220222 b var b ? b var c 1 b 1 c 2 b 2 c 2 1 b var b

2.20–2.22
b
Var
b
λ
=
b
Var
(
c
1
b
1
+
c
2
b
2
)
=
c
2
1
b
Var
(
b
1
) +
c
2
2
b
Var
(
b
2
) +
2 c
1
c
2
b
Cov
(
b
1
, b
2
)
, (3.9)
David Atkin (UCLA) Lecture Note 4C: Hypotheses Testing of Linear Combination of Parameters Spring, 2015 5

The standard error of
b
λ is the square root of the estimated variance
se
b
λ
=
se ( c
1
b
1
+
c
2
b
2
)
=
q
b
Var ( c
1
b
1
+
c
2
b
2
) (3.10)
David Atkin (UCLA) Lecture Note 4C: Hypotheses Testing of Linear Combination of Parameters Spring, 2015 6

If in addition SR6 holds, or if the sample is large, the least squares
estimators b
1
and b
2
have normal distributions.
It is also true that linear combinations of normally distributed
variables are normally distributed, so that
b
λ
=
c
1
b
1
+
c
2
b
2
∼
N λ , Var
b
λ .
David Atkin (UCLA) Lecture Note 4C: Hypotheses Testing of Linear Combination of Parameters Spring, 2015 7

We can estimate the average (or expected) expenditure on food as:
[
Food Exp
=
83.42
+
10.21
·
Income
If the household income is $ 2, 000, which is 20 since income is
measured in $ 100 units in this example, then the average expenditure
is:
b
E ( Food Exp | Income = 20 ) = b
1
+ b
2
20
=
83.42
+
10.21
·
20
=
287.61
We estimate that the expected food expenditure by a household with
$ 2, 000 income is $ 287.61 per week
David Atkin (UCLA) Lecture Note 4C: Hypotheses Testing of Linear Combination of Parameters Spring, 2015 8

The t -statistic for the linear combination is:
t
=
b
λ
-
λ
r
b
Var
b
λ
=
b
λ
-
λ
se
b
λ
=
( c
1
b
1
+
c
2
b
2
)
-
( c
1
β
1
+
c
2
β
2
)
se ( c
1
b
1
+ c
2
b
2
)
∼ t
(
N
-
2
)
David Atkin (UCLA) Lecture Note 4C: Hypotheses Testing of Linear Combination of Parameters Spring, 2015 9

Substituting the t value into Pr
( -
t
c
≤
t
≤
t
c
) =
1
-
α , we get:
Pr
((
c
1
b
1
+
c
2
b
2
) -
t
c
·
se
(
c
1
b
1
+
c
2
b
2
) ≤
( c
1
β
1
+
c
2
β
2
)
≤
( c