moreover, these authors have reported a decreasing behavior of PMSIII (θ) in speciﬁc cases. This is obviously not the case when one is leading with a GARCH(1, 1) target process and other types of simultaneous schemes. The estimated values of the PMS of Type III are summarized in tables A.5 and A.6. They are often larger than 0.5, for θ 1.1, 1.2, 1.3 or when the pairs (7), (8) and (9) are considered. In addition, ∈{ } sometimes the PMSIII can be larger than 0.95 for large shifts in the process variance, namely PMSIII (3) =

0.9531 (resp. PMSIII (3) = 0.9572) when the simultaneous modiﬁed EWMA scheme with a chart for σ 2 based on ln(Xt ) (scheme IV) is used with (λ1, λ2) = (1.0, 0.1) (pair (7)), to monitor the mean and variance of Process I (resp. II).

4.1.2 PMS of Type IV

In Figure 4.2, one ﬁnds on the right (resp. left) the plots of PMSIV (δ) for Process I (resp. II). It is worth mentioning that in this case, when one compares the several plots in Figure 4.2, it seems ∗ 2 that the simultaneous modiﬁed EWMA scheme with a chart for σ based on (σt ) (scheme III) should be used if one intends to avoid PMSIV values larger than 0.5 as much as possible, for both processes. (Further related comments can be found in Section 4.2.)

Table 4.3: Behavior of the estimated PMSIV (δ).

(λ1, λ2) Proc. (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10)(0.25,0.25)(0.50,0.50)(0.75,0.75)(1.00,1.00)(0.10,1.00)(1.00,0.10)(1.00,0.25)(1.00,0.50) Joint scheme based I 2 ↑↓ ↓↑↓ ↓ ↓ ↓ ↑ ↓ ↓ ↓ on Xt and X II t ↓↑↓ ↓ ↓ ↓ ↑ ↓ ↓ ↓ Joint scheme based I 2 ↓↑ ↓↑↓ ↓ ↓ ↓ ↑ ↓ ↓ ↓ on Xt and σ II t ↑ ↓↑↓ ↓ ↓ ↓ ↑ ↓ ↓ ↓ Joint scheme based I ∗ 2 ↓↑ ↓ ↓ ↓ ↓ ↓↑ ↓ ↓ ↓ on Xt and (σb ) II t ↓↑↓↑ ↓ ↓ ↓ ↓ ↓↑ ↓ ↓ ↓ Joint scheme based I 2 ↓↓↓↓↓↑↓↓↓ on Xt and ln(X ) II t ↓↓↓↓↓↑↓↓↓

The behavior of the estimates of PMSIV is summarized in Table 4.3. For the two processes one considered, the PMSIV value usually decreases with δ, as desired. Nonetheless, for the pairs (1), (2)

69 (1) (2) (3)

IV IV (4) (5) (6)

PMS PMS (7) (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 δ δ

2 2 (a) Joint scheme based on Xt and Xt (Process I). (b) Joint scheme based on Xt and Xt - Process II.

(1) (2) (3)

IV IV (4) (5) (6)

PMS PMS (7) (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 δ δ

2 2 (c) Joint scheme based on Xt and σt (Process I). (d) Joint scheme based on Xt and σt (Process II).

b b

(1) (2) (3)

IV IV (4) (5) (6)

PMS PMS (7) (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 δ δ

∗ 2 ∗ 2 (e) Joint scheme based on Xt and (σt ) (Process I). (f) Joint scheme based on Xt and (σt ) (Process II).

(1) (2) (3)

IV IV (4) (5) (6)

PMS PMS (7) (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 δ δ

2 2 (g) Joint scheme based on Xt and ln(Xt ) (Process I). (h) Joint scheme based on Xt and ln(Xt ) (Process II).

Figure 4.2: Estimated PMSIV (δ) for the pairs (λ1, λ2): (1) (0.10,0.10); (2) (0.25,0.25); (3) (0.50,0.50); (4) (0.75,0.75); (5) (1.00,1.00); (6) (0.10,1.00); (7) (1.00,0.10); (8) (1.00,0.25) and (9) (1.00,0.50).

70 and (6), the PMSIV has sometimes a non monotonous behavior. One also realized that the behavior of the PMSIV depends on the autocorrelation structure of the target process, as reported by Knoth et al. (2009) and Ramos (2013, p. 27) for simultaneous residual schemes for other target processes.

Additionally, the estimated PMSIV shown in Figure 4.2 and in tables A.7 and A.8 lead to the conclusion that these estimates can be larger than 0.75 for large shifts in the process mean, some combinations of control scheme and some pairs of (λ1, λ2). For example, PMSIV (3) = 0.7615 (resp. 2 PMSIV (3) = 0.8384) when the simultaneous modiﬁed EWMA scheme with chart for σ based on σt is used with (λ1, λ2)=(0.1, 1.0) (pair (6)), for Process I (resp. II), as reported in tables A.7–A.8. b 4.1.3 PUNS of Type III

The plots of the estimated PUNS of Type III found in Figure 4.3 give no clue as to which of the schemes should be used in order to maximize the PUNSIII . One can add though that they depend on the scheme and pair (λ1, λ2) one uses, and also on the process one is monitoring. The simultaneous scheme with 2 chart for σ based on ln Xt (scheme IV) should be avoided when one is dealing with Process II because the values of PUNSIII are rather small for all pairs (λ1, λ2), with the exception of the one associated with pair (6). (Further related comments can be found in Section 4.2.)

The overall behavior of the estimated PUNSIII is described in Table 4.4.

Table 4.4: Behavior of the estimated PUNSIII (θ).

(λ1, λ2) Proc. (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10)(0.25,0.25)(0.50,0.50)(0.75,0.75)(1.00,1.00)(0.10,1.00)(1.00,0.10)(1.00,0.25)(1.00,0.50) Joint scheme based I 2 ↓↑↓ ↓↑↓ ↓ ↓ ↓ ↓↑↓ ↓ ↓ ↓ on Xt and X II t ↓↑ ↓↑↓ ↓ ↓ ↓ ↓↑ ↓ ↓ ↓ Joint scheme based I 2 ↓↑↓ ↓↑↓ ↓↑↓ ↓ ↓ ↓↑↓ ↓ ↓ ↓ on Xt and σ II t ↓↑ ↓↑ ↓↑↓ ↓ ↓ ↓↑ ↓ ↓ ↓ Joint scheme based I ∗ 2 ↓↑↓ ↓↑↓ ↓ ↓ ↓ ↓↑↓ ↓ ↓ ↓ on Xt and (σb ) II t ↓↑↓ ↓↑↓ ↓↑↓ ↓ ↓ ↓↑ ↓ ↓ ↓ Joint scheme based I 2 ↓↑↓ ↓ ↓ ↓ ↓ ↓↑↓ ↓ ↓ ↓ on Xt and ln(X ) II t ↓ ↓ ↓ ↓ ↓ ↓↑ ↓ ↓ ↓

One can observe that the PUNSIII value usually decreases with θ, contrary to what one would expect and strongly wishes for.

It is worth mentioning that in tables A.9–A.10 one can unfortunately ﬁnd some values of the PUNSIII smaller than 0.01, for large shifts in the process variance. In particular, when the simultaneous modiﬁed 2 EWMA scheme with a chart for σ based on Xt (scheme I) is used to monitor Process I (resp. II) with

(λ1, λ2)=(1.0, 0.5) (pair (9)) one has PUNSIII (3) = 0.0084 (resp. PUNSIII (3) = 0.0013). This is not a surprising fact given the large values of the PMSIII in these circumstances.

4.1.4 PUNS of Type IV

Figure 4.4 contains the plots of the estimated PUNS of Type IV as function of δ. These plots and the estimates in tables A.11–A.12 suggest that the use of the simultaneous modiﬁed EWMA scheme with a ∗ 2 chart for σ based on (σt ) (scheme III) leads in general to the largest values of PUNSIV for processes I and II.

71 (1) (2) (3) III III (4) (5) (6) (7) PUNS PUNS (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 θ θ

2 2 (a) Joint scheme based on Xt and Xt (Process I). (b) Joint scheme based on Xt and Xt (Process II).

