D.K. Christopoulos r Energy Economics 22 2000 569]586 575
L L
L L
TC
M s
F M
q a q g
ln P q
d lnY
q m
T
Ý Ý
Ý Ý
i i k
i ,tyk
i i j
j ,tyk
i j tyk
T i tyk
i ,ks1
j ,ks0
j ,ks0
j ,ks0
Ž .
12 where in our application only the lag structure, k s 1 will be used.
L
Ž .
The equations in the system 12 can be transformed by subtracting F
M
Ý
k t
ks 1
Ž . Ž
. from both sides of 12
Wickens and Breusch, 1988; Kesavan et al., 1993 . By algebrical manipulation we obtain
L L
TC TC
M s
F D M
q a q g
ln P q d lnY q m T y b D
ln P
Ý Ý
Ý
i i k
k i
i i j
j i Y
T i i j
k j
i ,ks1
j j
,ks1
Ž .
y b D lnY y b D T
13
i Y k
iT k
where D refers to annual differences and i s K,L,E. Ž
. In a similar way we transform the share Eq. 10 derived from the aggregator
energy function, obtaining
L L
E E
Ž .
M s F
D M q b q
g ln P y
b D ln P
14
Ý Ý
Ý
i i
,K K
i i
i j j
i j K
j
i ,ks1
j j
,ks1
i s EL,D,M.
4. Statistical estimation
4.1. The estimation of the sub-energy model Ž
. Empirical implementation requires that the energy share Eq. 14 should be
statistically specified. Consequently, an error term is added to each equation because of lags of cost in response to the changes in exogenous variables. Since the
Ž .
Ž shares of the three energy inputs EL,D,M always sum to unity adding-up
E
. criterion
M s 1, i s EL,D,M the sum of the disturbances across the three
Ý
i
i
share energy equations is zero at each observation. This implies a singular disturbance covariance matrix. Therefore, in order to avoid singularity, one share
Ž .
equation must be dropped the equation of crude oil is deleted . Following Ž
. Anderson and Blundell 1982 for the invariance of results due to arbitrary
deletion of one equation the restriction that the adjustment coefficients must be equal across equations together with the one for long-run symmetry is considered
Ž .
for the general dynamic form 14 . Ž
. Non-linear iterative Zellner estimation see Zellner, 1962, 1963 is used to
Ž .
estimate the parameters of the dynamic model 14 . This procedure, which is Ž
. equivalent to maximum likelihood estimation Kmenta and Gilbert, 1968 , assures
D.K. Christopoulos r Energy Economics 22 2000 569]586 576
Table 1 Tests of dynamic structure for the energy-sub model
2 2
2 2
x Number of
x x
x
0.95 0.90
0.75
restrtion Partial adjustment
13.890 4
9.488 7.779
5.385 Static
12.428 5
11.070 9.236
6.625
that the estimates will be invariant to which the equation is deleted only under the Ž
acceptance of null hypothesis of error correlations across equations see Berndt .
and Savin, 1975; Christopoulos, 1995 . For the reasons presented above, the IZEF method is adopted for estimating the
Ž .
two dynamic equations in 14 , using annual data from the Greek manufacturing sector for the period 1970]1990.
4
Also, its partial adjustment and the static Ž
. specifications of the general dynamic form 14 are estimated and compared
Ž .
5
against the general specification 14 . Starting with the statistical comparison between the general dynamic, the partial
Ž .
adjustment and the static specification of the dynamic share energy system 14 , the likelihood ratio test of its dynamic structure is computed. The results are presented
in Table 1. The x
2
results in the second column of Table 1 reveal that the general dynamic Ž
. specification of the model 14 is superior to its corresponding partial adjustment.
Also, the general dynamic specification nests that of static at the 5 level of significance. Therefore, the dynamic specification is adopted to estimate and
analyse the own and cross price elasticities of demand for energy components. Parameter estimates for this model are presented in Table 2.
Next the own and cross price elasticities are estimated to measure the price Ž .
responsiveness. It can be shown, that for the translog energy cost function 9 Ž
. under the dynamic specification of share Eq. 14 , these estimates can be calcu-
lated as follow: g
i i E
ˆ
E s q M y
1 i s
EL,D,M
i i i
E
ˆ
M
i
Ž .
15 g
i j E
ˆ
E s q M
i , j s EL,D,M
i j j
E
ˆ
M
i
where i j
4
For the nature and the structure of the variables see Appendix A.
