R.M. Shrestha r Energy Economics 22 2000 297]308 300
Ž .
Ž .
where, L l ,l and L l ,l
are the maximum likelihood ratio values of the
1 2 UR
1 2 R
unrestricted and restricted models, respectively.
3. Data and their sources
Cross-sectional data from 1988 to 1990 for 41 countries were used in the estimation of output]energy relationship with GNP
and GDP as output
ME R PPP
measures while the data set involving GDP was smaller and involved 31
ME R
countries only as GDP data for some countries were obviously questionable
ME R
and were therefore excluded. The countries included in this study are listed in Appendix A.
Data on GNP per capita and GDP
per capita were obtained from UNDP
ME R PPP
Ž .
Ž .
1991, 1992, 1993 while data on GDP were taken from IMF 1993 . The source
ME R
Ž for per capita commercial energy consumption data was World Bank 1990, 1991,
. 1992 . Traditional energy consumption data for 1989 were obtained from WRI
Ž .
1992 .
4. Results
4.1. Effects of using alternati
¨
e output measures Ž
Tables 1]3 present the maximum likelihood estimates of parameters a , b, l ,
1
. l
of the generalized Box]Cox model along with the ordinary least squares
2
estimates of parameters a and b of double-log, semi-log and linear models when the output is measured by GNP
, GDP and GDP
, respectively. The
ME R MER
PPP
tables also show the associated values of R
2
, the maximum of log likelihood w
Ž .x
2
w function i.e. L l , l
and the calculated values of x test statistic as defined in
1 2
Ž .x
2
Eq. 5 . Note that R values across the models here are not directly comparable as the dependent variables are not the same for each model. Following Kennedy
Ž .
Ž .
2
1992, p. 102 and Gujarati 1995, pp. 209]210 , we have also calculated R values Ž .
Ž . for Models 1 and 4 in a manner that makes them comparable to the values
Ž . Ž .
1
calculated by Models 2 and 3 . These are also presented in the tables. As can be seen from the tables, of the three restricted models considered, the
calculated values of x
2
statistic in the case of the double-log model were less than Ž
. the critical value at the 5 significance level i.e. 5.991 for sample data of each of
Ž .
2
the 3 years considered i.e. 1988, 1989, and 1990 . However, the calculated x values were greater than the critical values at both 5 and 1 levels for both
linear and semi-log models. Thus, the double-log model could not be rejected
1
The following procedure was used in the derivation of the comparable value of R
2
. First, the estimated
ˆ ˆ
Ž .
Ž .
values of Y i.e. Y from the ‘optimal’ i.e. Box]Cox and linear models are converted into ln Y . Then
i i
i 2
ˆ
R between ln Y and ln Y is computed for each of these models.
i i
R.M. Shrestha r Energy Economics 22 2000 297]308 301
Table 1
a
Estimates of parameters of output]energy relationship with GNP as the output measure
ME R 2
2 b 2
Ž .
Ž .
Model a
b l
l R
R L l
,l x
1 2
1 2
2
1988
c
Optimal 0.721
0.959 0.059
0.088 0.880
0.880 y
364.381 Ž
. Ž
. Ž
. Ž
. 3.760
0.860 0.096
0.128 Double-log
0.033 1.120
0.879 0.879
y 364.765
0.768 Ž
. Ž
. 0.492
0.066 Semi-log
7.070 0.00044
1 0.668
0.668 y
385.518 42.274
Ž .
Ž .
0.181 0.00005
Linear 1050.0
2.580 1
1 0.696
0.722 y
401.0 73.238
Ž .
Ž .
987.0 0.273
1989
c
Optimal 1.40
0.769 0.024
0.081 0.860
0.861 y
372.064 Ž
. Ž
. Ž
. Ž
. 3.360
0.778 0.100
0.158 Double-log
y 0.054
1.140 0.859
0.859 y
372.325 0.522
Ž .
Ž .
0.549 0.074
Semi-log 7.110
0.00045 1
0.659 0.659
y 390.395
36.662 Ž
. Ž
. 0.190
0.000052 Linear
1180.0 2.80
1 1
0.658 0.681
y 408.0
71.872 Ž
. Ž
. 1190.0
0.323 1990
c
Optimal 0.903
0.901 0.041
0.081 0.858
0.858 y
376.309 Ž
. Ž
. Ž
. Ž
. 3.920
0.901 0.104
0.160 Double-log
y 0.115
1.150 0.857
0.857 y
376.576 0.534
Ž .
Ž .