(1) (2) (3) III III (4) (5) (6) (7) PUNS PUNS (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 θ θ

2 2 (c) Joint scheme based on Xt and σt (Process I). (d) Joint scheme based on Xt and σt (Process II).

b b

(1) (2) (3) III III (4) (5) (6) (7) PUNS PUNS (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 θ θ

∗ 2 ∗ 2 (e) Joint scheme based on Xt and (σt ) (Process I). (f) Joint scheme based on Xt and (σt ) (Process II).

(1) (2) (3) III III (4) (5) (6) (7) PUNS PUNS (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 θ θ

2 2 (g) Joint scheme based on Xt and ln(Xt ) (Process I). (h) Joint scheme based on Xt and ln(Xt ) (Process II).

Figure 4.3: Estimated PUNSIII (θ) for the pairs (λ1, λ2): (1) (0.10,0.10); (2) (0.25,0.25); (3) (0.50,0.50); (4) (0.75,0.75); (5) (1.00,1.00); (6) (0.10,1.00); (7) (1.00,0.10); (8) (1.00,0.25) and (9) (1.00,0.50).

72 (1) (2) (3) IV IV (4) (5) (6) (7) PUNS PUNS (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 δ δ

2 2 (a) Joint scheme based on Xt and Xt (Process I). (b) Joint scheme based on Xt and Xt (Process II).

(1) (2) (3) IV IV (4) (5) (6) (7) PUNS PUNS (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 δ δ

2 2 (c) Joint scheme based on Xt and σt (Process I). (d) Joint scheme based on Xt and σt (Process II).

b b

(1) (2) (3) IV IV (4) (5) (6) (7) PUNS PUNS (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 δ δ

∗ 2 ∗ 2 (e) Joint scheme based on Xt and (σt ) (Process I). (f) Joint scheme based on Xt and (σt ) (Process II).

(1) (2) (3) IV IV (4) (5) (6) (7) PUNS PUNS (8) (9) 0.0 0.4 0.8 0.0 0.4 0.8 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 δ δ

2 2 (g) Joint scheme based on Xt and ln(Xt ) (Process I). (h) Joint scheme based on Xt and ln(Xt ) (Process II).

Figure 4.4: Estimated PUNSIV (δ) for the pairs (λ1, λ2): (1) (0.10,0.10); (2) (0.25,0.25); (3) (0.50,0.50); (4) (0.75,0.75); (5) (1.00,1.00); (6) (0.10,1.00); (7) (1.00,0.10); (8) (1.00,0.25) and (9) (1.00,0.50).

73 The behavior of the estimated PUNSIV has been systematized in Table 4.5, for processes I and II, each pair (λ1, λ2) and the four simultaneous modiﬁed EWMA schemes.

Table 4.5: Behavior of the estimated PUNSIV (δ).

(λ1, λ2) Proc. (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10)(0.25,0.25)(0.50,0.50)(0.75,0.75)(1.00,1.00)(0.10,1.00)(1.00,0.10)(1.00,0.25)(1.00,0.50) Joint scheme based I 2 ↓ ↑↓ ↓ ↓ ↓ ↓ ↑↓ ↑↓ ↑↓ on Xt and X II t ↓↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↓ ↑↓ ↑↓ ↑↓ Joint scheme based I 2 ↓ ↑↓ ↑↓ ↑↓ ↓ ↓ ↑ ↑ ↑↓ on Xt and σ II t ↓ ↓ ↓ ↓ ↓ ↓ ↑↓ ↑↓ ↓ Joint scheme based I ∗ 2 ↑↓ ↑ ↑ ↑↓ ↑↓ ↑↓ ↑ ↑ ↑ on Xt and (σb ) II t ↑↓↑↓ ↑ ↑ ↑↓ ↑↓ ↑↓ ↑ ↑ ↑ Joint scheme based I 2 ↑ ↑ ↑ ↑↓ ↑↓ ↓ ↑ ↑ ↑ on Xt and ln(X ) II t ↑ ↑ ↑ ↑ ↑↓ ↓ ↑ ↑ ↑

This performance measure seems to behave heterogeneously, can sadly decrease and depends on the parameters of the target process.

One can also notice that, for large short-lived shifts in process mean, the PUNSIV can be smaller than 0.20 (see tables A.11–A.12). In fact, PUNSIV (3) = 0.1877 (resp. PUNSIV (3) = 0.1976), when the 2 simultaneous modiﬁed scheme with a chart for σ based on Xt (scheme I) with (λ1, λ2) = (0.75, 0.75) (pair (4)) is adopted to monitor the mean and variance of Process I (resp. II).

4.2 Summary of ﬁndings

As mentioned before, it is desirable that a control scheme triggers a signal as soon as possible when the process is out-of-control, emits MS sporadically and triggers UNS as often as possible. Therefore when several simultaneous control schemes are compared, one ought to choose the one that is associated with the minimum out-of-control ARL or the minimum PMS or the maximum PUNS.

The minimum ARL, the minimum PMSIII and the maximum PUNSIII estimated for Process I (resp. II) in the presence of several sustained shifts in scale can be found in Table 4.6 (resp. 4.7). The minimum

ARL were taken from Table 3.5 (resp. 3.6), the minimum PMSIII were selected in Table A.5 (resp. A.6) and the maximum PUNSIII were identiﬁed in Table A.9 (resp. A.10), for Process I (resp. II) and for each simultaneous scheme, over all the pairs (λ1, λ2).

Similarly, Table 4.8 (resp. 4.9) summarizes the minimum ARL, the minimum PMSIV and the maximum PUNSIV computed for Process I (resp. II) in the presence of several short-lived shifts in location. For Process I (resp. II) and for each simultaneous scheme, the minimum ARL were copied from Table

3.5 (resp. 3.6), the minimum PMSIV were identiﬁed in Table A.7 (resp. A.8) and the maximum PUNSIV were obtained from Table A.11 (resp. A.12).

The estimated ARL and PMS (resp. PUNS) values are followed by the label of the pair (λ1, λ2) at which the minimum (resp. maximum) is attained. For each value of θ (resp. δ), the ARL, the PMSIII

(resp. PMSIV ) and the PUNSIII (resp. PUNSIV ) estimated for the simultaneous modiﬁed EWMA scheme with the chart for σ listed in order corresponding to: squared observations, conditional variance, exponentially weighted variance and logarithm of squared observations.

74 The values corresponding to the minimum ARL, the minimum PMSIII (resp. PMSIV ) and the maximum PUNSIII (resp. PUNSIV ) for a given θ (resp. δ) when the values obtained for the four charts are compared are in bold. Tables 4.6 to 4.9 surely deserve some comments. When there is a sustained shift in the process variance, the simultaneous EWMA control scheme with the minimum ARL values, most of the times, is the one with the chart for σ based on the logarithm of squared observations (resp. the squared observations), when Process I (resp. II) is being monitored.

The minimum PMSIII were attained, in most cases, when one uses the simultaneous EWMA control scheme with a chart for σ based on the conditional variance (resp. the logarithm of squared observations) with λ2 =1.0, for Process I (resp. II).

The largest PUNSIII values are usually observed, for both processes, when one adopts the simultaneous EWMA schemes with a chart for σ based on the conditional variance. It is worth mentioning that, for some small shifts, the maximum values of PUNSIII in bold are regretfully smaller than 0.5.

Table 4.6: Minimum ARL, minimum PMSIII (θ) and maximum PUNSIII (θ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process I.