5
The partial adjustment specification results from the imposition of the restrictions b s 0 on the
i j
parameters of the dynamic specification. The restrictions required to obtain the static model are the partial adjustment restriction with the addition of F
s 1.
i K
D.K. Christopoulos r Energy Economics 22 2000 569]586 577
Table 2 Parameter estimates of translog-energy sub model for Greek manufacturing with homogeneity and
a
symmetry imposed in the long-run: 1970]1990 Ž
. Ž
. Equation I electricity
Equation II diesel Parameters
Coefficients Parameters
Coefficients
U U
Y 0.669
Y 0.038
EL D
Ž .
Ž .
0.031 0.004
U
Y 0.232
Y y
0.0126
EL EL DD
Ž .
Ž .
0.057 0.0110
UUU U
Y 0.0205
Y 0.0205
EL D DEL
Ž .
Ž .
0.0104 0.0104
U U
F 0.807
F 0.807
EL D
Ž .
Ž .
0.186 0.186
b y
0.029 b
y 0.004
EL EL DD
Ž .
Ž .
0.130 0.018
b 0.164
b 0.008
EL D DEL
Ž .
Ž .
0.149 0.016
2 2
R 0.404
R 0.565
SSR 0.103
SSR 0.002
OBS 19
OBS 19
Ž . Ž .
Q 5
10.042 Q
5 1.151
Ž .
Ž .
Q 10
13.348 Q
10 1.947
a
The numbers in parentheses are the standard errors.
U
,
UUU
Indicate statistical significance at 1 and 10, respectively. Q denotes the Box]Pierce Q statistic for serial correlation in the residuals. The
figures in parentheses are the degrees of freedom for x
2
statistics. The 5 critical values are 11.070 and 18.307, respectively, for 5 and 10 d.f.s.
Ž .
Ž .
6
The estimates of own E and cross price E
elasticities evaluated at the
i i i j
mean values in the period 1970]1990 are presented in Table 3. It can be seen from Table 3 that all long-run own price elasticites are negative
except for crude oil. However, the only result that appears to be statistically different from zero is that for diesel. For this energy component the estimated
Table 3
a
Estimates of own and cross long-run price elasticities Ž
. Ž .
Ž . Electricity EL
Diesel D Crude oil M
U
Ž .
Ž .
Ž .
Ž .
Electricity EL y
0.02 0.10 1.12
0.27 y
0.07 0.15
U U
Ž . Ž
. Ž
. Ž
. Diesel D
0.08 0.02
y 1.29
0.29 0.02 0.03
Ž . Ž
. Ž
. Ž
. Crude oil M
y 0.05 0.11
0.19 0.32 0.05 0.15
a
The effect of a change in the price of electricity on each of the components is contained in the first row, etc. The figures in parentheses are the standard errors.
U
, denotes significance at 1.
6
The long-run own and cross price elasticities for energy components are derived under the assumption that the total energy cost is held constant.
D.K. Christopoulos r Energy Economics 22 2000 569]586 578
long-run own price elasticity is higher, in absolute value, than unity. This means that the demand for diesel is very responsive to a change in its own price. So, an
increase in the price of diesel by 10 will decrease the amount of diesel demanded by 12.9.
Turning to long-run cross price elasticities, the results suggest that they are positive but statistically zero, apart from cross price elasticities of diesel and
electricity which also exhibit a positive sign but, are statistically different from zero at the 1 level. The size of substitution between electricity and diesel is high
7
with respect to the price of electricity and low with respect to the price of diesel. This
means that a 5 increase in price of electricity, given the price of other inputs, will lead to a 5.6 increase in the relative share of diesel whereas a 5 increase in
price of diesel, given the price of other inputs, will lead to a 0.4 increase in the relative share of electricity.
Overall the results suggest that the demand for diesel, holding constant the total energy cost, is highly sensitive to price movement, while for electricity and crude oil
the high standard errors make it impossible to draw any conclusion about their price responsiveness. Finally, there is no evidence for inter-energy substitution
except for high level of substitutability of diesel for electricity.
Next the aggregate total cost is considered in order to determine the demand for total energy.
4.2. Estimation of the total cost function Before proceeding to the estimation of the general dynamic system of share Eq.
Ž .
13 the aggregate price index for energy has to be generated. For this reason the Ž
. estimated long-run parameters from 14 are used as starting values for the energy
Ž . Ž .
cost function 9 and then the energy cost function 9 is estimated using maximum
ˆ
Ž .
likelihood techniques. Thus, an aggregate price index P is obtained which serves
E
Ž .
as an instrumental variable for the price of energy P in the estimation of the
E
Ž .
dynamic system 13 of the shares of total cost. Ž
. The test for dynamic structure of 13 is shown in Table 4.