0.564 0.075
Semi-log 7.190
0.00045 1
0.651 0.651
y 394.870
37.122 Ž
. Ž
. 0.194
0.000053 Linear
1360.0 3.030
1 1
0.651 0.662
y 412.0
71.382 Ž
. Ž
. 1300.0
0.355
a
Figures inside parentheses represent standard errors of the estimated parameters.
b
These figures represent the values of R
2
that are comparable to those of double and semi-log Ž
. models i.e. with log Y as the dependent variable .
c
‘Optimal’ here refers to the Box]Cox model.
irrespective of the output measures considered while both the linear and semi-log models were rejected.
Ž .
The estimated coefficient of energy consumption per capita i.e. b for the double-log model with GNP
as the output measure was found to range from
ME R
Ž .
1.12 to 1.15 all statistically significant at 5 . Note that the estimated values of b Ž
. in Table 1 are higher than those reported by MSV i.e. 0.890]0.914 . The values of
b with GDP
as the output measure varied from 0.925 to 0.934 while those in
ME R
the case of GDP were in the range 0.614]0.658.
ppp
The hypothesis of non-diminishing returns to energy per capita was not rejected at the 5 significance level for all 3 years in the GNP
case as in MSV. In the
ME R
GDP case, the hypothesis could not be rejected at the 5 significance level
ME R
R.M. Shrestha r Energy Economics 22 2000 297]308 302
Table 2
a
Estimates of parameters of output]energy relationship with GDP as the output measure
ME R 2
2 b 2
Ž .
Ž .
Model a
b l
l R
R L l
,l x
1 2
1 2
2
1988
c
Optimal y
1.050 1.700
0.123 0.055
0.950 0.950
y 258.463
Ž .
Ž .
Ž .
Ž .
5.190 1.290
0.128 0.129
Double-log 1.320
0.934 0.951
0.951 y
259.244 1.562
Ž .
Ž .
0.297 0.039
Semi-log 7.120
0.00038 1
0.714 0.714
y 286.597
56.268 Ž
. Ž
. 0.180
0.000044 Linear
1030.0 1.830
1 1
0.853 0.869
y 280.0
43.074 Ž
. Ž
. 575.0
0.141 1989
c
Optimal 0.097
1.380 0.081
0.034 0.937
0.939 y
263.039 Ž
. Ž
. Ž
. Ž
. 4.450
1.230 0.132
0.148 Double-log
1.370 0.925
0.939 0.939
y 263.340
0.602 Ž
. Ž
. 0.333
0.044 Semi-log
7.120 0.00038
1 0.716
0.716 y
287.182 48.286
Ž .
Ž .
0.182 0.000044
Linear 1040
1.820 1
1 0.825
0.860 y
283.0 39.922
Ž .
Ž .
646.0 0.156
1990
c
Optimal 0.013
1.400 0.084
0.036 0.933
0.935 y
264.484 Ž
. Ž
. Ž
. Ž
. 4.550
1.240 0.122
0.144 Double-log
1.350 0.930
0.935 0.935
y 264.793
0.618 Ž
. Ž
. 0.347
0.046 Semi-log
7.150 0.00037
1 0.706
0.706 y
288.114 47.26
Ž .
Ž .
0.184 0.000044
Linear 1130
1.82 1
1 0.810
0.847 y
285.0 41.032
Ž .
Ž .
678.0 0.164
a
Figures inside parentheses represent standard errors of the estimated parameters.
b
These figures represent the values of R
2
that are comparable to those of double and semi-log Ž
. models i.e. with log Y as the dependent variable .
c
‘Optimal’ here refers to the Box]Cox model.
except with the data set of 1989. The hypothesis in this case was, however, rejected for all 3 years at the 10 significance level.
Interestingly, in the GDP case, the hypothesis of non-decreasing output
ppp
returns to energy per capita was consistently rejected for all 3 years even at the 1 significance level. This is contrary to the finding of MSV in the GNP
case.
ME R
Clearly, output returns to energy is found to be sensitive to the choice of the output measure. Note that for the reasons stated earlier, GDP
is considered to
PP P
be a better measure of output for cross country studies and the present analysis shows the output returns to energy use to be diminishing when GDP
is used as
PP P
the output measure. It may be interesting to ask why the estimated value of b in the GDP
case is
PP P
R.M. Shrestha r Energy Economics 22 2000 297]308 303
Table 3
a
Estimates of parameters of output]energy relationship with GDP as the output measure
PP P 2
2 b 2
Ž .
Ž .
Model a
b l
l R
R L l
,l x
1 2
1 2
2
1988
c
Optimal 7.140
0.766 0.133
0.130 0.897
0.896 y
357.324 Ž
. Ž
. Ž
. Ž
. 4.710
1.090 0.209
0.135 Double-log
4.250 0.614
0.894 0.894
y 358.027
1.404 Ž
. Ž
. 0.250
0.034 Semi-log
8.10 0.00025
1 0.703
0.703 y
379.233 43.816
Ž .
Ž .
0.093 0.000026
Linear 3140.0
1.880 1
1 0.793
0.828 y
377.0 39.350
Ž .