Performance θ measures 0.7 0.8 0.9 1.1 1.2 1.3 1.5 2.0 3.0 29.64 (7) 44.41 (7) 58.81 (7) 51.66 (5) 34.14 (5) 24.06 (5) 13.45 (5) 5.52 (5) 2.62 (5) 27.05 (5) 42.47 (5) 56.90 (1) 49.21 (5) 34.99 (5) 24.51 (5) 13.28 (5) 5.33 (5) 2.61 (9) min(ARL) 30.15 (5) 43.83 (5) 58.15 (7) 49.38 (7) 35.18 (7) 24.70 (7) 13.68 (5) 5.56 (5) 2.68 (5) 46.68 (7) 62.33 (1) 67.97 (1) 42.27 (9) 29.55 (9) 21.28 (9) 12.35 (9) 5.26 (9) 2.60 (9) 0.0163 (7) 0.0754 (7) 0.2032 (9) 0.3778 (9) 0.3802 (9) 0.3600 (9) 0.2692 (1) 0.1175 (1) 0.0338 (1) 0.0140 (5) 0.0701 (5) 0.2078 (5) 0.4665 (5) 0.4345 (6) 0.3559 (6) 0.2243 (6) 0.0754 (6) 0.0146 (6) min(PMSIII ) 0.0163 (5) 0.0744 (5) 0.2206 (5) 0.5145 (6) 0.4729 (6) 0.4002 (6) 0.2726 (6) 0.1285 (6) 0.0416 (6) 0.0229 (7) 0.1065 (7) 0.2396 (5) 0.3434 (5) 0.3493 (5) 0.3355 (5) 0.2811 (6) 0.1339 (6) 0.0409 (6) 0.9810 (7) 0.9092 (7) 0.7170 (7) 0.4738 (6) 0.4912 (6) 0.5219 (6) 0.5818 (6) 0.6696 (6) 0.6317 (6) 0.9840 (8) 0.9177 (7) 0.7365 (7) 0.4465 (6) 0.4724 (6) 0.5278 (6) 0.6322 (6) 0.7303 (6) 0.6812 (6) max(PUNSIII ) 0.9827 (8) 0.9150 (8) 0.7357 (7) 0.4190 (6) 0.4327 (6) 0.4856 (6) 0.5706 (6) 0.6348 (6) 0.5783 (6) 0.9764 (7) 0.8889 (7) 0.7089 (7) 0.4748 (6) 0.4899 (6) 0.5209 (6) 0.5854 (6) 0.6706 (6) 0.6323 (6)

Table 4.7: Minimum ARL, minimum PMSIII (θ) and maximum PUNSIII (θ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process II.

Performance θ measures 0.7 0.8 0.9 1.1 1.2 1.3 1.5 2.0 3.0 40.47 (8) 50.68 (8) 57.89 (7) 51.08 (6) 42.95 (6) 35.99 (6) 24.27 (5) 10.38 (5) 3.69 (5) 39.42 (5) 50.03 (7) 58.39 (1) 60.29 (5) 53.00 (5) 44.35 (5) 29.30 (5) 11.32 (5) 3.69 (5) min(ARL) 44.83 (9) 51.88 (9) 58.14 (9) 58.83 (1) 52.33 (1) 44.89 (1) 29.02 (7) 11.05 (7) 3.82 (9) 46.13 (7) 55.48 (7) 61.52 (7) 51.58 (6) 43.64 (4) 36.25 (5) 24.58 (5) 10.37 (5) 3.71 (5) 0.0752 (9) 0.1354 (9) 0.2131 (9) 0.3190 (5) 0.3477 (5) 0.3628 (5) 0.3685 (5) 0.3122 (6) 0.1274 (6) 0.0740 (5) 0.1356 (5) 0.2177 (5) 0.3849 (5) 0.4374 (5) 0.4530 (5) 0.4422 (5) 0.3084 (6) 0.1052 (6) min(PMSIII ) 0.0936 (5) 0.1614 (5) 0.2551 (5) 0.4685 (5) 0.5345 (5) 0.5632 (5) 0.5212 (6) 0.3547 (6) 0.1573 (6) 0.1147 (7) 0.1707 (5) 0.2251 (5) 0.3093 (5) 0.3351 (5) 0.3513 (5) 0.3565 (5) 0.3057 (5) 0.1271 (6) 0.8686 (7) 0.7688 (7) 0.6291 (7) 0.4553 (6) 0.4282 (6) 0.4194 (6) 0.4267 (6) 0.4997 (6) 0.5557 (6) 0.8758 (9) 0.7743 (8) 0.6367 (7) 0.4128 (6) 0.3641 (6) 0.3533 (6) 0.3835 (6) 0.5177 (6) 0.6195 (6) max(PUNSIII ) 0.8711 (5) 0.7736 (5) 0.6426 (8) 0.3911 (6) 0.3316 (6) 0.3132 (6) 0.3417 (6) 0.4371 (6) 0.4853 (6) 0.8721 (7) 0.7724 (7) 0.6429 (7) 0.4520 (6) 0.4282 (6) 0.4180 (6) 0.4253 (6) 0.4997 (6) 0.5572 (6)

75 In the presence of a short-lived shift in the process mean, the minimum ARL values are associated, in most cases, to the simultaneous EWMA control scheme with the chart for σ based on the logarithm of squared observations (resp. the squared observations), for Process I (resp. II).

The smallest PMSIV were obtained, for both processes, when one uses a simultaneous Shewhart control scheme with a chart for σ based on the squared observations.

Moreover, the maximum PUNSIV were attained, in nearly all the cases, when the simultaneous

EWMA scheme with a chart for σ based on the exponentially weighted variance was applied with λ2 =0.1, for both processes.

Table 4.8: Minimum ARL, minimum PMSIV (δ) and maximum PUNSIV (δ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process I.

Performance δ measures 0.5 1.0 1.5 2.0 2.5 3.0 60.66 (3) 58.59 (4) 71.34 (5) 58.41 (5) 42.04 (5) 25.51 (5) 60.84 (5) 58.79 (5) 54.61 (5) 45.74 (5) 33.82 (5) 21.07 (5) min(ARL) 60.34 (7) 58.62 (7) 53.77 (7) 45.55 (7) 33.64 (7) 21.34 (7) 60.51 (7) 57.65 (5) 52.69 (5) 43.21 (5) 30.94 (5) 18.94 (5) 0.1554 (5) 0.1475 (5) 0.1320 (5) 0.1097 (5) 0.0786 (5) 0.0448 (5) 0.4598 (5) 0.4398 (5) 0.4009 (5) 0.3314 (5) 0.2430 (5) 0.1511 (5) min(PMSIV ) 0.4640 (5) 0.4421 (5) 0.4059 (5) 0.3327 (5) 0.2455 (5) 0.1500 (5) 0.3663 (5) 0.3484 (5) 0.3114 (5) 0.2553 (5) 0.1840 (5) 0.1087 (5) 0.4719 (6) 0.4624 (1) 0.4543 (1) 0.4645 (7) 0.4621 (7) 0.4047 (7) 0.5161 (7) 0.5355 (7) 0.5736 (7) 0.6440 (7) 0.7408 (7) 0.8345 (7) max(PUNSIV ) 0.5159 (7) 0.5375 (7) 0.5765 (7) 0.6488 (7) 0.7416 (7) 0.8407 (7) 0.4972 (1) 0.5007 (1) 0.5303 (7) 0.6071 (7) 0.7112 (7) 0.8208 (8)

Table 4.9: Minimum ARL, minimum PMSIV (δ) and maximum PUNSIV (δ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process II.

