Both my meta-analysis of interfuel elasticities of substitution and Koetse

*et al*.'s meta-analysis of the capital-energy elasticities of substitution show that elasticity estimates are dependent on the type of data – time series, panel, or cross-section – and the estimators used. Estimates that use time series data tend to be smallest in absolute value and those using cross-section data tend to be largest.

We review the econometric research that discusses how best to get long-run elasticity estimates from panel data. One suggestion is to use the between estimator, which is equivalent to an OLS regression on the average values over time for each country, firm etc. in the panel. Alternatively, Chirinko

*et al*. (2011) argued in favor of estimating long-run elasticities of substitution using a long-run difference estimator, which is very similar to the "growth rates estimator" we have used recently.

We apply both these estimators to a Chinese dataset we have put together from both public and non-public data sources. We have data for 30 Chinese provinces over 11 years from 2000 to 2010. We estimate models for choice of fuels - interfuel substitution - and for the choice between capital, labor, and energy - interfactor substitution.

A big issue with the between estimator, which has made it relatively unpopular, is that it is particularly vulnerable to omitted variables bias. The big omitted variable in most production analysis is the state of technology. There is a lot of variation across provinces in productivity and prices and it seems that the two are correlated:

The first graph shows the price index for aggregate coal input that we constructed. Generally, coal is more expensive in Eastern China. The second graph shows an index of provincial total factor productivity, relative to Shanghai, which is the most productive province. Coastal provinces are the most productive - their distance to the technological frontier is low. To address this potential omitted variables bias, we add province level inefficiency and national technological change terms to the cost function equation. Chirinko

*et al*. (2011) instead used instrumental variables estimation, but we found that their proposed instruments in many cases have very low or negative correlations with the targeted variables. We do use instrumental variables estimation, but this is due to the endogeneity inherent in our constructed coal and energy prices indices. We use Pindyck's (1979) approach to this. We also impose concavity on the cost function, if necessary.

The results show that demand for coal and electricity in China is very inelastic, while demand for diesel and gasoline is elastic. With the exception of gasoline and diesel, there are limited substitution possibilities among the fuels. Substitution possibilities are greater between energy and labor than between energy and capital. These results seem very intuitive to us. However, they are quite different to some previous studies for China, in particular the estimates in the paper by Hengyun Ma

*et al*. (2008) Their estimates of the elasticities of substitution are negatively correlated with ours. Their study uses similar but older data, though we have improved the calculation of some variables. They use fixed effects estimation and don't impose concavity. These might be some of the reasons why our results differ. We also provide traditional fixed effects estimates with concavity imposed. These estimates are mostly close to zero. This suggests that the between and difference estimators are picking up longer-run behavior.

Which of these two estimators should we use in future? We can't give a definitive answer to that question but the difference estimator does seem to have some advantages. In particular, it allows cross-equation restrictions on the bias of technical change, which should result in better estimates of those parameters. So, that would be my first preference, though I am kind of reluctant to ignore the between variation in the data.

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