<s> Pairs
TradingQuantitative Methodsand Analysis
GANAPATHY VIDYAMURTHY
John Wiley & Sons, Inc.
Contents
Preface
ix
Acknowledgments
xi
PART ONEBackground MaterialCHAPTER 1IntroductionThe CAPM ModelMarket Neutral StrategyPairs TradingOutlineAudience
CHAPTER 2Time SeriesOverviewAutocorrelationTime Series ModelsForecastingGoodness of Fit versus BiasModel ChoiceModeling Stock Prices
CHAPTER 3Factor ModelsIntroductionArbitrage Pricing TheoryThe Covariance MatrixApplication: Calculating the Risk on a PortfolioApplication: Calculation of Portfolio BetaApplication: Tracking Basket DesignSensitivity Analysis
3358910
1414151624252630
3737394244474950v
CONTENTS
vi
CHAPTER 4Kalman FilteringIntroductionThe Kalman FilterThe Scalar Kalman FilterFiltering the Random WalkApplication: Example with the Standard & Poor Index
525254576064
PART TWOStatistical Arbitrage PairsCHAPTER 5OverviewHistoryMotivationCointegrationApplying the ModelA Trading StrategyRoad Map for Strategy Design
CHAPTER 6Pairs Selection in Equity MarketsIntroductionCommon Trends Cointegration ModelCommon Trends Model and APTThe Distance MeasureInterpreting the Distance MeasureReconciling Theory and Practice
CHAPTER 7Testing for TradabilityIntroductionThe Linear RelationshipEstimating the Linear Relationship: The MultifactorApproachEstimating the Linear Relationship: The Regression ApproachTesting Residual for Tradability
73737475808283
85858790939497
104104106107108112
CHAPTER 8Trading Design
118
Introduction
118
Contents
Band Design for White NoiseSpread DynamicsNonparametric ApproachRegularizationTying Up Loose Ends
vii
119122126130135
PART THREERisk Arbitrage PairsCHAPTER 9Risk Arbitrage Mechanics
139
IntroductionHistoryThe Deal ProcessTransaction TermsThe Deal SpreadTrading StrategyQuantitative Aspects
139140141142145147149
CHAPTER 10Trade ExecutionIntroductionSpecifying the OrderVerifying the ExecutionExecution During the Pricing PeriodShort Selling
CHAPTER 11The Market Implied Merger ProbabilityIntroductionImplied Probabilities and Arrow-Debreu TheoryThe Single-Step ModelThe Multistep ModelReconciling Theory and PracticeRisk Management
CHAPTER 12Spread InversionIntroductionThe Prediction EquationThe Observation Equation
151151152155161166
171171173175177180184
189189190192
CONTENTS
viii
Applying the Kalman FilterModel SelectionApplications to Trading
193194197
Index
205
Preface
ost book readers are likely to concur with the idea that the least readportion of any book is the preface. With that in mind, and the fact thatthe reader has indeed taken the trouble to read up to this sentence, we promise to leave no stone unturned to make this preface as lively and entertaining as possible. For your reading pleasure, here is a nice story with a picturethrown in for good measure. Enjoy!Once upon a time, there were six blind men. The blind men wished toknow what an elephant looked like. They took a trip to the forest and withthe help of their guide found a tame elephant. The first blind man walkedinto the broadside of the elephant and bumped his head. He declared thatthe elephant was like a wall. The second one grabbed the elephant’s tusk andsaid it felt like a spear. The next blind man felt the trunk of the elephant andwas sure that elephants were similar to snakes. The fourth blind man huggedthe elephant’s leg and declared the elephant was like a tree. The next onecaught the ear and said this is definitely like a fan. The last blind man felt thetail and said this sure feels like a rope. Thus the six blind men all perceivedone aspect of the elephant and were each right in their own way, but noneof them knew what the whole elephant really looked like.
