Micro-Econometrics: Methods of Moments and Limited Dependent Variables

789 Pages·2010·5.17 MB·English

by  Myoung-jae Lee (auth.)

Micro-Econometrics
Second Edition
Myoung-jae Lee
Micro-Econometrics
Methods of Moments and Limited
Dependent Variables
Second Edition
123
Myoung-jaeLee
DepartmentofEconomics
KoreaUniversity
Anam-dong,Sungbuk-gu
Seoul,Korea136-701
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ISBN978-0-387-95376-2 e-ISBN978-0-387-68841-1
DOI10.1007/b60971
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To my “Little Women”
Hyun-joo, Hyun and Young,
full of life, love and the cool
PREFACE
WhenIwrotethebookMethodsofMomentsandSemiparametricEcono-
metrics for Limited Dependent Variable Models published from Springer in
1996, my motivation was clear: there was no book available to convey the
latest messages in micro-econometrics. The messages were that most econo-
metric estimators can be viewed as method-of-moment estimators and that
inferences for models with limited dependent variables (LDV) can be done
without going fully parametric.
Time has passed and there are now several books available for the same
purpose.Thesedays,methodsofmomentsarethemainstayineconometrics,
not just in micro-, but also in macro-econometrics. Many papers have been
published for semiparametric methods and LDV models. I, myself, learned
much over the years since 1996, so much so that my own view on what
should be taught, and how, has changed much. Particularly, my exposure to
the“sampleselection”and“treatmenteffect”literaturehaschangedtheway
Ilookateconometricsnow.WhenIsetouttowritethesecondeditionofthe
1996 book, these changes prompted me to re-title, reorganize, and re-focus
the book.
This book, or the second edition of the book from Springer in 1996,
differs greatly from the 1996 book in three aspects. First, I tried to write
the book more as a textbook than as a monograph, so that the book can
be used as a first year textbook in graduate econometrics courses. Second,
differently from the 1996 book, many empirical examples have been added
and estimators that work well in practice are given more coverage than the
others. Third, the literature have been updated, or at least, the relevant new
papers have been cited so that the reader can consult them if he/she desires
so. These changes resulted in more than doubling the book length.
Onemayclassifyeconometricsintotwo:micro-econometricsdealingwith
individualdata,andmacro-econometricsdealingwith(aggregate)time-series
data. Micro-econometrics may befurtherclassified into“cross-section micro-
econometrics” and “panel-data micro- econometrics”; an analogous classi-
fication can be done for macro-econometrics. In 2002, I published a book
entitled Panel Data Econometrics: Methods of Moments and Limited De-
pendent Variables from Academic Press; for me, this leaves “cross-section
micro-econometrics”tocoverinmicro-econometrics,whichiswhatthisbook
is mainly about, although panel data models are also examined occasionally.
One of the “buzz word” in micro-econometrics these days is “treatment
effect.” This topic has been studied extensively in epidemiology and medical
science as well as in some social science disciplines. Treatment effect frame-
work is, in fact, nothing but “switching regression” in micro-econometrics
thatwaspopularsometimeago:theeffectofabinarytreatmentisofinterest,
and if there is any treatment effect, we get to see two different (i.e., switch-
ing)regimesdependingonthetreatment.In2005,Ipublishedabookentitled
vii
viii Preface
Micro-EconometricsforPolicy,Program,andTreatmentEffects fromOxford
University Press. Hence, despite its prominence, treatment effect will be dis-
cussed at minimum, if at all, in this book.
Closelyrelatedtotreatmenteffectis“sampleselection”wherethesample
at hand comes from only one regime while our interest is on both or on the
“averaged” regime. I am planning to write a book on sample selection in
the near future, and thus the coverage of sample selection in this book will
not be extensive. Sample selection is a fairly well-confined topic in micro-
econometrics, and the non-extensive coverage in this book would not distort
the overall picture of micro-econometrics.
The book consists of three parts in the main text, with each part hav-
ing a number of chapters, and three appendices. The first part (Chapters
1 and 2) in the main text is for methods of moments for linear models, the
second part (Chapter 3–6) is for nonlinear models and parametric methods
for LDV models, and the third part (Chapter 7–9) is for semiparametric and
nonparametric methods. Appendix I contains one section on mathematical
andstatisticalbackgrounds,andeightmoresectionsofappendicesforChap-
ters 2–9. Appendix II has further supporting materials. Both appendices are
technical, digressive or tentative, and Appendix II is more so than Appendix
I in this regard. Most things the reader may feel missing while reading the
main text can be found in the appendices, although what is available in the
appendix is not specifically mentioned in the main text. Some interesting
topicsareputintheappendicestoavoidlengtheningthemaintexttoomuch
and thus discouraging the reader prematurely.
