Introduction

A great deal of empirical microeconometrics research uses linear regression and its various extensions. Before moving to nonlinear models, the emphasis of this book, we provide a summary of some important results for the single-equation linear regression model with cross-section data. Several different estimators in the linear regression model are presented.

Ordinary least-squares (OLS) estimation is especially popular. For typical microeconometric cross-section data the model error terms are likely to be heteroskedastic. Then statistical inference should be robust to heteroskedastic errors and efficiency gains are possible by use of weighted rather than ordinary least squares.

The OLS estimator minimizes the sum of squared residuals. One alternative is to minimize the sum of the absolute value of residuals, leading to the least absolute deviations estimator. This estimator is also presented, along with extension to quantile regression.

Various model misspecifications can lead to inconsistency of least-squares estimators. In such cases inference about economically interesting parameters may require more advanced procedures and these are pursued at considerable length and depth elsewhere in the book. One commonly used procedure is instrumental variables regression. The current chapter provides an introductory treatment of this important method and additionally addresses the complication of weak instruments.

Section 4.2 provides a definition of regression and presents various loss functions that lead to different estimators for the regression function. An example is introduced in Section 4.3.

Introduction

A nonlinear estimator is one that is a nonlinear function of the dependent variable. Most estimators used in microeconometrics, aside from the OLS and IV estimators in the linear regression model presented in Chapter 4, are nonlinear estimators. Nonlinearity can arise in many ways. The conditional mean may be nonlinear in parameters. The loss function may lead to a nonlinear estimator even if the conditional mean is linear in parameters. Censoring and truncation also lead to nonlinear estimators even if the original model has conditional mean that is linear in parameters.

Here we present the essential statistical inference results for nonlinear estimation. Very limited small-sample results are available for nonlinear estimators. Statistical inference is instead based on asymptotic theory that is applicable for large samples. The estimators commonly used in microeconometrics are consistent and asymptotically normal.

The asymptotic theory entails two major departures from the treatment of the linear regression model given in an introductory graduate course. First, alternative methods of proof are needed since there is no direct formula for most nonlinear estimators. Second, the asymptotic distribution is generally obtained under the weakest distributional assumptions possible. This departure was introduced in Section 4.4 to permit heteroskedasticity-robust inference for the OLS estimator. Under such weaker assumptions the default standard errors reported by a simple regression program are invalid. Some care is needed, however, as these weaker assumptions can lead to inconsistency of the estimator itself, a much more fundamental problem.

Introduction

In this chapter we consider two closely related topics: regression when the dependent variable of interest is incompletely observed and regression when the dependent variable is completely observed but is observed in a selected sample that is not representative of the population. This includes limited dependent variable models, latent variable models, generalized Tobit models, and selection models.

All these models share the common feature that even in the simplest case of population conditional mean linear in regressors, OLS regression leads to inconsistent parameter estimates because the sample is not representative of the population. Alternative estimation procedures, most relying on strong distributional assumptions, are necessary to ensure consistent parameter estimation.

Leading causes of incompletely observed data are truncation and censoring. For truncated data some observations on both the dependent variable and regressors are lost. For example, income may be the dependent variable and only low-income people are included in the sample. For censored data information on the dependent variable is lost, but not data on the regressors. For example, people of all income levels may be included in the sample, but for confidentiality reasons the income of high-income people may be top-coded and reported only as exceeding, say, $100,000 per year. Truncation entails greater information loss than does censoring. A leading example of truncation and censoring is the Tobit model, named after Tobin (1958), who considered linear regression under normality.

Introduction

This book provides a detailed treatment of microeconometric analysis, the analysis of individual-level data on the economic behavior of individuals or firms. A broader definition would also include grouped data. Usually regression methods are applied to cross-section or panel data.

Analysis of individual data has a long history. Ernst Engel (1857) was among the earliest quantitative investigators of household budgets. Allen and Bowley (1935), Houthakker (1957), and Prais and Houthakker (1955) made important contributions following the same research and modeling tradition. Other landmark studies that were also influential in stimulating the development of microeconometrics, even though they did not always use individual-level information, include those by Marschak and Andrews (1944) in production theory and by Wold and Jureen (1953), Stone (1953), and Tobin (1958) in consumer demand.

