In 1966, Bill Phillips delivered chapter 45 as the Walras-Bowley Lecture at the North American Meeting of the Econometric Society in San Francisco. The paper set out the maximum likelihood estimation of parameters of a system of simultaneous equations when there are lagged endogenous variables and moving average (MA) errors. At that point in time, what work there was on estimating equations with MA errors was largely restricted to a single equation, so that a paper purporting to describe how to produce estimators within a simultaneous equations context was very advanced. One can see how advanced by noting that the first paper to appear in the statistics literature on estimating systems of ARMA equations in the time domain was Wilson (1973), although Hannan (1970) had proposed frequency domain estimators. To this day the paper is probably the clearest analysis of how to construct a full information estimator of the parameters of simultaneous equation models with moving average errors.

A number of interesting questions arise over the origins and fate of the paper. There were, I think, two reasons why Bill was interested in this problem. Both of these reasons are consistent with lectures he delivered at the ANU in 1969. First, there was the fact that rational approximations to dynamic systems, of the sort in Jorgenson (1966), ended up inducing moving average errors into the specifications to be estimated. Griliches' (1967) survey reflects this concern very well.

Chapters 16, 17, 18 and 19 summarise Bill Phillips' early thoughts on the problems of regulating an economy. There can be little doubt that the ideas were sourced from his background and interest in electrical engineering and were essentially a transcription of the methods of ‘classical ’ control to the economic system. Concepts such as the multiplier, accelerator, real balance (Pigou) effects, etc. were used to describe the system in terms whereby the engineering concepts might be applied. These essays are still cited forty years after their publication. It is worth exploring why this is so, as few essays written in that era would have such longevity. To do so, it is useful to spell out what I perceive of as the innovations in the essays.

First, policy should not be thought of in a static but dynamic mode. This idea may seem trite but the accepted analysis of the time was largely one of comparative statics. ‘Policy’ consisted of shifting the aggregate demand curve up and down the ‘45° line’. The theoretical foundation of policy laid out in Tinbergen's (1952) classic work was also static. Where the past was recognised it was simply taken as predetermined. The objective of policy was either to manipulate instruments to hit a specified target value or to get as close as possible to this value in the event that some trade-off between targets was needed due to a lack of instruments.

Introduction

Although never as popular in econometrics as in fields such as sociology or psychology, there has always been some interest in the application of factor models to economic data. This interest has been heightened in the past two decades by the rise of quantitative finance, where thinking in factor terms has been found to be very useful, with one representation, the arbitrage pricing theory of Ross (1976), becoming a key approach in the modeling of asset returns. Some of the lack of enthusiasm for factor models might be explained by a concern that the factors were unobservable, and such a qualification would naturally lead to a clear preference for identifying influences upon economic variables from forces that were capable of being directly measured. In this respect things have changed a good deal in many parts of economics, nowhere more so than in macroeconomics. Today unobserved factors such as technology shocks are prominent in many models, while the common trends apparent in many time series have become the basis of the expanding literature on cointegration between them.

Section 2 of the chapter sets out in more detail various applications of factor models. Such a review emphasizes the common structure underlying these models, making it possible to consider estimation issues from a general perspective, rather than through the extant specialized approaches. As we describe, direct estimation of the parameters of such models, θ, is frequently very difficult, and this leads us to propose the indirect estimation methods of Gourieroux, Monfort, and Renault (1993) and Gallant and Tauchen (1996) as good ways of finding estimates of θ at reasonable computational cost.

Introduction

Chapters 3 and 4 have been concerned with the estimation of the conditional moments of Y given that X equals some value x and the derivatives of this measure. We have referred to the techniques for doing this as nonparametric, even though it was either necessary to prescribe “parameters” such as the window width for kernel estimators or the number of approximating terms for series estimators. Our use of the term “nonparametric” in that context was prompted by the fact that the “parameters” mentioned did not have any economic meaning. This chapter looks at a different scenario, distinguished by the feature that the models considered contain parameters with some economic meaning and which are the fundamental concern of an investigator, but which also have other aspects that may be difficult to describe convincingly in a parametric fashion. We will refer to these as the class of semiparametric (SP) models, and we will explore a number of members of this class in the following sections. An alternative description would have been seminonparametric (SNP), depending upon where one wishes to place the emphasis, and both terms have appeared in the literature.

Section 5.2 explores the situation when the relationship governing yi can be written as a linear function of some variables x1i and a nonlinear function of other variables x2i, where the nonlinearity is unknown and it is the coefficients of x1i that are of primary interest. By adapting the estimators of Chapter 3, Robinson (1988a) has shown that it is possible to construct an estimator of the parameters of the linear part that exhibits √n consistency.