M
ix
x
PREFACE
Oftentimes, the market poses itself as the elephant. There are peoplewho say that predicting the market is like predicting the weather, becauseyou can do well in the short term, but where the market will be in the longrun is anybody’s guess. We have also heard from others that predicting themarket short term is a sure way to burn your fingers. “Invest for the longhaul” is their mantra. Some will assert that the markets are efficient, and yetsome others would tell you that it is possible to make extraordinary returns.While some swear by technical analysis, there are some others, the so-calledfundamentalists, who staunchly claim it to be a voodoo science. Multiplevaluation models for equities like the dividend discount model, relative valuation models, and the Merton model (treating equity as an option on firmvalue) all exist side by side, each being relevant at different times for different stocks. Deep theories from various disciplines like physics, statistics,control theory, graph theory, game theory, signal processing, probability,and geometry have all been applied to explain different aspects of marketbehavior.It seems as if the market is willing to accommodate a wide range ofsometimes opposing belief systems. If we are to make any sense of this smorgasbord of opinions on the market, we would be well advised to draw comfort from the story of the six blind men and the elephant. Under thesecircumstances, if the reader goes away with a few more perspectives on themarket elephant, the author would consider his job well done.
PART
One
Background Material
CHAPTER
1
Introduction
W
e start at the very beginning (a very good place to start). We begin withthe CAPM model.
THE CAPM MODELCAPM is an acronym for the Capital Asset Pricing Model. It was originallyproposed by William T. Sharpe. The impact that the model has made in thearea of finance is readily evident in the prevalent use of the word beta. Incontemporary finance vernacular, beta is not just a nondescript Greek letter, but its use carries with it all the import and implications of its CAPMdefinition.Along with the idea of beta, CAPM also served to formalize the notionof a market portfolio. A market portfolio in CAPM terms is a portfolio ofassets that acts as a proxy for the market. Although practical versions ofmarket portfolios in the form of market averages were already prevalent atthe time the theory was proposed, CAPM definitely served to underscore thesignificance of these market averages.Armed with the twin ideas of market portfolio and beta, CAPM attempts to explain asset returns as an aggregate sum of component returns.In other words, the return on an asset in the CAPM framework can be separated into two components. One is the market or systematic component,and the other is the residual or nonsystematic component. More precisely, ifrp is the return on the asset, rm is the return on the market portfolio, and thebeta of the asset is denoted as b, the formula showing the relationship thatachieves the separation of the returns is given asrp = βrm + θ p
(1.1)
Equation 1.1 is also often referred to as the security market line (SML). Notethat in the formula, brm is the market or systematic component of the return.b serves as a leverage number of the asset return over the market return. For3
BACKGROUND MATERIAL
4
instance, if the beta of the asset happens to be 3.0 and the market moves1 percent, the systematic component of the asset return is now 3.0 percent.This idea is readily apparent when the SML is viewed in geometrical termsin Figure 1.1. It may also be deduced from the figure that b is indeed theslope of the SML.qp in the CAPM equation is the residual component or residual returnon the portfolio. It is the portion of the asset return that is not explainableby the market return. The consensus expectation on the residual componentis assumed to be zero.Having established the separation of asset returns into two components,CAPM then proceeds to elaborate on a key assumption made with respect tothe relationship between them. The assertion of the model is that the market component and residual component are uncorrelated. Now, many ascholarly discussion on the import of these assumptions has been conductedand a lot of ink used up on the significance of the CAPM model since its introduction. Summaries of those discussions may be found in the referencesprovided at the end of the chapter. However, for our purposes, the preceding introduction explaining the notion of beta and its role in the determination of asset returns will suffice.Given that knowledge of the beta of an asset is greatly valuable in theCAPM context, let us discuss briefly how we can go about estimating itsvalue. Notice that beta is actually the slope of the SML. Therefore, beta maybe estimated as the slope of the regression line between market returns andthe asset returns. Applying the standard regression formula for the estimation of the slope we have
rpAsset Return
qp
brmβ
rmMarket Return
FIGURE 1.1
The Security Market Line.
