# Category Introduction to the Mathematical and Statistical Foundations of Econometrics

## Introduction to the Mathematical and Statistical Foundations of Econometrics

This book is intended for use in a rigorous introductory Ph. D.-level course in econometrics or in a field course in econometric theory. It is based on lec­ture notes that I developed during the period 1997-2003 for the first-semester econometrics course “Introduction to Econometrics” in the core of the Ph. D. program in economics at the Pennsylvania State University. Initially, these lec­ture notes were written as a companion to Gallant’s (1997) textbook but have been developed gradually into an alternative textbook. Therefore, the topics that are covered in this book encompass those in Gallant’s book, but in much more depth. Moreover, to make the book also suitable for a field course in econometric theory, I have included various advanced topics as well...

## Outer Measure

Any subset A of [0,1] can always be completely covered by a finite or countably infinite union of sets in the algebra ^0: A c U°=j Aj, where Aj є ^0; hence, the “probability” of A is bounded from above by P(Aj). Taking the infimum of 2^= P (Aj) over all countable sequences of sets Aj є such that A c Ucj=1 Aj then yields the outer measure:

Definition 1.7: Let Ж0 be an algebra of subsets of The outer measure of an

arbitrary subset A of ^ is

Note that it is not required in (1.21) that Uj=1 Aj є ^c.

Because a union of sets Aj in an algebra can always be written as a union of disjoint sets in the algebra (see Theorem 1.4), we may without loss of generality assume that the infimum in (1.21) is taken over all disjoint sets Aj in such that A c Uj>=1 Aj. This implies that

If A є...

## Probability and Measure

1.1. The Texas Lotto

1.1.1. Introduction

Texans used to play the lotto by selecting six different numbers between 1 and 50, which cost \$1 for each combination.[1] Twice a week, on Wednesday and Saturday at 10 p. m., six ping-pong balls were released without replacement from a rotating plastic ball containing 50 ping-pong balls numbered 1 through 50. The winner of the jackpot (which has occasionally accumulated to 60 or more million dollars!) was the one who had all six drawn numbers correct, where the order in which the numbers were drawn did not matter. If these conditions were still being observed, what would the odds of winning by playing one set of six numbers only?

To answer this question, suppose first that the order of the numbers does matter...

## Lebesgue Integral

The Lebesgue measure gives rise to a generalization of the Riemann integral. Recall that the Riemann integral of a nonnegative function f (x) over a finite interval (a, b] is defined as

where the Im’s are intervals forming a finite partition of (a, b] – that is, they are disjoint and their union is (a, b] : (a, b] = unm=1 Im — and A.(Im) is the length of Im; hence, X(Im) is the Lebesgue measure of Im, and the supremum is taken over all finite partitions of (a, b]. Mimicking the definition of Riemann integral, the Lebesgue integral of a nonnegative function f (x) over a Borel set A can be defined as

j f (x)dx = sup£ (xinf f (x)^ X(Bm),

where now the Bm’s are Borel sets forming a finite partition of A and the supre – mum is taken over all such partitions.

If the function...