It is
not
about the probability of
β
1
being any particular value.
β
1
is not a random variable. It is some unknown number. The
data is what is random. In particular, the pvalue is
not
the
probability that
H
0
is false given the data.
Hypothesis testing: we must make a decision (usually reject or
fail to reject
H
0
)
Choose significance level
α
(usually 0.05 or 0.10)
Construct procedure such that if
H
0
is true, we will incorrectly
reject with probability
α
Reject null if pvalue less than
α
Simple Regression
January 8, 2018
57 / 74
Growth and GDP
Simple Regression
January 8, 2018
58 / 74
Caution: economic significance
6
=
statistical
significance
Simple Regression
January 8, 2018
59 / 74
Other issues: Reporting Results
There are two formats to report results
1
Table
Appropriate for multiple Specifications
Refer ”Growth and GDP” table
2
Equation
\
Growth
= 0
.
42
(0
.
96)
+ 0
.
09
(0
.
25)
yearsschool
n
= 65
,
R
2
= 0
.
11
, Adj

R
2
= 0
.
1
Simple Regression
January 8, 2018
60 / 74
Other issues: Regression Interpretations
1
Semilogarithmic form
log(
wage
) =
β
0
+
β
1
educ
+
Percentage
change in
Y
associated with
unit
change in
X
β
1
=
∂
log(
wage
)
∂educ
=
∂
log(
wage
)
wage
∂educ
2
Loglogarithmic form
log(
salary
) =
β
0
+
β
1
log(
sales
) +
Percentage
change in
Y
associated with
percentage
change in
X
β
1
=
∂
log(
salary
)
∂
log(
sales
)
=
∂
log(
salary
)
salary
∂
log(
sales
)
sales
Simple Regression
January 8, 2018
61 / 74
Estimating
σ
2
I
Recall that
Var(
ˆ
β

x
1
, ..., x
n
) =
σ
2
∑
n
i
=1
(
x
i

¯
x
)
2
=
σ
2
n
d
Var(
x
)
σ
2
unknown
We estimate
σ
2
using the residuals,
ˆ
σ
2
=
1
n

2
n
X
i
=1
ˆ
2
i
{z}
=(
y
i

ˆ
β
0

ˆ
β
1
x
i
)
2
If SLR.1SLR.5,
E[ˆ
σ
2
] =
σ
2
Using
1
n

2
instead of
1
n
makes
ˆ
σ
2
unbiased
ˆ
i
depends on 2 estimated parameters,
ˆ
β
0
and
ˆ
β
1
, so only
n

2
degrees of freedom
Simple Regression
January 8, 2018
62 / 74
Estimating
σ
2
II
Estimate
Var(
ˆ
β
1

x
1
, ..., x
n
)
by
d
Var(
ˆ
β
1

x
1
, ..., x
n
) =
ˆ
σ
2
∑
n
i
=1
(
x
i

¯
x
)
2
=
1
n

2
∑
n
i
=1
ˆ
2
i
∑
n
i
=1
(
x
i

¯
x
)
2
Standard error of
ˆ
β
1
is
q
d
Var(
ˆ
β
1

x
1
, ..., x
n
)
If SLR.1SLR.6, tstatistic with estimated
d
Var(
ˆ
β
1

x
1
, ..., x
n
)
has
a
t
(
n

2)
distribution instead of
N
(0
,
1)
t
=
ˆ
β
1

β
1
q
d
Var(
ˆ
β
1

x
1
, ..., x
n
)
∼
t
(
n

2)
Simple Regression
January 8, 2018
63 / 74
Confidence intervals I
ˆ
β
1
is random
d
Var(
ˆ
β
1
)
, pvalues, and hypthesis tests are ways of expressing
how random is
ˆ
β
1
Confidence intervals are another
A
1

α
confidence interval,
CI
1

α
= [
LB
1

α
, UB
1

α
]
is an
interval estimator for
β
1
such that
P(
β
1
∈
CI
1

α
) = 1

α
(
CI
1

α
is random;
β
1
is not)
Recall: if SLR.1SLR.6, then
ˆ
β
1
∼
N
β
1
,
Var(
ˆ
β
1
)
Simple Regression
January 8, 2018
64 / 74
Confidence intervals II
Implies
P
ˆ
β
1
< β
1
+
q
Var(
ˆ
β
1
)Φ

1
(
α/
2)
=
α/
2
P
ˆ
β
1

q
Var(
ˆ
β
1
)Φ

1
(
α/
2)
< β
1
)
=
α/
2
and
P
ˆ
β
1
> β
1
+
q
Var(
ˆ
β
1
)Φ

1
(1

α/
2)
=
α/
2
P
ˆ
β
1

q
Var(
ˆ
β
1
)Φ

1
(1

α/
2)
> β
1
=
α/
2
Simple Regression
January 8, 2018
65 / 74
Confidence intervals III
so
P
ˆ
β
1
+
q
Var(
ˆ
β
1
)Φ

1
(
α/
2)
< β
1
β
1
<
ˆ
β
+
q
Var(
ˆ
β
1
)Φ

1
(1

α/
2)
=
=1

P
ˆ
β
1
+
q
Var(
ˆ
β
1
)Φ

1
(
α/
2)
< β
1


P
ˆ
β
1
+
q
Var(
ˆ
β
1
)Φ

1
(1

α/
2)
> β
1
=1

α
For
α
= 0
.
05
,
Φ

1
(0
.
025)
≈ 
1
.
96
,
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