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
of Y = (Yb Y2)t is
Then lima4... Read More
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 ... Read More
Recall that, for real-valuedx the exponential function ex, also denoted by exp(x), has the series representation
The property ex+y = ex ey corresponds to the equality
^ (x + y)k
= E 1 Z(k)xk—mym
k= 0 m = 0
Ж k xk—m ym
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
cose») _E (-12_.^, Srn(b) _£ (-1>, + . (Ш.13)
and thus (III.12) becomes
ga+ib _ ga [cos(b) + i • sin(b)].
Setting... Read More
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;
E[X ■ Y] = E[ Uo U1 U2 ]
1 1 1
/// u0u1U2du0 du1 du2
J u0 du0 J u1du1 J u2du2
(1/3) x (1/2) x (1/2) = 1/12,
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... Read More
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... Read More