Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-27T04:02:08.578Z Has data issue: false hasContentIssue false
  • English
  • Français

IDENTIFICATION OF LINEAR REGRESSIONS WITH ERRORS IN ALL VARIABLES

Published online by Cambridge University Press:  29 June 2020

Dan Ben-Moshe
Dan Ben-Moshe*
Affiliation:
The Hebrew University of Jerusalem
*
Address correspondence to Dan Ben-Moshe, The Hebrew University of Jerusalem, Jerusalem, Israel; e-mail: danbm@huji.ac.il.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper analyzes the classical linear regression model with measurement errors in all the variables. First, we provide necessary and sufficient conditions for identification of the coefficients. We show that the coefficients are not identified if and only if an independent normally distributed linear combination of regressors can be transferred from the regressors to the errors. Second, we introduce a new estimator for the coefficients using a continuum of moments that are based on second derivatives of the log characteristic function of the observables. In Monte Carlo simulations, the estimator performs well and is robust to the amount of measurement error and number of mismeasured regressors. In an application to firm investment decisions, the estimates are similar to those produced by a generalized method of moments estimator based on third to fifth moments.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 4 , August 2021 , pp. 633 - 663
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The paper is loosely based on the second chapter of my PhD thesis. I am grateful to Rosa Matzkin and Jinyong Hahn for their generous support, advice, and guidance. I thank the editor Liangjun Su as well as Xavier D’Haultfœuille, Jinyong Hahn, Allyn Jackson, Arthur Lewbel, Rosa Matzkin, Yannay Spitzer, and Elie Tamer for reading various parts of drafts and providing feedback. I benefited from comments at Aarhus, Boston College, CREST, Harvard, Hebrew University of Jerusalem, and Penn State, and discussions with David Genesove, Patrik Guggenberger, Kei Hirano, Stefan Hoderlein, Max Kasy, Saul Lach, Áureo de Paula, Anna Simoni, James Stock, Daniel Wilhelm, and Martin Weidner.

