This
paper provides methods for assessing the precision of cost elasticity estimates
when the underlying regression function is assumed to be polynomial.
Specifically, the paper adapts two well-known methods for computing
confidential intervals for ratios: the delta-method and the Fieller method. We
show that performing the estimation with mean-centered explanatory variables
provides a straightforward way to estimate the elasticity and compute a
confidence interval for it. A theoretical discussion of the proposed methods is
provided, as well as an empirical example based on publicly available postal
data. Possible areas of application include postal service providers worldwide,
transportation and electricity.
References
[1]
United States Postal Service (2014) Report on the City Carrier Street Time Study. Postal Regulatory Commission, Washington DC, 25-26.
https://www.prc.gov/docs/90/90869/prop.13.city%20carrier.report.pdf
[2]
Bradley, D.M. (2016) Research on Estimating the Variability of Purchased Highway Transportation Capacity with Respect to Volume. Postal Regulatory Commission, Washington DC, 2
https://www.prc.gov/docs/96/96940/Research.Report.Proposal.Four.pdf
[3]
Bradley, D.M. (2019) A New Study of Special Purpose Route Carrier Costs. Postal Regulatory Commission, Washington DC.
https://www.prc.gov/docs/109/109484/spr.public.study.report.pdf
[4]
United States Postal Service (2012) Postal Service Report Regarding Cost Studies: Response to PRC Order No. 1626 (April 18, 2012). Postal Regulatory Commission, Washington DC, 6.
https://www.prc.gov/docs/86/86858/Report_Response_Order_1626.pdf
[5]
Cazals, C., Duchemin, P., Florens, J-P., Roy, B. and Vialaneix, O. (2002) An Econometric Study of Cost Elasticity in the Activities of Post Office Counters. In: Crew, M.A. and Kleindorfer, P.R., Eds., Postal and Delivery Services, Kluwer Academic Publishers, Boston, 161-170. https://doi.org/10.1007/978-1-4613-0253-7_9
[6]
Echambadi, R. and Hess, J.D. (2007) Mean-Centering Does Not Alleviate Collinearity Problems in Moderated Multiple Regression Models. Marketing Science, 26, 438-445. https://doi.org/10.1287/mksc.1060.0263
[7]
Lehmann, E.L. and Casella, G. (1998) Theory of Point Estimation. 2nd Edition, Springer, New York, 150. https://doi.org/10.1007/b98854
[8]
Miller, S.E., Capps Jr., O. and Wells, G.J. (1984) Confidence Intervals for Elasticities and Flexibilities from Linear Equations. American Journal of Agricultural Economics, 66, 392-396. https://doi.org/10.2307/1240807
[9]
Anderson, R.G. and Thursby, J.G. (1986) Confidence Interval for Elasticity Estimators in Translog Models. The Review of Economics and Statistics, 68, 647-656.
[10]
Hirschberg, J.G., Lye, N. and Slottje, D.J. (2008) Inferential Methods for Elasticity Estimates. Journal of Econometrics, 147, 299-315.
[11]
Christensen, L.R., Jorgenson, D.W. and Lau, L.J. (1973) Transcendental Logarithmic Production Frontiers. The Review of Economics and Statistics, 55, 28-45.
https://doi.org/10.2307/1927992
[12]
Borovkov, A.A. (1998) Mathematical Statistics. First Edition, Theorem 1, Gordon and Breach Science Publishers, New York, 13.
[13]
White, H. (1980) A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity. Econometrica, 48, 817-838.
https://doi.org/10.2307/1912934
[14]
Gleser, L.J. and Hwang, J.T. (1987) The Nonexistence of 100(1-α)% Confidence Sets of Finite Expected Diameter-in-Errors in Variables and Related Models. The Annals of Statistics, 15, 1351-1362. https://doi.org/10.1214/aos/1176350597
[15]
Dufour, J.-M. (1997) Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models. Econometrica, 65, 1365-1387.
https://doi.org/10.2307/2171740
[16]
Koschat, M.A. (1987) A Characterization of the Fieller Solution. The Annals of Statistics, 15, 462-468. https://doi.org/10.1214/aos/1176350282
[17]
Fieller, E.C. (1940-1941) The Biological Standardization of Insulin. Supplement to the Journal of the Royal Statistical Society, 7, 1-64. https://doi.org/10.2307/2983630
[18]
Radhakrishna, R.C. (1973) Linear Statistical Inference and Its Applications. 2nd Edition, Wiley, Hoboken, 241.
[19]
Hirschberg, J.G.J., Lye, N. and Slottje, D.J. (2008) Inferential Methods for Elasticity Estimates. Journal of Econometrics, 147, 299-315.
https://doi.org/10.1016/j.jeconom.2008.09.037
[20]
Scheffé, H. (1970) Multiple Testing versus Multiple Estimation. Improper Confidence Sets. Estimation of Directions and Ratios. The Annals of Mathematical Statistics, 41, 1-29. https://doi.org/10.1214/aoms/1177697184
