slide 6.1 linear hypotheses mathematicalmarketing in this chapter we will cover deductions we can...
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Slide 6.Slide 6.11Linear HypothesesLinear Hypotheses
MathematicalMathematicalMarketingMarketing
In This Chapter We Will Cover
Deductions we can make about even though it is not observed. These include
Confidence Intervals
Hypotheses of the form H0: i = c
Hypotheses of the form H0: i c
Hypotheses of the form H0: a′ = c
Hypotheses of the form A = c
We also cover deductions when V(e) 2I (Generalized Least Squares)
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Slide 6.Slide 6.22Linear HypothesesLinear Hypotheses
MathematicalMathematicalMarketingMarketing
The Variance of the Estimator
.)(ˆ 1 yXXXβ
V(y) = V(X + e) = V(e) = 2I
12
112
121
)(
)()(
])[(])[()ˆ(V
XX
XXIXXXX
XXXIXXXβ
From these two raw ingredients and a theorem:
we conclude
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Slide 6.Slide 6.33Linear HypothesesLinear Hypotheses
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What of the Distribution of the Estimator?
As
1nn1 ab
n
normal
Central Limit Property of Linear Combinations
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Slide 6.Slide 6.44Linear HypothesesLinear Hypotheses
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So What Can We Conclude About the Estimator?
])(,[N~)(ˆ 121 XXβyXXXβ
From the Central Limit Theorem
From the V(linear combo) +assumptions about e
From Ch 5- E(linear combo)
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Slide 6.Slide 6.55Linear HypothesesLinear Hypotheses
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Steps Towards Inference About
df~)q(V
)q(Eqt
kn
i
ii ~)ˆ(V
ˆ
t
In general
In particular
(X′X)-1X′y But note the hat on the V!
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Slide 6.Slide 6.66Linear HypothesesLinear Hypotheses
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Lets Think About the Denominator
thatsokn
e
kn
SSsˆ
n
i
2
iError22
ii2
i ds)ˆ(V
where dii are diagonal elements of
D = (XX)-1 = {dij}
ii2
i d)ˆ(V
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Slide 6.Slide 6.77Linear HypothesesLinear Hypotheses
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Putting It All Together
knii2
ii ~ds
ˆ
t
Now that we have a t, we can use it for two types of inference about :
Confidence Intervals
Hypothesis Testing
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Slide 6.Slide 6.88Linear HypothesesLinear Hypotheses
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A Confidence Interval for i
A 1 - confidence interval for i is given by
ii2
kn,2/i dstˆ
1dstˆdstˆPr ii2
kn,2/ii
ii2
kn,2/i
which simply means that
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Slide 6.Slide 6.99Linear HypothesesLinear Hypotheses
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Graphic of Confidence Interval
)ˆPr( i
ii2
kn,2/i dstˆ
i
1.0
0
ii2
kn,2/i dstˆ
1 -
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Slide 6.Slide 6.1010Linear HypothesesLinear Hypotheses
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Statistical Hypothesis Testing: Step One
H0: i = c
HA: i ≠ c
Generate two mutually exclusive hypotheses:
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Slide 6.Slide 6.1111Linear HypothesesLinear Hypotheses
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Statistical Hypothesis Testing Step Two
Summarize the evidence with respect to H0:
ii2
i
i
ii
ds
cˆ
)ˆ(V
ˆˆ
t
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Slide 6.Slide 6.1212Linear HypothesesLinear Hypotheses
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Statistical Hypothesis Testing Step Three
,t| ˆ| k-n /2,t
reject H0 if the probability of the evidence given H0 is small
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Slide 6.Slide 6.1313Linear HypothesesLinear Hypotheses
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One Tailed Hypotheses
Our theories should give us a sign for Step One in which case we might have
k-n ,tt
H0: i c
HA: i < c
In that case we reject H0 if
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Slide 6.Slide 6.1414Linear HypothesesLinear Hypotheses
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A More General Formulation
Consider a hypothesis of the form
H0: a´ = c
so if c = 0…
00110a
00110a
012
1
2
10a
tests H0: 1= 2
tests H0: 1 + 2 = 0
tests H0: 3
21
2
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Slide 6.Slide 6.1515Linear HypothesesLinear Hypotheses
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A t test for This More Complex Hypothesis
aXXa
aβaβa
12 )(
)ˆ(V)ˆ(V
.)(s
cˆˆ
12 aXXa
βa
t
We need to derive the denominator of the t using the variance of a linear combination
which leads to
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Slide 6.Slide 6.1616Linear HypothesesLinear Hypotheses
