FINANCE Corporate financial policy and R and D Management

We still need to say something about the covariance matrix of u·iand u·j, i≠j.
The two vectors need not be uncorrelated. Clearly, if

Cov(u·i, u·j) ≠ 0 (5.35)

then the ith equation conveys some information about the jth equation.
One should use all statistical information in estimated equations.
In general, since the error terms of the system may be interpreted as re-
flecting, in part, the impact of many relevant influences that are not indi-
vidually accounted for, it would be reasonable to assume that uitis
correlated with utj. Finally, if the observations are interpreted as being a
random sample on the multivariate vector (y 1 ,y 2 ,...,ym), then, of course,
ut′jis uncorrelated with, indeed independent of, ut′jfor t≠t′. Hence let us
solve the estimation problem posed by the system in (5.33) under these as-
sumptions. Specifically, we assume

Cov(u·i, u·j) = σijIE(X′ju·i) = 0

E(u·i) = 0 i, j= 1, 2,...,m (5.36)

Then the entire system in (5.33) can be written more compactly (and re-
vealingly) as:

y= Xβ+ u (5.37)

The problem of efficient estimation of the parameter vector in (5.37)
has already been solved in the preceding discussion; however, the solution
depends on the form of the covariance matrix of the error vector u, which
may be written as:

(5.38)

(5.38)

The reader is referred to Greene (1997) and Dhrymes (1974) for complete
statistical treatments of regression estimations.

Cov

()uEuu E( )

uu uu uu

uu uu uu

uu uu uu

II I

II

m

mm mm

m

= ′ =

′′ ′

′′ ′

′′ ′



















     =

⋅⋅ ⋅⋅ ⋅⋅

⋅⋅ ⋅⋅ ⋅⋅

⋅⋅ ⋅⋅ ⋅⋅

11 12 1

21 22 22

12

11 12 1

21 22

K

K

MM M

K

K

K

σσ σ

σσ σσ

σσ σ

2

12

m

mm mm

I

II I

MM M

K





























Estimation of Simultaneous Equations Systems 85