Ž .
554.0 0.154
1989
c
Optimal 9.560
0.749 0.185
0.197 0.893
0.891 y
361.009 Ž
. Ž
. Ž
. Ž
. 7.770
0.993 0.2
0.164 Double-log
4.240 0.620
0.886 0.886
y 362.478
2.938 Ž
. Ž
. 0.266
0.036 Semi-log
8.120 0.00025
1 0.723
0.723 y
380.635 39.252
Ž .
Ž .
0.092 0.000025
Linear 3090.0
2.040 1
1 0.818
0.843 y
378.0 33.982
Ž .
Ž .
566.0 0.154
1990
c
Optimal 6.260
1.380 0.190
0.130 0.865
0.861 y
371.792 Ž
. Ž
. Ž
. Ž
. 7.070
2.0 0.226
0.172 Double-log
4.040 0.658
0.862 0.862
y 372.632
1.680 Ž
. Ž
. 0.315
0.042 Semi-log
8.190 0.00026
1 0.665
0.665 y
390.780 37.976
Ž .
Ž .
0.108 0.000029
Linear 3690.0
2.190 1
1 0.735
0.861 y
390.0 36.416
Ž .
Ž .
775.0 0.211
a
Figures inside parentheses represent standard errors of the estimated parameters.
b
These figures represent the values of R
2
that are comparable to those of double and semi-log Ž
. models i.e. with log Y as the dependent variable .
c
‘Optimal’ here refers to the Box]Cox model.
less than that in the GDP case. In most low income developing countries which
ME R
are at the lower range of energy consumption per capita, the reported values of GDP
are higher than the corresponding value of GNP while the opposite is
PP P MER
the case for most high income industrialized countries which have a higher level of Ž
. energy consumption see Fig. 1 . Similar observations can be made between
GDP and GDP
cases. This indicates that, with other things remaining the
PP P MER
Ž .
same, the estimated value of b with the ‘correct’ measure of output i.e. GDP
PP P
would be smaller than those in cases with GNP and GDP
.
ME R MER
4.2. Effects of inclusion of traditional energy consumption In the preceding analysis, like MSV, we considered only the use of commercial
R.M. Shrestha
r Energy
Economics 22
2000 297
] 308
304
Fig. 1. Output per capita vs. energy consumption per capita in 1990.
R.M. Shrestha r Energy Economics 22 2000 297]308 305
Ž .
i.e. ‘modern’ fuels in the energy input. However, it is well-known that unlike industrialized countries, developing countries have high dependence on traditional
fuels. Although the data on traditional fuel consumption are far less reliable than Ž
. the commercial energy data, especially in developing countries
a study of output]energy relationship cannot totally ignore the use of traditional fuels given
the latter’s importance in developing countries. In order to gain some insight into the order of magnitude of the effects of traditional fuel consumption, we estimated
Ž . Ž . the parameters of Eqs. 1 ] 4 with the 1989 data of 31 countries by including the
use of both commercial and traditional fuels in the input measure. The results are presented in Table 4. The log-linear model was again preferred to semi-log and
linear models. The estimated values of b in GNP GDP
and GDP cases
ME R, MER
PPP
are 1.300, 1.060 and 0.711, respectively. Two observations are interesting to make: first, for each of the output measures considered, the estimated values of b are
now consistently higher than the corresponding figures when only commercial fuels were included in energy consumption. Secondly, the hypothesis of increasing
output returns to energy consumption is rejected even at the 1 significance level in the GDP
case while it is not rejected in both GNP and GDP
cases at
PP P MER
MER
the 5 significance level. In other words, with correct measures of output and energy consumption, the results of this study give a strong indication that output
per capita exhibits non-increasing returns to energy consumption contrary to the findings of MSV.
4.3. Robustness of the estimated parameters As in MSV, the stability of the estimated parameter values over the selected
years in the study were examined using the Chow-test. The calculated values of F
in the GNP , GDP
and GDP cases are 0.2642, 0.0096 and 1.1485,
4,117 MER
MER PPP
respectively, well below the critical value of 2.45 at the 5 significance level. Thus, the hypothesis that the estimated values of parameters were robust over the years
could not be rejected in any of the cases considered. As normality of residuals is w
assumed in the Chow-test, it was examined through the Jarque]Bera test see e.g. Ž
. x
Ž Kmenta 1986 , pp. 265]267 . The values of the test statistic for this purpose which
2
. is x distributed for the data sets of 1988, 1989 and 1990 for each type of output
2
measure and commercial energy consumption are presented in Table 5. Note that
2
Ž
2
the calculated values of x are all - 5.991 i.e. the critical value of x at the 5
2 2
. significance level . Thus, the hypothesis of the normality of residuals could not be
rejected for any of the data sets.
5. Conclusions and final remarks