Performance δ measures 0.5 1.0 1.5 2.0 2.5 3.0 60.04 (6) 59.15 (9) 53.43 (9) 41.14 (5) 26.38 (5) 13.36 (5) 61.20 (2) 60.92 (9) 55.38 (9) 39.07 (5) 29.23 (9) 11.22 (5) min(ARL) 61.01 (7) 59.25 (9) 54.65 (8) 43.45 (9) 29.47 (8) 15.39 (7) 61.75 (7) 59.84 (8) 53.38 (9) 41.47 (5) 26.74 (5) 13.39 (5) 0.4053 (5) 0.3857 (5) 0.3451 (5) 0.2639 (5) 0.1683 (5) 0.0771 (5) 0.4423 (5) 0.4314 (8) 0.3879 (9) 0.3048 (9) 0.1985 (9) 0.1006 (9) min(PMSIV ) 0.4639 (1) 0.4590 (9) 0.4150 (9) 0.3298 (9) 0.2146 (9) 0.1085 (9) 0.4140 (5) 0.3982 (5) 0.3535 (5) 0.2745 (5) 0.1722 (5) 0.0836 (5) 0.4538 (1) 0.4533 (1) 0.4551 (1) 0.4855 (7) 0.5061 (7) 0.4385 (7) 0.4912 (1) 0.5029 (7) 0.5358 (7) 0.5878 (7) 0.6110 (7) 0.5455 (7) max(PUNSIV ) 0.5182 (1) 0.5192 (1) 0.5582 (7) 0.6449 (7) 0.7667 (7) 0.8752 (7) 0.5002 (1) 0.5085 (1) 0.5151 (7) 0.6126 (7) 0.7439 (7) 0.8676 (7)

76 Chapter 5

Final thoughts

Know when to stop And you will meet with no danger. Lao Tzu

The main purpose of this dissertation was to assess the phenomenon of misleading and unambiguous signals (MS and UNS), in simultaneous schemes for the process mean and variance of GARCH processes. The reader should be reminded that these types of valid signals can result in money and time spent in attempting to identify inexistent causes of unnatural variation or misapplying a trading strategy. One wraps up with a few recommendations for further work to complement the findings of this thesis. The impact of falsely assuming independence, when the process mean and variance refer to a linear ARCH(1) and to a GARCH(1, 1) model, on several RL based performance measures was thoroughly illustrated for a Shewhart chart for individual measurements. This study could be extended to a traditional simultaneous scheme for individual observations comprising that Shewhart chart and a moving range chart to control the process variance. In doing so, it would be possible not only to assess the impact of that false assumption on the RL based performance measures, but also on the PMS and PUNS types III and IV. After reading Ramos (2013, p. 111) it seems reasonable to study the impact of falsely assuming a simpler model (e.g., an ARMA(1, 1)) when the output is drawn from a more complex process (e.g., a linear ARCH(1) or a GARCH(1, 1) model). The ARL, PMS and PUNS of types III and IV of the four simultaneous modified EWMA schemes introduced by Schipper (2001) were estimated only for two different GARCH(1, 1) processes. Further results on the influence of the model structure on those performance measures require the use of a sophisticated program to compute the critical values of the simultaneous scheme for any GARCH process.1 Other simultaneous control schemes can be considered, namely a different chart to monitor the variance, such as the one based on the squared residuals (X µ )2/σ2 (Schipper and Schmid, 2001a), or a chart t − 0 t for µ based on a more complex function of Xt. b In addition, we could have also computed the probability of a simultaneous signal (PSS)2 to provide a further insight on the out-of-control performance of the four simultaneous control schemes. Finally, it is important to refer that the study made in this dissertation for the two GARCH(1, 1)

1Dr. S. Schipper was willing to provide its code but could not ﬁnd it. 2A simultaneous signal (SS) occurs when both µ and σ are out-of-control and both individual charts, that constitute the simultaneous scheme, trigger a signal (Ralha, 2014).

77 processes can also be extended for GARCH processes of higher order and for the univariate and multivariate extensions of these models that were previously considered in the SPC literature, namely, for the TGARCH models,3 the multivariate GARCH(p, q)4 and the constant conditional correlation model.5

3The in-control ARL of a modiﬁed Shewhart chart to control the mean of a TARCH process was studied by Gon¸calves et al. (2013). 4Okhrin and Schmid (2008) analyzed surveillance procedures for the variance (resp. covariance matrix) of univariate (resp. multivariate) GARCH processes. 5Garthoﬀ et al. (2015) introduced control charts to monitor simultaneously the mean vector and covariance matrix of CCC processes.

78 References

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84 Appendix A

Additional plots and tables

In this appendix one finds a few additional plots and tables that support and complement the content of this dissertation. Section A.1 consists of plots of the estimates of four RL based performance measures obtained as described in Section 3.1. These plots are different from the ones in figures 3.1–3.12: the estimates of a given RL performance measure are all together in the same plot. Section A.2 comprises four tables with the critical values used to compute the control limits of the four simultaneous schemes described in Section 3.2; these critical values have been taken from Schipper (2001, tables A.3 to A.6, pp. 109–110). Finally, the estimates of the PMS and PUNS of types III and IV obtained for processes I and II and for each pair (λ1, λ2) are summarized in the eights tables in Section A.3. Those estimates have been also plotted in figures 4.1 to 4.4, in Section 4.1.

85 A.1 Estimates of RL based performance measures ARCH(1) model Sustained shift in the process variance

0.7 0.8 0.9 1.0 1.1 1.2 ARL

SDRL 1.3 1.5 2.0 3.0 0 500 1000 0 500 1000 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

0.7 0.8 0.9 1.0 1.1 1.2 CVRL MdRL 1.3 1.5 2.0

0.80 0.90 1.00 3.0 0 400 800 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

Figure A.1: Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a sustained shift in scale — ARCH(1) model.

Short-lived shift in the process mean

0.0 0.5 1.0 1.5 2.0 ARL

SDRL 2.5 3.0 20 60 100 140 40 80 120 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

0.0 0.5 1.0 1.5 2.0 CVRL MdRL 2.5 3.0 1 2 3 4 0 20 60 100 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

Figure A.2: Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a short-lived shift in location — ARCH(1) model.

86 Short-lived shift in the process mean and sustained shift in the process variance

(1.5,0.7) (2.0,0.8) (0.0,1.0) (1.0,1.1) (3.0,1.1) (1.5,1.5) ARL

SDRL (2.0,2.0) (0.5,3.0) (3.0,3.0) 0 500 1000 0 400 800 1400 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

(1.5,0.7) (2.0,0.8) (0.0,1.0) (1.0,1.1) (3.0,1.1) (1.5,1.5) CVRL MdRL (2.0,2.0) (0.5,3.0) (3.0,3.0) 1.0 2.0 3.0 4.0 0 400 800 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

Figure A.3: Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a short-lived shift in location and a sustained shift in scale — ARCH(1) model.

GARCH(1,1) model Sustained shift in the process variance

0.7 0.8 0.9 1.0 1.1 1.2 ARL

SDRL 1.3 1.5 2.0 3.0 0 500 1000 0 500 1000 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

0.7 0.8 0.9 1.0 1.1 1.2 CVRL MdRL 1.3 1.5 2.0 3.0 0.8 1.0 1.2 0 400 800 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

Figure A.4: Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a sustained shift in scale — GARCH(1,1) model.

87 Short-lived shift in the process mean

0.0 0.5 1.0 1.5 2.0 ARL

SDRL 2.5 3.0 50 150 50 100 150 200

0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

0.0 0.5 1.0 1.5 2.0 CVRL MdRL 2.5 3.0 1 2 3 4 5 0 50 100 150 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

Figure A.5: Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a short-lived shift in location — GARCH(1,1) model.

Short-lived shift in the process mean and sustained shift in the process variance

(1.5,0.7) (2.0,0.8) (0.0,1.0) (1.0,1.1) (3.0,1.1) (1.5,1.5) ARL

SDRL (2.0,2.0) (0.5,3.0) (3.0,3.0) 0 500 1000 0 400 800 1400 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

(1.5,0.7) (2.0,0.8) (0.0,1.0) (1.0,1.1) (3.0,1.1) (1.5,1.5) CVRL MdRL (2.0,2.0) (0.5,3.0) (3.0,3.0) 1 2 3 4 5 0 400 800 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8

α1 α1

Figure A.6: Estimates of ARL, SDRL, CVRL and MdRL, in the presence of a short-lived shift in location and a sustained shift in scale — GARCH(1,1) model.

88 A.2 Critical values of the simultaneous modiﬁed EWMA schemes

(I) (I) Table A.1: Critical values c1, c2 and c3 — simultaneous modiﬁed EWMA scheme with chart for σ based on the squared observations (Schipper, 2001, p. 109).

