Unformatted text preview: mal random variables. We can show that zj jx and wj jx are independent RVs.
Putting all these results together we conclude:
ˆ
βj βj
z
h
i=q j
t(n K 1 )
wj
ˆ
se βj jx
nK1 M elissa Tartari (Yale) Econometrics 15 / 41 Hypothesis Testing: Food for Thought I
Observe that ∑n=1 ui2 is the sum of n normal random variables.
ˆ
i
However, fewer than n are independent and this is the reason for the
degrees of freedom of the χ2 distribution. Let us verify this lack of
independence by considering the simple case in which Y = βo + U .
We know that
ˆ
ˆ
βo = y ) βo = βo + u . Melissa Tartari (Yale) Econometrics 16 / 41 Hypothesis Testing: Food for Thought I
Observe that ∑n=1 ui2 is the sum of n normal random variables.
ˆ
i
However, fewer than n are independent and this is the reason for the
degrees of freedom of the χ2 distribution. Let us verify this lack of
independence by considering the simple case in which Y = βo + U .
We know that
ˆ
ˆ
βo = y ) βo = βo + u .
We also know (recall chapter 3) that
ui
b Melissa Tartari (Yale) = ui
= ui ˆ
βo βo u. Econometrics 16 / 41 Hypothesis Testing: Food for Thought I
Observe that ∑n=1 ui2 is the sum of n normal random variables.
ˆ
i
However, fewer than n are independent and this is the reason for the
degrees of freedom of the χ2 distribution. Let us verify this lack of
independence by considering the simple case in which Y = βo + U .
We know that
ˆ
ˆ
βo = y ) βo = βo + u .
We also know (recall chapter 3) that
ui
b = ui
= ui ˆ
βo βo u. We are now in the position of deriving cov (ui , uj ) for j 6= i and if we
bb
…nd that it is nonzero we conclude that ui and uj are not
b
b
independent.
Melissa Tartari (Yale) Econometrics 16 / 41 Hypothesis Testing: Food for Thought II cov (ui , uj ) = cov (ui
bb u , uj = cov (ui , uj ) =0
=
= M elissa Tartari (Yale) 2cov u) cov (ui , u ) cov (u , uj ) + cov (u , u )
!
∑N=1 uk
+ Var (u )
ui , k
N σ2
σ2
+
N
N
σ2
6= 0.
N 2 Econometrics 17 / 41 Hypothesis Testing
Testing Simple Hypothesis Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing
Testing Simple Hypothesis
Testing Hypothesis About a Single Parameter Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing
Testing Simple Hypothesis
Testing Hypothesis About a Single Parameter
onesided (e.g. Ho : βj > 0) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing
Testing Simple Hypothesis
Testing Hypothesis About a Single Parameter
onesided (e.g. Ho : βj > 0)
twosided (e.g. Ho : βj = 0) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing
Testing Simple Hypothesis
Testing Hypothesis About a Single Parameter
onesided (e.g. Ho : βj > 0)
twosided (e.g. Ho : βj = 0)
other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing
Testing Simple Hypothesis
Testing Hypothesis About a Single Parameter
onesided (e.g. Ho : βj > 0)
twosided (e.g. Ho : βj = 0)
other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Testing Hypothesis About a Linear Combination of Parameters (e.g.
Ho : β j = β i ) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing
Testing Simple Hypothesis
Testing Hypothesis About a Single Parameter
onesided (e.g. Ho : βj > 0)
twosided (e.g. Ho : βj = 0)
other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Testing Hypothesis About a Linear Combination of Parameters (e.g.
Ho : β j = β i )
(Aside: we also review the notion of Con…dence Interval) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing
Testing Simple Hypothesis
Testing Hypothesis About a Single Parameter
onesided (e.g. Ho : βj > 0)
twosided (e.g. Ho : βj = 0)
other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Testing Hypothesis About a Linear Combination of Parameters (e.g.
Ho : β j = β i )
(Aside: we also review the notion of Con…dence Interval) Testing Joint Hypothesis Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing
Testing Simple Hypothesis
Testing Hypothesis About a Single Parameter
onesided (e.g. Ho : βj > 0)
twosided (e.g. Ho : βj = 0)
other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Testing Hypothesis About a Linear Combination of Parameters (e.g.
Ho : β j = β i )
(Aside: we also review the notion of Con…dence Interval) Testing Joint Hypothesis
that combine multiple oneparameter simple hypothesis (e.g.
Ho : βj = 0& βi = 1) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing
Testing Simple Hypothesis
Testing Hypothesis About a Single Parameter
onesided (e.g. Ho : βj > 0)
twosided (e.g. Ho : βj = 0)
other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Testing Hypothesis About a Linear Combination of Parameters (e.g.
Ho : β j = β i )
(Aside: we also review the notion of Con…dence Interval) Testing Joint Hypothesis
that combine multiple oneparameter simple hypothesis (e.g.
Ho : βj = 0& βi = 1)
about multiple linear restrictions (e.g. Ho : 2 βj βk = 0& βi = 1) Melissa Tartari (Yale) Econometrics 18 / 41 A Simple Hypothesi...
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 Fall '10
 DonaldBrown
 Econometrics, Normal Distribution, Yale, Jx, Melissa Tartari

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