 |
IS BAYESIAN ANALYSIS SUPERIOR? ABENCHMARKED COMPARISON OF REGRESSION AND BAYESIAN ANALYSIS ONINCOMPLETE STATE-LEVEL DATA
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| | Unformatted Document Text:
9
convergence and signifies a counting variable between convergence points. The
standard error of b
k, ,average
is:
(
)
(
)
2
)
(
)
1
(
2
,
,
,
,
−
−
−
=
∑
+
=
λ
θ
θ
λ
θ
θ
i
k
average
k
average
k
B
b
b
se
. Convergence can
then be determined by assessing
1
)
(
,
,
,
,
*
,
−
−
−
=
λ
θ
θ
θ
θ
average
k
k
average
k
k
b
se
B
b
t
. Since )
2
(
~
*
−
−
λ
θ
t
t
,
testing the hypotheses: H
0
: b
k, ,average
= B
k
; H
1
: b
k, ,average
≠
B
k
. Convergence
9
exists when
a failure to reject H
0
occurs at
α
= 0.95 which is the same as stating with 95% confidence
that b
k, ,average
= B
k
. More instances of convergence demonstrate higher degrees of
consistency and efficiency in b
k
. Table 1 lists number of times b
k, ,average
converged to B
k
(via t
*
hypothesis testing). The number of convergences is statistically significant
Table 1: Convergence Count by Variable and Method
MCMC
OLS
Scheme
1 2 3 1 2 3
X
1
0
1
0
2
0
0
X
2
0
0
0
1
1
0
X
3
0
0
0
0
0
2
X
4
1
0
2
1
1
1
X
5
0
1
0
1
1
0
X
6
0
0
0
1
0
0
X
7
0
0
0
0
0
1
X
8
0
0
0
2
1
1
Totals
1 1 2 8*
4*
5*
*p-value < 0.05
9
Typically, hypothesis testing tests whether or not some parameter is different from some other parameter
or value. When testing hypotheses in this manner, our critical regions of rejection are very near the tails of the distribution, hence we choose a small
α
like
α
= 0.05. Since the interest is in concluding the
convergence to the population parameter, our critical regions must be very near the middle of our distribution. This is why
α
= 0.95.
|
| | Authors: Granberg-Rademacker, Scott. |
|
| |
|
|
9
convergence and signifies a counting variable between convergence points. The
standard error of b
k, ,average
is:
(
)
(
)
2
)
(
)
1
(
2
,
,
,
,
−
−
−
=
∑
+
=
λ
θ
θ
λ
θ
θ
i
k
average
k
average
k
B
b
b
se
. Convergence can
then be determined by assessing
1
)
(
,
,
,
,
*
,
−
−
−
=
λ
θ
θ
θ
θ
average
k
k
average
k
k
b
se
B
b
t
. Since )
2
(
~
*
−
−
λ
θ
t
t
,
testing the hypotheses: H
0
: b
k, ,average
= B
k
; H
1
: b
k, ,average
≠
B
k
. Convergence
9
exists when
a failure to reject H
0
occurs at
α
= 0.95 which is the same as stating with 95% confidence
that b
k, ,average
= B
k
. More instances of convergence demonstrate higher degrees of
consistency and efficiency in b
k
. Table 1 lists number of times b
k, ,average
converged to B
k
(via t
*
hypothesis testing). The number of convergences is statistically significant
Table 1: Convergence Count by Variable and Method
MCMC
OLS
Scheme
1 2 3 1 2 3
X
1
0
1
0
2
0
0
X
2
0
0
0
1
1
0
X
3
0
0
0
0
0
2
X
4
1
0
2
1
1
1
X
5
0
1
0
1
1
0
X
6
0
0
0
1
0
0
X
7
0
0
0
0
0
1
X
8
0
0
0
2
1
1
Totals
1 1 2 8*
4*
5*
*p-value < 0.05
9
Typically, hypothesis testing tests whether or not some parameter is different from some other parameter
or value. When testing hypotheses in this manner, our critical regions of rejection are very near the tails of
the distribution, hence we choose a small
α
like
α
= 0.05. Since the interest is in concluding the
convergence to the population parameter, our critical regions must be very near the middle of our
distribution. This is why
α
= 0.95.
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