Critical (λ1, λ2) values (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50)

c1 0.7090 1.3110 2.0710 2.7620 3.5220 0.6970 3.6190 3.5950 3.5690 (I) Process I c2 8.0650 4.0640 1.5360 0.4860 0.0001 0.0006 8.3660 4.2370 1.5950 (I) c3 40.6600 61.5900 92.0200 120.7000 148.3000 159.1000 39.2300 59.5300 89.6700

c1 1.9930 3.8360 6.1810 8.2600 10.3500 1.9530 10.6600 10.6000 10.5300 (I) Process II c2 3.9860 1.8170 0.6990 0.2330 0.0007 0.0005 4.2770 1.9260 0.7280 (I) c3 42.7900 65.3600 93.3000 116.8000 134.3000 154.1000 38.3400 60.0300 88.1900

(II) (II) Table A.2: Critical values c1, c2 and c3 — simultaneous modiﬁed EWMA scheme with chart for σ based on the conditional variance (Schipper, 2001, p. 109).

Critical (λ1, λ2) values (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50)

c1 0.7110 1.3180 2.0960 2.8190 3.6320 0.7160 3.6410 3.6400 3.6370 (II) Process I c2 15.6600 14.7700 14.3600 14.2100 14.1300 13.9800 15.8000 14.8500 14.4100 (II) c3 24.9900 26.9500 28.2700 29.0000 29.8600 30.6100 24.6000 26.6600 28.0900

c1 1.9910 3.8280 6.2140 8.3160 10.6400 2.0100 10.7100 10.6600 10.6600 (II) Process II c2 7.1200 5.5980 4.9410 4.7150 4.6110 4.4990 7.3770 5.7040 4.9920 (II) c3 33.8400 42.9600 50.3500 55.3600 59.9300 67.5200 30.8900 40.1000 48.4600

(III) (III) Table A.3: Critical values c1, c2 and c3 — simultaneous modiﬁed EWMA scheme with chart for σ based on the exponentially weighted variance (Schipper, 2001, p. 110).

Critical (λ1, λ2) values (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50)

c1 0.7090 1.3160 2.0940 2.8230 3.6340 0.7090 3.6370 3.6420 3.6380 (III) Process I c2 12.7500 11.5300 11.0000 10.7900 10.6900 10.4000 13.0700 11.7200 11.0900 (III) c3 27.4400 29.8000 31.1500 31.9200 32.7500 33.7000 26.7700 29.3100 30.9000

c1 1.9990 3.8770 6.2570 8.3720 10.7500 2.0000 10.7700 10.7500 10.7200 (III) Process II c2 7.2080 6.3590 6.1170 6.0580 6.0340 5.6630 7.6710 6.6330 6.2680 (III) c3 26.1900 29.0400 30.1000 30.4700 30.9800 33.9300 24.2700 27.3700 29.1200

(IV) (IV) Table A.4: Critical values c1, c2 and c3 — simultaneous modiﬁed EWMA scheme with chart for σ based on the logarithm of squared observations (Schipper, 2001, p. 110).

Critical (λ1, λ2) values (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50)

c1 0.7060 1.3020 2.0620 2.7620 3.5200 0.6960 3.6490 3.6280 3.5870 (IV) Process I c2 -1.9530 -3.2590 -5.2530 -7.2200 -9.1700 -9.6530 -1.9190 -3.2370 -5.2160 (IV) c2 0.4660 1.0890 1.7080 2.1760 2.6960 2.7660 0.4450 1.0780 1.6980

c1 1.9900 3.8250 6.1220 8.1430 10.3900 1.9570 10.8100 10.6700 10.5200 (IV) Process II c2 -0.3240 -1.5910 -3.4380 -5.2930 -7.2110 -7.6730 -0.2570 -1.5150 -3.3630 (IV) c3 2.7620 3.4890 4.1110 4.5190 4.9010 5.0390 2.6790 3.4300 4.0680

89 A.3 Estimates of PMS and PUNS via Monte Carlo simulation PMS of Type III

Table A.5: Estimated PMSIII (θ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process I.

θ (λ1, λ2) (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50) 0.0451 0.0348 0.0321 0.0285 0.1282 0.1728 0.0163 0.0175 0.0207 0.0417 0.0284 0.0213 0.0167 0.0140 0.0386 0.0161 0.0156 0.0158 0.7 0.0442 0.0310 0.0231 0.0185 0.0163 0.0439 0.0173 0.0170 0.0169 0.0664 0.0627 0.0611 0.0523 0.0467 0.1696 0.0229 0.0318 0.0415 0.1393 0.1251 0.1168 0.1065 0.2856 0.3218 0.0754 0.0777 0.0827 0.1318 0.1101 0.0928 0.0781 0.0701 0.1303 0.0813 0.0817 0.0785 0.8 0.1384 0.1136 0.0971 0.0837 0.0744 0.1387 0.0855 0.0833 0.0816 0.1895 0.1910 0.1795 0.1580 0.1345 0.3188 0.1065 0.1286 0.1438 0.3082 0.2868 0.2630 0.2336 0.3882 0.4307 0.2219 0.2116 0.2032 0.3095 0.2837 0.2622 0.2328 0.2078 0.2929 0.2581 0.2592 0.2462 0.9 0.3161 0.2905 0.2641 0.2422 0.2206 0.3035 0.2604 0.2583 0.2474 0.3631 0.3540 0.3298 0.2888 0.2396 0.4279 0.2763 0.2937 0.2937 0.5108 0.4841 0.4418 0.3931 0.4131 0.4622 0.5023 0.4336 0.3778 0.5766 0.5881 0.5866 0.5342 0.4665 0.4846 0.6905 0.6670 0.6089 1.1 0.5942 0.6100 0.6122 0.5753 0.5228 0.5145 0.6991 0.6792 0.6350 0.5542 0.5383 0.5032 0.4340 0.3434 0.4605 0.5869 0.5507 0.4915 0.4638 0.4675 0.4402 0.3943 0.3898 0.4231 0.5139 0.4423 0.3802 0.5520 0.5902 0.5975 0.5408 0.4578 0.4345 0.7572 0.7228 0.6427 1.2 0.5829 0.6181 0.6348 0.6004 0.5354 0.4729 0.7761 0.7398 0.6831 0.5584 0.5596 0.5283 0.4549 0.3493 0.4252 0.6573 0.6081 0.5314 0.3935 0.4173 0.4068 0.3762 0.3625 0.3760 0.4857 0.4172 0.3600 0.4996 0.5496 0.5621 0.5032 0.4137 0.3559 0.7749 0.7287 0.6265 1.3 0.5344 0.5925 0.6177 0.5826 0.5120 0.4002 0.8009 0.7615 0.6878 0.5485 0.5608 0.5308 0.4511 0.3355 0.3777 0.7018 0.6392 0.5499 0.2692 0.3142 0.3282 0.3200 0.3068 0.2825 0.4263 0.3475 0.3021 0.4160 0.4765 0.4808 0.4093 0.3160 0.2243 0.7907 0.7132 0.5684 1.5 0.4739 0.5481 0.5796 0.5400 0.4598 0.2726 0.8361 0.7781 0.6776 0.5209 0.5475 0.5193 0.4260 0.2943 0.2811 0.7645 0.6805 0.5601 0.1175 0.1609 0.1879 0.2021 0.2056 0.1352 0.3393 0.2408 0.1990 0.3504 0.3892 0.3559 0.2719 0.1829 0.0754 0.8548 0.6961 0.4603 2.0 0.4622 0.5347 0.5414 0.4790 0.3887 0.1285 0.9205 0.8328 0.6638 0.5168 0.5449 0.5010 0.3484 0.2011 0.1339 0.8665 0.7543 0.5679 0.0338 0.0605 0.0775 0.1035 0.1121 0.0398 0.2397 0.1545 0.1137 0.3184 0.3026 0.2325 0.1626 0.0996 0.0146 0.8307 0.5877 0.3235 3.0 0.5221 0.5289 0.4536 0.3709 0.2858 0.0416 0.9520 0.8005 0.5585 0.6103 0.6099 0.4780 0.2442 0.1131 0.0409 0.9531 0.8311 0.5496