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Multiple Degree of Freedom Hypotheses
q
2
1
.q
.2
.1
0
1q0
c
c
c
:H
:H
β
a
a
a
cAβ
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Slide 6.Slide 6.1717Linear HypothesesLinear Hypotheses
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Examples of Multiple df Hypotheses
0
0
1000
0100:H
3
2
1
0
0
0
0
1010
0110:H
3
2
1
0
0
tests H0: 2 = 3 = 0
tests H0: 1 = 2 = 3
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Slide 6.Slide 6.1818Linear HypothesesLinear Hypotheses
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Testing Multiple df Hypotheses
)ˆ()()ˆ(SS11
HcβAAXXAcβA
kn,q
Error
H F~kn/SS
q/SS
yXX)XX(yyy 1
ErrorSS
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Slide 6.Slide 6.1919Linear HypothesesLinear Hypotheses
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Another Way to Think About SSH
0
0
1000
0100:H
3
2
1
0
0
We could calculate the SSH by running two versions of the model: the full modeland a model restricted to just 1
SSH = SSError (Restricted Model) – SSError (Full Model)
so F is
Assume we have an A matrix as below:
kn/)Full(SS
2/)Full(SS)stricted(ReSSF
Error
ErrorError
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Slide 6.Slide 6.2020Linear HypothesesLinear Hypotheses
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A Hypothesis That All ’s Are Zero
0:H *k210
kn/)Full(SS
*k/)Full(SS)tostricted(ReSSF
Error
Error0Error
)tostricted(ReSS
)Full(SS)tostricted(ReSSR
0Error
Error0Error2
If our hypothesis is
Then the F would be
Which suggests a summary for the model
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Slide 6.Slide 6.2121Linear HypothesesLinear Hypotheses
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Generalized Least Squares
f = eV-1e
yVXXVXβ 111 ][ˆ
When we cannot make the Gauss-Markov Assumption that V(e) = 2I
Suppose that V(e) = 2V. Our objective function becomes
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Slide 6.Slide 6.2222Linear HypothesesLinear Hypotheses
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SSError for GLS
kn
SSs Error2
)ˆ(V)ˆ(SS 1
Error βXyβXy
with
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Slide 6.Slide 6.2323Linear HypothesesLinear Hypotheses
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GLS Hypothesis Testing
ii
2
i
ds
cˆt
H0: i = 0 where dii is the ith diagonal element of (XV-1X)-1
H0: a = c aXVXa
a112 )(s
cˆˆ
β
t
H0: A - c = 0 )ˆ(])([)ˆ(SS 111
H cβAAXVXAcβA
kn,q
Error
H F~kn/SS
q/SS )ˆ()ˆ(SSError βXyβXy
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Slide 6.Slide 6.2424Linear HypothesesLinear Hypotheses
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Accounting for the Sum of Squares of the Dependent Variable
e′e = y′y - y′X(X′X)-1X′y
SSError = SSTotal - SSPredictable
y′y = y′X(X′X)-1X′y + e′e
SSTotal = SSPredictable + SSError
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Slide 6.Slide 6.2525Linear HypothesesLinear Hypotheses
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SSPredicted and SSTotal Are a Quadratic Forms
PyyyXXXXy 1)(
And SSTotal
yy = yIy
SSPredicted is
Here we have defined P = X(X′X)-1X′
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Slide 6.Slide 6.2626Linear HypothesesLinear Hypotheses
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The SSError is a Quadratic Form
Having defined P = X(XX)-1 X, now define M = I – P, i. e. I - X(XX)-1X.
The formula for SSError then becomes
.
][
)( 1
Myy
yPIy
PyyIyy
yXXXXyyyee
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Slide 6.Slide 6.2727Linear HypothesesLinear Hypotheses
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Putting These Three Quadratic Forms Together
SSTotal = SSPredictable + SSError
yIy = yPy + yMy
I = P + M
here we note that
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Slide 6.Slide 6.2828Linear HypothesesLinear Hypotheses
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M and P Are Linear Transforms of y
y = Py and
e = My
so looking at the linear model:
and again we see that
I = P + M
eyy ˆ
Iy = Py + My
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Slide 6.Slide 6.2929Linear HypothesesLinear Hypotheses
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The Amazing M and P Matrices
y = Py and yy ˆˆ = SSPredicted = y′Py
e = My and = SSError = y′My
What does this imply about M and P?
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Slide 6.Slide 6.3030Linear HypothesesLinear Hypotheses
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The Amazing M and P Matrices
y = Py and yy ˆˆ = SSPredicted = y′Py
e = My and = SSError = y′My
PP = P
MM = M
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Slide 6.Slide 6.3131Linear HypothesesLinear Hypotheses
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In Addition to Being Idempotent…
.nn
n1nnn1
n1nnn1
0PM
0P1
0M1