As important as the above earlier cited work is on household budgets and demand analysis, the material covered in this book has stronger connections with the work on discrete choice analysis and censored and truncated variable models that saw their first serious econometric applications in the work of McFadden (1973, 1984) and Heckman (1974, 1979), respectively. These works involved a major departure from the overwhelming reliance on linear models that characterized earlier work. Subsequently, they have led to significant methodological innovations in econometrics. Among the earlier textbook-level treatments of this material (and more) are the works of Maddala (1983) and Amemiya (1985).

Part 1 emphasized that microeconometric models are frequently nonlinear models estimated using large and heterogeneous data sets drawn from surveys that are complex and subject to a variety of sampling biases. A realistic depiction of the economic phenomena in such settings often requires the use of models for which estimation and subsequent statistical inference are difficult. Advances in computing hardware and software now make it feasible to tackle such tasks. Part 3 presents modern, computerintensive, simulation-based methods of estimation and inference that mitigate some of these difficulties. The background required to cover this material varies somewhat with the chapter, but the essential base is least squares and maximum likelihood estimation.

Chapter 11 presents bootstrap methods for statistical inference. These methods have the attraction of providing a simple way to obtain standard errors when the formulae from asymptotic theory are complex, as is the case, for example, for some two-step estimators. Furthermore, if implemented appropriately, a bootstrap can lead to a more refined asymptotic theory that may then lead to better statistical inference in small samples.

Chapter 12 presents simulation-based estimation methods. These methods permit estimation in situations where standard computational methods may not permit calculation of an estimator, because of the presence of an integral over a probability distribution that leads to no closed-form solution.

Chapter 13 surveys Bayesian methods that provide an approach to estimation and inference that is quite different from the classical approach used in other chapters of this book.

Example 1

In Chapter 7, section 7.4 we learnt the basics of participles:

Luke 18.22: ἀκουσας δε ὁ Ἰησους εἰπεν αὐτῳ Ἐτι ἑν σοι λειπει

1. – having heard Jesus said to him, ‘One thing still remains for you…’

2. – when Jesus heard (this) he said to him, ‘You still lack one thing…’

ἀκουσας is a participle from ἀκουω. It agrees with ὀ Ἰησους (nom. masc. sing.), which tells us that it is Jesus who is doing the hearing. It is in the Aorist to convey the ‘sequence’ meaning (present would be ‘simultaneous’), i.e the action in the participle is happening before that in the main verb: first Jesus hears, then he speaks.

Example 2

Luke 7.9: ἀκουσας δε ταυτα ὁ Ἰησους ἐθαυμασεν αὐτον.

1. – when he heard these things Jesus was amazed at him.

Once again, ἀκουσας is a participle, but this time it has its own object ταυτα – these things.

Thus the participle has some of the features of a verb, and some of an adjective (grammarians call it a ‘verbal adjective’).

Like adjectives:

Like verbs:

Up to now, we have only dealt with participles that are in the nominative – qualifying the subject. However, participles can qualify any noun.

KEY GRAMMAR

Participles must agree with the noun they qualify in gender, case and number

KEY GRAMMAR

Participles have a tense (Present or Aorist) and may have an object

This book has a single aim:

To help you learn enough Greek to read the New Testament.

This might seem obvious for a book entitled The Elements of New Testament Greek. However, there are many books designed for those beginning to study New Testament Greek that do not focus exclusively on this aim. The point will become clearer if I highlight certain things that this book does not aim at.

This book does not set out to present my understanding of New Testament Greek. It is a book for you, not for me. If I want to impress my colleagues with my Greek expertise, I will do that elsewhere. You deserve a book written to help you. In the same way it is not a ‘Greek Grammar’, as if my work was merely to set out Greek grammar, and it is then up to you to understand it and learn it. This is a textbook, written to help you in the process of learning.

This book does not try to teach you Christianity. It assumes that you want to read the New Testament in Greek in order to understand the New Testament better. For many the reason for wanting to understand it better will be a religious motivation, and that is great – I personally share that motivation. But for others it will be different. You may be unsure about Christianity, or indeed negative towards it.