Introduction to the Mathematical and Statistical Foundations of Econometrics

The Nonlinear Case

If we denote G (x) = Ax + b, G-i(y) = A-i( y - b), then the result of Theo­rem 4.3 reads h(y) = f (G-i(y))|det(9G-i(y)/9y)|. This suggests that Theo­rem 4.3 can be generalized as follows:

Theorem 4.4: Let X be k-variate, absolutely continuously distributed with joint density f(x),x = (xi,...,xk)T, and let Y = G(X), where G(x) = (gi(x),..., gk(x))T is a one-to-one mapping with inverse mapping x = G-i(y) = (gj(y),...,gl(y))T whose components are differentiable in the components ofy = (yi;yk)T. Let J(y) = dx/dy = dG-i(y)/dy, that is, J(y) is the matrix with i, j’s element dgf(y)/dyj, which is called the Jacobian. Then Y is k-variate, absolutely continuously distributed with joint density h(y) = f (G-i(y))|det(J(y))| for y in the set G(Rk) = {y є Rk :y = G(x), f (x) > 0, x є Kk} and h(y) = 0 elsewhere.

This conjecture is indeed true. Its formal proof is given in Appendix 4.B. An application of Theorem 4.4 is the following problem. Consider the function

f (x) = c ■ exp(-x2/2) if x > 0,

= 0 if x < 0. (4.26)

For which value of c is this function a density?

To solve this problem, consider the joint density f (x, x2) = c2 exp[-(x2 + xf)/2], xi > 0, x2 > 0, which is the joint distribution of X = (Xj, X2)T, where Xi and X2 are independent random drawings from the distribution with density (4.26). Next, consider the transformation Y = (Yi, Y2)T = G(X) defined by

Yi = УX2 + X2 є (0, то)

Y2 = arctan(Xi/X2) є (0, л/2).

The inverse X = G-i(Y) of this transformation is

Xi = Yi sin(Y2),

X2 = Yi cos(Y2)

with Jacobian

J (Y)= ( dXi/д Yi д Xi/д Y2 ( sin(Y2) Yicos(Y2)

J ( ) І, д X2/д Yi д X2/д Y2J cos(Y2) - Yisin(Y2^l '

Note that det[J(Y)] = — Yi. Consequently, the density h(y) = h(yi, y2) = f(G-i(y))|det(J(y))| is

h(Уь y2) = c2yi exp (—yjV2) for yi > 0 and 0 < y2 < л/2,

= 0 elsewhere;

hence,

TO n/2

i = j j c2yi exp (— y2 /2) ф>2Фт 00

TO

= c2(x/2) J yi exp (— y2 /2) dyi 0

= c2 л/2.

Thus, the answer is c = V2/л:

Note that this result implies that

TO

I л = 1.

J V2n