Introduction
5
β =
cov(rp rm )var(rm )
(1.2)
that is, beta is the covariance between the asset and market returns dividedby the variance of the market returns.To see the typical range of values that the beta of an asset is likely to assume in practice, we remind ourselves of an oft-quoted adage about themarkets, “A rising tide raises all boats.” The statement indicates that whenthe market goes up, we can typically expect the price of all securities to goup with it. Thus, a positive return for the market usually implies a positivereturn for the asset, that is, the sum of the market component and the residual component is positive. If the residual component of the asset return issmall, as we expect it to be, then the positive return in the asset is explainedalmost completely by its market component. Therefore, a positive return inthe market portfolio and the asset implies a positive market component ofthe return and, by implication, a positive value for beta. Therefore, we canexpect all assets to typically have positive values for their betas.
MARKET NEUTRAL STRATEGYHaving discussed CAPM, we now have the required machinery to definemarket neutral strategies: They are strategies that are neutral to market returns, that is, the return from the strategy is uncorrelated with the market return. Regardless of whether the market goes up or down, in good times andbad the market neutral strategy performs in a steady manner, and results aretypically achieved with a lower volatility. This desired outcome is achievedby trading market neutral portfolios. Let us therefore define what we meanby a market neutral portfolio.In the CAPM context, market neutral portfolios may be defined as portfolios whose beta is zero. To examine the implications, let us apply a betavalue of zero to the equation for the SML. It is easy to see that the return onthe portfolio ceases to have a market component and is completely determined by qp, the residual component. The residual component by the CAPMassumption happens to be uncorrelated with market returns, and the portfolio return is therefore neutral to the market. Thus, a zero beta portfolioqualifies as a market neutral portfolio.In working with market neutral portfolios, the trader can now focus onforecasting and trading the residual returns. Since the consensus expectationor mean on the residual return is zero, it is reasonable to expect a strongmean-reverting behavior (value oscillates back and forth about the mean
BACKGROUND MATERIAL
6
value) of the residual time series.1 This mean-reverting behavior can then beexploited in the process of return prediction, leading to trading signals thatconstitute the trading strategy.Let us now examine how we can construct market neutral portfoliosand what we should expect by way of the composition of such portfolios.Consider a portfolio that is composed of strictly long positions in assets. Weexpect that beta of the assets to be positive. Then positive returns in themarket result in a positive return for the assets and thereby a positive returnfor the portfolio. This would, of course, imply a positive beta for the portfolio. By a similar argument it is easy to see that a portfolio composed ofstrictly short positions is likely to have a negative beta. So, how do we construct a zero beta portfolio, using securities with positive betas? This wouldnot be possible without holding both long and short positions on differentassets in the portfolio. We therefore conclude that one can typically expecta zero beta portfolio to comprise both long and short positions. For this reason, these portfolios are also called long–short portfolios. Another artifact oflong–short portfolios is that the dollar proceeds from the short sale are usedalmost entirely to establish the long position, that is, the net dollar value ofholdings is close to zero. Not surprisingly, zero beta portfolios are alsosometimes referred to as dollar neutral portfolios.
ExampleLet us consider two portfolios A and B, with positive betas bA and bB andwith returns rA and rBrA = β A rm + θ A
(1.3)
rB = β Brm + θ BWe now construct a portfolio AB, by taking a short position on r units ofportfolio A and a long position on one unit of portfolio B. The return on thisportfolio is given as rAB = –r.rA + rB. Substituting for the values of rA and rB,we haverAB = (− rβ A + β B ).rm + (− r.θ A + θ B )
1
(1.4)
The assertion of CAPM that the expected value of residual return is zero is ratherstrong. It has been discussed extensively in academic literature as to whether this prediction of CAPM is indeed observable. It is therefore recommended that we explicitly verify the mean-reverting behavior of the spread time series. In later chapters wewill discuss methods to statistically check for mean-reverting behavior.
