Ftest and Joint Hypotheses
Supplementary Lecture Notes:
Spring 2007
Joint Hypothesis Test:
Statistical procedure for testing more than one coefficient at a time. This involves an Ftest.
Consider the following regression model:
E(Y)
=
β
0
+
β
1
X
1
+
β
2
X
2
+
β
3
X
3
+
β
4
X
4
+
β
5
X
5
Suppose we want to test the following joint hypotheses.
Ho:
β
2
= 0,
β
4
= 0,
β
5
= 0
H
1
:
At least one coefficient in Ho
≠
0
Note:
The null states that the parameters,
β
2
,
β
4
, and
β
5
, are simultaneously (i.e., jointly) equal to zero, i.e.,
β
2
= 0 and
β
4
= 0 and
β
5
= 0.
The alternative hypothesis states that at least
one of the parameters in the null is not zero.
“At least” means either one of the parameters is not
zero or a combination of any of the parameters is not zero or all the parameters stated in the null are not zero.
In other words, the alternative
hypothesis says that
β
2
≠
0
OR
β
4
≠
0
OR
β
5
≠
0
OR
(
β
2
≠
0 and
β
4
≠
0)
OR
(
β
2
≠
0 and
β
5
≠
0)
OR (
β
4
≠
0 and
β
5
≠
0)
OR (
β
2
≠
0
and
β
4
≠
0
and
β
5
≠
0 )
The null hypothesis imposes parameter restrictions on the model.
As such, the null implies what is called a Restricted Model. In
this case since the parameters
β
2
,
β
4
, and
β
5
, are being restricted to zero values, the restricted model excludes three terms from
the original model.
In other words we are left with a model with only two explanatory variables.
Restricted Model:
E(Y)
=
β
0
+
β
1
X
1
+
β
2
X
2
The model without restrictions is simply the original model and is called the Unrestricted Model.
Unrestricted Model:
E(Y)
=
β
0
+
β
1
X
1
+
β
2
X
2
+
β
3
X
3
+
β
4
X
4
+
β
5
X
5
Thus the test involves estimating two regressions; the restricted and unrestricted model. The basic idea underlying the test is to
see whether these two models are not statistically different.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 Francisco
 Statistics, Econometrics, Null hypothesis, Statistical hypothesis testing, fstat, unrestricted model

Click to edit the document details