Overview

Empirical work on the CAPM and APT has two main objectives: (i) to test whether or not the theories should be rejected; and (ii) to provide information that can aid financial decisions. The two aims are clearly complementary: only theories that are compatible with the evidence are likely to be helpful in making reliable decisions.

To accomplish objective (i), tests are conducted that could – potentially, at least – reject the model. The model is deemed to pass the test if it is not possible to reject the hypothesis that it is true. Such a methodology imposes a severe standard, for it is invariably possible to find evidence that contradicts the predictions of any testable economic theory. Hence, the methods of statistical inference need to be applied in order to draw sensible conclusions about just how far the data support the model. Definitive judgements are never possible in applied work. This need not be an excuse for despair but should serve as a counsel for cautious scepticism.

Tests are almost never as clear-cut as they at first seem. They are typically tests of joint hypotheses, so that care is necessary to recognize what is, or is not, being tested. Also, the relevant alternative to the hypothesis being tested often remains vague, or, even more commonly, is ignored. For example, if the CAPM is rejected, which theory is it rejected in comparison with? No simple answer may be available.

Overview

Options contracts are used in a multitude of different ways for different purposes. For example, an investor who plans to acquire shares in a company but considers that the current price is too high might choose to write put options on the shares. If the share price remains high during the life of the options, the options are not likely to be exercised and the investor pockets the option premium, without buying the shares. Alternatively, if the share price falls and the options are exercised against the investor, the shares are acquired, as the investor intended, at the exercise price. (In addition, the investor keeps the premium, of course.)

Rather than attempt to catalogue all these policies, this chapter studies several applications that illustrate different aspects of options analysis. Section 20.1 begins with a review of stock index options. Sections 20.2 and 20.3 introduce options on futures contracts, together with a variety of applications. In particular, section 20.3 explains how options on interest rate futures can be used to construct caps and floors on the effective interest rate for borrowing or lending.

Section 20.4 outlines how the inclusion of options in portfolios can mitigate the impact of uncertainty about future asset prices. Hedging, introduced in chapter 15, is re-examined using options (rather than futures) as hedge instruments.

A successful hedge reduces the risks associated with asset price fluctuations. It is not designed to reap the benefits of asset price increases while also protecting the investor against losses when asset prices fall.

Overview

In chapter 18 bounds were obtained on the range of option prices compatible with the absence of arbitrage opportunities. No attempt was made, however, to predict the level of an option price. This is the purpose of the present chapter. While attention concentrates throughout on the arbitrage principle, extra assumptions are required about the determinants of the underlying asset price in order to obtain the option price itself. Armed with these extra assumptions, the objective is to obtain a formula for an option price, where the arguments of the formula comprise a set of explanatory variables including, among other things, the option's exercise price and its time to expiry.

Very often the aim of the analysis is expressed in terms of determining the ‘fair’ option price, or of option ‘valuation’. This approach typically makes most sense for an over-the-counter option that is not exchange traded, where the goal is to calculate the option's price as if the option were openly traded in the absence of arbitrage opportunities – and together with the other assumptions needed to make the calculation. It should be obvious that the ‘fair’ price depends on the assumptions of a model, but in practice it is often overlooked that the computed value may well be sensitive to the model on which it is based.

Section 19.1 outlines the assumptions common to most option price theories and describes the method of analysis.

How can yet another book on finance be justified? The field is already well served with advanced works, many of impressive technical erudition. And, towards the other end of the academic spectrum, an abundance of mammoth texts saturates the MBA market. For the general reader, manuals confidently promising investment success compete with sensational diagnoses of financial upheavals to attract attention from the gullible, avaricious or unwary.

Alas, no one can expect to make a fortune as a consequence of reading this book. It has a more modest objective, namely to explore the economics of financial markets, at an ‘intermediate’ level – roughly that appropriate for advanced undergraduates. It is a work of exposition, not of original research. It unashamedly follows Keynes's immortal characterization of economic theory as ‘an apparatus of the mind, a technique of thinking’. Principles – rather than assertions of doctrine, policy pronouncements or institutional description – are the focus of attention. If the following chapters reveal no get-rich-quick recipes, they should at least demonstrate why all such nostrums merit unequivocal disbelief.

This book evolved, over more years than the author cares to admit, from lecture notes for a course in financial economics taught at the University of Essex. For reasons of space, one topic – corporate finance – has been omitted from the book, though its core insight – the Modigliani–Miller theorem – is slipped in under options (chapter 18, section 6).

Overview

Perhaps the commonest equation in the whole of finance is the one that sets the value of an asset equal to the net present value (or ‘present discounted value’) of a sequence of its payoffs. The equation plays a central role in corporate finance, where NPV criteria constitute the basis for the selection of investment projects. In particular, the NPV rule is applied to value assets (projects) the market prices of which may not be readily observed.

