Consider an example. (ut , xt ) are iid sequences, mutually independent. E [ut ] = 0.
E [u2 ] = 2 . E [x4 ] < . Denote:
t
t
xt ut
.
x2
t
t = +
Let Vt =
1
T
xt ut and Wt =
1
T
x2 . Thus,
t
E [Vt ] = 0, and V ar[Vt ] =
E [x 2 ] 2
t
0 as T .
T
V ar[x2 ]
t
9
Lecture 9: March 1, 2005
9.1
More on Hypothesis Testing
F -tests and t-tests
Recall, under the null of the restricted model:
F=
(ESSR ESSU )/G
F (G, T K ).
ESSU /(T K )
See G-9.1. If F is large, maybe the restriction is NOT valid so we should reject t
Two estimators might be:
= c ,
= c .
So,
V ar() = E [( )2 ]
= E [(c c )2 ]
= E [(c ( )2 ]
scalar
=
=
=
=
E [(c ( )(c ( ) ]
E [(c ( )( ) c)]
c E [( )( ) ]c
c c
Similarly,
V ar() = c c.
So, is ecient relative to if:
V ar() V ar() = c c c c = c ( )c 0.
De
Note that if Ab = a, then b = A1 a if A can be inverted. If not, we denote the
generalize inverse, or Moor-Penrose Inverse as A+ such that:
AA+ A = A,
A+ AA+ = A+ ,
AA+ symmetric,
A+ A symmetric.
Moreover the Moor-Penrose inverse is unique, and if A is s
Economics 624: Econometrics
Matthew Chesnes
Updated: May 12, 2005
These are Matthew Chesnes notes from a course taught by Ingmar Prucha.
1
Lecture 1: January 27, 2005
1.1
The Nature of Econometric Modelling
3 input streams:
Economic Theory Economic Model