Introduction
7
Thus, the combined portfolio has an effective beta of –rbA + bB. This valuebecomes zero, when r = bB/bA. Thus, by a judicious choice of the value of rin the long–short portfolio we have created a market neutral portfolio.
COCKTAIL CORNERIn cocktail situations involving investment professionals, it is fairlycommon to hear the terms long–short, market neutral, and dollar neutral investing bandied about. Very often they are assumed to mean thesame thing. Actually, that need not be the case. You could be long–short and dollar neutral but still have a nonzero beta to the market. Inwhich case you have a nonzero market component in the portfolioreturn and therefore are not market neutral.If you ever encountered such a situation, you could smile to yourself. Tempting as it might be, I strongly urge that you restrain yourself.But, of course, if you are looking to be anointed the “resident nerd,”you could go ahead and launch into an exhaustive explanation of thesubtle differences to people with cocktails in hand not particularlylooking for a lesson in precise terminology.
8
BACKGROUND MATERIAL
PAIRS TRADINGPairs trading is a market neutral strategy in its most primitive form. Themarket neutral portfolios are constructed using just two securities, consisting of a long position in one security and a short position in the other, in apredetermined ratio. At any given time, the portfolio is associated with aquantity called the spread. This quantity is computed using the quoted pricesof the two securities and forms a time series. The spread is in some ways related to the residual return component of the return already discussed. Pairstrading involves putting on positions when the spread is substantially awayfrom its mean value, with the expectation that the spread will revert back.The positions are then reversed upon convergence. In this book, we will lookat two versions of pairs trading in the equity markets; namely, statistical arbitrage pairs and risk arbitrage pairs.Statistical arbitrage pairs trading is based on the idea of relative pricing.The underlying premise in relative pricing is that stocks with similar characteristics must be priced more or less the same. The spread in this case maybe thought of as the degree of mutual mispricing. The greater the spread, thehigher the magnitude of mispricing and greater the profit potential.The strategy involves assuming a long–short position when the spread issubstantially away from the mean. This is done with the expectation that themispricing is likely to correct itself. The position is then reversed and profits made when the spread reverts back. This brings up several questions:How do we go about calculating the spread? How do we identify stockpairs for which such a strategy would work? What value do we use for theratio in the construction of the pairs portfolio? When can we say that thespread has substantially diverged from the mean? We will address thesequestions and provide some quantitative tools to answer them.Risk arbitrage pairs occur in the context of a merger between two companies. The terms of the merger agreement establish a strict parity relationship between the values of the stocks of the two firms involved. The spreadin this case is the magnitude of the deviation from the defined parity relationship. If the merger between the two companies is deemed a certainty,then the stock prices of the two firms must satisfy the parity relationship,and the spread between them will be zero. However, there is usually a certain level of uncertainty on the successful completion of a merger after theannouncement, because of various reasons like antitrust regulatory issues,proxy battles, competing bidders, and the like. This uncertainty is reflectedin a nonzero value for the spread. Risk arbitrage involves taking on this uncertainty as risk and capturing the spread value as profits. Thus, unlike thecase of statistical arbitrage pairs, which is based on valuation considerations,risk arbitrage trade is based strictly on a parity relationship between theprices of the two stocks.
Introduction
9
The typical modus operandi is as follows. Let us call the acquiring firmthe “bidder” and the acquired firm the “target.” On the eve of merger announcement, the bidder shares are sold short and the target shares arebought. The position is then unwound on completion of the merger. Thespread on merger completion is usually lower than when it was put on. Therealized profit is the difference between the two spreads. In this book, wediscuss how the ratio is determined based on the details of the merger agreement. We will develop a model for the spread dynamics that can be used toanswer questions like, “What is the market expectation on the odds ofmerger completion?” We shall also demonstrate how the model may be usedfor risk management. Additionally, we will focus on trade timing and provide some quantitative tools for the process.
