G.E. Halkos, E.G. Tsionas r Energy Economics 23 2001 191᎐210 196
ity problems.
1
If indeed the EKC hypothesis is not accurately described by a quadratic but by a switching regime model, it is reasonable to expect that simple
quadratics are heavily misspecified as it turns out to be the case in our sample. Our econometric models, proposed next, are non-linear. If they are more faithful
Ž .
to the data compared to the linear models that previous work has employed then OLS applied to linear models would yield biased results. We also condition on
several economic, social and demographic factors and we try to verify the existence of an EKC conditionally on these factors.
3. Econometric models
Our modeling strategy is based on the concept of switching regimes. Switching Ž
regimes are prominent in modeling time series with a change in regime e.g. .
Hamilton, 1989, 1994, ch. 22 . Bayesian work in the field includes Albert and Chib Ž
. Ž
. Ž
. 1993 , Geweke and Terui 1993 and Muller et al. 1997 .
Two formulations will be used that are versions of a switching regime model with different assumptions about the switching process. In the first approach, we have
y s x
X
 q u ,
if z F z
U
,
i i
1 i
1 i
Ž . 1
U
y s x⬘  q u ,
if z z
, i s
1, . . . ,n
i i
2 i
2 i
Ž .
where x is a k = 1 vector containing data on exogenous variables see Section 5 ;
i
z
U
is a break point and z is a certain variable that defines the structural change.
i
Ž .
We call this model an exogenous break switching regime model EBSR . If z is
i
Ž income, then we assume in advance that income induces a break i.e. we impose
. Kuznets’ hypothesis and that z is the critical level of income. A posteriori, it is
U
Ž possible to reject the validity of this model if we find that z is too close to min z ,
i
. i s
1, . . . ,n . Second, assume that there is a separating variable S such that when S exceeds a
Ž .
certain limit which can be normalized to zero the model undergoes a structural change. In other words,
y s x
X
 q u ,
if S
i i
1 i
1 i
Ž . 2
X
y s x  q u ,
if S F
i i
2 i
2 i
where
X
Ž . S s z ␥ q
¨
, i s
1, . . . ,n 3
i i
i
Ž
2
. Ž
2
. Ž
2
. u
, u and
¨
are independent N 0, , N 0, and N 0, , respectively, x
i 1
i 2
i 1
2 i
Ž and z are k = 1 and m = 1 vectors of explanatory variables having some, none or
i
. all variables in common , y is the dependent variable and  , 
and ␥ are
i 1
2
parameter vectors conformable with x and z . Without loss of generality we may
i i
1
Results using CO emissions as dependent variable were equally poor.
2
G.E. Halkos, E.G. Tsionas r Energy Economics 23 2001 191᎐210 197
set s 1 in order to identify the ␥s. We call this model a normal separating Ž
. hyperplane switching regime model N-SHSR . Later on we will consider a Student’s
t distribution as an alternative.
The EKC hypothesis holds for this model provided income can be included in the separating function. If this is not the case, the transition from one model to the
other does not depend on the level of income, violating Kuznets’ law. One can think of the SHSR model as a generalization of the EBSR model, in the sense that
the switching point is made a stochastic function of certain explanatory variables.
The likelihood function for the SHSR, is given by
n
y 1 r2
2 X
2 2
2 2
Ž .
Ž .
Ž .
L  , ,␥, , ;y,X s
2 exp y y y x 
r 2
Ł
½
1 2
1 2
1 i
i 1
1
is 1
w Ž
X
.x =
1 y ⌽ yz ␥
i y
1 r2 2
X X
2 2
Ž .
Ž .
Ž .
Ž . q
2 exp y y y x 
r 2
⌽ yz ␥ 4
5
2 i
i 2
2 i
where
x
y 1 r2
Ž . Ž
. Ž
. Ž .
⌽ x s
2 exp ytr2 dt
5
H
y⬁
Ž .
is the cumulative distribution function cdf of a standard normal random variable, w
x w
X X
x
X
y s y
, . . . , y and X s x , . . . , x . Alternatively, it can be assumed that S is
1 n
1 n
i
distributed according to a leptokurtic distribution to take account of the cross- section heteroskedastic fluctuations of a possible separating hyperplane.
2
To that Ž .
end we adopt a Student’s t distribution with fixed degrees of freedom , in which case the cdf is given by:
w Ž
. x
x
⌫ q 1 r2
Ž .
y q 1 r2
2
Ž . Ž
. Ž .
⌽ x s
1 q u r du
6
H
1r2
Ž . Ž
. ⌫
1r2 ⌫ r2
y⬁
Ž . Ž .
As ª ⬁, ⌽ x ª ⌽ x and we get the normal distribution. The corresponding
Ž .
model is called Student’s t separating hyperplane switching regime ST-SHSR . The likelihood function for the EBSR is given by:
n
y 1 r2
2 2
2
Ž .
Ž .
L  , ,␥, , ;y,X s
2
Ł
1 2
1 2
1
U
Ž .
igA z n
y 1 r2
2 X
2 2
Ž .
Ž .
exp y y y x  r
2 2
Ł
i i
1 1
2
U
Ž .
igB z
2 X
2
Ž .
Ž . exp y y y x 
r 2
7
i i
2 2
Ž
U
.
U
4 Ž
U
.
U
4 4
where A z s i g I
: z F z and B z
s i g I : z z , I s 1,2, . . . ,n .
i i
2
When a large number of different countries are pooled together, residuals might show evidence of Ž
. long tails longer than normal tails .
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
