# 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 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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