Chapter 2
Properties of the regression
coefficients and
hypothesis testing
Overview
Chapter 1 introduced least squares regression analysis, a mathematical technique for fitting a relationship given
suitable data on the variables involved.
It is a fundamental chapter because much of the rest of the text is
devoted to extending the least squares approach to handle more complex models, for example models with
multiple explanatory variables, nonlinear models, and models with qualitative explanatory variables.
However, the mechanics of fitting regression equations are only part of the story.
We are equally
concerned with assessing the performance of our regression techniques and with developing an understanding of
why they work better in some circumstances than in others.
Chapter 2 is the starting point for this objective and
is thus equally fundamental.
In particular, it shows how the three criteria for assessing the performance of
estimators, namely unbiasedness, efficiency, and consistency, are applied in the context of a regression model.
Further material
Derivation of the expression for the variance of the naïve estimator in Section 2.5
The variance of the naïve estimator in Section 2.5 and Exercise 2.10 is not of any great interest in itself but its
derivation provides an example of how one obtains expressions for variances of estimators in general.
In Section 2.5 we considered the naïve estimator of the slope coefficient derived by joining the first and last
observations in a sample and calculating the slope of that line:
1
1
2
X
X
Y
Y
b
n
n
−
−
=
.
It was demonstrated that the estimator could be decomposed as
1
1
2
2
X
X
u
u
b
n
n
−
−
+
=
β
and hence that
E
(
b
2
) =
β
2
.
The population variance of a random variable
X
is defined to be
E
([
X
–
μ
X
]
2
) where
μ
X
=
E
(
X
).
Hence the
population variance of
b
2
is given by
[
]
(
)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎭
⎬
⎫
⎩
⎨
⎧
−
−
+
=
−
=
2
1
1
2
2
1
1
2
2
2
2
2
2
X
X
u
u
E
X
X
u
u
E
b
E
n
n
n
n
b
β
β
β
σ
On the assumption that
X
is nonstochastic, this can be written
October 2007

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