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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)
Mathematical
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Slide 6.1
Linear Hypotheses
The Variance of the Estimator
From these two raw ingredients and a theorem:
βˆ  ( XX) 1 Xy.
V(y) = V(X + e) = V(e) = 2I
we conclude
V(βˆ )  [( XX) 1 X] 2I [( XX) 1 X]
  2 ( XX) 1 XIX( XX) 1
  2 ( XX) 1
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Slide 6.2
Linear Hypotheses
What of the Distribution of the Estimator?
As n  
1
bn n a1  normal
Central Limit Property of Linear Combinations
Mathematical
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Slide 6.3
Linear Hypotheses
So What Can We Conclude About the Estimator?
From the V(linear combo) +
assumptions about e
From the Central Limit Theorem
βˆ  ( XX) 1 Xy ~ N[β,  2 ( XX) 1 ]
From Ch 5- E(linear combo)
Mathematical
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Slide 6.4
Linear Hypotheses
Steps Towards Inference About 
In general
q  E (q )
V̂ (q )
~ t df
In particular
(X′X)-1X′y
ˆ i   i
~ t n k
ˆ
V̂ ( )
But note the
hat on the V!
i
Mathematical
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Slide 6.5
Linear Hypotheses
Lets Think About the Denominator
V(ˆ i )   2  d ii
where dii are diagonal elements of
D = (XX)-1 = {dij}
n
e i2

SS
ˆ 2  s 2  Error  i
nk nk
so that
V̂(ˆ i )  s 2  d ii
Mathematical
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Slide 6.6
Linear Hypotheses
Putting It All Together
ˆ i  i
ŝ 2  d ii
~ t n k
Now that we have a t, we can use it for two types of inference about :
 Confidence Intervals
 Hypothesis Testing
Mathematical
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Slide 6.7
Linear Hypotheses
A Confidence Interval for i
A 1 -  confidence interval for i is given by
ˆ i  t  / 2,n k s 2 d ii
which simply means that


Pr ˆ i  t  / 2,n  k s 2 d ii  i  ˆ i  t  / 2, n  k s 2 d ii  1  
Mathematical
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Slide 6.8
Linear Hypotheses
Graphic of Confidence Interval
1-
1.0
Pr(ˆ i )
0
ˆ i  t  / 2,n k s 2 d ii
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i
ˆ i  t  / 2,n k s 2 d ii
Slide 6.9
Linear Hypotheses
Statistical Hypothesis Testing: Step One
Generate two mutually exclusive hypotheses:
H0: i = c
HA: i ≠ c
Mathematical
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Slide 6.10
Linear Hypotheses
Statistical Hypothesis Testing Step Two
Summarize the evidence with respect to H0:
ˆ
ˆ
ˆt   i   i   i  c
s 2 d ii
V̂(ˆ i )
Mathematical
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Slide 6.11
Linear Hypotheses
Statistical Hypothesis Testing Step Three
reject H0 if the probability of the evidence given H0 is small
| tˆ|  t /2,n-k ,
Mathematical
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Slide 6.12
Linear Hypotheses
One Tailed Hypotheses
Our theories should give us a sign for Step One in which case we might have
H0: i  c
HA: i < c
In that case we reject H0 if
tˆ  t  , n-k
Mathematical
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Slide 6.13
Linear Hypotheses
A More General Formulation
Consider a hypothesis of the form
H0: a´ = c
so if c = 0…
a  0 1  1 0  0
a  0 1 1 0  0
1 1


a  0
1  0
2
2


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tests H0: 1= 2
tests H0: 1 + 2 = 0
tests H0:
1  2
 3
2
Slide 6.14
Linear Hypotheses
A t test for This More Complex Hypothesis
We need to derive the denominator of the t using the variance of a linear combination
V(aβˆ )  aV(βˆ ) a
 2a( XX) 1 a
which leads to
tˆ 
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aβˆ  c
.
s 2a( XX) 1 a
Slide 6.15
Linear Hypotheses
Multiple Degree of Freedom Hypotheses
H 0 : Aβ  q c1
 a1. 
 c1 
a 
c 
2.
2


H0 :
β 


 
 
aq . 
cq 
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Slide 6.16
Linear Hypotheses
Examples of Multiple df Hypotheses
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0 0 1 0 
H0 : 