90 Table A.6: Estimated PMSIII (θ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process II.

θ (λ1, λ2) (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50) 0.1351 0.1170 0.1059 0.0947 0.1212 0.2410 0.0909 0.0774 0.0752 0.1404 0.1243 0.1076 0.0893 0.0740 0.1270 0.1151 0.1063 0.0892 0.7 0.1607 0.1425 0.1259 0.1087 0.0936 0.1397 0.1291 0.1227 0.1149 0.1577 0.1599 0.1562 0.1398 0.1158 0.2421 0.1147 0.1207 0.1179 0.2262 0.2028 0.1847 0.1611 0.1789 0.3192 0.1593 0.1406 0.1354 0.2382 0.2200 0.1916 0.1609 0.1356 0.2157 0.2087 0.1902 0.1642 0.8 0.2609 0.2349 0.2136 0.1857 0.1614 0.2299 0.2208 0.2151 0.2006 0.2644 0.2559 0.2417 0.2101 0.1707 0.3230 0.2049 0.2023 0.1910 0.3381 0.3091 0.2799 0.2415 0.2355 0.3860 0.2527 0.2280 0.2131 0.3653 0.3405 0.3011 0.2582 0.2177 0.3287 0.3314 0.3142 0.2710 0.9 0.3880 0.3579 0.3272 0.2909 0.2551 0.3355 0.3492 0.3367 0.3152 0.3871 0.3626 0.3305 0.2816 0.2251 0.3934 0.3217 0.3084 0.2733 0.5358 0.5008 0.4452 0.3752 0.3190 0.4621 0.4687 0.4127 0.3633 0.5914 0.5667 0.5121 0.4527 0.3849 0.5026 0.6248 0.5832 0.4930 1.1 0.6241 0.6074 0.5732 0.5284 0.4685 0.5347 0.6358 0.6307 0.5766 0.5887 0.5345 0.4747 0.4016 0.3093 0.4645 0.5572 0.5074 0.4231 0.5820 0.5491 0.4916 0.4163 0.3477 0.4743 0.5404 0.4746 0.4168 0.6428 0.6213 0.5684 0.5055 0.4374 0.5359 0.7232 0.6704 0.5633 1.2 0.6848 0.6803 0.6480 0.6018 0.5345 0.5744 0.7434 0.7313 0.6718 0.6441 0.5885 0.5189 0.4410 0.3351 0.4764 0.6383 0.5760 0.4794 0.5851 0.5620 0.5086 0.4321 0.3628 0.4714 0.5741 0.5098 0.4463 0.6499 0.6450 0.5858 0.5278 0.4530 0.5345 0.7737 0.7134 0.5944 1.3 0.7007 0.7091 0.6823 0.6386 0.5632 0.5779 0.8028 0.7924 0.7211 0.6720 0.6246 0.5561 0.4732 0.3513 0.4722 0.6991 0.6339 0.5302 0.5374 0.5393 0.4985 0.4393 0.3685 0.4413 0.5791 0.5312 0.4580 0.6044 0.6150 0.5703 0.5137 0.4422 0.4851 0.7919 0.7261 0.5962 1.5 0.6704 0.6950 0.6770 0.6348 0.5538 0.5212 0.8350 0.8202 0.7355 0.6938 0.6615 0.5984 0.5140 0.3565 0.4424 0.7744 0.7138 0.6010 0.3782 0.4030 0.3886 0.3541 0.3128 0.3122 0.5104 0.4683 0.4021 0.4401 0.4681 0.4290 0.3795 0.3204 0.3084 0.7256 0.6396 0.4841 2.0 0.5685 0.6210 0.6128 0.5616 0.4731 0.3547 0.8493 0.8215 0.7056 0.6959 0.6892 0.6484 0.5319 0.3057 0.3122 0.8667 0.8180 0.6918 0.1685 0.2003 0.2044 0.2059 0.1871 0.1274 0.3699 0.3161 0.2477 0.2382 0.2486 0.2119 0.1747 0.1355 0.1052 0.5642 0.4314 0.2657 3.0 0.5494 0.5881 0.5253 0.4443 0.3441 0.1573 0.8864 0.8029 0.6126 0.7449 0.7430 0.6734 0.4390 0.1806 0.1271 0.9572 0.9093 0.7386

91 PMS of Type IV

Table A.7: Estimated PMSIV (δ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process I.

δ (λ1, λ2) (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50) 0.4973 0.4859 0.4529 0.4010 0.1554 0.4840 0.4470 0.4175 0.3888 0.4946 0.4884 0.4772 0.4652 0.4598 0.5166 0.4683 0.4642 0.4628 0.5 0.4830 0.4829 0.4752 0.4698 0.4640 0.4983 0.4719 0.4693 0.4689 0.4832 0.4736 0.4564 0.4251 0.3663 0.4881 0.5030 0.4855 0.4537 0.4972 0.4852 0.4428 0.3888 0.1475 0.4948 0.4307 0.4001 0.3711 0.4944 0.4854 0.4696 0.4488 0.4398 0.5222 0.4488 0.4461 0.4457 1.0 0.4806 0.4757 0.4663 0.4509 0.4421 0.4964 0.4504 0.4514 0.4491 0.4796 0.4695 0.4455 0.4112 0.3484 0.5014 0.4827 0.4659 0.4355 0.5039 0.4932 0.4236 0.3520 0.1320 0.5209 0.3890 0.3635 0.3375 0.5001 0.4900 0.4456 0.4155 0.4009 0.5437 0.4114 0.4056 0.4078 1.5 0.4759 0.4734 0.4408 0.4174 0.4059 0.4990 0.4123 0.4116 0.4059 0.4717 0.4612 0.4214 0.3748 0.3114 0.5199 0.4415 0.4243 0.3930 0.5253 0.5022 0.3798 0.2974 0.1097 0.5738 0.3236 0.2979 0.2803 0.5149 0.4913 0.4010 0.3525 0.3314 0.5983 0.3426 0.3374 0.3375 2.0 0.4763 0.4555 0.3927 0.3529 0.3327 0.5154 0.3412 0.3402 0.3355 0.4616 0.4407 0.3739 0.3170 0.2553 0.5765 0.3692 0.3550 0.3267 0.5713 0.5071 0.3081 0.2175 0.0786 0.6569 0.2343 0.2145 0.1987 0.5499 0.4798 0.3278 0.2627 0.2430 0.6749 0.2484 0.2480 0.2454 2.5 0.4860 0.4219 0.3149 0.2632 0.2455 0.5586 0.2505 0.2482 0.2462 0.4442 0.4013 0.3042 0.2338 0.1840 0.6593 0.2706 0.2591 0.2353 0.6432 0.4860 0.2194 0.1347 0.0448 0.7449 0.1424 0.1307 0.1198 0.5991 0.4431 0.2363 0.1698 0.1511 0.7615 0.1561 0.1519 0.1539 3.0 0.5016 0.3623 0.2237 0.1675 0.1500 0.6179 0.1543 0.1543 0.1536 0.4249 0.3349 0.2071 0.1472 0.1087 0.7469 0.1680 0.1582 0.1435

Table A.8: Estimated PMSIV (δ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process II.

