social sciences, although it has been the hallmark оГ the natural sciences for quite some time. It is one of the most rewarding aspects of financial econometrics, so much so thai we felt impelled to write this graduate-level textbook as a means ol introducing others to this exciting field.
Section 1.1 explains which topics we cover in this book, and how we have organized the material. We also suggest some ways in which the book might be used in a one-semester course on financial econometrics or empirical finance.
In Section 1.2, we describe the kinds of background material that are most useful for financial econometrics and suggest references for those readers who wish to review or learn such material along the way. In our experience, students are often more highly motivated lo pick tip the necessary background after they see how it is to be applied, so we encourage readers with a serious interest in financial econometrics but with somewhat less preparation to take a crack al ibis material anyway.
In a book of this magnitude, notation becomes a nontrivial challenge of coordination; hence Section 1.3 describes what method there is in our notalional madness. We urge readers to review this carefully to minimize the confusion that can arise when fi is mistaken for fi and X is incorrcctlv assumed to be the same as X.
Section 1 .1 extends our discussion of notation by presenting notalional conventions for and definitions ol some of the fundamental objects of om study: prices, returns, methods of compounding, and probability distributions. Although much of I his material is well-known to finance students and investment professionals, we thjnk a brief review will help many readers.
In Section 1.5, we turn our attention to quite a different subject: In-Efficient Markets Hypothesis. Because so much attention has been lavished on this hypothesis, often at the expense of other more substantive issues, we wish to dispense with this issue first. Much of the debate involves theological tenets that are empirically undecidable and, therefore, beyond the purview of this text. But lor completeness-no self-respecting finance text could omit market efficiency altogether-Section 1.5 briefly discusses the topic.
I.I Organization of the Book
In organizing this book, we have followed two general principles. First, the early chapters concentrate exclusively on stock markets. Although many of the methods discussed can be applied equally well to other asset markets, the empirical literature on slock markets is particularly large and by focusing on these markets we are able lo keep the discussion concrete. In later chapters, we cover derivative securities (Chapters (1 and 12) and fixed-income securi-
ties (Chapters 10 and 11). The last chapter of the book presents nonlinear methods, with applications to both stocks and derivatives.
Second, we start by presenting statistical models of asset returns, aitid then discuss more highly structured economic models. In Chapter 2, for example, we discuss methods for predicting stock returns from their otkn past history, without much attention to institutional detail; in Chapter 3 we show how the microstructure of stock markets affects the short-run behavior of returns. Similarly, in Chapter 4 we discuss simple statistical models of the cross-section of individual stock returns, and the application of these models to event studies; in Chapters 5 and 6 we show how the Capital Asset Pricing Model and multifactor models such as the Arbitrage Pricing Theory restrict the parameters of the statistical models. In Chapter 7 we discuss longer-run evidence on the predictability of stock returns from variables other than past stock returns; in Chapter 8 we explore dynamic equilibrium models which can generate persistent time-variation in expected returns. We use the same principle to divide a basic treatment of fixed-income securities in Chapter 10 from a discussion of equilibrium term-structure models in Chapter 11.
We have tried to make each chapter as self-contained as possible. While some chapters naturally go together (e.g., Chapters 5 and 6, and Chapters 10 and 11), there is certainly no need to read this book straight through from beginning to end. For classroom use, most teachers will find that there is too much material here to be covered in one semester. There are several ways to use the book in a one-semester course. For example one teacher might start by discussing short-run time-series behavior of stock prices using Chapters 2 and 3, then cover cross-sectional models in Chapters 4, 5, and 6, then discuss intertemporal equilibrium models using Chapter 8, and finally cover derivative securities and nonlinear methods as advanced topics using Chapters 9 and 12. Another teacher might first present the evidence on short- and long-run predictability of stock returns using Chapters 2 and 7, then discuss static and intertemporal equilibrium theory using Chapters 5, 6, and 8, and finally cover fixed-income securities using Chapters 10 and 11.
