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Theory, Econometric 14, 1998, 369 374+ Printed in the United States of America+ TOPICS IN ADVANCED ECONOMETRICS: ESTIMATION , TESTING, AND SPECIFICATION OF CROSS-SECTION AND TIME SERIES MODELS Herman J. Bierens Cambridge University Press, 1994 O A H YO 0 N -JA E WH A N G Ewha University This book, Topics in Advanced Econometrics, is written primarily as a textbook for an advanced graduate econometrics course+ The topics covered include consistent model specification testing, unit roots and cointegration, and nonparametric regression estimation; they are mainly the topics in which Professor Bierens has made significant contributions to the literature over the last 15 years+ This book is unusual as a textbook in the sense that it treats both cross-sectional and time series (i+e+, both subscript i and t ) issues in econometrics at an advanced level+ Most of the results given, other than those available in standard econometrics or statistics textbooks, are drawn from the published work of the author+ The book is very useful because it puts together a number of important current issues that have been treated separately in the literature and presents them systematically using a well-organized set of statistical tools+ Another advantage of this book is that the materials given are almost self-contained, making this book suitable for self-tuition+ This book thus ideally suits students who need tools for independent research especially in the area of nonlinear and nonparametric models and time series analysis+ It will also be useful to more advanced researchers who are interested in a thorough understanding of some of the author s original and influential work in these areas+ This book is presented in a theorem-proof style, which enables the author to make a clear and rigorous presentation but also tends to make this book a bit too dry for use as a sole textbook+ There is also insufficient discussion of the results and motivation, and there are very few examples+ Most of the results discussed are first-order asymptotic; no higher-order, exact, or simulation results are given+ This book consists of 10 chapters+ Each chapter contains exercises that are directly related to the materials in the preceding text+ The content of this book can be classified into six parts: (1) the basic concepts and statistical tools required for the discussion of the remaining parts (Chapters 1 3); (2) nonlinear parametric regression analysis and maximum likelihood theory (Chapter 4); (3) tests of model specification (Chapter 5); (4) linear and nonlinear time series models (Chapters I thank the editor, Alex Maynard, and Sangbuhm Hahn for helpful comments+ Address correspondence to: Yoon-Jae Whang, Department of Economics, Ewha University, Seoul 120-750, Korea+ 1998 Cambridge University Press 0266-4666098 $9+50 369 370 BOOK REVIEWS 6 8); (5) unit roots and cointegration (Chapter 9); and (6) nonparametric kernel regression estimation (Chapter 10)+ I now briefly discuss each of these parts and comment on the results contained therein+ The first part, Chapters 1 3, deals with the background materials for the remaining sections+ Chapter 1 introduces measure-theoretic probability tools including the concept of probability spaces, Borel fields, independence of random variables, Borel measurable random functions, mathematical expectation with respect to a general measure, and characteristic functions+ Chapter 2 presents the asymptotic theory regarding convergence of sequences of random variables (and random functions indexed by a finite dimensional parameter), pointwise and uniform laws of large numbers, and central limit theorems+ The results given in this chapter apply mostly to independent and (non)identically distributed random variables (and functions)+ Chapter 3 gives the definition of a conditional expectation relative to a random vector and summarizes some of the basic properties of conditional expectations+ Much of the material in Chapters 1 3 is standard and could be found in, for example, Billingsley (1979) or Chung (1974), though the level of presentation is less technical than that of these works+ Proofs are given for most of the results presented, and they are generally quite rigorous, taking into account the thorny issues of measurability, and are also very instructive+ On the other hand, there are some unique results in this book that do not appear in standard textbooks; for example, the uniqueness of the conditional expectation is proved using the results of Bierens (1982) rather than using the usual Radon Nikodym theorem+ Also, there are few results in Chapters 1 