0 0 0 1 
0 
  0
 1   
2  0
 
3 
tests H0: 2 = 3 = 0
0 1  1 0 
H0 : 

0 1 0  1
0 
  0
 1   
2  0
 
3 
tests H0: 1 = 2 = 3
Slide 6.17
Linear Hypotheses
Testing Multiple df Hypotheses
1
SSH  ( Aβˆ  c)A( XX) 1 A ( Aβˆ  c)
SSH / q
~ Fq ,n k
SSError / n  k
SSError  yy  yX( XX) 1 Xy
Mathematical
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Slide 6.18
Linear Hypotheses
Another Way to Think About SSH
Assume we have an A matrix as below:
0 0 1 0 
H0 : 

0 0 0 1 
 0 
  0
 1   
 2  0
 
 3 
We could calculate the SSH by running two versions of the model: the full model
and a model restricted to just 1
SSH = SSError (Restricted Model) – SSError (Full Model)
so F is
F̂ 
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SSError (Restricted )  SSError (Full ) / 2
SSError (Full ) / n  k
Slide 6.19
Linear Hypotheses
A Hypothesis That All ’s Are Zero
If our hypothesis is
H 0 : 1   2     k*  0
Then the F would be
F̂ 
SSError (Restricted to  0 )  SSError (Full) / k*
SSError (Full) / n  k
Which suggests a summary for the model
R2 
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SSError (Re stricted to 0 )  SSError (Full )
SSError (Re stricted to 0 )
Slide 6.20
Linear Hypotheses
Generalized Least Squares
When we cannot make the Gauss-Markov Assumption that V(e) = 2I
Suppose that V(e) = 2V. Our objective function becomes
f = eV-1e
βˆ  [ XV 1 X]1 XV 1y
Mathematical
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Slide 6.21
Linear Hypotheses
SSError for GLS
s2 
SSError
nk
with
SSError  (y  Xβˆ )V1 (y  Xβˆ )
Mathematical
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Slide 6.22
Linear Hypotheses
GLS Hypothesis Testing
H0: i = 0
H0: a = c
H0: A - c = 0
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t̂ 
tˆ 
ˆ i  c
s 2d
ii
where dii is the ith diagonal element of (XV-1X)-1
aβˆ  c
s 2a( XV 1 X) 1 a
SSH / q
~ Fq ,n k
SSError / n  k
SS H  ( Aβˆ  c)[ A( XV 1 X) 1 A]1 ( Aβˆ  c)
SS Error  (y  Xβˆ )(y  Xβˆ )
Slide 6.23
Linear Hypotheses
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
Mathematical
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Slide 6.24
Linear Hypotheses
SSPredicted and SSTotal Are a Quadratic Forms
SSPredicted is
And SSTotal
yX(XX) 1 Xy  yPy
yy = yIy
Here we have defined P = X(X′X)-1X′
Mathematical
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Slide 6.25
Linear Hypotheses
The SSError is a Quadratic Form
Having defined P = X(XX)-1 X, now define M = I – P, i. e. I - X(XX)-1X.
The formula for SSError then becomes
ee  y y  y X( XX) 1 Xy
 y Iy  y Py
 y [I  P] y
 y My.
Mathematical
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Slide 6.26
Linear Hypotheses
Putting These Three Quadratic Forms Together
SSTotal = SSPredictable + SSError
yIy = yPy + yMy
here we note that
I=P+M
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Slide 6.27
Linear Hypotheses
M and P Are Linear Transforms of y
ŷ = Py and
e = My
so looking at the linear model:
y  yˆ  e
Iy = Py + My
and again we see that
I=P+M
Mathematical
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Slide 6.28
Linear Hypotheses
The Amazing M and P Matrices
ŷ = Py and yˆ yˆ = SSPredicted = y′Py
What does this imply about M and P?
e = My and = SSError = y′My
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Slide 6.29
Linear Hypotheses
The Amazing M and P Matrices
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ŷ = Py and yˆ yˆ = SSPredicted = y′Py
PP = P
e = My and = SSError = y′My
MM = M
Slide 6.30
Linear Hypotheses
In Addition to Being Idempotent…
1
1n n M n 1 0n
1
1n n Pn 1 0n
PM  n 0 n.
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Slide 6.31
Linear Hypotheses