Category Introduction to the Mathematical and Statistical Foundations of Econometrics

Follows now from Theorem 5.2. Q. E. D

Note that this result holds regardless of whether the matrix BE BT is non­singular or not. In the latter case the normal distribution involved is called “singular”:

Definition 5.2: Ann x 1 random vector Y has a singular Nn (д, E) distribution if its characteristic function is of the form yY (t) = exp(i ■ t т д – 21TE t) with E a singular, positive semidefinite matrix.

Because of the latter condition the distribution of the random vector Y in­volved is no longer absolutely continuous, but the form of the characteristic function is the same as in the nonsingular case – and that is all that matters. For example, let n = 2 and

*=й -=(0 "0.

where a 2 > 0 but small. The density of the corresponding N2(*, —) distribution

Подпись: f (У1. У2|а ) Подпись: exp( - y2/2) .. exP ( - y|/(2a2)) V2n a V2n Подпись: (5.5)

of Y = (Yb Y2)t is

Then lima4...

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The Tobit Model

Let Zj = (Yj, XTj )T, j = 1,…, и be independent random vectors such that

Yj = max(Yj, 0), where Yj = a0 + всТXj + Uj

with Uj |Xj – N(0,o02). (8.16)

The random variables Yj are only observed if they are positive. Note that

P[Yj = 0|Xj] = P [ao + eoTXj + Uj < 0|Xj]

= P [Uj >a0 + вГXj |Xj] = 1 – Ф (ja0 + A0TXj)М>),


where Ф(х) = j exp(-u2/2)/V2ndu.

This is a Probit model. Because model (8.16) was proposed by Tobin (1958) and involves a Probit model for the case Yj = 0, it is called the Tobit model. For example, let the sample be a survey of households, where Yj is the amount of money household j spends on tobacco products and Xj is a vector of household characteristics. But there are households in which nobody smokes, and thus for these households Yj = 0.

In this case the setup of ...

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The Complex Exponential Function

image972 Подпись: (III.10)

Recall that, for real-valuedx the exponential function ex, also denoted by exp(x), has the series representation

image974 Подпись: (III.11)

The property ex+y = ex ey corresponds to the equality


^ (x + y)k

k=0 k!

= E 1 Z(k)xk—mym

k! m

k= 0 m = 0

Ж k xk—m ym

Подпись: a+і b df
Подпись: e image978

The first equality in (Ш.11) results from the binomial expansion, and the last equality follows easily by rearranging the summation. It is easy to see that (Ш.11) also holds for complex-valued x and y. Therefore, we can define the complex exponential function by the series expansion (Ш.10):


Moreover, it follows from Taylor’s theorem that

о / nm i2m со / nm u2„ +1

Подпись: (2„ + 1)! ’cose») _E (-12_.^, Srn(b) _£ (-1>, + . (Ш.13)

and thus (III.12) becomes

Подпись: (III.14)ga+ib _ ga [cos(b) + i • sin(b)].


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Expectations of Products of Independent Random Variables

Let X and Y be independent random variables, and let f and g be Borel – measurable functions on R. I will show now that

E [f(X)g(Y)] = (E [f(X)])(E [g(Y)]). (2.30)

In general, (2.30) does not hold, although there are cases in which it holds for dependent X and Y. As an example of a case in which (2.30) does not hold, let X = U0 ■ U1 and X = U0 ■ U2, where U0, U1, and U2 are independent and

uniformly [0, 1] distributed, and let f (x) = x, g(x) = x. The joint density of

U0, U1 and U2 is

h(u0, u1, u2) = 1 if (u0, u 1, u2)T є [0, 1] x [0, 1] x [0, 1], h(u0, u1, u2) = 0 elsewhere;


Подпись: E [f (X) g(Y)]E[X ■ Y] = E[ Uo U1 U2 ]

1 1 1

/// u0u1U2du0 du1 du2

Подпись: 000 1

J u0 du0 J u1du1 J u2du2


(1/3) x (1/2) x (1/2) = 1/12,





E [f(X)] = E [X]




and sim...

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Convergence in Probability and the Weak Law of Large Numbers

Let Xn be a sequence of random variables (or vectors) and let X be a random or constant variable (or conformable vector).

Definition 6.1: We say that Xn converges in probability to X, also de­noted as plimn^TO Xn = X or Xn ^ pX, if for an arbitrary є > 0 we have limn^TOP(|Xn — X| > є) = 0, or equivalently, limn^TOP(|Xn — X| < є) = 1.

In this definition, X may be a random variable or a constant. The latter case, where P(X = c) = 1 for some constant c, is the most common case in econometric applications. Also, this definition carries over to random vectors provided that the absolute value function |x | is replaced by the Euclidean norm

||x У = V x Tx.

The right panels of Figures 6.1-6.3 demonstrate the law of large numbers...

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Gaussian Elimination of a Square Matrix and the Gauss-Jordan Iteration for Inverting a Matrix

I.6.1. Gaussian Elimination of a Square Matrix

The results in the previous section are the tools we need to derive the following result:

Theorem I.8: Let A be a square matrix.

(a) There exists a permutation matrix P, possibly equal to the unit matrix I, a lower-triangular matrix L with diagonal elements all equal to 1, a diagonal matrix D, and an upper-triangular matrix U with diagonal elements all equal to 1 such that PA = LDU.

(b) If A is nonsingular and P = I, this decomposition is unique; that is, if A = LDU = L*D*U*, then L* = L, D* = D, and U* = U.

The proof of part (b) is as follows: LDU = L * D*U* implies

L-1L * D* = DUU-1. (I.21)

It is easy to verify that the inverse ofa lower-triangular matrix is lower triangu­lar and that the product of lower-triangular matrices is lower triangu...

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