This chapter's objective is somewhat different from, though consistent with, that of corporate finance. Here the NPV relationship appears as a market equilibrium condition that has testable implications for observed asset prices.

In its simplest and most broadly applicable form, studied in section 10.1, the NPV relationship is a consequence of the arbitrage principle. In this sense it is nothing more than the extension of the results of chapter 7 to a multiperiod framework.

While central to financial theory, arbitrage ideas on their own tend to yield few predictions. Stronger assumptions – in particular about investors' expectations – permit predictions about asset price volatility to be derived. Section 10.2 reviews these assumptions and discusses the degree to which empirical evidence casts doubt on the validity of a theory commonly interpreted as expressing rational investor behaviour. By implication, doubt is also cast on asset market efficiency.

Section 10.3 explores other models, also motivated by the NPV, that seek to provide more empirically acceptable explanations of asset price volatility.

Introduction

This chapter serves as an introduction to Bayesian econometrics. Bayesian regression analysis has grown in a spectacular fashion since the publication of books by Zellner (1971) and Leamer (1978). Application to routine data analysis has also expanded enormously, greatly aided by revolutionary advances in computer hardware and software technology. In the light of such major developments, a single chapter can never do adequate justice to the many facets of this subject. This chapter therefore has the very modest goal of providing a rough road map to the major ideas and developments in Bayesian econometrics. Despite this modest objective some parts are still quite technical.

The Bayesian approach, unlike the likelihood or frequentist or classical approach presented in previous chapters, requires the specification of a probabilistic model of prior beliefs about the unknown parameters, given an initial specification of a model. Many researchers are uncomfortable about this step, both philosophically and practically. This has traditionally been the basis of the concern that the Bayesian approach is subjective rather than objective. It will be shown that in large samples the role of the prior may be negligible, that relatively uninformative priors can be specified, and that there are methods available for studying the sensitivity of inferences to priors. Therefore, the charge of subjectivity may not always be as serious as many claim.

Bayesian approaches play a potentially large role in applied microeconometrics, especially when dealing with complex models that lack analytically tractable likelihood functions.

Introduction

Microeconometrics research is usually performed on data collected by survey of a sample of the population of interest. The simplest statistical assumption for survey data is simple random sampling (SRS), under which each member of the population has equal probability of being included in the sample. Then it is reasonable to base statistical inference on the assumption that the data (y i, x i) are independent over i and identically distributed. This assumption underlies the small-sample and asymptotic properties of estimators presented in this book, with the notable exception of sample selection models in Chapter 16.

In practice, however, SRS is almost never the right assumption for survey data. Alternative sampling schemes are instead used to reduce survey costs and to increase precision of estimation for subgroups of the population that are of particular interest.

For example, a household survey may first partition the population geographically into subgroups, such as villages or suburbs, with differing sampling rates for different subgroups. Interviews may be conducted on households that are clustered in small geographic areas, such as city blocks. The data (y i, x i) are clearly no longer iid. First, the distribution of (y i, x i) may vary across subgroups, so the identical distribution assumption may be inappropriate. Second, since data may be correlated for households in the same cluster, the assumption that (y i, x i) are independent within the cluster breaks down.

Introduction

The previous chapter focused on m-estimation, including ML and NLS estimation. Now we consider a much broader class of extremum estimators, those based on method of moments (MM) and generalized method of moments (GMM).

The basis of MM and GMM is specification of a set of population moment conditions involving data and unknown parameters. The MM estimator solves the sample moment conditions that correspond to the population moment conditions. For example, the sample mean is the MM estimator of the population mean. In some cases there may be no explicit analytical solution for the MM estimator, but numerical solution may still be possible. Then the estimator is an example of the estimating equations estimator introduced briefly in Section 5.4.

In some situations, however, MM estimation may be infeasible because there are more moment conditions and hence equations to solve than there are parameters. A leading example is IV estimation in an overidentified model. The GMM estimator, due to Hansen (1982), extends the MM approach to accommodate this case.

The GMM estimator defines a class of estimators, with different GMM estimators obtained by using different population moment conditions, just as different specified densities lead to different ML estimators. We emphasize this moment-based approach to estimation, even in cases where alternative presentations are possible, as it provides a unified approach to estimation and can provide an obvious way to extend methods from linear to nonlinear models.

Part 2 presents the core estimation methods – least squares, maximum likelihood and method of moments – and associated methods of inference for nonlinear regression models that are central in microeconometrics. The material also includes modern topics such as quantile regression, sequential estimation, empirical likelihood, semiparametric and nonparametric regression, and statistical inference based on the bootstrap. In general the discussion is at a level intended to provide enough background and detail to enable the practitioner to read and comprehend articles in the leading econometrics journals and, where needed, subsequent chapters of this book. We presume prior familiarity with linear regression analysis.

