### Problem Set 6: Linear Hypotheses II

```Hypothesis testing
Ec402
PS 6
Notes on the Wald Statistic
• Want to test hypotheses of the form 0 :  =  (with r restrictions
= number of rows in matrix R), based on our estimate .
• Note:
=
=
−    −
=    −   −
=   ′
=  2   ′
−1 ′

′ ′

′
• With A5N this implies:
~(,  2   ′
−1  ′ )
− ~(0,  2   ′
−1  ′ )
• And so under 0 :  =  we have:
− ~(0,  2   ′
−1 ′
) (1)
• If  2 is known, use a  2 distribution.
• Why?
•  2 () is the sum of r squared independent N(0,1)
variables by definition.
• From (1):
[ 2   ′
−1  ′ ]−1/2 (
− )~(0, )
And so from the definition of  2 ():
−
′
2 ′
−1  ′ −1
−  ~ 2
(2)
• If  2 is unknown, use an F distribution.
• Why?
• F distribution is defined in terms of 2 independent  2
distributions. Let 1 and 2 be independently
distributed  2 variables with 1 and 2 degrees of
freedom. Then:
1 /1
=
~(1 , 2 )
2 /2
• Can show that
′
2
~
(
2

− ) and is independent of
(see Johnston & DiNardo p.495)
• And combining this result with (2) we get:
′
−
′  −1 ′ −1  −  /
~(,  − )
′/( − )
(Note the  2 s cancel)
• Since
′
−
−
′
=  2 this gives us
2 ′
−1  ′ −1
−  / ~ (,  − )
• Note that this Wald test is equal to the
Likelihood Ratio test given by:
=
( − )/
/(−)
• See lecture notes for a proof
```