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SEQUENTIALLY ESTIMATING THE STRUCTURAL EQUATION BY POWER TRANSFORMATION

Published online by Cambridge University Press:  19 September 2022

Jaedo Choi
Affiliation:
University of Michigan
Hyungsik Roger Moon
Affiliation:
University of Southern California and Yonsei University
Jin Seo Cho*
Affiliation:
Yonsei University
*
Address correspondence to Jin Seo Cho, School of Economics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Korea; e-mail: jinseocho@yonsei.ac.kr

Abstract

This study provides an econometric methodology to test a linear structural relationship among economic variables. We propose the so-called distance-difference (DD) test and show that it has omnibus power against arbitrary nonlinear structural relationships. If the DD-test rejects the linear model hypothesis, a sequential testing procedure assisted by the DD-test can consistently estimate the degree of a polynomial function that arbitrarily approximates the nonlinear structural equation. Using extensive Monte Carlo simulations, we confirm the DD-test’s finite sample properties and compare its performance with the sequential testing procedure assisted by the J-test and moment selection criteria. Finally, through investigation, we empirically illustrate the relationship between the value-added and its production factors using firm-level data from the United States. We demonstrate that the production function has exhibited a factor-biased technological change instead of Hicks-neutral technology presumed by the Cobb–Douglas production function.

Type
ARTICLES
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

The Editor (Peter Phillips), the Co-Editor (Patrik Guggenberger), and two anonymous referees provided very helpful comments for which we are most grateful. The authors are also benefited from discussions with Juwon Seo. Cho further acknowledges with gratitude the research grant provided by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2018S1A5A2A01035256).

