FINANCE Corporate financial policy and R and D Management

In connection with equation (5.23), the following assumptions are made:

|xti| < χ E(u) = 0 Cov(u) = σ^2 IE(X′u) = 0 (5.24)

These assumptions, in order of appearance, mean:

1.The explanatory variables are uniformly bounded by the finite (but

perhaps very large) constant χ.

2.The disturbances, ut, have mean zero, are uncorrelated, and have com-

mon variance σ^2.

3.The explanatory variables are uncorrelated with the disturbances.

4.There are no linear dependencies among the explanatory variables; that

is, the correlations among the independent variables are less than 1.0.

The technique of ordinary least squares (OLS) obtains an estimator for β,
say b, by minimizing the sum of squared errors committed when we re-
place βby b—and thus “predict” yby Xb.
Thus, we minimize

S= (y– Xb)′(y– xb) = y′y– b′X′y– y′Xb+ b′X′Xb (5.25)

The first-order conditions are

(5.26)

Solving, we obtain

b= (X′X)–1X′y (5.27)

Estimators that are efficient with respect to the class of linear unbiased esti-
mators are said to be best linear unbiased estimators (BLUE), the so-called
Gauss-Markov theorem (Greene 1997).
To complete the estimation problem, we must derive an estimator for
the remaining parameter, namely, σ^2. Thus consider

(5.28)

where

û= y– Xb (5.29)

σˆ^2 = ˆˆ′

−

uu

Tk

∂

∂

S

b

=−22 0Xy′ + XXb′ =

Estimation of Simultaneous Equations Systems 83