References

REFERENCES

Abel, A.B. (1983) Optimal investment under uncertainty. The American Economic Review 73(1), 228–233.Google Scholar
Abel, A.B. (1990) Chapter 14 consumption and investment. In Handbook of Monetary Economics, Vol. 2, pp. 725–778. Elsevier.CrossRefGoogle Scholar
Abel, A.B. & Eberly, J.C. (1994) A unified model of investment under uncertainty. The American Economic Review 84(5), 1369–1384.Google Scholar
Aigner, D.J., Hsiao, C., Kapteyn, A., & Wansbeek, T. (1984) Latent variable models in econometrics. Handbook of Econometrics 2, 1321–1393.CrossRefGoogle Scholar
Bekker, P.A. (1986) Comment on identification in the linear errors in variables model. Econometrica 54(1), 215–217.CrossRefGoogle Scholar
Ben-Moshe, D., D’Haultfœuille, X., & Lewbel, A. (2017) Identification of additive and polynomial models of mismeasured regressors without instruments. Journal of Econometrics 200(2), 207–222.CrossRefGoogle Scholar
Bickel, P. & Ritov, Y. (1987) Efficient estimation in the errors in variables model. Annals of Statistics 15, 513–540.CrossRefGoogle Scholar
Blanchard, O.J., Lopez-de Silanes, F., & Shleifer, A. (1994) What do firms do with cash windfalls? Journal of Financial Economics 36(3), 337–360.CrossRefGoogle Scholar
Brainard, W.C. & Tobin, J. (1968) Pitfalls in financial model building. The American Economic Review 58(2), 99–122.Google Scholar
Buonaccorsi, J.P. (2010) Measurement Error: Models, Methods, and Applications. Chapman & Hall/CRC Interdisciplinary Statistics.CrossRefGoogle Scholar
Cao, D., Lorenzi, G., & Walentin, K. (2019) Financial frictions, investment, and Tobin’s q . Journal of Monetary Economics 103, 105–122.CrossRefGoogle Scholar
Carrasco, M. & Florens, J.-P. (2000) Generalization of GMM to a continuum of moment conditions. Econometric Theory 16(6), 797–834.CrossRefGoogle Scholar
Carrasco, M. & Kotchoni, R. (2017) Efficient estimation using the characteristic function. Econometric Theory 33(2), 479–526.CrossRefGoogle Scholar
Carroll, R.J., Ruppert, D., Stefanski, L.A., & Crainiceanu, C.M. (2006) Measurement Error in Nonlinear Models: A Modern Perspective. CRC Press.CrossRefGoogle Scholar
Chen, X., Hong, H., & Nekipelov, D. (2011) Nonlinear models of measurement errors. Journal of Economic Literature 49(4), 901–937.CrossRefGoogle Scholar
Cheng, C.-L. & Van Ness, J.W. (1999) Statistical Regression with Measurement Error, Vol. 6. Arnold.Google Scholar
Chesher, A. (1991) The effect of measurement error. Biometrika 78(3), 451–462.CrossRefGoogle Scholar
Cragg, J.G. (1997) Using higher moments to estimate the simple errors-in-variables model. Rand Journal of Economics 28, S71–S91.CrossRefGoogle Scholar
Dagenais, M.G. & Dagenais, D.L. (1997) Higher moment estimators for linear regression models with errors in the variables. Journal of Econometrics 76, 193–221.CrossRefGoogle Scholar
Erickson, T., Jiang, C.H., & Whited, T.M. (2014) Minimum distance estimation of the errors-in-variables model using linear cumulant equations. Journal of Econometrics 183(2), 211–221.CrossRefGoogle Scholar
Erickson, T., Parham, R., & Whited, T.M. (2017) Fitting the errors-in-variables model using high-order cumulants and moments. Stata Journal 17(1), 116–129.CrossRefGoogle Scholar
Erickson, T. & Whited, T.M. (2000) Measurement error and the relationship between investment and q . Journal of Political Economy 108(5), 1027–1057.CrossRefGoogle Scholar
Erickson, T. & Whited, T.M. (2002) Two-step GMM estimation of the errors-in-variables model using high-order moments. Econometric Theory 18(3), 776–799.CrossRefGoogle Scholar
Fazzari, S., Hubbard, R.G., & Petersen, B. (1988) Investment, financing decisions, and tax policy. The American Economic Review 78(2), 200–205.Google Scholar
Fuller, W.A. (2009) Measurement Error Models, Vol. 305. Wiley.Google Scholar
Geary, R.C. (1941) Inherent relations between random variables. In Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, pp. 63–76.Google Scholar
Gilchrist, S. & Himmelberg, C.P. (1995) Evidence on the role of cash flow for investment. Journal of Monetary Economics 36(3), 541–572.CrossRefGoogle Scholar
Gillard, J. (2010) An overview of linear structural models in errors in variables regression. REVSTAT–Statistical Journal 8(1), 57–80.Google Scholar
Hausman, J.A., Newey, W.K., Ichimura, H., & Powell, J.L. (1991) Identification and estimation of polynomial errors-in-variables models. Journal of Econometrics 50, 273–295.CrossRefGoogle Scholar
