G.E. Halkos, E.G. Tsionas r Energy Economics 23 2001 191᎐210 198
Non-informative priors are used throughout. These priors are of the form
y 1
2 2
Ž .
Ž .
Ž . 
, ,␥, , A
8
1 2
1 2
1 2
for the SHSR model and
y 1
U U
2 2
Ž .
Ž .
Ž .
Ž . 
, ,␥, , , z A
1 k F z F n y k q
1 9
1 2
1 2
1 2
Ž . Ž
for the EBSR model, where 1 denotes the indicator function. Informative e.g.
. normal priors may be used for regression coefficients but at this stage one could
argue that use of such priors biases the results for or against the EKC hypothesis. Therefore, their use is not advisable. Model posteriors are analyzed using Markov
Ž .
chain Monte Carlo MCMC methods, as detailed next.
4. Bayesian computations
The purpose of Markov Chain Monte Carlo methods is to produce a sample of
Ž i.
4 Ž
.
Ž i.
4 draws
for the parameters of a posterior kernel Data such that converges in distribution to . The Metropolis algorithm and the Gibbs sampler
are leading numerical posterior simulators that can be used to accomplish this task. The problem in its most general form can be described as generating random
Ž .
draws from a general density x , x g X.
For the SRSF, the posterior distribution was analyzed using a random walk Markov Chain Monte Carlo method. First, in the Metropolis᎐Hastings algorithm
Ž .
Metropolis et al., 1953; Hastings, 1970; Tierney, 1994; Tsionas, 1999 consider a Ž
.
candidate transition kernel with density q x,y , x,y g X which generates potential transitions for a discrete time Markov chain evolving on X. A candidate transition
Ž .
to y generated according to the density q x,. is then accepted with probability
Ž .
␣ x ,y given by
Ž . Ž .
y q y
,x
Ž .
Ž . Ž .
␣ x ,y s min 1,
if
x q x
,y 0
½ 5
Ž . Ž .
x q x
,y
Ž .
Ž . Ž .
Ž .
␣ x ,y s 1
if x
q x ,y s 0
10 Thus, actual transitions of the Hastings chain, take place according to a law with
Ž . Ž
. Ž .
transition probability q x,y ␣ x,y y x
and a probability of remaining at the same point given by
Ž . Ž
. w Ž
.x Ž
.
r x s q x
,y 1 y ␣ x,y dy 11
H
A particularly simple approach that can implement the above method is to Ž
.
symmetric transition density q x,y in which case
Ž .
Ž . Ž .4
Ž . Ž .
␣ x ,y s min 1, y r x
if x
q x ,y 0.
G.E. Halkos, E.G. Tsionas r Energy Economics 23 2001 191᎐210 199
A convenient transition density is a uniform distribution centered at the current
state x. In our implementation the range of the transition density is adjusted every 50 passes to ensure that acceptance rates are not too high or too low. This is the
Ž .
approach originally suggested by Metropolis et al. 1953 . Ž
For the EBSR model, a Gibbs sampler Gelfand and Smith, 1989; Tanner and .
Wong, 1987 has been used. The Gibbs sampler is an iterative Monte Carlo technique for numerical posterior integration in high-dimensional Bayesian mod-
Ž .
els. For a posterior distribution y,X the Gibbs sampler starts from an arbitrary
Ž0. Ž i.
4 initial parameter vector
and produces parameter draws , i s 1, . . . , M that Ž
. converge in distribution to the posterior y,X . These random drawings are
produced as follows. For i s 1,2, . . . , M:
Ž i.
Ž
Ž iy1.
. Draw
from
,y,X
1 1
y 1
Ž i. Ž iy1.
Ž .
Draw
from
,y,X
2 2
y 2
Ž .
12 ..
.
Ž i. Ž iy1.
Ž .
Draw
from
,y,X
k k
yk
w x
where s
. . .
. . . .
yi 1
iy 1
iq 1
k
This requires that univariate conditional distributions are in a form suitable for practical random variate generation. Generally, the degree to which this can be
accomplished varies greatly from application to application. The required conditional distributions of the parameters for the EBSR model,
are as follows:
y 1
U 2
2 2
ˆ
Ž .
  , , , z ,y,X ; N  , X⬘ X
1 2
1 2
1 1
1 1
Ž .
13
y 1
U 2
2 2
ˆ
Ž .
 
, , , z ,y,X ; N  , X⬘ X
2 1
1 2
2 2
2 2
X U
U 2
2 2
Ž . Ž
. Ž
.
y y X  y y X  r
is X
n given

, , , z ,y,X
1 1
1 1
1 1
1 1
2 2
X U
U 2
2 2
Ž . Ž
. Ž
.
y y X  y y X  r
is X
n y n given
 , , , z ,y,X
2 2
2 2
2 2
2 1
2 1
Ž
U
.
U
4
where y , X are the observations corresponding to A z
s z F z
, y , X are
1 1
i 2
2 U
U U
Ž .
4 Ž
. the observations corresponding to B z
s z z and n s 噛A z , and 
i j
Ž .
j s 1,2 are obvious least squares estimators within the corresponding subsam-
ples. Finally, the conditional distribution of z is Ž
U 2
2 U
.
z  , ,␥, , ,z ,y,X A
1 2
1 2
n
y 1 r2
2 X
2 2
Ž .
Ž .
2 exp = y y y x 
r 2
Ł
1 i
i 1
1
U
Ž .
igA z n
y 1 r2
2 X
2 2
Ž .
Ž .
2 exp y y y x 
r 2
Ł
2 i
i 2
2
U
Ž .
igB z
Ž .
14
G.E. Halkos, E.G. Tsionas r Energy Economics 23 2001 191᎐210 200
With the exception of the conditional distribution of z
U
, all other conditional distributions are in standard form and random sampling is particularly easy. To get
Ž . a random draw from z . we have used a griddy Gibbs sampler in the interval
w x
Ž .
z , z
, see Ritter and Tanner 1992 .
min max
4.1. Con
¨
ergence For the SREB model we have used 100 points for the griddy random number
generator of the conditional distribution of z
U
. We have used 5000 Gibbs sampler passes with an initial burn-in phase consisting of 3000 passes. For the SRSH model,
we have used 10 000 Metropolis᎐Hastings passes with a burn-in period consisting Ž
of 7000. Convergence has been assessed using standard criteria Gelman and .
Rubin, 1992; Geyer, 1992 . For the application of this paper, convergence was found to occur in the first few hundred passes of the Markov chain Monte Carlo
samplers. For the SREB we have used an informative but locally uniform prior for ␥
. This prior is of the form Ž
y 5
. ␥
is N
0, 10 I
Results were not sensitive to the choice of prior. 4.2. Computation of a
¨
erage posterior regime probabilities For each parameter draw, the probability that a given country belongs to regime
2 can be recorded. At the end of the MCMC sampling scheme there is a number of draws from the posterior distribution of this probability. Posterior means were
computed based on this distribution. One can compute other measures of the Ž
. distribution for example standard deviations and, of course, the exact distribution
of the probability. Such information, however, cannot be presented in a satisfying manner when the number of countries is large, as in our case. Based on estimates
of the posterior mean of the regime 2 probability, regime 1 otherwise, these separation results are available from the authors upon request. What is the
important aspect of these results, is how sensitive the results are to different distributional specifications and alternative dependent variables in EKC switching
regression models. These results are presented in Section 6.1.
5. Data