It took almost a decade to complete the work on this book. During this period Pagan moved from the University of Rochester to the Australian National University while Ullah moved to the University of California, Riverside from the University of Western Ontario. Such moves are always difficult and are made more so when the protagonists were trying to write about a field that at time seemed to be growing at an exponential rate. All of this disruption caused the book to be typed and retyped several times on two continents as well as ensured that the idea of completing it died and was reborn more than once. In the end our desire to see a completed text won out.

As anyone who performs applied work knows, many assumptions are made in coming up with answers. Mostly these involve assertions about the distributional features of variables and functional relations among them. It is not unusual for one to feel a little uncomfortable with some of these assumptions and to wonder what would happen if they could be relaxed. We were therefore rather intrigued to come across a literature in mathematical statistics that provided tools for doing some of the applied work with a minimal number of assumptions. This “nonparametrics” literature was found to be very useful in some of our early work on the risk premium, and that experience suggested that these methods could be quite useful more generally and that they should be more widely known to econometricians. Hence our objective became one of bringing this literature together in a systematic and integrated way within a single source.

The asymptotic theories of semiparametric and nonparametric econometrics heavily depend on several statistical concepts. These include probability theory, convergence, and conditioning. Accordingly, the objective of this appendix is to present results and definitions related to these concepts that are useful for the asymptotic theory covered in this book. To do so it is assumed that the reader has a basic knowledge of probability and statistics.

Probability Concepts

The axiomatic approach to probability theory is based on the concept of a probability space and an associated probability triple (Ω, F, P) defined below:

1. (i) The Sample Space, Ω

The set Ω is called the sample space. This contains the set of all possible outcomes (elementary events) of a random experiment E. For example, consider the experiment of tossing a fair coin twice. Then the sample space is

where ω1 = HT etc. Note that ωi, i = 1, …, 4, are the outcomes or elementary events.

1. (ii) The Class of Subsets of the Sample Space, F

Consider a nonempty collection of subsets (events) of Ω, F, satisfying the following two properties:

1. (a) If a subset F1 ∈ F then ∈ F, where is the complement of the subset F1 with respect to Ω and ∈ represents “belong to.”

2. By definition the subset or event F1 is the collection of elementary events in an experiment. For example, let F1 denote the event of obtaining one head in the example (A.1). F1 therefore consists of elementary events ω1, ω2, ω3, i.e., F1 = {ω1, ω2, ω3}.

3. […]

Introduction

In many data sets occurring in the social sciences the object of interest is whether a decision has been made or an action carried out. Examples would be whether a durable good is purchased, a bond is issued, or, perhaps, whether a corporation removes its CEO. Generally, an investigator is interested in what it is that precipitates the decision or action, and a range of possible influences (xi) is entertained. What makes this different from a regular regression problem is that the variable to be explained (yi) is qualitative rather than quantitative; or, treating the decision outcome as a random variable taking the value zero or unity (reflecting negative and positive decisions or actions), yi must be a discrete rather than a continuous random variable.

An enormous literature has built up on the types of models that can be employed to explain such data, a masterly exposition of which is Amemiya (1981). Economists imported many of these models from medical and biological statistics, and the traditional ways of estimating unknown parameters have revolved around formulating a suitable density function for the observed data. Section 7.2 reviews this literature, pointing to the fact that estimation has proceeded in either one of two ways. In the first, the discrete nature of the random variable enables one to formulate relationships as a heteroskedastic nonlinear regression. In the second, maximum likelihood methods are employed.

Introduction

In earlier chapters we have considered single-equation economic models where a dependent variable is a function of other variables. The basic assumption used in these models was that the right-hand side variables X are independent of the error term in the equation and therefore are not affected by the dependent variable Y. We did not allow for the possibility of a reverse causality, or feedback, where Y and X were jointly determined (endogenous) variables. There are a large number of economic models where some of the right-hand side variables are endogenous and these are referred to as a system of simultaneous equations models. Familiar examples are models of markets and the macro economy, as well as sets of factor demand equations. In all of these models, whether we are considering a single equation or a system as a whole, some of the right-hand side variables will be endogenous and hence correlated with the error term. For an overview of such system models, see Judge et al. (1985). In such systems direct application of the semiparametric and nonparametric regression estimators of Chapters 3 to 5 would lead to inconsistent estimators. Consequently, this chapter concentrates upon the modifications needed to those estimators when applied to random variables making up a system.

A primary division in the chapter is between those cases in which only a vector of parameters is unknown within the system and those when the system cannot be given a parametric form. In the first instance one wishes to devise an SP estimator of the parameters.

