Fixed Income Securities

Statistics and Econometrics (ECO00037)

Lecture 10: Simple Regression Analysis:

Inference

Lecturer: Takashi Yamagata (Room A/EC/018)

E-mail: takashi.yamagata@york.ac.uk

O¢ ce Hour: Tuesday 9.30-11.30

Reading: Dougherty Ch2

Autumn 2015

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Review: Assumptions

yi = β1 + β2 xi + ui

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A.1 The model (above) is correctly speci…ed.

A.2 There is some variation in the regressor in the

sample

A.3 E (ui ) = 0 for all i.

A.4 The disturbance term is homoscedastic: E (ui2 ) = σ2

for all i

A.5 ui is independent of uj for all i 6= j.

A.6 (For small sample) ui is normally distributed.

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Sampling Distribution of the OLS Estimator

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Since xi are nonstochastic, Assumptions A.1-A.6 imply

that

yi

i.i.d.N β1 + β2 xi , σ2 .

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In addition, we can prove that, under Assumptions

A.1-A.6 we have

b2

therefore,

where

N [ β2 , Var (b2 )] ,

b2 β2

p

Var (b2 )

Var (b2 ) =

N (0, 1),

σ2

∑N 1 (xi

i=

x )2

¯

(1)

.

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Sampling Distribution of Standardised Estimator

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Why? Recall that

N

b2 = β2 + ∑i =1 ai ui with ai =

I

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¯

(xi x )

.

¯

(xj x )2

∑N 1

j=

Since b2 is a linear function of ui

i.i.d.N (0, σ2 ), b2

should be normally distributed.

In practice the results (1) may not be very useful since

Var (b2 ) is usually not observable.

When we replace Var (b2 ) with its estimator,

d

ˆ

Var (b2 ) = σ2 / ∑N 1 (xi x )2 , we have the following

¯

i=

result:

b2 β2

q

tN 2 .

(2)

d (b2 )

Var

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Sampling Distribution of Standardised Estimator

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N denotes the sample size, and 2 is the number of

parameters to estimate. N 2 comes from the

N

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e2

i

ˆ

denominator of the estimator of σ2 = ∑N=1 2 i .

As discussed in the previous lecture, we call the square

root of population variance ‘

standard deviation (s.d.)’

,

and estimator of the latter ‘

standard error (s.e.)’ Then,

.

we can rewrite (1) and (2) as

b2 β2

sd (b2 )

N (0,...