Hayashi, F. (1982) Tobin’s marginal q and average q: A neoclassical interpretation. Econometrica 50, 213–224.CrossRefGoogle Scholar
Hong, H. & Tamer, E. (2003) A simple estimator for nonlinear error in variable models. Journal of Econometrics 117(1), 1–19.CrossRefGoogle Scholar
Hu, Y. & Sasaki, Y. (2015) Closed-form estimation of nonparametric models with non-classical measurement errors. Journal of Econometrics 185(2), 392–408.CrossRefGoogle Scholar
Kaldor, N. (1966) Marginal productivity and the macro-economic theories of distribution: Comment on Samuelson and Modigliani. The Review of Economic Studies 33(4), 309–319.CrossRefGoogle Scholar
Kapteyn, A. & Wansbeek, T. (1983) Identification in the linear errors in variables model. Econometrica 51, 1847–1849.CrossRefGoogle Scholar
Keynes, J.M. (1936) The general theory of money, interest and employment. Reprinted in The Collected Writings of John Maynard Keynes, p. 7.Google Scholar
Khatri, C. & Rao, C.R. (1972) Functional equations and characterization of probability laws through linear functions of random variables. Journal of Multivariate Analysis 2(2), 162–173.CrossRefGoogle Scholar
Klepper, S. & Leamer, E.E. (1984) Consistent sets of estimates for regressions with errors in all variables. Econometrica 52, 163–183.CrossRefGoogle Scholar
Koopmans, T.C. (1937) Linear Regression Analysis of Economic Time Series, Vol. 20. De erven F. Bohn nv.Google Scholar
Lewbel, A. (1997) Constructing instruments for regressions with measurement error when no additional data are available, with an application to patents and R&D. Econometrica 65, 1201–1213.CrossRefGoogle Scholar
Lewbel, A. (2012) Using heteroscedasticity to identify and estimate mismeasured and endogenous regressor models. Journal of Business & Economic Statistics 30(1), 67–80.CrossRefGoogle Scholar
Li, T. (2002) Robust and consistent estimation of nonlinear errors-in-variables models. Journal of Econometrics 110(1), 1–26.CrossRefGoogle Scholar
Marcinkiewicz, J. (1939) Sur une propriete de la loi de gauss. Mathematische Zeitschrift 44(1), 612–618.CrossRefGoogle Scholar
Mussa, M. (1977) External and internal adjustment costs and the theory of aggregate and firm investment. Economica 44(174), 163–178.CrossRefGoogle Scholar
Newey, W.K. & McFadden, D. (1994) Large sample estimation and hypothesis testing. Handbook of Econometrics 4, 2111–2245.CrossRefGoogle Scholar
Pal, M. (1980) Consistent moment estimators of regression coefficients in the presence of errors in variables. Journal of Econometrics 14(3), 349–364.CrossRefGoogle Scholar
Reiersøl, O. (1945) Confluence Analysis by Means of Instrumental Sets of Variables. PhD thesis. Almqvist & Wiksell.Google Scholar
Reiersøl, O. (1950) Identifiability of a linear relation between variables which are subject to error. Econometrica 18, 375–389.CrossRefGoogle Scholar
Schennach, S. (2007) Instrumental variables estimation of nonlinear errors-in-variables models. Econometrica 75, 201–239.CrossRefGoogle Scholar
Schennach, S. & Hu, Y. (2013) Nonparametric identification and semiparametric estimation of classical measurement error models without side information. Journal of the American Statistical Association 108, 177–186.CrossRefGoogle Scholar
Schennach, S.M. (2004) Estimation of nonlinear models with measurement error. Econometrica 72, 33–75.CrossRefGoogle Scholar
Schennach, S.M. (2016) Recent advances in the measurement error literature. Annual Review of Economics 8(1), 341–377.CrossRefGoogle Scholar
Spiegelman, C. (1979) On estimating the slope of a straight line when both variables are subject to error. Annals of Statistics 7(1), 201–206.CrossRefGoogle Scholar
Székely, G. & Rao, C. (2000) Identifiability of distributions of independent random variables by linear combinations and moments. Sankhyā: The Indian Journal of Statistics Series A 62, 193–202.Google Scholar
Tobin, J. (1969) A general equilibrium approach to monetary theory. Journal of Money, Credit and Banking 1(1), 15–29.CrossRefGoogle Scholar
Willassen, Y. (1979) Extension of some results by reiersøl to multivariate models. Scandinavian Journal of Statistics 6, 89–91.Google Scholar
Yu, J. (2004) Empirical characteristic function estimation and its applications. Econometric Reviews 23(2), 93–123.CrossRefGoogle Scholar