Historical data analysis
Before estimating the model described by Eq. 2, the time trend was split into three different periods. The historical development of final energy intensity (final energy use per passenger-km) for U.S. passenger travel shows a roughly constant development from 1950 to 1964, followed by a slight increase through 1978, and a subsequent decline through 2010 (Schäfer 2013). In freight transportation, energy intensity (final energy use per revenue tonne-km) has declined between 1950 and 1957 but remained roughly constant thereafter (Schäfer 2013). Taking these developments into account, an additional time trend was introduced in 1957 and another one in 1979 into Eq. 2 using dummy variables. Manual iteration for maximum R2 suggested that the first additional time trend should already be introduced in 1954 but confirmed the 1979 introduction of the second trend.
Equation 2 was estimated with ordinary least squares for the entire 1950–2010 data series; the results are shown in Table 2. All signs are consistent with economic theory and all coefficient estimates with the exception of the long-run multiplier of population are statistically significant at least at the 95 % confidence level. The adjusted R2 results in around 0.90. The residuals seem to have white noise properties (normality test) and there is no evidence of serial correlation (Portmanteau Q test).
Table 2 Parameter estimates, t-statistics (in parenthesis), and regression statistics for Eq. 2 (historical data) using annual data from 1950 to 2010
The short-run income elasticity of around 0.6 is slightly mitigated at any subsequent time step t+1 and t+2, but remains positive overall. The short-run fuel price elasticity is around −0.08. Among the dummy variables, only those associated with 1974, 1980, and 1991 turn out to be significant. Jointly, the three time trends roughly reproduce the energy intensity trends described above, i.e., a long-term decline by around 0.6 % per year with a slightly lower intermediate (1955–1978) reduction.
The rate of error correction ϕ suggests that around 8 % of the remaining deviation from equilibrium has been corrected in each subsequent year in response to any change in per person GDP, fuel price, and population. The resulting long-term effects include an income elasticity of around 3.1, with a 95 % confidence interval ranging from 1.8 to 4.4. This elasticity accounts for the increase in passenger and freight transportation with rising income, the shift towards larger and more energy intensive vehicles in passenger travel and the mode shift towards (more energy intensive) trucks in freight transportation, along with the rising relative importance of air travel in intercity transportation. Our estimated income elasticity is substantially higher than the values found in the literature for road transportation (mainly automobiles). Based on a review of around 50 studies, Goodwin et al. (2004) derive an income elasticity of 1.08 (standard deviation of 0.35). The long-run fuel price elasticity turns out to be an inelastic −0.33 and compares well with the gasoline long-run price elasticities of −0.31 and −0.38 for light-duty vehicles, estimated by Small and Van Dender (2007) using 1997–2001 data and 1966–2001 data, respectively. In addition, the associated 95 % confidence interval stretches from −0.66 to 0 and is also consistent with other estimates of long-run fuel price elasticities for passenger travel (see, e.g., Oum et al. 1990; Goodwin et al. 2004). Finally, the long-run population elasticity turns out to be 3.4. Intuition suggests that this value should be around unity: an extra person per household would most likely result in a less than proportional increase in transportation energy use as some of the additional trips are likely to be shared trips, while an extra household could lead to a roughly proportional increase in transportation energy use—an elasticity of around one. However, the lower end of the 95 % confidence interval is as small as 0.29 and the confidence interval thus includes a range of all plausible values.
Panel data analysis
Several estimation challenges exist with respect to the dynamic panel data model in Eq. 3. The error term u
i,t
can be decomposed into unobserved individual level effects (that are independent of time) and observation specific errors. First-differencing Eq. 3 then removes the unobserved individual level effects, and therefore eliminates the omitted variable bias in the model estimation. However, differencing the predetermined variables F
i,t-1
makes them endogenous because the first differenced observation specific errors are correlated with the first-differenced F
i,t-1
, thus violating the requirement for strict exogeneity. Ordinary least squares estimators would thus be biased. Therefore we use the estimator developed by Arellano and Bond (1991) that is derived from the Generalized Method of Moments and instruments those differenced variables that are not strictly exogenous with their available lags.