Appendix III provides some GAUSS programs. I tried to select only
simpleandnumericallystable(i.e.,reliable)programs.Allprogramsusesim-
ulated data. Although I wrote this book so that the readers can write their
own programs, STATA commands are occasionally referred to, in case the
reader may think that the procedure under consideration is difficult to im-
plement and not available in ready-made econometric packages.
As in my other books, small sample issues and matters of “second order
importance” will not be discussed much, because econometricians will be
making mistakes of large magnitude, if any. With this being the case, paying
attention to small sample improvement and low-order precision seems not so
meaningful. Of course, ideally, one should avoid mistakes of both large and
smallmagnitudes,butsayingthatwouldbeignoringeconometricians’budget
and time constraints; politicians might feel comfortable saying that, but not
most economists.
Some glaring omissions in this book’s coverage include weak instru-
ments, factor analysis, stochastic frontiers, measurement errors (or errors
invariables),semiparametricefficiency,auction-relatedeconometrics,spatial
dependence, demand system analysis, sampling, and missing data and impu-
tation which are closely related to sample selection. Also it would be nicer
to have more detailed coverages of duration analysis, multinomial choices,
“bandwidth-dependent” semiparametric methods for LDV models, and so
on. All of these require much more time and efforts on my side, and cover-
Preface ix
ing them would mean this book not seeing the daylight for another several
years—perhaps next time.
The target audience of this book are graduate students and researchers.
The entire book may be covered in two to four semesters—one semester
for each part plus the appendices—but covering essential topics selectively
while omitting the optional starred topics (and some others) may be done
in two semesters. Most estimators and tests have been tried with real or
simulateddataexceptsomeintheappendices.Thereaderwillfindintuitions
forhowestimators/testsworkaswellasvarioustipsforhand-onexperiences.
Aboutempiricalexamplesinthisbook,itwouldbeidealtochoosethe“best”
empirical examples for a given estimator/test. But, unfortunately, my time
constraintspreventedmefromdoingthat;rather,mostexampleswerechosen
moreorless“randomly”—i.e.,Ihappenedtoruninto,orjustremember,the
example when the topic was written about.
In this book, theoretically oriented readers will find an overview on
micro-econometrics, and applied researchers will find helpful informations
on how to apply micro-econometric techniques; there will be something for
everybody—atleastthatiswhatIhope.Thereadermayalsowanttoconsult
other good books with micro-econometric focus such as Wooldridge (2002),
Cameron and Trivedi (2005, 2009), and Green (2007). Compared with these
books,thetheoreticalcoverageofthisbookisrelativelyatahigherlevelwith
a semi-(non) parametric bent.
I am grateful to the Springer Statistics Editor John Kimmel for his
patience while this project was dragging on for eight-plus years after the
initialtalk.Iamalsogratefultotheanonymousreviewersfortheircomments
whichledtosubstantialimprovementsandre-organizationsofthebook.Juao˜
Santos-Silvaprovidedvaluablefeedbacksonmanyoccasions,andJing-young
Choi helped me much by proof-reading most parts of the book. Also Sang-
hyeokLee,Jong-hunChoiandYoung-minJuproof-readvariouschaptersand
gave me comments. I should admit, however, that I could not incorporate all
the comments/feedbacks due to the book-length/time constraints, and also
duetothefactthatmakingtoomanychangesnearthefinalstageisarather
risky thing to do. Without implicating any reviewer or anybody for that
matter, I will be solely responsible for any errors in the book.
REMARKS ON EXPRESSIONS AND NOTATIONS
Manyacronymswillbeusedinlower/uppercaseletters:“rv”forrandom
variable,“cdf”or“df”for(cumulative)distributionfunction,“rhs”forright-
hand side, “lhs” for left-hand side, “dof” for degree of freedom, “wrt” for
“with respect to”, “cov” for covariance, “cor” for correlation, and so on.
For matrix A, “p.d.” (n.d.) stands for “positive definite” (negative defi-
nite)and“p.s.d.”(n.s.d.)standsfor“positivesemidefinite”(negativesemidef-
inite). tr(A) denotes its trace, and |A| (or det(A)) denotes its determinant;
||A|| will be then the absolute value of the determinant. But sometimes, |A|
or ||A|| may mean the matrix norm{tr(A(cid:2)A)}1/2. For matrices a ,...,a ,
1 M
“diag(a ,...,a )” is the block diagonal matrix with a ,...,a along the di-
1 M 1 M
agonal.
