Category Introduction to the Mathematical and Statistical Foundations of Econometrics

Borel Measurability

Let g be a real function and let X be a random variable defined on the probability space {^, P}. For g(X) to be a random variable, we must have that

It is possible to construct a real function g and a random variable X for which this is not the case. But if

For all Borel sets B, Ab = {x є R : g(x) є B} is a Borel set itself,

(2.5)

then (2.4) is clearly satisfied because then, for any Borel set B and A B defined in (2.5),

{ш є ^ : g(X(o)) є B} = {o є ^ : X(o) є AB}є

Moreover, if (2.5) is not satisfied in the sense that there exists a Borel set B for which A B is not a Borel set itself, then it is possible to construct a random variable X such that the set

{o є ^ : g(X(ш)) є B} = {o є ^ : X(ш) є AB} /

hence, for such a random variable X, g(X) is not a random variable itself...

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Least-Squares Estimation Observe that

E[(Y – X0)T(Y – X0)] = E[(U + X(00 – 0))T(U + X0 – 0))]

= E[UTU] + 2(00 – 0)TE(XTE[U|X])

+ (00 – 0 )T(E [XTX])(00 – 0)

= n • a2 + (00 – 0)T(E[XTX])(00 – 0).

(5.33)

Hence, it follows from (5.33) that[16]

в0 = argmin E[(Y – XQ)T(Y – XQ)] = (E[XTX])-1 E[XTY]

в eRk

(5.34)

provided that the matrix E [XTX] is nonsingular. However, the nonsingularity of the distribution of Zj = (Yj, Xj)T guarantees that E [XTX] is nonsingular because it follows from Theorem 5.5 that the solution (5.34) is unique if YXX = Var(Xj) is nonsingular.

The expression (5.34) suggests estimating в0 by the ordinary[17] least-squares! estimator

в = argmin(Y – XQ)T(Y – XQ) = (XTX)-1XTY. (5.35)

в eRk

It follows easily from (5.32) and (5.35) that

в – во = (XTX)-1XTU; (5.36)

hence, в is conditionally unbiased: E [в|X]...

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Appendix I – Review of Linear Algebra

I.1. Vectors in a Euclidean Space

A vector is a set of coordinates that locates a point in a Euclidean space. For example, in the two-dimensional Euclidean space K2 the vector

image755(I.1)

is the point whose location in a plane is determined by moving a1 = 6 units away from the origin along the horizontal axis (axis 1) and then moving a2 = 4 units away parallel to the vertical axis (axis 2), as displayed in Figure I.1. The distances a1 and a2 are called the components of the vector a involved.

An alternative interpretation of the vector a is a force pulling from the origin (the intersection of the two axes). This force is characterized by its direction (the angle of the line in Figure I.1) and its strength (the length of the line piece between point a and the origin)...

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The Binomial Distribution A random variable X has a binomial distribution if

P(X = k) =^n^jpk(1 – p)n-k for k = 0, 1, 2,…, n,

P(X = k) = 0 elsewhere, (4.3)

where 0 < p < 1. This distribution arises, for example, if we randomly draw n balls with replacement from a bowl containing K red balls and N – K white balls, where K /N = p. The random variable X is then the number of red balls in the sample.

We have seen in Chapter 1 that the binomial probabilities are limits of hy­pergeometric probabilities: If both N and K converge to infinity such that K/N ^ p, then for fixed n and k, (4.1) converges to (4.3). This also suggests that the expectation and variance of the binomial distribution are the limits of the expectation and variance of the hypergeometric distribution, respectively:

E [ X] = np,

(4.4)

var(X) = np(1 – p).

(4.5)

As we will see in Chapte...

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B.2. Slutsky’s Theorem

Theorem 6.B.1 can be used to prove Theorem 6.7. Theorem 6.3 was only proved for the special case that the probability limit Xis constant. However, the general result of Theorem 6.3 follows straightforwardly from Theorems 6.7 and 6.B.3. Let us restate Theorems 6.3 and 6.7 together:

Theorem 6.B.4: (Slutsky’s theorem). Let Xn a sequence of random vectors in Kk converging a. s. (inprobability) to a (random or constant) vectorX. Let Ф(x) be an Rm-valued function on Kk that is continuous on an open (Borel) set B in Rk for which P(X є B) = 1). Then Ф(Xn) converges a. s. (inprobability) to Ф( X).

Proof: Let Xn ^ X a. s. and let {^, tX, P} be the probability space in­volved. According to Theorem 6.B.1 there exists a null set N1 such that lim„^Z Xn(ш) = X(of pointwise in ш є N1...

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Eigenvalues and Eigenvectors

I.15.1. Eigenvalues

Eigenvalues and eigenvectors play a key role in modern econometrics – in par­ticular in cointegration analysis. These econometric applications are confined to eigenvalues and eigenvectors of symmetric matrices, that is, square matrices A for which A = AT. Therefore, I will mainly focus on the symmetric case.

Definition I.21: The eigenvalues11 ofann x n matrix A are the solutions for X of the equation det( A — X In) = 0.

It follows from Theorem I.29 that det(A) = J2 ±a1,i1 a2,i2… an, in, where the summation is over all permutations i1, i2,…,in of 1, 2,…,n. Therefore, if we replace A by A — XIn it is not hard to verify that det(A — XIn) is a polynomial of order n in X, det(A — XIn) = J^=o ckXk, where the coefficients ck are functions of the elements of A.

For ex...

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