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.
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 Spring '07
 Francisco
 Statistics, Econometrics, Null hypothesis, Statistical hypothesis testing, fstat, unrestricted model

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