OUTLINEThe book provides an overview of two different versions of pairs trading inthe equity markets. The first version is based on the idea of relative valuation and is called statistical arbitrage pairs trading. The second involvespairs trading that arises in the context of mergers and is called risk arbitrage.Even though they are commonly called arbitrage strategies in the industry,they are by no means risk-free. In this book we take an in-depth look at thevarious aspects of these strategies and provide quantitative tools to assist intheir analysis.I must also quickly point out at this juncture that the methodologies discussed in the book are not by any measure to be construed as the only wayto trade pairs because, to put it proverbially, there is more than one way toskin a cat. We do, however, strive to present a compelling point of view attempting to integrate theory and practice. The book is by no means meantto be a guarantee for success in pairs trading. However, it provides a framework and insights on applying rigorous analysis to trading pairs in the equity markets.The book is in three parts. In the first part, we present preliminary material on some key topics. We concede that there are books entirely devotedto each of the topics addressed, and the coverage of the topics here is not exhaustive. However, the discussion sets the context for the rest of the bookand helps familiarize the reader with some important ideas. It also introduces some notation and definitions. The second part is devoted to statistical arbitrage pairs, and the third part is on risk arbitrage.The book assumes some knowledge on the part of the reader of algebra,probability theory, and calculus. Nevertheless, we have strived to make thematerial accessible and the reader could choose to pick up the backgroundalong the way. As a refresher, the appendix at the end of this chapter lists the
BACKGROUND MATERIAL
10
basic probability formulas that the reader can expect to encounter in thecourse of reading the book.In terms of the sequence of chapters, we highly recommend that readersfamiliarize themselves with the chapters on time series and multifactor models before getting on to statistical arbitrage pairs, as those ideas and technical terms are referenced quite frequently in the course of the discussions.Concepts from Chapter 4, on Kalman filtering, are used in Chapter 12, related to smoothing risk arbitrage spreads. Other than the preceding dependencies, the rest of the material is mostly self-contained.
AUDIENCEThis book is written to appeal to a broad audience spanning students, practitioners, and self-study enthusiasts. It is written in an easy reading style, firstpresenting the broad ideas and concepts and subsequently delving into thedetails. The idea is to provide readers with the flexibility to revisit aspects ofthe details on their own timetable. To further facilitate this, a bullet summary highlighting the key points is provided at the end of every chapter. Thebook could serve as a reference text for students pursuing a degree in mathematical finance or be used as part of an advanced course for MBA students.Also, the topics addressed in the book would be of keen interest not only toacademicians but also to traders and quantitative analysts in hedge fundsand brokerage houses.The background material in Part 1 provides a primer on various subjectsthat are drawn on in the course of the analysis. The background materialand the analysis methodology appear as a recurring theme in strategy analysis and are generally applicable to other asset classes as well. Given this andthe easy readable style of the book, we hope that this book serves as a reference for investment professionals.
SUMMARYThe CAPM model helps separate out portfolio returns into a marketcomponent and a residual component.Portfolios with a zero market component are called market neutralportfolios.Market neutral strategies involve the trading of market neutral portfolios, and the returns generated by such strategies are uncorrelated withthe market.Pairs trading is a genre of market neutral strategies in which a portfoliohas only two assets.
Introduction
11
In the book, we will discuss two classes of pairs trading strategies;namely, risk arbitrage and statistical arbitrage.
FURTHER READING MATERIALCAPMElton, Edwin J. and Martin J. Gruber. Modern Portfolio Theory and InvestmentAnalysis, 4th Edition. (New York: John Wiley & Sons, Inc., 1991).Fama, Eugene F. and Kenneth R. French. “The Cross-Section of Expected Stock Returns.” Journal of Finance 47, no. 2 (June 1992): 427–465.Market Neutral StrategiesNicholas, Joseph G. Market Neutral Investing: Long/Short Hedge Fund Strategies.(New York: Bloomberg Press, 2000).