References

REFERENCES

Acemoglu, D. (2008) Introduction to Modern Economic Growth. Princeton University Press.Google Scholar
Acemoglu, D. & Restrepo, P. (2018) The race between man and machine: Implications of technology for growth, factor shares, and employment. American Economic Review 108, 14881542.CrossRefGoogle Scholar
Ackerberg, D.A., Caves, K., & Frazer, G. (2015) Identification properties of recent production function estimators. Econometrica 83, 24112451.CrossRefGoogle Scholar
Ai, C. & Chen, X. (2003) Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica 71, 17951843.CrossRefGoogle Scholar
Akaike, H. (1973) Information theory and an extension of the maximum likelihood principle. In Petrov, B. and Csake, F. (eds), 2nd International Symposium on Information Theory, 1973, pp. 268281. Akademiai Kiado.Google Scholar
Andrews, D.W.K. (1992) Generic uniform convergence. Econometric Theory 8, 241257.CrossRefGoogle Scholar
Andrews, D.W.K. (1999) Consistent moment selection procedures for generalized method of moments estimation. Econometrica 67, 543563.CrossRefGoogle Scholar
Andrews, D.W.K. & Cheng, X. (2015) Estimation and inference with weak, semi-strong, and strong identification. Econometrica 80, 21532211.Google Scholar
Angrist, J.D. & Krueger, A.B. (1991) Does compulsory school attendance affect schooling and earnings? Quarterly Journal of Economics 106, 9791014.CrossRefGoogle Scholar
Antràs, P. (2004) Is the U.S. aggregate production function Cobb–Douglas? New estimates of the elasticity of substitution. The Berkeley Electronic Journal of Macroeconomics 4, 136.Google Scholar
Baek, Y.I., Cho, J.S., & Phillips, P.C.B. (2015) Testing linearity using power transforms of Regressors. Journal of Econometrics 187, 376384.CrossRefGoogle Scholar
Balassa, B. (1964) The purchasing-power parity doctrine: A reappraisal. Journal of Political Economy 72, 584596.CrossRefGoogle Scholar
Bartlett, M.S. (1947) Multivariate analysis. Supplement to the Journal of the Royal Statistical Society 50, 176197.CrossRefGoogle Scholar
Bierens, H.J. (1982) Consistent model specification tests. Journal of Econometrics 20, 105134.CrossRefGoogle Scholar
Bierens, H.J. (1990) A consistent conditional moment test of functional form. Econometrica 58, 14431458.CrossRefGoogle Scholar
Breunig, C. (2015) Goodness-of-fit tests based on series estimators in nonparametric instrumental regression. Journal of Econometrics 184, 328346.CrossRefGoogle Scholar
Card, D. & Krueger, A.B. (1992) Does school quality matter? Returns to education and the characteristics of public schools in the United States. Journal of Political Economy 100, 140.CrossRefGoogle Scholar
Chen, X. & Liao, Z. (2014) Sieve M inference on irregular parameters. Journal of Econometrics 182, 7086.CrossRefGoogle Scholar
Chen, X. & Pouzo, D. (2015) Sieve Wald and QLR inferences on semi/nonparametric conditional moment models. Econometrica 83, 10131079.CrossRefGoogle Scholar
Cho, J.S., Cheong, T.U., & White, H. (2011) Experience with the weighted bootstrap in testing for unobserved heterogeneity in exponential and Weibull duration models. Journal of Economic Theory and Econometrics 22, 6091.Google Scholar
Cho, J.S. & Phillips, P.C.B. (2018) Sequentially testing polynomial model hypotheses using power transforms of Regressors. Journal of Applied Econometrics 33, 141159.CrossRefGoogle Scholar
Cragg, J.G. & Donald, S.G. (1993) Testing identifiability and specification in instrumental variable models. Econometric Theory 33, 222240.CrossRefGoogle Scholar
Cragg, J.G. & Donald, S.G. (1996) On the asymptotic properties of LDU-based tests of the rank of a matrix. Journal of the American Statistical Association 91, 176197.CrossRefGoogle Scholar
Davies, R.B. (1977) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64, 247254.CrossRefGoogle Scholar
Davies, R.B. (1987) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 3343.Google Scholar
Dhyne, E., Petrin, A., Smeets, V., and Warzynski, F. (2017) Multi Product Firms, Import Competition, and the Evolution of Firm-Product Technical Efficiencies. NBER Working paper 23637, National Bureau of Economic Research.CrossRefGoogle Scholar
Elliott, G., Müller, U., & Watson, M. (2015) Nearly optimal tests when a nuisance parameter is present under the null hypothesis. Econometrica 83, 771811.CrossRefGoogle Scholar
Griliches, Z. & Mairesse, J. (1995) Production Functions: The Search for Identification. NBER Working paper 5067, National Bureau of Economic Research.CrossRefGoogle Scholar
Hall, A.R. & Inoue, A. (2003) The large sample behaviour of the generalized method of moments estimator in misspecified models. Journal of Econometrics 114, 361394.CrossRefGoogle Scholar
Hannan, E.J. & Quinn, B.G. (1979) The determination of the order of an autoregression. Journal of the Royal Statistical Society: Series B (Methodological) 41, 190195.Google Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413430.CrossRefGoogle Scholar
Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Econometrica 50, 10291054.CrossRefGoogle Scholar
Hong, Y. & White, H. (1995) Consistent specification testing via nonparametric series regression. Econometrica 63, 11331159.CrossRefGoogle Scholar
Horowitz, J.L. (2006) Testing a parametric model against a nonparametric alternative with identification through instrumental variables. Econometrica 74, 521538.CrossRefGoogle Scholar
Hosoya, Y. (1989) Hierarchical statistical models and a generalized likelihood ratio test. Journal of the Royal Statistical Society: Series B (Methodological) 51, 435447.Google Scholar
İmrohoroğlu, A. & Tüzel, Ş. (2014) Firm-level productivity, risk, and return. Management Science 60, 20732090.CrossRefGoogle Scholar
Karabarbounis, L. & Neiman, B. (2014) The global decline of the labor share. Quarterly Journal of Economics 129, 61103.CrossRefGoogle Scholar
Kleinbergen, F. & Papp, R. (2006) Generalized reduced rank tests using the singular value decomposition. Journal of Econometrics 133, 97126.CrossRefGoogle Scholar
Krusell, P., Ohanian, L.E., Ríos-Rull, J.V., & Violante, G.L. (2000) Capital-skill complementarity and inequality: A macroeconomic analysis. Econometrica 68, 10291053.CrossRefGoogle Scholar
Leeb, H. & Pötscher, B. (2005) Model selection and inference: Facts and fiction. Econometric Theory 21, 2159.CrossRefGoogle Scholar
Levinsohn, J. & Petrin, A. (2003) Estimating production functions using inputs to control for unobservables. Review of Economic Studies 70, 317341.CrossRefGoogle Scholar
Mincer, J. (1958) Investment in human capital and personal income distribution. Journal of Political Economy 66, 281302.CrossRefGoogle Scholar
Mincer, J. (1997) Changes in wage inequality, 1970–1990. Research in Labor Economics 16, 118.Google Scholar
Newey, W.K. (1985) Generalized method of moments specification testing. Journal of Econometrics 29, 229256.CrossRefGoogle Scholar
Newey, W.K. & Powell, J.L. (2003) Instrumental variable estimation of nonparametric models. Econometrica 71, 15651578.CrossRefGoogle Scholar
Oberfield, E. & Raval, D. (2021) Micro data and macro technology. Econometrica 89, 703732.CrossRefGoogle Scholar
Olley, G.S. & Pakes, A. (1996) The dynamics of productivity in the telecommunications equipment industry. Econometrica 64, 12631297.CrossRefGoogle Scholar
Piketty, T. (2014) Capital in the 21st Century. Harvard University Press.CrossRefGoogle Scholar
Piterbarg, V.I. (1996) Asymptotic Methods in the Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs, vol. 148. American Mathematical Society.Google Scholar
Raval, D.R. (2019) The micro elasticity of substitution and non-neutral technology. RAND Journal of Economics 50, 142167.CrossRefGoogle Scholar
Robin, J.M. & Smith, R.J. (2000) Tests of rank. Econometric Theory 16, 151175.CrossRefGoogle Scholar
Samuelson, P.A. (1964) Theoretical notes on trade problems. The Review of Economics and Statistics 46, 145154.CrossRefGoogle Scholar
Sargan, J.D. (1958) The estimation of economic relationships using instrumental variables. Econometrica 26, 393415.CrossRefGoogle Scholar
Sargan, J.D. (1988) Lectures on Advanced Econometric Theory. Basil Blackwell.Google Scholar
Schwarz, G. (1978) Estimating the dimension of a model. Annals of Statistics 6, 461464.CrossRefGoogle Scholar
Scott, D.J. (1973) Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach. Advances in Applied Probability 5, 119137.CrossRefGoogle Scholar
Staiger, D. & Stock, J.H. (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557586.CrossRefGoogle Scholar
Stock, J.H. & Yogo, M. (2005) Testing for weak instruments in linear IV regression. In Andrews, D.W.K. and Stock, J.H. (eds), Identification and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg, pp. 80108. Cambridge University Press.CrossRefGoogle Scholar
Wald, A. (1943) Tests of statistical hypotheses concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society 54, 426482.CrossRefGoogle Scholar
Wooldridge, J.M. (2009) On estimating firm-level production functions using proxy variables to control for unobservables. Economics Letters 104, 112114.CrossRefGoogle Scholar
Zhu, Y. (2020) Inference in nonparametric/semiparametric moment equality models with shape restrictions. Quantitative Economics 11, 609636.CrossRefGoogle Scholar