3 that are not referred to in the later parts of this book; that is, the selection of the basic material is very efficient for the purpose of this book+ A consequence of this efficiency is that some concepts, such as outer measure and stochastic equicontinuity, that have much use in the recent nonlinear and semiparametric econometric literature (see Andrews, 1994a, 1994b; Newey, 1991) are not discussed+ The reader who needs a more extensive review of recent statistical tools may refer to other textbooks such as Davidson (1994)+ Some results seem to be unnecessarily specific; for example, to verify a multivariate convergence in distribution using a univariate convergence result, the Cramer Wold device might have been presented instead of Theorem 2+3+7, which applies only when the asymptotic distribution is normal+ Also, some results are derived using concepts defined later on; for example, the concept of null set is used in Chapter 1 (p+ 17) but is defined in Chapter 2 (Definition 2+1+2)+ Finally, Section 3+3 consists of material that is not basic and is more relevant to the specification testing results; it might have more appropriately been placed somewhere in Chapter 5+ The second part, Chapter 4, discusses the asymptotic properties of nonlinear lest squares and maximum likelihood estimators, along with those of the trinity of tests of (non)linear restrictions Wald, likelihood ratio, and Lagrange multiplier (LM) tests+ It starts with the definition and motivation of nonlinear regression models and proves weak and strong consistency and asymptotic normality of an BOOK REVIEWS 371 estimator that minimizes a smooth criterion function under a set of high-level assumptions+ It then gives more primitive assumptions for the special cases where the criterion functions are those of nonlinear least squares and maximum likelihood estimators, respectively+ The arguments used in the proofs are standard, but they are rigorous and instructive+ The results given, however, are restrictive as a result of the assumption of a twice continuously differentiable criterion function; it rules out a number of important estimators in econometrics such as, for example, least absolute deviation (LAD) estimator, censored LAD estimator (see Powell, 1984), and Huber M-estimators (see Huber, 1982)+ The results also do not cover most semiparametric estimators+ In these cases, there are general results for the asymptotic properties of extremum estimators that minimize a criterion function that is not smooth and0or depends on an estimate of an infinite-dimensional nuisance parameter+ Such results include Andrews (1994a) and Newey and McFadden (1994), among others+ On the other hand, no local power and optimality results of the trinity of the tests are discussed+ The third part, Chapter 5, is about tests of model specification+ The tests considered are those of the orthogonality condition that the conditional expectation of the error relative to the regressors in the nonlinear regression model equals zero, without a well-specified alternative+ Specifically, Hausman tests, M-tests of Newey (1985) and Tauchen (1985), and several versions of consistent tests of Bierens (1982, 1991) are considered, with a particular emphasis on the consistent tests, which is not surprising given the author s interest and expertise in this subject+ There is, however, little general discussion of specification tests; the general framework of Newey (1985) and Tauchen (1985) (hereafter N&T) is introduced only to motivate the consistent conditional M-tests discussed later+ A framework more general than that of N&T is also available; Whang and Andrews (1993), among others, extended the results of N&T by allowing the criterion function to be possibly nondifferentiable and indexed by an infinite-dimensional nuisance parameter+ The latter framework, therefore, justifies a number of specification tests in both parametric and nonparametric models+ The consistent specification tests (discussed on pp+ 96 109), on the other hand, assume that the probability limits of the estimator un of the parameter of interest u0 are the same under both the null and alternative hypotheses (see, e+g+, Assumption 5+2+3 and Theorem 5+2+4)+ This assumption does not hold in general, because un may be inconsistent for u0 if the model is misspecified, and it is not strictly necessary for the consistency of the tests+ (For a general treatment of this case, see Whang and Andrews, 1993+) The fourth part, Chapters 6 8, deals with some of the (stationary) time series issues+ Chapter 6 contains basic materials for Chapters 7 and 8 and extends some of the results in Chapters 2 and 3 to