δ (λ1, λ2) (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50) 0.4883 0.4867 0.4764 0.4434 0.4053 0.4964 0.4552 0.4367 0.4246 0.4739 0.4725 0.4732 0.4564 0.4423 0.5055 0.4671 0.4515 0.4495 0.5 0.4639 0.4782 0.4832 0.4757 0.4789 0.4951 0.4842 0.4787 0.4772 0.4687 0.4779 0.4756 0.4519 0.4140 0.4938 0.5021 0.4846 0.4554 0.4860 0.4808 0.4650 0.4286 0.3857 0.4997 0.4406 0.4185 0.4088 0.4743 0.4722 0.4677 0.4468 0.4323 0.5195 0.4457 0.4314 0.4320 1.0 0.4628 0.4724 0.4778 0.4632 0.4632 0.4931 0.4697 0.4611 0.4590 0.4614 0.4728 0.4622 0.4362 0.3982 0.4969 0.4829 0.4671 0.4409 0.4836 0.4799 0.4452 0.3938 0.3451 0.5220 0.3956 0.3778 0.3628 0.4857 0.4842 0.4586 0.4177 0.4047 0.5683 0.4048 0.3916 0.3879 1.5 0.4576 0.4685 0.4521 0.4252 0.4187 0.4924 0.4261 0.4186 0.4150 0.4564 0.4648 0.4386 0.3952 0.3535 0.5223 0.4421 0.4221 0.3945 0.4915 0.4709 0.3899 0.3213 0.2639 0.5704 0.3119 0.2916 0.2844 0.5218 0.5048 0.4211 0.3531 0.3374 0.6542 0.3211 0.3072 0.3048 2.0 0.4584 0.4504 0.4011 0.3510 0.3310 0.4981 0.3413 0.3357 0.3298 0.4452 0.4440 0.3838 0.3187 0.2745 0.5708 0.3528 0.3323 0.3083 0.5137 0.4440 0.2942 0.2116 0.1683 0.6580 0.2025 0.1894 0.1805 0.5810 0.5116 0.3376 0.2513 0.2337 0.7624 0.2091 0.2025 0.1985 2.5 0.4553 0.4061 0.3037 0.2383 0.2176 0.5217 0.2227 0.2186 0.2146 0.4306 0.3953 0.2856 0.2107 0.1722 0.6563 0.2317 0.2171 0.1981 0.5581 0.3807 0.1793 0.1125 0.0771 0.7351 0.1002 0.0933 0.0882 0.6644 0.4664 0.2283 0.1435 0.1288 0.8384 0.1098 0.1029 0.1006 3.0 0.4586 0.3286 0.1906 0.1276 0.1095 0.5619 0.1166 0.1114 0.1085 0.3983 0.3151 0.1724 0.1082 0.0836 0.7332 0.1189 0.1110 0.0953

92 PUNS of Type III

Table A.9: Estimated PUNSIII (θ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations, conditional variance, exponentially weighted variance and logarithm of squared observations — Process I.

θ (λ1, λ2) (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50) 0.9543 0.9636 0.9636 0.9628 0.8019 0.8240 0.9810 0.9777 0.9709 0.9582 0.9715 0.9782 0.9821 0.9835 0.9609 0.9838 0.9840 0.9835 0.7 0.9557 0.9688 0.9764 0.9805 0.9821 0.9556 0.9825 0.9827 0.9825 0.9331 0.9350 0.9318 0.9331 0.9282 0.8277 0.9764 0.9658 0.9502 0.8570 0.8648 0.8581 0.8460 0.5181 0.6677 0.9092 0.8934 0.8702 0.8673 0.8879 0.9028 0.9130 0.9146 0.8662 0.9177 0.9158 0.9164 0.8 0.8608 0.8850 0.8995 0.9092 0.9143 0.8581 0.9137 0.9150 0.9144 0.8071 0.7999 0.7933 0.7886 0.7737 0.6711 0.8889 0.8588 0.8230 0.6757 0.6778 0.6607 0.6288 0.2835 0.5437 0.7170 0.6861 0.6507 0.6862 0.7079 0.7182 0.7283 0.7299 0.6918 0.7365 0.7302 0.7298 0.9 0.6802 0.7031 0.7207 0.7280 0.7311 0.6826 0.7357 0.7333 0.7342 0.6278 0.6222 0.6093 0.5924 0.5573 0.5472 0.7089 0.6728 0.6314 0.4198 0.3847 0.3261 0.2499 0.0885 0.4738 0.2536 0.2311 0.2126 0.4009 0.3681 0.3210 0.2941 0.2730 0.4465 0.2806 0.2785 0.2805 1.1 0.3877 0.3543 0.3124 0.2862 0.2692 0.4190 0.2788 0.2781 0.2776 0.4149 0.3982 0.3643 0.3202 0.2378 0.4748 0.3669 0.3651 0.3424 0.4388 0.3626 0.2618 0.1676 0.0546 0.4912 0.1695 0.1380 0.1148 0.4143 0.3471 0.2714 0.2234 0.1950 0.4724 0.2017 0.2018 0.2022 1.2 0.3896 0.3305 0.2606 0.2133 0.1873 0.4327 0.1929 0.1990 0.1967 0.3995 0.3586 0.3108 0.2493 0.1561 0.4899 0.2847 0.2912 0.2674 0.4868 0.3748 0.2419 0.1250 0.0356 0.5219 0.1411 0.1008 0.0680 0.4574 0.3705 0.2696 0.2024 0.1673 0.5278 0.1751 0.1760 0.1783 1.3 0.4289 0.3423 0.2523 0.1919 0.1590 0.4856 0.1618 0.1658 0.1661 0.3997 0.3416 0.2804 0.2089 0.1043 0.5209 0.2315 0.2432 0.2205 0.5745 0.4177 0.2281 0.0884 0.0172 0.5818 0.1155 0.0755 0.0353 0.5207 0.4069 0.2811 0.1929 0.1528 0.6322 0.1444 0.1529 0.1591 1.5 0.4742 0.3626 0.2456 0.1669 0.1327 0.5706 0.1211 0.1330 0.1354 0.4089 0.3304 0.2499 0.1600 0.0553 0.5854 0.1608 0.1791 0.1629 0.6531 0.4419 0.1983 0.0577 0.0053 0.6696 0.0684 0.0486 0.0176 0.5306 0.3925 0.2394 0.1454 0.1122 0.7303 0.0670 0.0879 0.1056 2.0 0.4542 0.3218 0.1833 0.1029 0.0709 0.6348 0.0445 0.0588 0.0680 0.3794 0.2855 0.1911 0.1011 0.0170 0.6706 0.0644 0.0870 0.0888 0.5971 0.3457 0.1250 0.0299 0.0016 0.6317 0.0202 0.0194 0.0084 0.4092 0.2579 0.1304 0.0691 0.0480 0.6812 0.0128 0.0245 0.0380 3.0 0.3246 0.1913 0.0839 0.0345 0.0192 0.5783 0.0049 0.0103 0.0163 0.2546 0.1743 0.1070 0.0520 0.0058 0.6323 0.0111 0.0250 0.0316

93 Table A.10: Estimated PUNSIII (θ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations, conditional variance, exponentially weighted variance and logarithm of squared observations — Process II.

θ (λ1, λ2) (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50) 0.8506 0.8609 0.8581 0.8422 0.7526 0.7305 0.8686 0.8669 0.8532 0.8525 0.8631 0.8696 0.8721 0.8717 0.8576 0.8748 0.8744 0.8758 0.7 0.8352 0.8512 0.8609 0.8670 0.8711 0.8470 0.8674 0.8702 0.8689 0.8341 0.8207 0.8011 0.7809 0.7604 0.7281 0.8721 0.8466 0.8101 0.7491 0.7582 0.7484 0.7262 0.6329 0.6387 0.7688 0.7600 0.7376 0.7477 0.7569 0.7648 0.7668 0.7613 0.7557 0.7730 0.7743 0.7723 0.8 0.7317 0.7537 0.7621 0.7719 0.7736 0.7466 0.7731 0.7724 0.7711 0.7224 0.7114 0.6897 0.6681 0.6428 0.6339 0.7724 0.7445 0.6978 0.6230 0.6264 0.6156 0.5872 0.5167 0.5590 0.6291 0.6086 0.5847 0.6114 0.6206 0.6256 0.6205 0.6109 0.6251 0.6367 0.6253 0.6228 0.9 0.6005 0.6218 0.6332 0.6386 0.6414 0.6267 0.6396 0.6426 0.6388 0.5908 0.5905 0.5758 0.5554 0.5243 0.5513 0.6429 0.6161 0.5764 0.3848 0.3737 0.3610 0.3255 0.3151 0.4553 0.3010 0.2818 0.2819 0.3618 0.3547 0.3448 0.3137 0.2924 0.4128 0.3094 0.2972 0.2981 1.1 0.3513 0.3504 0.3453 0.3327 0.3259 0.3911 0.3403 0.3251 0.3285 0.3721 0.3909 0.3879 0.3684 0.3246 0.4520 0.3830 0.3781 0.3601 0.3221 0.2998 0.2803 0.2361 0.2420 0.4282 0.1855 0.1706 0.1737 0.2987 0.2797 0.2563 0.2142 0.1832 0.3641 0.1941 0.1852 0.1896 1.2 0.2836 0.2658 0.2480 0.2219 0.2113 0.3316 0.2271 0.2130 0.2109 0.3100 0.3269 0.3281 0.2999 0.2517 0.4282 0.2931 0.2961 0.2848 0.3018 0.2654 0.2291 0.1725 0.1876 0.4194 0.1144 0.0992 0.1043 0.2802 0.2421 0.2111 0.1563 0.1193 0.3533 0.1280 0.1182 0.1233 1.3 0.2606 0.2276 0.1969 0.1580 0.1419 0.3132 0.1612 0.1418 0.1390 0.2750 0.2826 0.2781 0.2455 0.1936 0.4180 0.2278 0.2304 0.2246 0.3224 0.2501 0.1831 0.1031 0.1117 0.4267 0.0630 0.0395 0.0381 0.3051 0.2368 0.1783 0.1088 0.0692 0.3835 0.0824 0.0689 0.0690 1.5 0.2788 0.2240 0.1727 0.1196 0.0956 0.3417 0.1193 0.0984 0.0976 0.2455 0.2333 0.2155 0.1768 0.1169 0.4253 0.1531 0.1511 0.1447 0.4138 0.2854 0.1539 0.0527 0.0364 0.4997 0.0435 0.0171 0.0046 0.4070 0.2909 0.1830 0.0977 0.0604 0.5177 0.0686 0.0504 0.0502 2.0 0.3515 0.2601 0.1733 0.1025 0.0738 0.4371 0.0930 0.0764 0.0722 0.2285 0.1855 0.1431 0.0907 0.0392 0.4997 0.0747 0.0701 0.0616 0.4717 0.2762 0.1116 0.0279 0.0084 0.5557 0.0192 0.0078 0.0013 0.4544 0.2926 0.1581 0.0772 0.0577 0.6195 0.0357 0.0307 0.0347 3.0 0.3190 0.2088 0.1122 0.0520 0.0327 0.4853 0.0319 0.0279 0.0303 0.1721 0.1258 0.0817 0.0411 0.0088 0.5572 0.0177 0.0201 0.0203