The essential estimation theory is presented in three chapters. Chapter 4 begins with the linear regression model. It then covers at an introductory level quantile regression, which models distributional features other than the conditional mean. It provides a lengthy expository treatment of instrumental variables estimation, a major method of causal inference. Chapter 5 presents the most commonly-used estimation methods for nonlinear models, beginning with the topic of m-estimation, before specialization to maximum likelihood and nonlinear least squares regression. Chapter 6 provides a comprehensive treatment of generalized method of moments, which is a quite general estimation framework that is applicable for linear and nonlinear models in single-equation and multi-equation settings. The chapter emphasizes the special case of instrumental variables estimation.

Introduction

The problem of missing data in survey data is one of long standing, arising from nonresponse or partial response to survey questions. Reasons for nonresponse include unwillingness to provide the information asked for, difficulty of recall of events that occurred in the past, and not knowing the correct response. Imputation is the process of estimating or predicting the missing observations.

In this chapter we deal with the regression setup with data vector (y i, x i), i = 1, …, N. For some of the observations some elements of x i or of both (y i, x i) are missing. A number of questions are considered. When can we proceed with an analysis of only the complete observations, and when should we attempt to fill the gaps left by the missing observations? What methods of imputation are available? When imputed values for missing observations are obtained, how should estimation and inference then proceed?

If a data set has missing observations, and if these gaps can be filled by a statistically sound procedure, then benefit comes from a larger and possibly more representative sample and, under ideal circumstances, more precise inference. The cost of estimating missing data comes from having to make (possibly wrong) assumptions to support a procedure for generating proxies for the missing observations, and from the approximation error inherent in any such procedure. Further, statistical inference that follows data augmentation after imputed values replace missing data is more complicated because such inference must take into account the approximation errors introduced by imputation.

In empirical work data frequently present not one but multiple complications that need to be dealt with simultaneously. Examples of such complications include departures from simple random sampling, clustering of observations, measurement errors, and missing data. When they occur, individually or jointly, and in the context of any of the models developed in Parts 4 and 5, identification of parameters of interest will be compromised. Three chapters in Part 6 – Chapters 24, 26, and 27 – analyze the consequences of such complications and then present methods that control for these complications. The methods are illustrated using examples taken from the earlier parts of the book. This feature gives points of connection between Part 6 and the rest of the book.

Chapter 24, which deals with several features of data from complex surveys, notably stratified sampling and clustering, complements various topics covered in Chapters 3, 5, and 16. Chapter 26 which deals with measurement errors in models studied in Chapters 4, 14, and 20. Chapter 27 is a stand-alone chapter on missing data and multiple imputation, but its use of the EM algorithm and Gibbs sampler also gives it points of contact with Chapters 10 and 13, respectively.

Chapter 25 presents treatment evaluation. Treatment is a broad term that refers to the impact of one variable, e.g. schooling, on some outcome variable, e.g. earnings. Treatment variables may be exogenously assigned, or may be endogenously chosen.

Introduction

In this chapter we consider tests of hypotheses, possibly nonlinear in the parameters, using estimators appropriate for nonlinear models.

The distribution of test statistics can be obtained using the same statistical theory as that used for estimators, since test statistics like estimators are statistics, that is, functions of the sample. Given appropriate linearization of estimators and hypotheses, the results closely resemble to those for testing linear restrictions in the linear regression model. The results rely on asymptotic theory, however, and exact t- and F-distributed test statistics for the linear model under normality are replaced by test statistics that are asymptotically standard normal distributed (z-tests) or chi-square distributed.

There are two main practical concerns in hypothesis testing. First, tests may have the wrong size, so that in testing at a nominal significance level of, say, 5%, the actual probability of rejection of the null hypothesis may be much more or less than 5%. Such a wrong size is almost certain to arise in moderate size samples as the underlying asymptotic distribution theory is only an approximation. One remedy is the bootstrap method, introduced in this chapter but sufficiently important and broad to be treated separately in Chapter 11. Second, tests may have low power, so that there is low probability of rejecting the null hypothesis when it should be rejected. This potential weakness of tests is often neglected. Size and power are given more prominence here than in most textbook treatments of testing.

This book provides a detailed treatment of microeconometric analysis, the analysis of individual-level data on the economic behavior of individuals or firms. This type of analysis usually entails applying regression methods to cross-section and panel data.

The book aims at providing the practitioner with a comprehensive coverage of statistical methods and their application in modern applied microeconometrics research. These methods include nonlinear modeling, inference under minimal distributional assumptions, identifying and measuring causation rather than mere association, and correcting departures from simple random sampling. Many of these features are of relevance to individual-level data analysis throughout the social sciences.

