FINANCE Corporate financial policy and R and D Management

Two-Stage Least Squares (2SLS)

Thus let us examine again the problem of estimating the parameters of

y·1= Y 1 β·1+ X 1 γ·1+ u·1 (5.39)

which is the first equation in a system of m-structural equations as dis-
cussed in the previous section. As before, let X(of rank G) be the T×G
matrix of observations on all the predetermined variables appearing in the
entire system and consider the transformed equation

X′y·1= X′Y 1 β·1+ X′X 1 γ·1+ X′u·1 (5.40)

The new explanatory variables consist essentially of sample cross
moments between the current endogenous and the predetermined vari-
ables—the former as they appear in the first equation, the latter as they
appear in the entire system. As the sample size increases, the new ex-
planatory variables are nonstochastic and are uncorrelated with the error
term appearing in (5.40). Thus, if one applies an “efficient” estimation
technique to that equation, one obtains at least consistent estimators of
the vectors β·1and γ·1. Here we should caution the reader that, in general,
it is not true that

Cov(X′u·1) = E(X′u·1u′·1X) = σ 11 X′X (5.41)

The errors in the system are jointly normally distributed or the errors
at time tare independent of errors at time t′, for t≠t′. If we do make the
normal distribution or independence assumptions, then

(5.42)

and thus for large samples we would have approximately

Cov(X′u·1) = σ 11 X′X (5.43)

It appears that an “efficient” procedure for estimating β·1and γ·1
from (5.40) would be the application of Aitken techniques, where the co-
variance matrix of the error vector X′u·1is taken as σ 11 (X′X). The Aitken
estimation has some optimal properties, so it is reasonable to conjecture

Eyuuytstt t s Ey

t

T

t

T

ts

t

T

− ′′−

′= =

−

=

∑∑∑











111 1=

1 1

11 1

2

1

,,σ (),

86 AN INTRODUCTION TO STATISTICAL ANALYSIS AND SIMULTANEOUS EQUATIONS