There are some important topics that we have not been able to include in this text. Most obviously, our focus is almost exclusively on US domestic asset markets. We say very little about asset markets in other countries, anc} we do not try to cover international topics such as exchange-fate behave ior or the home-bias puzzle (the tendency for each countrys investors to hold a disproportionate share of their own countrys assets in their portfolios). We also omit such important econometric subjects as Bayesian analysis and frequency-domain methods of time-series analysis. In many cases our choice of topics has been influenced by the dual objectives of the book: lo explain the methods of financial econometrics, and to review the em pirical literature in finance. We have tended to concentrate on topics that
involve econometric issues, sometimes a! the expense of oilier equally inter-cstiig malerial-including much recent work in behavioral finance-dial is etbnomctrically more straightforward.
1.2 Useful Background
The many rewards of financial econometrics come al a price. A solid background in mathematics, probability and statistics, and finance theory is necessary for the practicing financial cconoiiieirician, for precisely the reasons that make financial econometrics such an engaging endeavor. To assist readers in obtaining this background (since only the most focused and directed of students will have it already), Ave outline in this section the topics in mathematics, probability, statistics, and finance theory that have become indispensable to financial econometrics. We hope that this outline can serve as a self-study guide for the more enterprising readers and that it will be a partial substitute for including background malerial in this book.
1.2.1 Mathematics Background
The mathematics background most useful for financial econometrics is not unlike the background necessary for econometrics in general: multivariate calculus, linear algebra, and matrix analysis. References for each of these topics are Lang (1973), Strang (1976), and Magnus and Neudecker (1988), respectively. Key concepts include
multivariate constrained optimization
basic rules of matrix differentiation.
In addition, option- and other derivative-pricing models, and continuous-time asscl pricing models, require some passing familiarity with the Ito or stochastic calculus. A lucid and thorough treatment is provided by Merlon (1990), who pioneered the application of stochastic calculus to financial economics. More mathematically inclined readers may also Wish to consult Chimftancl Williams (1990).
1.2.2 Trohal/ilily and Statistics Huchpound
Basic probability theory is a prerequisite for any discipline in which uncertainty is involved. Although probability theory has varying degrees of mathematical sophistication, from coin-flipping calculations to measure-theoretic foundations, perhaps ihe most useful approach is one thai emphasizes the
intuition and subtleties of elemental у probabilistic reasoning. An amazingly durable classic that takes just this approach is Keller (1968). Brieman (1992) provides similar intuition but at a measure-theoretic level. Key concepts include
definition of a random variable
distribution and density functions
modes of convergence
laws of large numbers
central limit theorems.
Statistics is, of course, the primary engine which drives the inferences that financial econometricians draw from the data. As with probability theory, statistics can be taught at various levels of mathematical sophistication. Moreover, unlike die narrower (and some would say purer ) focus of probability theory, statistics has increased its breadth as it has matured, giving birth to many well-defined subdisciplines such as multivariate analysis, nonpara-metrics, time-series analysis, order .statistics, analysis of variance, decision theory, Bayesian statistics, etc. Each of these subdisciplines has been drawn upon by financial econometricians al one time or another, making it rather difficult to provide a single reference for all of these topics. Amazingly, such a reference does exist: Stuart and Orels (1987) three-volume tour de Joyce. A more compact reference thai contains most of the relevant material for our purposes is die elegant monograph by Silvey (1975). For topics in time-series analysis, Hamilton (1994) is an excellent comprehensive text. Key concepts include
Neynuin-Pcarson hypothesis testing
basic lime-series analysis (stationarity, autoregressive and ARMA processes, vector autoregressions, unit roots, etc.)
elementary Bayesian inference.
For continuous-lime financial models, an additional dose of stochastic processes is a must, at least at the level of Cox and Miller (1965) and Hoel, Port, and Stone (1972).
