Category Introduction to the Mathematical and Statistical Foundations of Econometrics

Distributions and Transformations

This chapter reviews the most important univariate distributions and shows how to derive their expectations, variances, moment-generating functions (if they exist), and characteristic functions. Many distributions arise as transformations of random variables or vectors. Therefore, the problem of how the distribution of Y = g(X) is related to the distribution of X for a Borel-measure function or mapping g(x) is also addressed.

4.1. Discrete Distributions

In Chapter 1 I introduced three “natural” discrete distributions, namely, the hypergeometric, binomial, and Poisson distributions. The first two are natural in the sense that they arise from the way the random sample involved is drawn, and the last is natural because it is a limit of the binomial distribution...

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Asymptotic Normality of M-Estimators

This section sets forth conditions for the asymptotic normality ofM-estimators in addition to the conditions for consistency An estimator в of a parameter в0 є Km is asymptotically normally distributed if an increasing sequence of positive numbers an and a positive semidefinite m x m matrix £ exist such that an(в — в0) ^d Nm [0, £]. Usually, an = – Jn, but there are exceptions to this rule.

Asymptotic normality is fundamental for econometrics. Most of the econo­metric tests rely on it. Moreover, the proof of the asymptotic normality theorem in this section also nicely illustrates the usefulness of the main results in this chapter.

Given that the data are a random sample, we only need a few additional conditions over those of Theorems 6.10 and 6.11:

Theorem 6...

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Inverse of a Matrix in Terms of Cofactors

Theorem I.31 now enables us to write the inverse of a matrix A in terms of cofactors and the determinant as follows. Define

image874 Подпись: cofn,1( A)^ cofn,n(A) Подпись: (I.57)

Definition I.20: The matrix

is called the adjoint matrix of A.

Note that the adjoint matrix is the transpose of the matrix of cofactors with typical (i, j)’s element cof;,j(A). Next, observe from Theorem I.31 that det( A) — Jfk—1 ai, k cofijk (A) is just the diagonal element i of A ■ Aadjoint. More­over, suppose that row j of A is replaced by row i, and call this matrix B. This has no effect on cofj, k(A), but ^nk—1 ai, kcofj, k(A) — Yl—1 a, kcof, k(B) is now the determinant of B. Because the rows of B are linear dependent, det(B) — 0. Thus, we have

ТИ—1 ai, k cofj, k (A) — det( A) if i — j,

— 0 if i — j;

hence,

Theorem I...

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The Uniform Distribution and Its Relation to the Standard Normal Distribution

As we have seen before in Chapter 1, the uniform [0,1] distribution has density

f (x) = 1 for 0 < x < 1, f (x) = 0 elsewhere.

image267
More generally, the uniform [a, b] distribution (denoted by U[a, b]) has density

where U and U2 are independent U[0, 1] distributed. Then Xj and X2 are

independent, standard normally distributed. This method is called the Box – Muller algorithm.

2.7. The Gamma Distribution

Подпись: g(x) Подпись: x“ 1exp(-x/в) Г (а) ва Подпись: x > 0, a > 0, в > 0.

The x2 distribution is a special case of a Gamma distribution. The density of the Gamma distribution is

This distribution is denoted by Г (а, в). Thus, the x2 distribution is a Gamma distribution with a = n/2 and в = 2.

The Gamma distribution has moment-generating function

тг(а, в)(0 = [1 – ві]-a, t < 1/в (4.44)

and characteristic function (рГ(а, в'() = [1 – в • i • t]—a ...

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Maximum Likelihood Theory

8.1. Introduction

Consider a random sample Z ь…, Zn from a ^-variate distribution with density f (z0), where в0 є © c 1” is an unknown parameter vector with © a given parameter space. As is well known, owing to the independence of the Zj’s, the joint density function of the random vector Z = (ZT, zj )T is the product of the marginal densities, ПП=1 f (zj в0). The likelihood function in this case is defined as this joint density with the nonrandom arguments zj replaced by the corresponding random vectors Zj, and в 0 by в:

n

L n (в) = П f (Zjв). (8.1)

j=1

The maximum likelihood (ML) estimator of в0 is now в = argmaxeє©Ln (в), or equivalently,

в = argmaxln(L n (в)), (8.2)

в є©

where “argmax” stands for the argument for which the function involved takes its maximum...

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The Mean Value Theorem

Consider a differentiable real function f (x) displayed as the curved line in Figure II.1. We can always find a point c in the interval [a, b] such that the slope of f (x) at x = c, which is equal to the derivative f(c), is the same as the slope of the straight line connecting the points (a, f(a)) and (b, f (b)) simply by shifting the latter line parallel to the point where it be­comes tangent to f (x). The slope of this straight line through the points (a, f (a)) and (b, f (b)) is (f (b) – f (a))/(b – a). Thus, at x = c we have f”(c) = (f (b) – f (a))/(b – a), or equivalently, f (b) = f (a) + (b – a)f(c). This easy result is called the mean value theorem...

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