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Hypothesis Testing in the Multiple regression
model
Testing that individual coefficients take a specific value such aszero or some other value is done in exactly the same way aswith the simple two variable regression model.
Now suppose we wish to test that a number of coefficients orcombinations of coefficients take some particular value.
In this case we will use the so called āF-testā
Suppose for example we estimate a model of the form
We may wish to test hypotheses of the form {H0:
b
=0 and
b2=0 against the alternative that one or more are wrong} or{H0:
b
=1 and
b
b
=0 against the alternative that one or more
are wrong} or {H0:
b
b
=1 and
a
=0 against the alternative that
one or more are wrong}
This lecture is inference in this more general set up.
We will not outline the underlying statistical theory for this. Wewill just describe the testing procedure.
i i i i i i i
u X b X b X b X b X b a Y
5 5 4 4 3 3 2 2 1 1
Example 1
Suppose we want to test that :
H0:
b
=0 and b2=
against the
alternative that one or more are wrong in:
The above is the
unrestricted model
The
Restricted Model would be
i i i i i i i
u X b X b X b X b X b a Y
5 5 4 4 3 3 2 2 1 1
i
i
i
i
i
u X b X b X b a Y
5 5 4 4 3 3
Example 2
Suppose we want to test that :
H
b
=1 and
b
b
against the
alternative that one or more are wrong :
The above is the
unrestricted model
The
Restricted Model would be
Rearranging we get a model that uses new variables as functionsof the old ones:
i i i i i i i
u X b X b X b X b X b a Y
5 5 4 4 3 3 2 2 1 1
i i i i i i i
u X b X b X b X b X a Y
5 5 4 4 3 2 2 2 1
i i i i i i i
u X b X b X X b a X Y
5 5 4 4 3 2 2 1
)
(
)
(
Further Definitions
Define the
U
nrestricted R
esidual R
esidual S
um of S
quares (URSS)
as the residual sum of squares obtained from estimating theunrestricted model.
Define the
R
estricted R
esidual R
esidual S
um of S
quares (RRSS)
as
the residual sum of squares obtained from estimating the restrictedmodel.
Note that according to our argument above
Define the
degrees of freedom
as
N-k
where
N
is the sample size and
k
is the number of parameters estimated in the unrestricted model (I.e
under the alternative hypothesis)
Define by
q
the number of restrictions imposed (in both our examples
there were two restrictions imposed
URSS
RRSS
ā„
The F-Statistic
The Statistic for testing the hypothesis we discussed is
The test statistic is always positive. We would like this to beāsmallā. The smaller the
F
-statistic the less the loss of fit due to
the restrictions
Defining āsmallā and using the statistic for inference we need toknow its distribution.
)
/(
/
)
(
K
N
URSS
q
URSS
RRSS
F
ā
ā
=
Since the smaller the test statistic the better and since the teststatistic is always positive we only have one critical value.
For a test at the
level of significance we choose a critical
value of
If the test statistic is below the critical value we accept the nullhypothesis.
Otherwise we reject.
Ī±
k
N
q
F
Ī±
Examples
Examples of Critical values for 5% tests in a regression modelwith 6 regressors under the alternative
- Sample size 18. One restriction to be tested: Degrees of
freedom 1, 12:
- Sample size 24. Two restrictions to be tested: degrees of
freedom 2, 18:
- Sample size 21. Three restrictions to be tested: degrees of
freedom 3, 15:
75
.
4
)
12 ,
1 ( ,
05 .
0
1
=
ā
F
55
.
3
=
F
)
15 ,
3 ( ,
05 .
0
1
ā
F
Examples
Examples of Critical values for 5% tests in a regression modelwith 6 regressors under the alternative. Inference based on large samples:
- One restriction to be tested: Degrees of freedom 1. :ā Two restrictions to be tested: degrees of freedom 2:ā Three restrictions to be tested: degrees of freedom 3:
84
.
3
=
Ļ
99
.
5
=
Ļ
81
.
7
=
Ļ
.
regr lbp lpbr lpsmr lryae ltba lrma
Source |
SS
df
MS
Number of obs =
F( 5,
Model |.
Prob > F
Residual |.
R-squared
Adj R-squared = 0.
Total |.
Root MSE
log butter purchases
lbp |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+----------------------------------------------------------------log price of butter
lpbr | -.
log price of margarine
lpsmr |
log real income
lryae |
log butter advertising
ltba | -.
log margarine advertising
lrma | -.
Constant
_cons |
Unrestricted Model
Example: The Demand for butter:
Hypothesis to be tested
: Butter and margarine advertising do not change
demand and income elasticity of butter is one:
Three restrictions
The Test
The value of the test statistic is
The critical value for a 5% test wit (3,45) degrees of freedom is2.
We accept the null hypothesis
since 0.71<2.81.
F
A Large sample example: Testing for seasonality in
fuel expenditure
Alternative form of the F-statistic using the R
squared
So long as the Total sum of squares is kept the same betweenmodels we can also write the F-statistic as
where U refers to the unrestricted model and R to the restrictedmodel
This will not work if we compute the R squared with differentdependent variables in each case (e.g. because oftransformations.
k
N
R
q
R
R
F
U
R
U
Heteroskedasticity
Heteroskedasticity means that the variance of the errors is notconstant across observations.
In particular the variance of the errors may be a function ofexplanatory variables.
Think of food expenditure for example. It may well be that theādiversity of tasteā for food is greater for wealthier people thanfor poor people. So you may find a greater variance ofexpenditures at high income levels than at low income levels.