94 PUNS of Type IV

Table A.11: Estimated PUNSIV (δ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process I.

δ (λ1, λ2) (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50) 0.4634 0.4309 0.3876 0.3439 0.4184 0.4719 0.4027 0.3612 0.3234 0.4934 0.4881 0.4703 0.4323 0.3815 0.4426 0.5161 0.5062 0.4743 0.5 0.5076 0.4973 0.4821 0.4527 0.4132 0.4641 0.5159 0.5072 0.4822 0.4972 0.4820 0.4453 0.3894 0.3126 0.4687 0.4660 0.4566 0.4233 0.4624 0.4321 0.3839 0.3388 0.4164 0.4621 0.4126 0.3694 0.3275 0.4927 0.4886 0.4723 0.4345 0.3780 0.4385 0.5355 0.5240 0.4876 1.0 0.5094 0.5036 0.4921 0.4683 0.4309 0.4657 0.5375 0.5249 0.5012 0.5007 0.4843 0.4559 0.4037 0.3170 0.4567 0.4870 0.4763 0.4440 0.4543 0.4160 0.3648 0.3353 0.4018 0.4389 0.4414 0.3813 0.3308 0.4858 0.4816 0.4813 0.4332 0.3681 0.4168 0.5736 0.5649 0.5129 1.5 0.5138 0.5052 0.5173 0.5004 0.4547 0.4611 0.5765 0.5657 0.5472 0.5076 0.4946 0.4833 0.4337 0.3128 0.4401 0.5303 0.5233 0.4978 0.4316 0.3864 0.3256 0.3092 0.3681 0.3891 0.4645 0.3831 0.3180 0.4661 0.4722 0.4890 0.4251 0.3393 0.3658 0.6440 0.6300 0.5484 2.0 0.5115 0.5243 0.5624 0.5474 0.4937 0.4425 0.6488 0.6403 0.6156 0.5182 0.5175 0.5424 0.4619 0.2961 0.3871 0.6071 0.6004 0.5806 0.3850 0.3224 0.2614 0.2567 0.2993 0.3088 0.4621 0.3532 0.2788 0.4235 0.4616 0.4785 0.3833 0.2843 0.2918 0.7408 0.7071 0.5586 2.5 0.5010 0.5579 0.6271 0.5883 0.5028 0.3967 0.7416 0.7362 0.6974 0.5371 0.5606 0.6266 0.4681 0.2479 0.3066 0.7112 0.7079 0.6937 0.3042 0.2336 0.1804 0.1877 0.2141 0.2121 0.4047 0.2867 0.2131 0.3632 0.4553 0.4309 0.3051 0.2071 0.2001 0.8345 0.7596 0.5092 3.0 0.4830 0.6163 0.6806 0.5847 0.4622 0.3271 0.8407 0.8327 0.7483 0.5557 0.6323 0.7343 0.4253 0.1815 0.2097 0.8205 0.8208 0.8015

Table A.12: Estimated PUNSIV (δ) — simultaneous modiﬁed EWMA scheme with chart for σ listed in order corresponding to: squared observations (I), conditional variance (II), exponentially weighted variance (III) and logarithm of squared observations (IV) — Process II.

δ (λ1, λ2) (1) (2) (3) (4) (5) (6) (7) (8) (9) (0.10,0.10) (0.25,0.25) (0.50,0.50) (0.75,0.75) (1.00,1.00) (0.10,1.00) (1.00,0.10) (1.00,0.25) (1.00,0.50) 0.4538 0.4165 0.3719 0.3155 0.2850 0.4348 0.3712 0.3287 0.2961 0.4912 0.4689 0.4153 0.3606 0.3062 0.4286 0.4842 0.4575 0.3887 0.5 0.5182 0.4906 0.4581 0.4202 0.3686 0.4481 0.4994 0.4889 0.4534 0.5002 0.4604 0.4087 0.3509 0.2742 0.4357 0.4489 0.4176 0.3580 0.4533 0.4211 0.3727 0.3213 0.2924 0.4343 0.3871 0.3464 0.3106 0.4878 0.4617 0.4057 0.3449 0.2899 0.4171 0.5029 0.4689 0.3857 1.0 0.5192 0.4959 0.4613 0.4324 0.3837 0.4512 0.5134 0.5063 0.4727 0.5085 0.4655 0.4249 0.3715 0.2808 0.4337 0.4698 0.4368 0.3799 0.4551 0.4150 0.3634 0.3242 0.2978 0.4143 0.4289 0.3792 0.3359 0.4738 0.4378 0.3760 0.3115 0.2521 0.3747 0.5358 0.4872 0.3701 1.5 0.5231 0.4999 0.4880 0.4704 0.4213 0.4490 0.5582 0.5508 0.5203 0.5124 0.4744 0.4529 0.4249 0.2911 0.4134 0.5151 0.4914 0.4472 0.4464 0.3937 0.3416 0.3180 0.2929 0.3699 0.4855 0.4274 0.3550 0.4322 0.3843 0.3179 0.2473 0.1899 0.2989 0.5878 0.4944 0.3269 2.0 0.5222 0.5188 0.5409 0.5332 0.4763 0.4415 0.6449 0.6389 0.6078 0.5249 0.4982 0.5244 0.5123 0.2841 0.3699 0.6126 0.5982 0.5643 0.4187 0.3446 0.2869 0.2781 0.2459 0.2826 0.5061 0.4260 0.3335 0.3648 0.2985 0.2278 0.1597 0.1130 0.1957 0.6110 0.4545 0.2460 2.5 0.5242 0.5641 0.6294 0.5962 0.4938 0.4125 0.7667 0.7611 0.7090 0.5405 0.5530 0.6432 0.6034 0.2394 0.2843 0.7439 0.7365 0.7168 0.3502 0.2576 0.2086 0.1976 0.1668 0.1806 0.4385 0.3493 0.2521 0.2533 0.1880 0.1285 0.0810 0.0551 0.1025 0.5455 0.3439 0.1494 3.0 0.5199 0.6445 0.7014 0.5804 0.4303 0.3467 0.8752 0.8661 0.7546 0.5739 0.6435 0.7831 0.6199 0.1620 0.1836 0.8676 0.8635 0.8595

95 96