LECTURE 1
1.
Let ˆ
y
=
E
(
y

x
) be the conditional expectation of
y
given
x
. Prove that
E
{
(
y
−
ˆ
y
)
2
} ≤
E
{
(
y
−
π
)
2
}
, where
π
=
π
(
x
) is any other function of
x
.
Show that
E x
(
y
−
ˆ
y
) = 0 and give an interpretation of this condition.
Demonstrate that, if
E
(
y

x
) is a linear function of
x
, then we have
E
(
y

x
) =
E
(
y
) +
β x
−
E
(
x
)
,
where
β
=
C
(
x, y
)
/V
(
x
).
2.
A simple model of the economy comprises two equations in income
y
,
consumption
c
and investment
i
:
y
=
c
+
i,
and
c
=
α
+
βy
+
ε.
Also,
s
=
y
−
c
or
s
=
i
, where
s
is savings.
Demonstrate that the
probability limit of the ordinary leastsquares estimate of
β
the marginal
propensity to consume, which is attained as the sample size increases, is
given by
plim
ˆ
β
=
β
+
σ
2
(1
−
β
)
m
ii
+
σ
2
=
βm
ii
+
σ
2
m
ii
+
σ
2
.
When could the bias in this estimator be safely ignored and under which
circumstances is it liable to become severe?
LECTURE 2
3.
Let
P
=
X
(
X X
)
−
1
X
. Show that the conditions
P
=
P
=
P
2
, which
indicate that
P
is a symmetric idempotent matrix, are equivalent to the
condition that
P
(
I
−
P
) = 0. Prove that, if
Xb
=
Py
, then
y
−
Xb
≤
y
−
Xγ
, where
γ
is any vector of the appropriate order.
Show that, if
y
∼
(
Xβ, σ
2
I
), then
b
= (
X X
)
−
1
X y
has
E
(
b
) =
β
and
D
(
b
) =
σ
2
(
X X
)
−
1
. Prove that, if
Ay
=
β
∗
has
E
(
β
∗
) =
β
, then
V
(
q
ˆ
β
)
≤
V
(
q β
∗
), where
q
is an arbitrary non stochastic vector.
4.
Derive the leastsquares estimator of
β
in the restricted regression model
(
y
;
Xβ

Rβ
=
r, σ
2
I
) under the assumption that the matrix
X
has full
column rank. Show that the restricted estimator has the same structure
as the conditional expectation of the vector
y
given the value of a vector
x
with which it is jointly distributed:
E
(
y

x
) =
E
(
y
) +
C
(
y, x
)
D
−
1
(
x
)
{
x
−
E
(
x
)
}
.
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 Spring '12
 D.S.G.Pollock
 Linear Regression, Regression Analysis, regression model, Conditional expectation, Ordinary LeastSquares, ordinary leastsquares estimators

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