Advanced Econometrics Takeshi Amemiya

Distribution Theory

The theorems listed in this appendix, as well as many other results concerning the distribution of a univariate continuous random variable, can be found in Johnson and Kotz (1970a, b). “Rao” stands for Rao (1973) and “Plackett” for Plackett (1960).

1. Chi-Square Distribution. (Rao, p. 166.) Let an «-component random vector z be distributed as N(0,1). Then the distribution of z'z is called the chi-square distribution with n degrees of freedom. Symbolically we write z'z ~ Xn - Its density is given by

f(x) = 2~n/2T{n/2)e~x/2x~n/2~

where Г(р) = f qXp~ 1 e~x dk is called a gamma function. Its mean is n and its variance 2 n.

2. (Rao, p. 186). Let z ~ ЛГ(0,1) and let A be a symmetric and idempotent matrix with rank n. Then z' Az ~ xl-

3.

Student’s t Distribution. (Rao, p. 170.) If z ~ N(0, 1) and w~xl and if z and w are independent, nxazw~xa has the Student’s t distribution with и degrees of freedom, for which the density is given by

where

Symbolically we write

4. F Distribution. (Rao, p. 167.) If w, ~ xl, and w2 ~ xl2 and if w, and w2 are independent, «71w1«2w'21 has the F distribution with n, and n2 degrees of freedom, for which the density is given by

(nl/n2)n'/2xn'/2~l

f(X) = MnJ2),(n2/2)](l + піХ/п2Г^

Symbolically we write

~F(nlfn2).

5. (Plackett, p. 24.) Let z ~ iV(0,1). Let A and В be symmetric matrices. Then z' Az is independent of z'Bz if and only if AB = 0.

6. (Plackett, p. 30.) Let z ~ iV(0,1). Then c'z and z'Az are independent if and only if Ac = 0.

7. (Plackett, p. 30.) Let z ~ N(0,1) and let A be a symmetric matrix. Then z'Az/z'z and z'z are independent.