dependent data+ Notably, the of concept v-stability due to McLeish (1975) and Bierens (1983) is defined and assumed frequently to control the amount of dependence of functions of mixing random variables+ The related concept of functions of mixing processes or near epoch 372 BOOK REVIEWS dependency (see Billingsley, 1968; Gallant and White, 1988; Andrews, 1988) is not introduced, although it is common in the literature+ It is also surprising that this chapter does not contain the ergodic theorem+ The focus of Chapter 7 is modeling the conditional expectation of a time series based on its entire past history+ A linear AMRA(X) time series model is justified using the Wold decomposition (Theorem 7+2+1) and nonlinear ARMAX models are briefly discussed+ Most of the discussion in this chapter, however, is given to the ARMA memory index model suggested in the author s earlier work (see Bierens, 1988)+ The reader might be referred to Ploberger and Deistler (1988) and Sims (1988) for a further discussion of this model+ Chapter 8 starts with estimation of linear and nonlinear ARMAX models and proves consistency and asymptotic normality of such estimators+ It also presents the consistent test of Bierens (1987a) of parametric time series models that has the standard normal limit distribution+ The ARMA memory index modeling theory of Chapter 7 plays a key role in justifying this test+ The last section of this chapter discusses an autocorrelation test of the errors of nonlinear ARMAX models+ The test, however, is consistent against the alternative of autocorrelation of an order fixed a priori+ Given the author s particular interest in consistent tests, however, it is worth noting that tests of autocorrelation that are powerful against more general alternatives are also available in the literature+ Examples of such tests include Hong (1996) and Andrews and Ploberger (1996), among others+ The fifth part, Chapter 9, is on unit roots and cointegration+ The content of this chapter is now quite standard: the concept of weak convergence, the functional central limit theorem, the unit root tests of Dickey and Fuller (1979), Phillips (1987), and Phillips and Perron (1988), the concept of cointegration, error correction models, and inference of cointegrated system using the approach of Engle and Granger (1987) and Johansen (1988, 1991)+ It is surprising that this book has no discussion about spurious regression+ Given the explosion of the literature on nonstationary time series over the last decade, the coverage of this chapter is far from extensive+ In this respect, the reader might refer to Phillips (1995, 1997) and Stock (1994), among others, for a more up-to-date survey and to Hamilton (1994) for a textbook treatment that has more extensive results and is more oriented toward applied workers+ The last part, Chapter 10, reviews the asymptotic properties of the Nadaraya Watson kernel estimator of a nonparametric regression function+ Most of the results given are those in the influential work of Bierens (1987b), with an exception that a proof of strong consistency is now added (see Theorem 10+1+1)+ The asymptotic results given are mostly based on a fixed sequence of bandwidth parameter; no general discussion is provided on the theory of optimal datadependent bandwidth parameters, which is important for a good performance of nonparametric estimators+ A generalization of the consistency results given in this chapter was made by Andrews (1995), among others, who allowed datadependent bandwidth parameters and relaxed several other assumptions+ This chapter also does not discuss nonparametric estimators other than the kernel type; BOOK REVIEWS 373 for a more extensive review of nonparametric estimators, the reader might refer to H rdle (1990) and Eubank (1988)+ In conclusion, this book aims, and largely succeeds, at providing the student with tools useful for independent (theoretical) research in some recent econometric areas+ The choice of subjects is such that this book might not be appropriate as the stand-alone textbook for most advanced topics courses; many parts of this book, however, could certainly be effectively used as a supplementary text+ As a whole, this book is remarkable; it will be a very useful source for those that are interested in nonlinear, nonparametric regression and time series analysis+ REFERENCES Andrews, D+W+K+ (1988) Laws of large numbers for dependent nonidentically distributed random variables+ Econometric Theory 4, 458 467+ Andrews, D+W+K+ (1994a) Asymptotics for semiparametric econometric models via stochastic equicontinuity+ Econometrica 62, 43 72+ Andrews, D+W+K+ (1994b) Empirical process