BACKGROUND MATERIAL
12
APPENDIXBelow are a few formulas on random variables that we are likely to encounter throughout the book.
DEFINITIONSLet X, Y, and Z be random variables. Let (x1, y1, z1),(x2, y2, z2),...,(xN, yN,zN) be N realization 3-tuples for these random variables.
MeanThe mean or expected value of X is denoted by E[X] = mx.The estimated value of the mean of a random variable is known as theaverage.NThe formula for the average is xavg = N1xi .
∑i =1
Variance
(
)
2The variance of X is var(X) = E  x − µ x  .The estimated value of the square root of variance is the familiar standard deviation.N
Its value is calculated using the formula xstddev =
1N
∑ (xi − xavg )2 .i =1
CovarianceThe covariance between X and Y is denoted as
[(
)(
)]
cov(X, Y ) = E x − µ x y − µ y .An estimation of the covariance may be calculated using the formulaN
1N
∑ (xi − xavg )(yi − yavg ) .i =1
CorrelationThe correlation between X and Y is corr(X, Y ) =
cov(X, Y )var(X) var(Y )
The formula for the estimate of correlation is given asN
1N
∑ (xi − xavg )(yi − yavg )i =1
(X
stddev
)(Y
stddev
)
Introduction
13
The correlation between any two random variables is always a value between +1 and –1.Every random variable is perfectly correlated with itself, that is, the correlation is 1.0.Two random variables are said to be uncorrelated when the correlationbetween them is 0.
FORMULASIf a, b are nonrandom numbers, then the following formulas hold:E[aX + bY] = aE[X] + bE[Y]var(aX + b) = a2 var(X)var(X + Y) = var(X) + var(Y) + 2cov(X,Y)var(X – Y) = var(X) + var(Y) – 2cov(X,Y)cov(aX, bY) = abcov(X,Y)cov(X,Y + Z) = cov(X,Y) + cov(X,Z)corr(aX, bY) = corr(X,Y)
CHAPTER
2
Time Series
OVERVIEWA time series is a sequence of values measured over time. These values maybe derived from a fixed deterministic formula, in which case they are referred to as a deterministic time series. Alternately, the value may be obtained by drawing a sample from a probability distribution, in which casethey may be termed as probabilistic or stochastic time series. In this chapter,we will focus on stochastic time series.Now, if the value at each instance in a stochastic time series is drawnfrom a probability distribution, how is it different from repeated drawingsfrom a probability distribution? The added twist is that the probability distributions used for the drawings can themselves vary with time. The formalspecification prescribing ways in which the distributions could vary withtime and the discipline of analyzing stochastic time series was pioneeredand popularized by Nobert Weiner.1 For this reason, the subject area is alsoreferred to at times as Weiner filtering.In the early days of Weiner filtering, the ideas were in theorem form,and to use them in practical applications one had to work through the rigorous mathematical definitions and theorems. Along came George Box andGwilym Jenkins in the early 1970s, who formulated the application ofWeiner filtering concepts into a recipe-like format. Their step-by-step prescription to the process of model building not only had great intuitive appealbut also managed to transform what was considered an esoteric science intoa robust engineering discipline. The approach could now be readily appliedto forecasting problems. The methodology gained instant popularity withtime series analysts and has become the staple by far for the analysis of sto-
1
Nobert Weiner is also credited with coining the word cybernetics, the shortenedversion of which is the ubiquitous cyber, which has by usage become a prefix for alot of terms associated with the Internet.
14
Time Series
15
chastic time series. Fittingly, their methodology for time series forecasting isreferred to as the Box-Jenkins approach. In this chapter, we will describe theBox-Jenkins approach. Instead of doing this by definition, we will attemptto do this by way of construction and examples.We begin by introducing some basic notation. Throughout the chapterthe value of a time series at time t is denoted as yt. It then follows that the general time series is the set of values yt, t = 0, 1, 2, 3...T. We denote this as yt.
