Basic Econometrics, Gujarati and Porter
30
CHAPTER 4:
CLASSICAL NORMAL LINEAR REGRESSION MODEL (CNLRM)
Appendix 4A Exercises
4.1 Given that the coefficient of correlation between Y
1
and Y
2
,
ρ
, is
zero, the bivariate normal PDF reduces to:
4.2
To ensure that the maximum likelihood estimators maximize the
likelihood function, the second derivatives from Eq. (5) in App. 4A
must be less than zero, which will ensure that RSS is minimized.
2
22
ln
0
LF n
σ
β
∂
= − <
∂
4.3
Since X follows the exponential distribution, its PDF is:
f(X) = 1
( )
i
X
i
f X e
θ
θ
−
=
n
4.4 Since p follows a Bernoulli distribution, its PDF is:
f(p)=p
(
)
X
1−p
(
)
1−X
Therefore, the LF will be
Basic Econometrics, Gujarati and Porter
32
Differentiating the preceding function with respect to
p
, we obtain: