in this chapter we will cover
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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 H 0 : i = c Hypotheses of the form H 0 : i c Hypotheses of the form H 0 : a ′ = c Hypotheses of the form A = c - PowerPoint PPT PresentationTRANSCRIPT
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Slide 6.Slide 6.11Linear HypothesesLinear Hypotheses
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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
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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