1.2.3 finance Theory Hackjrnmnd
Since the raison detre of financial econometrics is the empirical implementation and evaluation of financial models, a solid background in finance theory is the most important of all. Several lexis provide excellent coverage
of (his material: Ditflie (1992), Huang and Litzenberger (1988), Ingersoll
ari.-f Mr-i -*- }>. :rz>u include
. isk .ivcrsiou .iiiiI ,-\pei led-iitiniv tlicoty
static mean-vatiant e portfolio theory
the Capital Asset Pricing Model (СЛРМ) and the Arbitrage Pricing The-0 ory (APT)
dynamic asset pricing models J option pricing theory.
I We have found that it is far from simple to devise a consistent notalional scheme for a book of this scope. The difficulty comes from the fad that financial econometrics spans several very different strands of the finance literature, each replete with its own firmly established set of notalional conventions. Rut the conventions in one literature often conllict with the conventions in another. Unavoidably, then, we must .sacrifice either internal notalional consistency across different chapters of this text or external
i consistency with the notation used in the professional literature. We have
J chosen die former as the lesser evil, but we do maintain the following con-
I ventions throughout the book:
! We use boldface for vectors and matrices, and regular face for scalars.
Where possible, we use bold uppercase for matrices and bold lowercase for vectors. Thus x is a vector while X is a matrix.
Where possible, we use uppercase letters for the levels of variables ai:d lowercase letters for the natural logarithms (logs) of the same variables. Thus if P is an asset price, p is the log asset price.
Our standard notation for an innovation is the Creek letter t. Where we need to define several different innovations, we use the alternative Creek letters I/, £, and <\
Where possible, we use Creek letters to denote parameters or parameter vectors.
We use the Creek letter <, lo denote a vector of ones.
We use hats lo denote sample estimates, so if ft is a parameter, is an estimate of ft.
When we use subscripts, we always use uppercase letters for the upper limits of the subscripts. Where possible, we use the same letters for upper limits as for the subscripts themselves. Thus subscript / runs from I lo 7, subscript к runs from I to K, and so on. An exception is thai we will let subscript / (usually denoting an asset) run from I to Л because this notation is so common. We use I and r for time subscripts:
i for asset subscripts; k, m, and я for lead and lag subscripts; and j as a generic subscript.
the end of period (. Thus R, denotes a return on an asset held from the end of period (-1 to the end of period /.
In writing variance-covariance matrices, we use ft for the variance-covariance matrix of asset returns, £ for the variance-covariance matrix of residuals from a time-series or cross-sectional model, and V for the variance-covariance matrix of parameter estimators.
We use script letters sparingly. ftf denotes the normal distribution, and С denotes a log likelihood function.
We use Pr() to denote the probability of an event.
The professional literature uses many specialized terms. Inevitably ve also use these frequently, and we italicize them when they first appear in the book.
1.4 Prices, Returns, and Compounding
Virtually every aspect of financial economics involves returns, and there are at least two reasons for focusing our attention on returns rather than on prices. First, for the average investor, financial markets may be considered close to perfectly competitive, so that the size of the investment does not affect price changes. Therefore, since the investment technology is constant-returns-lo-scale, the return is a complete and scale-free summary of the investment opportunity.
Second, for theoretical and empirical reasons that will become apparent below, returns have more attractive statistical properties than prices, such as stationarity and ergodicity. In particular, dynamic general-equilibritjim models often yield nonstationary prices, but stationary returns (see, for example, Chapter 8 and Lucas ). i
1.4.1 Definitions and Conventions
Denote by P, the price of an asset at date ( and assume for now that this asset pays no dividends. The simple net return, R on the asset between dates / - 1 and ( is defined as
Pi = ~--L (1.4.1)
The simple gross return on the asset is just one plus the net return, 1 + R/.
From this definition it is apparent that the assets gross return over the most recent к periods from date / - k to date /, written 1 + Рч(к), is simply
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