Exercises in Econometrics, Vols. I and II

P.C.B. Phillips and M.R. Wickens
Philip Allan Publishers Ltd./Ballinger Publishing Co., 1978

Preface

This book provides a set of worked and unworked exercises to supplement the main textbook material in econometrics. it is written partly for students who are commencing undergraduate work in econometrics and who have some prior knowledge of statistics, and partly for students who are undertaking more advanced undergraduate and graduate instruction. We also hope that the book will prove useful to teachers by providing material for classroom discussion and to research workers by going a small way towards bridging the gap between the textbook and the rapidly expanding literature in this field.

The book attempts to supplement the existing econometric literature in two ways. First, in our experience, students very often find that the textbook they are assigned does not prepare them adequately to solve the sort of problems they face in examinations. They are not shown in a direct way how the theory can be used to solve such problems and they are not provided with similar questions which they can attempt on their own. Second, with the increasing output of econometric research, the coverage of the established econometrics textbooks is seen to be less complete. This problem is of particular concern to advanced students searching for a recent overview of the subject and to beginning research workers who often feel the need for a simpler introduction to the recent literature, more particularly in the technical areas.

In this book we provide exercises both on econometric theory and applied econometrics that cover most of the material in introductory and advanced econometric textbooks. In addition, we have constructed problems based on more recent research in order to introduce the reader to some new results. As far as possible, we have included more than one question on each topic. We have provided a fairly detailed solution to (at least) one of these questions; the other questions are intended to be supplementary and are left either for the reader to answer on his own or for classroom discussion.

The material within each chapter is organised in the same way. At the start of each chapter we have a brief introduction to the subject matter of the chapter and, where this is appropriate, an outline of the notation that is to be used. The next two sections contain all of the questions in the chapter: first, those questions for which worked solutions are provided; and, second, the supplementary questions. The final section in each chapter contains the solutions. The questions and solutions are separated to encourage the reader to attempt the exercises alone before looking at the solutions that are provided. The questions are arranged and the solutions are written to enable the reader to proceed naturally from one question to the next. Knowledge of topics which are treated later in the book is not usually required. Where appropriate, the solution includes some discussion of the relevant econometric theory; and we have made such discussions more detailed in those solutions which relate to more recent research. Naturally, when the theory has already been explained in an earlier solution, it is not repeated, but reference is made to that previous solution. Moreover, to assist those readers who will wish to use the book in conjunction with a textbook, we have given references, as far as possible, to the major textbooks.

Although the questions are arranged in order of their topic, they are not always arranged in order of difficulty. But, when a topic is first introduced, it is usually done through simpler questions; and we have sometimes used numerical exercises to clarify the manipulations that are involved in a particular procedure. Subsequent questions on the same topic then tend to be of increasing difficulty. Therefore, both beginning and advanced students should find questions of interest throughout the book. Some questions deal with various aspects of applied work in econometrics and these questions are included towards the end of each chapter.

The book is divided into two volumes and this has enabled us to cover a fairly wide range of topics in the exercises. Volume I covers most of the usual textbook material dealing with regression techniques and their application in econometrics. Chapter I of this volume is concerned with a number of methodological issues that arise in the use of econometric techniques including the underlying concept of an econometric model and fundamental problems such as aggregation, causality and the distinction between recursive and interdependent systems. We also consider the problem of extracting and comparing the sampling distribution of least squares and other estimators in simple bivariate models allowing for serial correlation and errors of measurement. A large part of the remainder of the volume concentrates on methods of estimation and inference that apply to models that are linear in both variables and parameters (Chapters 2 and 4). There are many exercises illustrating standard results on the linear regression model. Most of these are contained in Chapter 2 and a number of extensions dealing with autocorrelated errors, multicollinearity, missing observations and seasonal adjustment are given in Chapter 4. We also consider multiple equation models with across equation parameter restrictions (Chapter 3) and covariance and error component models (Chapter 4). In Chapter 5 we deal with nonlinear regression models and models with errors in variables.

Volume II contains two chapters and deals with models of simultaneous equations (Chapter 6) and dynamic models (Chapter 7). Chapter 6 contains a number of introductory questions on identifiability and the use of single equation estimators such as two stage least squares and limited information maximum likelihood. A number of numerical questions have been included, as in Volume I, to lay out the sequence of manipulations needed to compute estimates and confidence intervals. The remaining questions in Chapter 6 deal with more advanced topics such as identifiability in the presence of cross equation parameter restrictions, systems methods of estimation, some simpler finite sample theory (Nagar's moment approximations and Edgeworth approximations) and an introduction to nonlinear simultaneous equations models. Chapter 7 covers problems such as the identification of parameters in dynamic models with serially correlated errors, the consistent estimation of dynamic models with serially correlated errors, distributed lag models, continuous time models and models of markets in disequilibrium, as well as a number of applied questions. Much of the material in Chapter 7 has not yet appeared in textbooks and will, we hope, be of particular interest to advanced students and research workers.

Two major omissions from the book should also be noted: time series regression by spectral methods and the use of Bayesian methods in econometrics. These omissions were made with reluctance through pressure of space and time.

We are greatly indebted to the University of Auckland, the Australian National University and the Universities of Birmingham, Bristol, Essex, London and York for their permission to use questions which have appeared in their examinations. Where we have used questions from these examinations, or adapted them for our purpose here, this has been clearly indicated. Although it is impossible to give due credit to particular people in such cases, we acknowledge with special thanks that the following are among the authors of some of these questions: A.R. Bergstrom, R. Bowden, A. Chesher, J. Durbin, L.G. Godfrey, D.F. Hendry, G. Mizon, A.R. Pagan, J. Richmond, J.D. Sargan and K.F. Wallis. We are also grateful to J. Richmond, W. Barnett, V.B. Hall, E. Maasoumi and M. Prior for their comments on earlier versions of some of the questions and solutions. They are, of course, absolved from blame for any of the errors that remain.

Finally, it is with great pleasure that we thank Mrs Lucy Lowther, Mrs Sheila Ogden and Mrs Phyllis Pattenden for their skill, patience and good spirits in preparing the typescript.