IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i11p1849-d1670290.html
   My bibliography  Save this article

Identification and Estimation in Linear Models with Endogeneity Through Time-Varying Volatility

Author

Listed:
  • Shih-Tang Hwu

    (Department of Economics, California State Polytechnic University, 3801 W. Temple Ave., Pomona, CA 91768, USA)

Abstract

This paper proposes a novel control function approach to identify and estimate linear models with endogenous variables in the absence of valid instrumental variables. The identification strategy exploits time-varying volatility to address the multicollinearity problem that arises in conventional control function methods when instruments are weak. We establish the identification conditions and show that the proposed method is T -consistent and asymptotically normal. We apply the proposed approach to estimate the elasticity of intertemporal substitution, a key parameter in macroeconomics. Using quarterly data on aggregate stock returns across eleven countries, we find that the data exhibit substantial time variation in volatility, supporting the identifying assumptions. The proposed method yields confidence intervals that are broadly consistent with the general findings in the literature and are substantially narrower than those obtained using weak-instrument robust methods.

Suggested Citation

  • Shih-Tang Hwu, 2025. "Identification and Estimation in Linear Models with Endogeneity Through Time-Varying Volatility," Mathematics, MDPI, vol. 13(11), pages 1-17, June.
  • Handle: RePEc:gam:jmathe:v:13:y:2025:i:11:p:1849-:d:1670290
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/11/1849/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/13/11/1849/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Motohiro Yogo, 2004. "Estimating the Elasticity of Intertemporal Substitution When Instruments Are Weak," The Review of Economics and Statistics, MIT Press, vol. 86(3), pages 797-810, August.
    2. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037, Decembrie.
    3. Galvao, Antonio F. & Montes-Rojas, Gabriel, 2015. "On the equivalence of instrumental variables estimators for linear models," Economics Letters, Elsevier, vol. 134(C), pages 13-15.
    4. Hausman, Jerry, 2015. "Specification tests in econometrics," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 38(2), pages 112-134.
    5. Larry G. Epstein & Stanley E. Zin, 2013. "Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 12, pages 207-239, World Scientific Publishing Co. Pte. Ltd..
    6. repec:bla:jfinan:v:44:y:1989:i:5:p:1115-53 is not listed on IDEAS
    7. John H. Cochrane, 2008. "The Dog That Did Not Bark: A Defense of Return Predictability," The Review of Financial Studies, Society for Financial Studies, vol. 21(4), pages 1533-1575, July.
    8. Epstein, Larry G & Zin, Stanley E, 1991. "Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis," Journal of Political Economy, University of Chicago Press, vol. 99(2), pages 263-286, April.
    9. Newey, Whitney K, 1994. "The Asymptotic Variance of Semiparametric Estimators," Econometrica, Econometric Society, vol. 62(6), pages 1349-1382, November.
    10. Jinyong Hahn & Jerry Hausman, 2002. "A New Specification Test for the Validity of Instrumental Variables," Econometrica, Econometric Society, vol. 70(1), pages 163-189, January.
    11. Frank Kleibergen, 2002. "Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Econometrica, Econometric Society, vol. 70(5), pages 1781-1803, September.
    12. Feng, Yuanhua, 2004. "Simultaneously Modeling Conditional Heteroskedasticity And Scale Change," Econometric Theory, Cambridge University Press, vol. 20(3), pages 563-596, June.
    13. Pagan, Adrian, 1984. "Econometric Issues in the Analysis of Regressions with Generated Regressors," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 25(1), pages 221-247, February.
