Robust Methods and Asymptotic Theory in Nonlinear Econometrics
This Lecture Note deals with asymptotic properties, i.e. weak and strong consistency and asymptotic normality, of parameter estimators of nonlinear regression models and nonlinear structural equations under various assumptions on the distribution of the d
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PRELIMINARY MATHEMATICS For understanding linear econometrics, a good background in calculus,
statis,tics and linear algebra may be sufficient, but for nonlinear econometrics we need additional knowledge of abstract probability theory. Since this book is mainly written for econometricians and not for mathematicians, it is presumed that the reader is not completely familiar with the measure-theoretical approach of probability theory. We shall therefore review and explain this additional mathematics in the sections 2.1 and 2.2, in order to make this study (nearly) self-contained. Most of the material can be found in textbooks like Chung (1974) and Feller (1966), but the theorems 2.2.15 through 2.2.17 are our own elaborations of known results. Section 2.3 deals with uniform convergence of random functions on compact spaces. Especially the theorems 2.3.3 and 2.3.4, which are further elaborations of results of Jennrich (1969), are very important for us. Section 2.4 gives a brief review of some results on characteristic functions and stable distributions. Moreover, we state there the famous central limit theorem of Liapounov. Finally, section 2.5 is devoted to properties of (symmetric) unimodal distributions.
2.1
Random variables, independence, Borel measurable functions and mathematical expectation
2.1.1 Measure theoretical foundation of probability theory In this section we shall give a brief outline of the measure-theoretical foundation of probability theory. Dealing with convergence of random variables and uniform convergence of random functions, measure-theoretical arguments are unavoidable. These convergence concepts will playa key role in this study. The basic concept of probability theory is the probability space. This is a triple {n,~,p} consisting of: 1. an abstractnQn empty set
n,
called the sample space. We do not impose any
eonditions on this set.
2. a non empty collection 9 of subsets of
n, having the following two properties:
- if EE~, then ECEf, (2.1.1) C where E denotes the complement of the subset E with respect to n: EC =~E - i f E.E' peE) n=l P(
U E.)~
j:r1
J
rJ·: 1 P(E J.) ,
where all sets involved are members of function F( t) is right continuous:
JC.
Moreover, the distribution
F( t) = lim F( t+e: ),as is easily verified, e:"0
and it satisfies F(oo) = lim F(t) = 1, F(-oo) = lim F(t) = O.
*) We recall that random variables,- vectors and - functions are underlined.
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Furthermore, by F(t-) we denote
= lim
F ( t-)
dO
F( t- e:) ,
which clearly satisfies F(t-)
~
F(t).
A finite dimensional random vector can now be defined as a vector with random variables as components, where these random components are assumed to be defined on a common probability space. A complex random variable, z=x + iy ~=z(
is now a complex valued functions
. ) = x( .) + iy(.)
on n with real valued random variables
~
and y, i.e.
for any real number t. Next we shall construct a Borelfield
@of
subsets of Rk such that for every
set B E~ and any k-dimensional random vector x on a probability spac
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