AUTOCORRELATIONLet us begin the discussion by introducing the notion of the autocorrelation.Given a stochastic time series, the first question one tends to ask in theprocess of analysis is, “Is there a relationship between the value now and thevalue observed one time step in the past?” We can choose to answer thequestion by measuring the correlation between the time series values onetime interval apart. The strength of the (linear) relationship is reflected in thecorrelation number. And what about the relationship of the current value tothe value two time steps in the past? What about three time steps in the past?The question seems to repeat itself naturally for the whole range of timesteps. The answer to these questions, spanning the entire range of time steps,could very well be the autocorrelation function.The autocorrelation function is the plot of the correlation between values in the time series based on the time interval between them. The x-axis denotes the length of the time lag between the current value and the value inthe past. The y-axis value for a time lag t, (x = t) is the correlation betweenthe values in the time series t time units apart. This correlation is estimatedusing the formulaT
ρˆ (τ ) =
1T
∑ [yt − y ][yt −τ − y ]
t = τ +1
T
1T
(2.1)
∑ (yt − y)
2
t =1
where y– is the calculated average of variable y.The plot of the estimated correlation against time intervals forms an estimation of the autocorrelation function, called the correlogram. It serves asa proxy for the autocorrelation function of the time series.We shall see in the ensuing discussions that the autocorrelation functionserves as a signature or fingerprint for a time series and plays a key role incharacterizing various cases of the time series that we describe in the following sections.
16
BACKGROUND MATERIAL
TIME SERIES MODELSThe approach we will adopt in the description of time series models is to startwith the special cases and eventually build up to the generalized version.
White NoiseThe white noise is the simplest case of a probabilistic time series. It is constructed by drawing a value from a normal distribution at each time instance. Furthermore, the parameters of the normal distribution are fixed anddo not change with time. Thus, in this case, the time series is equivalent todrawing samples repeatedly from a probability distribution. If we denote thevalue from the drawing at time t as et, the value of the time series at time tis then yt = et.Note that there is no special requirement in the definition of white noisethat the invariant distribution be a normal or Gaussian distribution. This is,however, the most widely used version of white noise in practice and is referred to as Gaussian white noise.A plot of a white noise series is shown in Figure 2.1a. The correlogramfor that time series is calculated as is shown in Figure 2.1b. Note that at thelag value of zero, the correlation is unity; that is, every sample is perfectlycorrelated with itself. At all the other lag values the measured correlation isnegligible. Let us see why that is. At all time steps, the values are drawn fromidentical independent normal distributions. It is also a fact that the correlation between independent random variables is zero; that is, they are uncorrelated. Therefore, for a white noise series, the correlation between thevalues for all time intervals is zero, and this is reflected in the correlogram.But what is the genesis of the term white noise? It has to do with the Fouriertransform of the autocorrelation function. A discussion of that is a little beyond the scope of this introduction, so for that we direct the reader to otherbooks written in the area, as noted in the reference section.Let us now focus on the predictability of the white noise time series. Thequestion we ask is as follows: Does knowledge of the past realization help inthe prediction of the time series value in the next time instant? It does helpto some extent. Knowledge of the past realization helps us to estimate thevariance of the normal distribution. This enables us to arrive at some intelligent conclusions about the odds of the next realization of the time seriesbeing greater than or less than some value.Summing up, in a white noise series, the variance of the value at eachpoint in the series is the variance of the normal distribution used for drawing the white noise values. This distribution with a specific mean and variance is time invariant. Thus, a white noise series is a sequence of uncorrelatedrandom variables with constant mean and variance.
Time Series
17
3210–1–210
30
50
FIGURE 2.1A
70
90
White Noise Series.
1.0Auto Correlation
0.80.60.40.2–0.0–0.20
10