Category Introduction to the Mathematical and Statistical Foundations of Econometrics

Appendix III – A Brief Review of Complex Analysis

III.1. The Complex Number System

Complex numbers have many applications. The complex number system allows computations to be conducted that would be impossible to perform in the real world. In probability and statistics we mainly use complex numbers in dealing with characteristic functions, but in time series analysis complex analysis plays a key role. See for example Fuller (1996).

Complex numbers are actually two-dimensional vectors endowed with arith­metic operations that make them act as numbers. Therefore, complex numbers are introduced here in their “real” form as vectors in K2.

In addition to the usual addition and scalar multiplication operators on the elements of K2 (see Appendix I), we define the vector multiplication operator

image946(III1)

image947

Observe that

image948
image949

Moreover,...

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Liapounov’s Inequality

Liapounov’s inequality follows from Holder’s inequality (2.22) by replacing Y with 1:

E(|X|) < (E(|X |p))1/p, where p > 1.

2.6.3. Minkowski’s Inequality

If for some p > 1, E[|X|p] < to and E [| Y|p] < to, then

E(|X + Y|) < (E(|X |p))1/p + (E(| Y |p))1/p. (2.23)

This inequality is due to Minkowski. For p = 1 the result is trivial. There­fore, let p > 1. First note that E[|X + Y|p] < E[(2 ■ max(|X|, |Y|))p] = 2pE[max(|X|p, |Y|p)] < 2pE[|X|p + |Y|p] < to; hence, we may apply Liapounov’s inequality:

E(|X + Y|) < (E(|X + Y|p))1/p. (2.24)

Next, observe that

E(|X + Y|p) = E(|X + Y|p-1|X + Y|) < E(|X + Y|p-1|X|)

+ E (| X + Y | p-1|Y |). (2.25)

Let q = p/(p — 1). Because 1/q + 1/p = 1 it follows from Holder’s inequality that

E(|X + Y|p—1|X|) < (E(|X + Y|(p—1)q))1/q(E(|X|p))1/p

< (E

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Modes of Convergence

5.3. Introduction

Toss a fair coin n times, and let Yj = 1 if the outcome of the j th tossing is heads and Yj = — 1 if the outcome involved is tails. Denote Xn = (1/n)Yl j= Yj. For the case n = 10, the left panel of Figure 6.1 displays the distribution function Fn(x)1 of Xn on the interval [—1.5, 1.5], and the right panel displays a typical plot of Xk for k = 1, 2,…, 10 based on simulated Yj’s 2

Now let us see what happens if we increase n: First, consider the case n = 100 in Figure 6.2. The distribution function Fn(x) becomes steeper for x close to zero, and Xn seems to tend towards zero.

These phenomena are even more apparent for the case n = 1000 in Figure

6.3.

What you see in Figures 6.1—6...

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Elementary Matrices and Permutation Matrices

Let A be the m x n matrix in (I.14). An elementary m x m matrix E is a matrix such that the effect of EA is the addition of a multiple of one row of A to another row of A. For example, let Ei, j (c) be an elementary matrix such that the effect

of E,, j(c)A is that c times row j is added to row i < j:

a1,1

• . . a1,n ^

ai-1,1

. . . ai — 1,n

ai, 1 + caj, 1

• ♦ ♦ ai, n + caj, n

ai+1,1

. • • ai +1,n

aj,1

••• a j, n

am,1

• • • am, n /

 

Ei, j (c) A

 

(1.19)

 

Then E+j (c)6 is equal to the unit matrix Im (compare (1.18)) except that the zero in the (i, j)’s position is replaced by a nonzero constant c. In particular, if i = 1 and j = 2 in (I...

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Transformations of Discrete Random Variables and Vectors

In the discrete case, the question Given a random variable or vector X and a Borel measure function or mapping g(x), how is the distribution of Y = g(X) related to the distribution of X? is easy to answer. If P[X є {хь x2,…}] = 1 and
g(x1), g(x2),… are all different, the answer is trivial: P(Y = g(xj)) = P(X = Xj). If some of the values g(x1), g(x2),… are the same, let {y1, y2,…} be the set of distinct values of g(x1), g(x2),… Then

TO

P(Y = yj) = £ I[y = g(Xi)]P(X = xi). (4.13)

i=1

It is easy to see that (4.13) carries over to the multivariate discrete case.

For example, if X is Poisson(X)-distributed and g(x) = sin2(nx) = (sin(nx))2 – and thus for m = 0, 1, 2, 3,…, g(2m) = sin2(nm) = 0 and g(2m + 1) = sin2(nm + n/2) = 1 – then P(Y = 0) = e-XY°TO=0 xlj/(2j)! and P(Y = 1) = e-kJ2j=0 X2j+1...

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Dependent Laws of Large Numbers and Central Limit Theorems

Chapter 6 I focused on the convergence of sums of i. i.d. random variables – in particular the law of large numbers and the central limit theorem. However, macroeconomic and financial data are time series data for which the indepen­dence assumption does not apply. Therefore, in this chapter I will generalize the weak law of large numbers and the central limit theorem to certain classes of time series.

7.1. Stationarity and the Wold Decomposition

Chapter 3 introduced the concept of strict stationarity, which for convenience will be restated here:

Definition 7.1: A time series process Xt is said to be strictly station­ary if, for arbitrary integers m < m2 < ••• < mn, the joint distribution of Xt—m1Xt—mn does not depend on the time index t.

A weaker version of stationarity is cova...

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