Table 3 summarizes the regression results of the panel data models, which were estimated with 2010–2050 data from all Groups 1–4. The sign of all coefficients is consistent with economic theory and all coefficients are highly significant. The null hypothesis of the Wald statistic (all coefficients except the constant are zero) is clearly rejected. For comparison, the long-run results estimated from Eq. 2 based on the entire time horizon 1950–2010 data are also shown.
Table 3 Parameter estimates, z-statistic (in parenthesis), and regression statistics for Eq. 3 (dynamic panel data) compared with key results from Table 2 (with t-statistics in parenthesis)
The elasticity of the lagged dependent variable (λ) of GCAM and REMIND over a 5-year period results in 0.542 and 0.504, respectively. Although both coefficients are slightly lower than that implied in the annual historical (1950–2010) dataset over 5 years of (1–0.078)5 = 0.666, thus suggesting a slightly smaller inertia compared to that observed historically, they are still within the confidence interval of the latter. The roughly comparable inertia and thus speed of error correction of the model outputs to those underlying the historical data accumulated over 5 years would imply comparing the long-run elasticities of the annual historical data to the short-run elasticities of the panel data model. However, because the long-run elasticities reflect the consumer and industry responsiveness in transportation energy use to changes in any of the RHS variables to have fully materialized, we also compare these estimates for both the historical and the projected data.
The intrinsic short- and long-run income elasticities of both energy-economy-models are significantly smaller than that observed historically, a fact we already observed when comparing the historical increase in final energy use to the projected levels of GCAM and REMIND in Fig. 1a. As already suspected from Fig. 1a, the income elasticity of REMIND (0.300[short-run: SR], 0.606[long-run: LR]) is larger than that of GCAM (0.158[SR], 0.345[LR]).
In contrast, the intrinsic short- and long-run price elasticities of both energy-economy-models are within the 95 % confidence interval of the long-run elasticity (of −0.327) estimated using the 1950–2010 time series. Thereby, the intrinsic price elasticity of the REMIND model (−0.257[SR], −0.519[LR]) turns out to be larger than that of GCAM (−0.125[SR], −0.272[LR]). The intrinsic short- and long-run population elasticities of GCAM and REMIND are also plausible and within the confidence interval of the rather large estimate of the historical data.
The estimated time coefficient of around 0.5 % autonomous reduction of final energy use per year (0.5 % for GCAM and 0.4 % for REMIND) is slightly smaller than those estimated over the historical time period of 0.6 % per annum but well within the confidence interval of the latter. (For GCAM, the estimated autonomous reduction of final energy use of 0.5 % per year is identical to the exogenously imposed autonomous reduction of final energy intensity; similarly, the inputs into the production functions underlying the REMIND model are reduced by scaling factors that change over time and the weighted average of these scaling factors results in our estimated 0.4 % decline in transportation sector final energy use). Between the two energy models, price elasticity and autonomous improvements seem to trade. Reductions of final energy use result to a larger extent from non-price induced changes in GCAM and due to a larger sensitivity to price changes in REMIND.
Figure 3 compares the 95 % confidence intervals of the estimated long-run elasticities from the 1950–2010 historical development and the short- and long-run coefficients from both GCAM and REMIND. In addition, literature survey-based 95 % confidence intervals related to road transportation (mainly automobiles) are shown, which also include other countries than the U.S. (Goodwin et al. 2004). As can be seen, the short- and long-run elasticity-based confidence intervals of the two energy-economy-models overlap with all those underlying the historical development and the literature survey-based studies, except for the income elasticity. There, the short-run elasticities of GCAM and REMIND are outside the 95 % confidence intervals of both the literature survey-based study and our historical estimate. However, these models’ intrinsic long-run elasticities still overlap with the confidence interval of the literature-based study, which itself is not within the confidence interval of our own estimate using the historical U.S. data. This apparent inconsistency may suggest that our own estimate of the long-run income elasticity with respect to final energy is at the higher end.