The results in Table 4 suggest that the general dynamic specification nests such of partial adjustment as the static at the 25 level of significance while the static
specification is also nested by the general dynamic at the 10 level of significance.
Table 4 Tests of dynamic structure for the total cost share equations
2 2
2 2
x Number of
x x
x
0.95 0.90
0.75
restrtions Partial adjustment
9.187 6
12.592 10.644
7.841 Static
12.173 7
14.067 12.017
9.037
7
Ž .
This result is in accordance with the findings of a previous paper see Palaskas et al., 1999a .
D.K. Christopoulos r Energy Economics 22 2000 569]586 579
Table 5 Parameter estimates of translog total cost function for the Greek manufacturing with homogeneity and
a
symmetry imposed in the long-run: 1970]1990 Ž
. Ž
. Equation I capital
Equation II labour Parameters
Coefficients Parameters
Coefficients
U U
a 0.652
a 0.289
K L
Ž .
Ž .
0.013 0.010
U U
Y 0.169
Y 0.162
KK LL
Ž .
Ž .
0.014 0.021
U U
Y y
0.128 Y
y 0.128
KL LK
Ž .
Ž .
0.014 0.014
U UUU
d y
0.107 d
0.051
YK LY
Ž .
Ž .
0.036 0.027
U U
m 0.009
m y
0.006
TK LT
Ž .
Ž .
0.001 0.001
U U
F 0.559
F 0.559
K L
Ž .
Ž .
0.141 0.141
U U
b 0.116
b 0.099
KK LL
Ž .
Ž .
0.038 0.029
UU U
b y
0.080 b
y 0.105
KL LK
Ž .
Ž .
0.031 0.031
b 0.012
b y
0.028
KY LY
Ž .
Ž .
0.077 0.059
2 2
R 0.962
R 0.944
SSR 0.002
SSR 0.0009
OBS 19
OBS 19
Ž . Ž .
Q 5
3.345 Q
5 5.149
Ž .
Ž .
Q 10
5.778 Q
10 7.766
a
The numbers in parentheses are the standard errors.
U
,
UU
,
UUU
Indicate statistical significance at 1, 5 and 10, respectively. Q denotes the Box]Pierce Q statistic for serial correlation in the
residuals. The figures in the parentheses are the degrees of freedom for x
2
statistics. The critical values at the 5 level are 11.070 and 18.307, respectively, for 5 and 10 d.f.s.
Parameter estimates using the IZEF method for the general dynamic specification are presented in Table 5.
Ž .
Ž .
Using the parameter estimates from 13 and the formulas 15 the long-run own and cross price elasticities for the total cost function were computed. The results
are tabulated in Table 6. As seen from Table 6 all own point elasticities for the aggregate factors are
negative, as one would expect theoretically. However, only the capital and labour elasticities are significant at conventional levels of statistical significance. The
common feature of these elasticities is that they are inelastic with the price of labour more elastic than the price elasticity of capital, as would be expected, given
the gestation period required for capital investment. The estimated own demand price elasticity for aggregate energy is not statistical different from zero. An
obvious interpretation of this result is that the demand for aggregate energy
D.K. Christopoulos r Energy Economics 22 2000 569]586 580
Table 6
a
Own and cross long-run price elasticity estimates Ž .
Ž . Ž .
Capital K Labour L
Energy E
U U
UU
Ž . Ž
. Ž
. Ž
. Capital K
y 0.15
0.03 0.17
0.04 0.14
0.07
U U
Ž . Ž
. Ž
. Ž
. Labour L
0.13 0.03
y 0.19
0.06 0.02 0.10
Ž . Ž
. Ž
. Ž
. Energy E
0.03 0.02 0.005 0.03
y 0.19 0.17
a
The effect of a change in the price of capital on each of the other inputs is contained in the first row, etc. The figures in parentheses are the standard errors.
U
,
UU
denote significance at 1 and 5, respectively.
remains invariant to change in its price. This might be attributed to the fact that the share of energy did not gain a dominant share in the total cost of production.
Regarding the cross price elasticities, our results strongly suggest that the possibilities of substitution between capital, labour and energy in Greek manufac-
turing are extremely limited. This finding is also confirmed by the computed Allen-Uzawa partial elasticities of substitution. These are s
s 0.33, s
s 0.25
KL KE
and s s
0.05. It can therefore be concluded that the Greek entrepreneur is
LE
faced with difficulties in obtaining the best combination of inputs, which minimise total cost. For example an increase in the price of aggregate energy will not
Ž decrease considerably the amount of energy the own price elasticity of energy is
. statistically zero and as a result will not increase considerably the demand for
Ž .