Nonparametric (NP) and semiparametric (SP) methods potentially offer a very high return to applied researchers, owing to their ability to adapt to many unknown features of the data. Moreover, it can now be said with some confidence that there exists an NP or SP technique capable of handling any situation an applied researcher would encounter. Such breadth may not be fully evident from this book in that its table of contents reflects the typical set of topics covered in a first-year graduate course in econometrics. But, just as such courses aim to establish principles in parametric econometrics that can be applied in a more general context, so too does this book attempt to perform the same function for the NP and SP approaches. Topics dealt with, such as kernel and series estimation, choice of window widths, bias adjustments, and efficiency bounds, recur in many areas that we have left untouched; examples include the literatures on broad areas such as panel data and hazard function estimation as well as some more restricted topics, such as frontier production functions and dynamic optimal decisions. A knowledge of the basic principles of NP and SP estimation sets a vital foundation for understanding this latter literature.

As should be apparent from the book there is a well-defined asymptotic theory for most estimators, and one can generally find one that is consistent and asymptotically normal, at least under particular assumptions. How useful this feature really is remains to be decided. Although there have been simulation studies of estimators, these tend to be in the original papers justifying them, and little comparative work has appeared.

Introduction

When individual data are used to estimate the parameters of a model, a persistent question that has to be addressed is whether the sample of individuals has been randomly selected. If their presence in the sample is a consequence of some choice made, and selection has been a function of some economic variables, then it is possible that a failure to recognize the endogeneity of inclusion in the sample can lead to inconsistent parameter estimators. The presence of such endogenous sample-selection bias has been recognized for many years, for example Roy (1951), but its implication for estimation theory was left until the papers by Gronau (1974) and Heckman (1974). Since that time it has been recognized that the problems caused by sample selection are pervasive in the analysis of microeconomic data and some adjustment for its effects needs to be made.

Conceptually, the effects of sample selection are simple; selection transforms a linear model with errors having a zero conditional mean and being homoskedastic into one where the errors have a nonzero conditional mean and are heteroskedastic. Somehow an adjustment needs to be performed to compensate for these changes to the error term. Parametric approaches, in particular that of Heckman (1976), have proceeded by finding expressions for the conditional means and variances of the error term under self-selection, and then making corrections to compensate for these effects. Section 8.2 outlines this estimator. It involves a two-step procedure that is simple to use and is available in many computer packages. An alternative, also discussed in this section, is to do maximum likelihood estimation, and this is also a standard option in programs such as LIMDEP.

Introduction

This chapter describes various methods of estimating the univariate density function of a random variable, closing with extensions to the multivariate case. Some motivation needs to be given for why we should be interested in density estimation at all. An important reason is that the techniques used in, and the complications arising from, the nonparametric estimation of densities recur many times in later chapters, and it pays to study them in a simplified setting first. But, apart from this pragmatic purpose, the need to estimate densities does arise in practice sufficiently often to make a study of this literature of interest in its own right.

Broadly, one can distinguish three areas in which the need to estimate densities arises. First, density estimates can be important in capturing the stylized facts that need explanation and for judging how well a potential model is likely to fit the data. For example, if it is known that the variable being examined has a density with fat tails, or strong peaks, any model of data corresponding to such a variable needs to be capable of generating a density with this characteristic. In other instances, one can efficiently learn about interrelationships between variables in large data sets from joint density estimates – a feature well illustrated in Deaton's (1989) work on rice subsidies in Thailand, in Marron and Schmitz's (1992) work on the U.K. income distribution, in Dinardo et al.'s (1996) study on the U.S. distribution of wages conditional on labor market institutions, and in Quah's (1997) cross-country analysis of the growth and convergence of economies.

Introduction

Perhaps the main use of econometric analysis is for counterfactual purposes. Economists are regularly asked to advise on what the impact upon employment or sales in an industry would be if a tax was imposed or lifted; what the likely effect upon unemployment would be if employment subsidies were introduced; or what might have happened if a different policy had been followed in some historical episode. Steps toward answering such questions involve the formulation of some theoretical relationship that is presumed to exist between the impacted and policy variables, the choice of some specific functional form relating these variables and any others needed for “control purposes,” and the selection of a formula that will provide the requisite information “accurately.”

Cast at this level of generality it is not clear how one should proceed, and in practice extra restrictions are placed upon the problem. Historically, the most prevalent has been that the relationship is a linear one, or that it be linear after some suitable transformation. Johnston (1984, pp. 61–74) for example discusses simple transformations such as logarithms or the inverse of a variable that might be employed. If yi is the target and xi the policy variable and E|yi| < ∞, the relation among them might be expressed in terms of the conditional moment E(yi|xi)= m(xi), where the dependence upon xi is made explicit. Now the central issue becomes the determination of the function m(·). The two functions mentioned above would sustain m(xi)=α + xiβ or perhaps m(xi)=α+β log xi.