methods in econometrics+ In R+F+ Engle & D+ McFadden (eds+), Handbook of Econometrics, vol+ IV, pp+ 2247 2294+ New York: North-Holland+ Andrews, D+W+K+ (1995) Nonparametric kernel estimation for semiparametric models+ Econometric Theory 11, 560 596+ Andrews, D+W+K+ & W+ Ploberger (1996) Testing for serial correlation against an ARMA(1,1) process+ Journal of the American Statistical Association 91, 1331 1342+ Bierens, H+J+ (1982) Consistent model specification tests+ Journal of Econometrics 20, 105 134+ Bierens, H+J+ (1983) Uniform consistency of kernel estimators of a regression function under generalized conditions+ Journal of the American Statistical Association 77, 699 707+ Bierens, H+J+ (1987a) ARMAX model specification testing with an application to unemployment in the Netherlands+ Journal of Econometrics 35, 161 190+ Bierens, H+J+ (1987b) Kernel estimators of regression functions+ In T+F+ Bewley (ed+) Advances in Econometrics: Fifth World Congress, vol+ I, pp+ 99 144+ New York: Cambridge University Press+ Bierens, H+J+ (1988) ARMA memory index modeling of economic time series+ Econometric Theory 4, 35 59+ Bierens, H+J+ (1991) A consistent conditional moment test of functional form+ Econometrica 58, 1443 1458+ Billingsley, P+ (1968) Convergence of Probability Measures+ New York: Wiley+ Billingsley, P+ (1979) Probability and Measure+ New York: Wiley+ Chung, K+L+ (1974) A Course in Probability Theory+ New York: Academic Press+ Davidson, J+ (1994) Stochastic Limit Theory+ New York: Oxford University Press+ Dickey, D+A+ & W+A+ Fuller (1979) Distribution of the estimators for autoregressive time series with a unit root+ Journal of the American Statistical Association 74, 427 431+ Engle, R+F+ & C+W+J+ Granger (1987) Cointegration and error correction: Representation, estimation, and testing+ Econometrica 55, 251 276+ Eubank, R+ (1988) Spline Smoothing and Nonparametric Regression+ New York: Marcel Dekker+ Gallant, A+R+ & H+ White (1988) A Unified Theory of the Estimation and Inference for Nonlinear Dynamic Models+ New York: Basil Blackwell+ Hamilton, J+D+ (1994) Time Series Analysis+ Princeton: Princeton University Press+ H rdle, W+ (1990) Applied Nonparametric Regression+ Cambridge: Cambridge University Press+ Hong, Y+ (1996) Consistent testing for serial correlation of unknown form+ Econometrica 64, 837 864+ Huber, P+J+ (1982) Robust Statistics+ New York: Wiley+ Johansen, S+ (1988) Statistical analysis of cointegrated vectors+ Journal of Economic Dynamics and Control 12, 231 254+ 374 BOOK REVIEWS Johansen, S+ (1991) Estimation and hypothesis testing of cointegrated vectors in Gaussian vector autoregressive models+ Econometrica 59, 1551 1580+ McLeish, D+L+ (1975) A maximal inequality and dependent strong laws+ Annals of Probability 3, 829 839+ Newey, W+K+ (1985) Maximum likelihood specification testing and conditional moment tests+ Econometrica 53, 1047 1070+ Newey, W+K+ (1991) Uniform convergence in probability and stochastic equicontinuity+ Econometrica 59, 1161 1167+ Newey, W+K+ & D+ McFadden (1994) Large sample estimation and hypothesis testing+ In R+F+ Engle & D+ McFadden (eds+), Handbook of Econometrics, vol+ IV, pp+ 2111 2245+ New York: NorthHolland+ Phillips, P+C+B+ (1987) Time series regression with a unit root+ Econometrica 56, 1065 1083+ Phillips, P+C+B+ (1988) Testing for a unit root in time series regression+ Biometrika 75, 335 346+ Phillips, P+C+B+ (1995) Cointegration and unit roots: Recent books and themes for the future+ Journal of Applied Econometrics 10, 87 94+ Phillips, P+C+B+ (1997) Unit root tests+ In S+ Kotz, C+B+ Read, & D+L+ Banks (eds+), Encyclopedia of Statistical Science, update vol+ 1, pp+ 531 542+ New York: Wiley+ Phillips, P+C+B+ & P+ Perron (1988) Testing for a unit root in time series regression+ Biometrika 75, 335 346+ Ploberger, W+ & M+ Deistler (1988) Comment on ARMA memory index modeling of economic time series+ Econometric Theory 4, 60 61+ Powell, J+L+ (1984) Least absolute deviations estimation for the censored regression model+ Journal of Econometrics 25, 303 325+ Sims, C+A+ (1988) Comment on ARMA memory index modeling of economic time series+ Econometric Theory 4, 64 69+ Stock, J+H+ (1994) Unit roots, structural breaks, and trends+ In R+F+ Engle & D+ McFadden (eds+), Handbook of Econometrics, vol+ IV, pp+ 2739 2841+ New York: North-Holland+ Tauchen, G+ (1985) Diagnostic testing and evaluation of maximum likelihood models+ Journal of Econometrics 30, 415 443+ Whang, Y+-J+ & D+W+K+ Andrews (1993) Tests of model specification for parametric and semiparametric models+ Journal of Econometrics 57, 277 318+

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