Category Introduction to the Mathematical and Statistical Foundations of Econometrics

Applications to Statistical Inference under Normality

5.6.1. Estimation

Statistical inference is concerned with parameter estimation and parameter in­ference. The latter will be discussed next in this section.

In a broad sense, an estimator of a parameter is a function of the data that serves as an approximation of the parameter involved. For example, if X1, X2,Xn is a random sample from the N(д, a2)-distribution, then the sample mean X = (1 /n) Хд = Xj may serve as an estimator of the un­known parameter д (the population mean). More formally, given a data set {X1, X2,.Xn } for which the joint distribution function depends on an un­known parameter (vector) в, an estimator of в is a Borel-measurable function 6 = gn (X1,.Xn) of the data that serves as an approximation of в ...

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Asymptotic Efficiency of the ML Estimator

The ML estimation approach is a special case of the M-estimation approach discussed in Chapter 6. However, the position of the ML estimator among the M – estimators is a special one, namely, the ML estimator is, under some regularity conditions, asymptotically efficient.

To explain and prove asymptotic efficiency, let

n

в = argmax(1/n) V’ g( 1j, в) (8.43)

ве© j=1

be an M-estimator of

Подпись: (8.44)§o = argmax E[g( 11, §)],

в ев

where again 11,…, 1n is a random sample from a k-variate, absolutely con­tinuous distribution with density f (z|§0), and © c Rm is the parameter space. In Chapter 6, I have set forth conditions such that

4П(§ _ §0) ^dNm [0, A_1 BA-1], (8.45)

Подпись: AE Подпись: d 2 g( Z 1,§0) d§0d§0T image709 Подпись: (8.46)

where

and

B = E [(dg( 11, §0)/d§0T) (dg( 11, §0>/9§0>]

Подпись: (8.47)= J {dg(z, §0)^) (dg(z, §0)/d§0) f (z| §0)dz.

R...

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Conditional Probability Measures and Conditional Independence

The notion of a probability measure relative to a sub-a – algebra can be defined as in Definition 3.1 using the conditional expectation of an indicator function:

Definition 3.2: Let {2, X, P} be a probability space, and let X0 C X be a a-algebra. Then for any set A in IX, P(A|X0) = E[IA|X0], where IA(ш) = I(ш e A).

In the sequel I will use the shorthand notation P(Y e B|X) to indicate the conditional probability P({ш e 2 : Y(ш) e B}|XX), where B is a Borel set and XX is the a – algebra generated by X, and P (Y e B|X0) to indicate P ({ш e 2 : Y(ш) e B}|X0) for any sub-a-algebra X0 of X. The event Y e B involved may be replaced by any equivalent expression.

Recalling the notion of independence of sets and random variables, vectors, or both (see Chapter 1), we can now define conditional...

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Convergence in Distribution

Let Xn be a sequence of random variables (or vectors) with distribution functions Fn (x), and let X be a random variable (or conformable random vector) with distribution function F(x).

Definition 6.6: We say that Xn converges to X in distribution (denoted by Xn ^d X) if limn^XlFn(x) = F(x) pointwise in x – possibly except in the dis­continuity points of F(x).

Alternative notation: If X has a particular distribution, for example N(0, 1), then Xn ^d X is also denoted by Xn ^d N(0, 1).

The reason for excluding discontinuity points of F(x) in the definition of convergence in distribution is that limn^TO Fn(x) may not be right-continuous in these discontinuity points. For example, let Xn = X + 1/n. Then Fn(x) = F(x – 1/n)...

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Projections, Projection Matrices, and Idempotent Matrices

Consider the following problem: Which point on the line through the origin and point a in Figure I.3 is the closest to point b? The answer is point p in Figure

I.4. The line through b and p is perpendicular to the subspace spanned by a, and therefore the distance between b and any other point in this subspace is larger than the distance between b and p. Point p is called the projection of b on the subspace spanned by a. To find p, let p = c ■ a, where c is a scalar. The distance between b and p is now ||b – c ■ a\; consequently, the problem is to find the scalar c that minimizes this distance. Because ||b – c ■ a\ is minimal if and only if

||b — c ■ a||2 = (b — c ■ a)T(b — c ■ a) = bTb — 2c ■ aTb + c2aTa

is minimal, the answer is c = aTb/aTa; hence, p = (aTb/aTa) ■ a.

Similarly, we...

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Distributions Related to the Standard Normal Distribution

The standard normal distribution generates, via various transformations, a few other distributions such as the chi-square, t, Cauchy, and F distributions. These distributions are fundamental in testing statistical hypotheses, as we will see in Chapters 5, 6, and 8.

4.6.1. The Chi-Square Distribution

Let X1,Xn be independent N(0, 1)-distributed random variables, and let

n

Yn = £ X2. (4.30)

j=1

The distribution of Yn is called the chi-square distribution with n degrees of freedom and is denoted by x2 or x2(n). Its distribution and density functions

can be derived recursively, starting from the case n = 1:

Gi(y) = P[71 < y] = P [X < y] = P[-Vy < Xi < Vt]

4у 4у

= j f (x)dx = 2 j f (x)dx for y > 0,

-Vt о

Gi(y) = 0 for y < 0,

where f (x) is defined by (4.28); hence,

gi(y) = G 1(y) = f (Vу) /...

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