Category Introduction to the Mathematical and Statistical Foundations of Econometrics

The Standard Cauchy Distribution

The ti distribution is also known as the standard Cauchy distribution. Its density is

Г(1)

VnT(1/2)(1 + x2)

 

1

n (1 + x2)’

 

h1(x)

 

(4.39)

 

where the second equality follows from (4.36), and its characteristic function is

Vh(t) = exp(-|t |).

The latter follows from the inversion formula for characteristic functions:

TO

2П f exp<

— TO

 

1

n (1 + x2)

 

i ■ t ■ x)exp(—|t |)dt

 

(4.40)

 

See Appendix 4.A. Moreover, it is easy to verify from (4.39) that the expectation of the Cauchy distribution does not exist and that the second moment is infinite.

4.6.2. The F Distribution

Let Xm ~ x2 and Yn ~ x2, where Xm and Yn are independent. Then the distri­bution of the random variable

F _ Xm / m Yn/n

is said...

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A.5. Proof of the Wold Decomposition

Let Xt be a zero-mean covariance stationary process and E[X2] = a2. Then the Xt’s are members of the Hilbert space U0 defined in Section 7.A.2. Let S—TO be the subspace spanned by Xt_j, j > 1, and let Xt be the projection of Xt on S——1,. Then Ut = Xt — Xt is orthogonal to all Xt _ j, j > 1, that is, E[UtXt —j] = 0 for j > 1. Because Ut —j є S—TO for j > 1, the Ut’s are also orthogonal to each other: E[UtUt—j] = 0 for j > 1.

Note that, in general, Xt takes the form Xt = fit, jXt—j, where the

coefficients et, j are such that WYt ||2 = E [Yt2] < to. However, because Xt is covariance stationary the coefficients fit, j do not depend on the time index t, for they are the solutions of the normal equations

TO

Y(m) = E[XtXt—m] = J2 вjE[Xt—jX— m]

j=1

TO

= Y! вj Y (j—m),...

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Uniform Continuity

A function g on Кк is called uniformly continuous if for every є > 0 there exists a 8 > 0 such that |g(x) – g(y)| < є if ||x – уУ <8. In particular,

Theorem II.7: If a function g is continuous on a compact subset © ofRk, then it is uniformly continuous on ©.

Proof: Let є > 0 be arbitrary, and observe from the continuity of g that, for each x in ©, there exists a 8(x) > 0 such that |g(x) – g(y)| < є/2 if \x — y\ < 2S(x). Now let U(x) = {y є Rk : ||y – x У < 8(x)}. Then the col­lection {U(x), x є ©} is an open covering of ©; hence, by compactness of © there exists a finite number of points 6, ■■■,6n in © such that © c и"= U (Qj). Next, let 8 = mini< j<„ 8(Qj )■ Each point x є © belongs to at least one of the open sets U(Qj):x є U(Qj) for some j...

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Borel Measurability, Integration, and Mathematical Expectations

2.1. Introduction

Consider the following situation: You are sitting in a bar next to a guy who proposes to play the following game. He will roll dice and pay you a dollar per dot. However, you have to pay him an amount y up front each time he rolls the dice. Which amount y should you pay him in order for both of you to have equal success if this game is played indefinitely?

Let X be the amount you win in a single play. Then in the long run you will receive X = 1 dollars in 1 out of 6 times, X = 2 dollars in 1 out of 6 times, up to X = 6 dollars in 1 out of 6 times. Thus, on average you will receive (1 + 2 + + 6)/6 = 3.5 dollars per game; hence, the answer is y = 3.5.

Clearly, X is a random variable: X(a) = Y^j=i j ‘ I(ш є U}), where here, and in the sequel, I( ) denotes the indicator funct...

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Applications to Regression Analysis

5.1.1. The Linear Regression Model

Consider a random sample Zj = (Yj, Xj)T, j = 1, 2,…,n from a ^-variate, nonsingular normal distribution, where Yj є К, Xj є R-1. We have seen in Section 5.3 that one can write

Yj = a + Xj в + Uj, Uj – N (0, a2), j = 1,…,n, (5.31)

where Uj = Yj – E [Yj | Xj ] is independent of Xj. This is the classical linear regression model, where Yj is the dependent variable, Xj is the vector of in­dependent variables, also called the regressors, and Uj is the error term. This model is widely used in empirical econometrics – even in the case in which Xj is not known to be normally distributed.

Подпись: Y1 1 X1T Y= , X = Yn 1 XnT Подпись: 00 image357
image358

If we let

model (5.31) can be written in vector-matrix form as

Подпись: (5.32)Y = X00 + U, U|X – Nn [0, a2In],

where U|Xis a shorthand notation for “U conditional on X.”

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Selecting a Test

The Wald, LR, and LM tests basically test the same null hypothesis against the same alternative, so which one should we use? The Wald test employs only the unrestricted ML estimator 0, and thus this test is the most convenient if we have to conduct unrestricted ML estimation anyway. The LM test is entirely based on the restricted ML estimator 0, and there are situations in which we start with restricted ML estimation or where restricted ML estimation is much easier to do than unrestricted ML estimation, or even where unrestricted ML estimation is not feasible because, without the restriction imposed, the model is incompletely specified. Then the LM test is the most convenient test. Both the Wald and the LM tests require the estimation of the matrix Й...

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