The ambitious agenda has determined the characteristics of this book. First, although oriented to the practitioner, the book is relatively advanced in places. A cookbook approach is inadequate because when two or more complications occur simultaneously – a common situation – the practitioner must know enough to be able to adapt available methods. Second, the book provides considerable coverage of practical data problems (see especially the last three chapters). Third, the book includes substantial empirical examples in many chapters to illustrate some of the methods covered. Finally, the book is unusually long. Despite this length we have been space-constrained. We had intended to include even more empirical examples, and abbreviated presentations will at times fail to recognize the accomplishments of researchers who have made substantive contributions.

Introduction

Two important practical aspects of microeconometric modeling are determining whether a model is correctly specified and selecting from alternative models. For these purposes it is often possible to use the hypothesis testing methods presented in the previous chapter, especially when models are nested. In this chapter we present several other methods.

First, m-tests such as conditional moment tests are tests of whether moment conditions imposed by a model are satisfied. The approach is similar in spirit to GMM, except that the moment conditions are not imposed in estimation and are instead used for testing. Such tests are conceptually very different from the hypothesis tests of Chapter 7, as there is no explicit statement of an alternative hypothesis model.

Second, Hausman tests are tests of the difference between two estimators that are both consistent if the model is correctly specified but diverge if the model is incorrectly specified.

Third, tests of nonnested models require special methods because the usual hypothesis testing approach can only be applied when one model is nested within another.

Finally, it can be useful to compute and report statistics of model adequacy that are not test statistics. For example, an analogue of R2 may be used to measure the goodness of fit of a nonlinear model.

Ideally, these methods are used in a cycle of model specification, estimating, testing, and evaluation.

Introduction

This chapter surveys issues concerning the potential usefulness and limitations of different types of microeconomic data. By far the most common data structure used in microeconometrics is survey or census data. These data are usually called observational data to distinguish them from experimental data.

This chapter discusses the potential limitation of the aforementioned data structures. The inherent limitations of observational data may be further compounded by the manner in which the data are collected, that is, by the sample frame (the way the sample is generated), sample design (simple random sample versus stratified random sample), and sample scope (cross-section versus longitudinal data). Hence we also discuss sampling issues in connection with the use of observational data. Some of this terminology is new at this stage but will be explained later in this chapter.

Microeconometrics goes beyond the analysis of survey data under the assumptions of simple random sampling. This chapter considers extensions. Section 3.2 outlines the structure of multistage sample surveys and some common forms of departure from random sampling; a more detailed analysis of their statistical implications is provided in later chapters. It also considers some commonly occurring complications that result in the data not being necessarily representative of the population. Given the deficiencies of observational data in estimating causal parameters, there has been an increased attempt at exploiting experimental and quasi-experimental data and frameworks. Section 3.3 examines the potential of data from social experiments.

Introduction

This chapter extends the linear model panel data methods of Chapters 21 and 22 to the nonlinear regression models presented in Chapters 14–20. We focus on short panels and models with a time-invariant individual-specific effect that may be fixed or may be random. Both static and dynamic models are considered.

There is no one-size-fits-all prescription for nonlinear models with individual specific effects. If individual-specific effects are fixed and the panel is short then consistent estimation of the slope parameters is possible for only a subset of nonlinear models. If individual-specific effects are instead purely random then consistent estimation is possible for a wide range of models.

Section 23.2 presents general approaches that may or may not be implementable for particular models. Section 23.3 provides an application to a nonlinear model with multiplicative individual-specific effects. Specializations to the leading classes of nonlinear models – discrete data, selection models, transition data, and count data – are presented in Sections 23.4–23.7. Semiparametric estimation is surveyed in Section 23.8.

General Results

General approaches to extending the methods for linear models are presented in this section. We first present the various models – fixed effects, random effects, and pooled models, distinguishing parametric from conditional mean models. Methods to estimate these models and obtain panel-robust standard errors are then presented. Further details for specific nonlinear panel models are provided in subsequent sections.

Introduction

The topic of treatment evaluation concerns measuring the impact of interventions on outcomes of interest, with the type of intervention and outcome being defined broadly so as to apply to many different contexts. The treatment evaluation approach and some of its terminology comes from medical sciences where intervention frequently means adopting a treatment regime. Subsequently, one may be interested in measuring the response to the treatment relative to some benchmark, such as no treatment or a different treatment. In economic applications treatment and interventions usually mean the same thing.

Examples of treatments in the economic context are enrollment into a labor training program, being a member of a trade union, receipt of a transfer payment from a social program, changes in regulations for receiving a transfer from a social program, changes in rules and regulations pertaining to financial transactions, changes in economic incentives, and so forth; see Moffitt (1992), Friedlander, Greenberg, and Robbins (1997), and Heckman, Lalonde, and Smith (1999). If the treatment that is applied can vary in intensity or type, we use the term multiple treatments when referring to them collectively. Relative to a single type of treatment this does not create complications, but now the choice of a benchmark for comparisons is more flexible.