FINANCE Corporate financial policy and R and D Management

but

û′û= y′[I– X(X′X)–1X′]y= u′[I– X(X′X)–1X′]u (5.30)

Hence

E(û′û) = σ^2 tr[I– X(X′X)–1X′]

= σ^2 [trI– tr(X′X)–1X′X0] = (T– k)σ^2 (5.31)

The last equality holds, for Iis T×T, while (X′X)–1X′Xis the identity ma-
trix of order k.

Estimation of Parameters in Multiple

Equation (Regression) Models

Let us generalize the preceding problem so that we consider not a single
equation but rather a system of equations of the form

(5.32)

The system here contains kindependent (or exogenous) variables x 1 , x 2 ,...,
xkand mdependent variables y 1 ,y 2 ,...,ym. It might seem that all indepen-
dent variables appear in each equation of (5.32) but this need not be so, for
some of the βjimay be known to be zero. In general, we will assume that
only ki≤kindependent variables appear in the ith equation.
Let us denote by y·Ithe vector of observations on the dependent vari-
able yiand by Xithe matrix of observations on the (ki) independent vari-
ables actually appearing in the ith equation. Then the system in (5.32) may
be written more conveniently as

y·i= Xiβ·i+ u·i i= 1, 2,...,m (5.33)

The vector β·idiffers from (β 1 i,β 2 i,...,βki)′in that it is the subvector of the
latter resulting after deletion of elements βjiknown to be zero. Thus, in
(5.33), y·iis T×1, Xiis T×ki, β·iis ki×1, and u·iis T×1. Now each equa-
tion in (5.33) represents a general linear model of the type examined ear-
lier. The covariance vector of error terms is:

Cov(u·i) = σiiI (5.34)

yxut Ti mrk tj ji ti

j

k

=+= =

=

∑ β^12 12

1

,, , ,KK,, ,

84 AN INTRODUCTION TO STATISTICAL ANALYSIS AND SIMULTANEOUS EQUATIONS