The notation “→p” means “convergence in probability”, and “→ae” or
“→as” means “convergence almost surely (a.s.)” or “convergence almost ev-
erywhere(a.e.)”. The notation “(cid:2)”denotes “convergence indistribution (or
inlaw)”, and“∼”denotesthedistributionofarv;e.g.,“x∼N(0,1)” means
thatxfollowsthestandardnormaldistribution.Wewillalsouse“x∼(μ,σ2)”
tomeanthatE(x)=μ andV(x)=σ 2 withoutitsdistributionspecified.Fre-
quently, φ and Φ will be used to denote the N(0,1) density and distribution
function, respectively. Uniform distribution on [a,b] is denoted as U[a,b], ex-
ponential distribution with parameter θ is denoted as Expo(θ), and Poission
distribution with parameter λ is denoted as Poi(λ). Other distributions are
often denoted analogously; e.g., Weibull(α,θ).
In many textbooks, an uppercase letter and its lowercase letter are used
to denote, respectively, a rv and its realized value. In this case, for a rv
Y with distr(cid:2)ibution function F(y) ≡ P(Y ≤ y) and density f(y), we have
E{g(Y)}= g(y)f(y)dy for a function g(·). But this distinction between Y
and y will not be followed in most parts of this book, because upper case
letters are frequently used to denote matrices in this book. A downside of
n(cid:2)ot following the uppercase/lowercase convention can(cid:2) be seen in E{g(y)} =
g(y)f(y)dy where y(cid:2)in E{g(y)} is a rv but y in g(y)f(y)dy is just an
integration dummy— g(z)f(z)dz would mean just the same thing. In most
cases, it will be clear from the context whether y is a rv(cid:2)or not. But if
necessary,toavoidthiskindofconfusion,wemayalsowrite g(y )f(y )dy .
o o o
Also from the given context, it will be clear whether F(y) means P(Y ≤ y)
with Y random and y fixed, or F(·) taken on a rv y; if the meaning is not
clear, we may write F(y ) forP (y ≤y ) where y is a rv.
o o
TheconditionaldistributionfunctionforP(y ≤y |x=x )isdenotedas
o o
Fy|x(yo|xo), Fy|x=xo(yo), or Fy|xo(yo). But if not interested in any particular
values of yo and xo, we may just write Fy|x(y|x), Fy|x(y), or F(y|x). The
correspondingdensityfunctionwillbedenotedasfy|x(y|x),fy|x(y),orf(y|x),
respectively. In these cases, y and x in parentheses are not rv’s, but stand
for some values that those rv’s can take (just to indicate that F and f are
for those rv’s). Ex(·) and Ey|x(·) denote that the expected value is taken
x
Remarks on Expressions and Notations xi
for x and y|x, respectively. Med(y|x) and Mode(y|x) denote the conditional
medianandmode,respectively.Q (y|x)(orq (y|x))denotestheconditional
α α
αth quantile. The independence between two random vectors x and y is
denoted as x(cid:6)y, and the conditional independence between x and y given
z is denoted as x(cid:6)y|z. The ‘indicator function’ 1[·] is defined as 1[A] = 1
if A holds and 0 otherwise. The ‘sign function’ sgn(a) ≡ 2×1[a ≥ 0]−1
denotes the sign of a: sgn(a) = 1 if a ≥ 0 and−1 ifa <0. Sometimes the
sign function may be defined such that it becomes 0 or −1 when a=0.
WhenE(z)isused,itisimplicitlyassumedthatE(z)<∞;whenE−1(z)
is used (a shorthand for {E(z)}−1), it is also assumed that E(z) is invert-
ible. Most vectors in this book are column vectors, and for a m×1 vector
function g(b) where b has dimension k×1, its first derivative matrix g has
b
dimension k ×m; in comparison, gb(cid:2) ≡ gb(cid:2) is a m×k matrix. Rk denotes
the k-dimensional Euclidean space, and R = R1 denotes the real space; |·|
denotes the Euclidean norm in most cases. For a function g(·), “increasing”
means “non-decreasing,” and “strictly increasing” means increasing without
the equality; the analogous usages hold “decreasing” and “strictly decreas-
ing.” L(y|x) and L (y|x) denote the linear projection E(yx(cid:2))E−1(xx(cid:2))x and
N
itsestimator,respectively;incomparisontoL(y|x),E(y|x)maybecalledthe
(nonlinear) projection and E (y|x) denotes an estimator for E(y|x).
N
Since this book is mainly for cross-section micro-econometrics, unless
otherwise noted, we will assume that data, say z ,z ,...,z from N subjects
1 2 N
(individuals), are (independent and identically distributed) from a common
distribution.Theindividualswillbeindexedbyi=1,...,N,andwewilloften
drop the subscript i to write z just as z, if not interested in any particular
i
subject.Hence,whenz =(z ,...,z )(cid:2) is an“m-vector”(i.e., m×1vector),
i i1 im
its mth component z may be denoted as z with i omitted.
im m