Ž .
labour E s
0.005 and the demand for capital E s
0.03 .
LE KE
Finally to complete our analysis the total own and cross price elasticity are Ž
. calculated for each energy component see Fuss, 1977 .
Y E
E Y
Ž .
E s E q M ? E i
, j s K,L,E. 16
i j i j
j EE
where E
Y
the cross price elasticity of demand for energy component i with respect
i j E
Ž . Ž
to P }
Y held constant; E is the elasticity with energy cost E constant from
E j i j
.
E Y
Table 3 ; M is the defined share in the energy-sub model, and E
is the own
j EE
price elasticity of aggregate energy. The results are presented in Table 7. It is noteworthy that these total own and
cross price elasticities take into account that a change in the price of an energy
ˆ
Ž .
component also causes a change in the aggregate price energy index P . This
E
results in substitution between energy and other aggregate factors which affects the Ž
. demand for the energy component Fuss, 1977 .
The results in Table 7 suggest that changes in the price of energy components and to some extent in the price of aggregate energy do not affect substantially the
demand for energy component, apart from diesel. It can therefore be concluded that in Greek manufacturing the level of substitution among energy components is
low and is associated with a low substitution of aggregate energy for other non-energy aggregate inputs.
D.K. Christopoulos r Energy Economics 22 2000 569]586 581
Table 7 Total energy price elasticities
Ž .
Ž . Ž .
Electricity EL Diesel D
Crude oil M Ž
. Electricity EL
y 0.11
1.01 y
0.18 Ž .
Diesel D 0.07
y 1.30
0.11 Ž .
Crude oil M y
0.13 0.11
y 0.03
4.3. Cointegration analysis Decisions on factors of production may be confounded by growth in manufactur-
ing itself. In other words, there is a ‘spurious regression’ issue in connection with estimation of energy demand systems. A cointegration issue arises and the question
is, whether or not the estimated relationships are meaningful in the long-run.
8
Ž .
Following the cointegration approaches of Engle and Granger 1987 and Philips Ž
. and Ouliaris 1990 , we have performed unit root tests on the residuals from the
demand equations for the various types of energy, that is electricity, diesel and crude oil, and on the residuals from the demand equations for aggregate inputs, i.e.
capital, labour and energy. Following standard practice, we have used the aug-
Ž .
Ž .
mented Dickey]Fuller ADF test Dickey and Fuller, 1981 . If the ADF test indicates the presence of a unit root, demand system residuals are non-stationary
implying that the estimated relationships are not structural. The results in Table 8 suggest that residuals do not contain a unit root implying
Ž . that these series are I 0 . Therefore, there is evidence to support that the
estimated relationships are indeed structural and not spurious. 4.4. Structural stability
Since there are several epochs of energy prices in the data, an issue of structural stability arises. In other words, model parameters may have changed from epoch to
epoch. Full-blown Chow tests involving hypotheses about structural stability of all parameters are impossible to implement, because of the small number of observa-
tions. One can, however, examine the structural stability of the constant term by
Ž testing the significance of dummy variables one following 1973 and one following
. 1979 in the estimated relationships. These dummies turn out to be insignificant
Ž and, therefore, we do not have evidence against the stability of the model the
computed x
2
for the sub-energy model is 11.63 and for the aggregate model 10.10, .
while the critical value for 4 d.f. at the 1 level of statistical significance is 13.277 An independent test of model stability is provided by examining the forecasting
ability of the model. If the model were subject to structural changes its forecasting performance should deteriorate. From an inspection of Figs. 1 and 2, it turns out
that the dynamic forecasting performance of the model is excellent in the entire
8
This issue and the one in Section 4.4 was brought to my attention by an anonymous referee.
D.K. Christopoulos r Energy Economics 22 2000 569]586 582
Table 8
a
Unit root test Equation residual
Augmented Dickey]Fuller test
Sub-energy model Electricity
y 3.197
Diesel y
3.334 Crude oil
y 2.811
Aggregate model Capital
y 5.033
Labour y
3.752 Energy
y 3.275
a
Ž .
The critical value of the ADF statistic with no constant and no time trend is y1.96 at the 5 level. Number of lags was selected optimally using the Schwarz criterion. The ADF regression is
performed without constant term because it is known that residuals have mean zero.
sample. This provides additional evidence in favour of the model’s structural stability
5. Concluding observations
