Category Introduction to the Mathematical and Statistical Foundations of Econometrics

Mathematical Expectation

With these new integrals introduced, we can now answer the second question stated at the end of the introduction: How do we define the mathematical ex­pectation if the distribution of X is neither discrete nor absolutely continuous?

Definition 2.12: The mathematical expectation of a random variable X is defined as E(X) = f X(o)dP(o) or equivalently as E(X) = f xdF(x) (cf(2.15)), whereFis the distribution function ofX, provided that the integrals involved are defined. Similarly, if g(x) is a Borel-measurable function on Kk and Xis a random vector in Kk, then, equivalently, E[g(X)] = f g(X(o))dP(o) = f g(x )dF(x), provided that the integrals involved are defined.

Note that the latter part of Definition 2.12 covers both examples (2.1) and (2.3).

As motivated in the introduction, the mathemat...

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Hypotheses Testing

Theorem 5.19 is the basis for hypotheses testing in linear regression analysis. First, consider the problem of whether a particular component of the vector Xj of explanatory variables in model (5.31) have an effect on Yj or not. If not, the corresponding component of в is zero. Each component of в corresponds to a component ві 0, i > 0, of в0. Thus, the null hypothesis involved is

H : ві,0 = 0. (5.49)

Let ei be component i of 0, and let the vector ei be column i of the unit matrix Ik. Then it follows from Theorem 5.19(a) that, under the null hypothesis (5.49),

в

ti = —, = ~ tn-k■ (5.50)

SjeJ( XT X)-1ei

The statistic t i in (5.50) is called the t-statistic or t-value of the coefficient ei,0. If ei 0 can take negative or positive values, the appropriate alternative hypothesis is

Hi...

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The Inverse and Transpose of a Matrix

I will now address the question of whether, for a given m x n matrix A, there exists an n x m matrix B such that, with y = Ax, By = x. If so, the action of A is undone by B, that is, B moves y back to the original position x.

If m < n, there is no way to undo the mapping y = Ax. In other words, there does not exist an n x m matrix B such that By = x. To see this, consider the 1 x 2matrix A = (2, 1). Then, withxasin(I.12),Ax = 2x + x2 = y, butifwe knowy and A we only know that x is located on the line x2 = y – 2xi; however, there is no way to determine where on this line.

If m = n in (I.14), thus making the matrix A involved a square matrix, we can undo the mapping A if the columns3 of the matrix A are linear independent. Take for example the matrix A in (I.11) and the vector y in (I...

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Sets in Euclidean Spaces

An open e-neighborhood of a point x in a Euclidean space Kk is a set of the form

Ne(x) = {y e Kk : Уy — x У < e}, e > 0, and a closed e-neighborhood is a set of the form

Ne(x) = {y e Kk : Уy — x У < e}, e> 0.

A set A is considered open if for every x e A there exists a small open e-neighborhood Ne(x) contained in A. In shorthand notation: Vx e A 3e > 0: Ne(x) c A, where V stands for “for all” and 3 stands for “there exists.” Note that the e’s may be different for different x.

A point x is called a point of closure of a subset A of Kk if every open e – neighborhood Ne (x) contains a point in A as well as a point in the complement A of A. Note that points of closure may not exist, and if one exists it may not be contained in A...

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The Poisson Distribution

A random variable X isPoisson(X)-distributediffor k = 0, 1, 2, 3,… and some

X > 0,

X k

P (X = k) = exp(-X)(4.7) k!

Recall that the Poisson probabilities are limits of the binomial probabilities (4.3) for n and p 10 such that np ^ X. It is left as an exercise to show that the expectation, variance, moment-generating function, and characteristic function of the Poisson(X) distribution are

Подпись: (4.8) (4.9) (4.10) (4.11) E [ X] = X, var(X) = X, mp(t) = exp[X(et -1)],

(Pp(t) = exp[X(e!4 -1)],

respectively.

4.1.2. The Negative Binomial Distribution

Consider a sequence of independent repetitions of a random experiment with constant probability p ofsuccess. Let the random variable X be the total number of failures in this sequence before the Mth success, where m > 1...

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Convergence of Characteristic Functions and Distributions

In this appendix I will provide the proof of the univariate version of Theorem 6.22. Let Fn be a sequence of distribution functions on К with corresponding characteristic functions yn (t), and let F be a distribution function on К with characteristic function y(t) = limn—TO^n(t). Let

F(x) = limliminf Fn (x + 8 ), F (x) = lim limsup Fn (x + 8 ).

5^0 n——to 5^0 n — TO

The function F(x) is right continuous and monotonic nondecreasing in x but not necessarily a distribution function itself because limx^TO F(x) may be less than 1 or even 0. On the other hand, it is easy to verify that limx;—TO F(x) = 0. Therefore, if limx—TO F(x) = 1, then F is a distribution function. The same applies to F(x): If limx—TO F(x) = 1, then F is a distribution function.

I will first show that limxF(...

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