FINANCE Corporate financial policy and R and D Management

(5.19)

(5.19)

Ordinarily, however, a multiple regression is carried out on a computer,
which is programmed to print out the value of the multiple coefficient of
determination (or of the multiple correlation coefficient).

Estimation of Simultaneous Equations Systems

Let us generalize regression analysis using Kvariables. A certain class of esti-
mators for the parameters of a simultaneous equations (S.E.) system can be
shown to have an interpretation as an ordinary least squares (OLS) estima-
tor. In view of this fundamental unity of estimation procedures, it would be
desirable at this stage to review carefully the estimation problem in the con-
text of the general linear model and some of its (straightforward) extensions.
Let ybe a variable of interest and suppose that observations on it, at
time t, are generated by

(5.20)

Here yis said to be the dependent variable and the xi, i= 1,2,...,k, the in-
dependent or explanatory variables; the latter are usually assumed nonsto-
chastic. The βiare unknown parameters to be estimated.

Y = Xβ+ u (5.21)

where yis a T×1 vector of observations on the dependent variable,

X= (xti) t= 1,2,...,T, i= 1,2,...,k (5.22)

is the matrix of observations on the explanatory variables, and

β= (β 1 ,β 2 ,...,βk)′ u= (u 1 ,u 2 ,...,ur)′ (5.23)

yxut Tt i ti t

i

k

=+=

=

∑β^ 1 2

1

,, ,K

R

bXXYYb XXYY

Y

Y

n

iii i i

i

n

i

n

i

i

i

n

i

n

2

11 2 22

1 1

2 1

2

1

=

−−+ − −

−













= =

=

=

∑ ∑

∑

∑

ˆ ()()ˆ ()()

82 AN INTRODUCTION TO STATISTICAL ANALYSIS AND SIMULTANEOUS EQUATIONS