    14. Xu, Ke-Li & Phillips, Peter C.B., 2008. "Adaptive estimation of autoregressive models with time-varying variances," Journal of Econometrics, Elsevier, vol. 142(1), pages 265-280, January.
    15. Wu, De-Min, 1973. "Alternative Tests of Independence Between Stochastic Regressors and Disturbances," Econometrica, Econometric Society, vol. 41(4), pages 733-750, July.
    16. Andrew Harvey & Esther Ruiz & Neil Shephard, 1994. "Multivariate Stochastic Variance Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 61(2), pages 247-264.
    17. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
    18. Peter C. B. Phillips & Ke‐Li Xu, 2006. "Inference in Autoregression under Heteroskedasticity," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(2), pages 289-308, March.
    19. Shih‐Tang Hwu & Chang‐Jin Kim, 2019. "Estimating Trend Inflation Based on Unobserved Components Model: Is It Correlated with the Inflation Gap?," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 51(8), pages 2305-2319, December.
    20. repec:bla:jfinan:v:59:y:2004:i:4:p:1481-1509 is not listed on IDEAS
    21. Roberto Rigobon, 2003. "Identification Through Heteroskedasticity," The Review of Economics and Statistics, MIT Press, vol. 85(4), pages 777-792, November.
    22. Neely, Christopher J & Roy, Amlan & Whiteman, Charles H, 2001. "Risk Aversion versus Intertemporal Substitution: A Case Study of Identification Failure in the Intertemporal Consumption Capital Asset Pricing Model," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(4), pages 395-403, October.
    23. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Gomes, Fábio Augusto Reis & Paz, Lourenço S., 2013. "Estimating the elasticity of intertemporal substitution: Is the aggregate financial return free from the weak instrument problem?," Journal of Macroeconomics, Elsevier, vol. 36(C), pages 63-75.
    2. Doko Tchatoka, Firmin & Dufour, Jean-Marie, 2020. "Exogeneity tests, incomplete models, weak identification and non-Gaussian distributions: Invariance and finite-sample distributional theory," Journal of Econometrics, Elsevier, vol. 218(2), pages 390-418.
    3. Guilhem Bascle, 2008. "Controlling for endogeneity with instrumental variables in strategic management research," Post-Print hal-00576795, HAL.
    4. Guo, Zijian & Kang, Hyunseung & Cai, T. Tony & Small, Dylan S., 2018. "Testing endogeneity with high dimensional covariates," Journal of Econometrics, Elsevier, vol. 207(1), pages 175-187.
    5. Antoine, Bertille & Renault, Eric, 2020. "Testing identification strength," Journal of Econometrics, Elsevier, vol. 218(2), pages 271-293.
    6. Carlos Velasco & Xuexin Wang, 2021. "Instrumental variable estimation via a continuum of instruments with an application to estimating the elasticity of intertemporal substitution in consumption," Working Papers 2024-09-06, Wang Yanan Institute for Studies in Economics (WISE), Xiamen University.
    7. Inoue, Atsushi & Rossi, Barbara, 2011. "Testing for weak identification in possibly nonlinear models," Journal of Econometrics, Elsevier, vol. 161(2), pages 246-261, April.
    8. Doko Tchatoka, Firmin Sabro, 2012. "Specification Tests with Weak and Invalid Instruments," MPRA Paper 40185, University Library of Munich, Germany.
    9. Yacine Ait-Sahalia & Jonathan A. Parker & Motohiro Yogo, 2001. "Luxury Goods and the Equity Premium," NBER Working Papers 8417, National Bureau of Economic Research, Inc.
    10. Michael P. Murray, 2006. "Avoiding Invalid Instruments and Coping with Weak Instruments," Journal of Economic Perspectives, American Economic Association, vol. 20(4), pages 111-132, Fall.
    11. Christopher F Baum & Mark E. Schaffer & Steven Stillman, 2003. "Instrumental variables and GMM: Estimation and testing," Stata Journal, StataCorp LLC, vol. 3(1), pages 1-31, March.
    12. Russell Davidson & Victoria Zinde‐Walsh, 2017. "Advances in specification testing," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 50(5), pages 1595-1631, December.
    13. Donald W.K. Andrews & James H. Stock, 2005. "Inference with Weak Instruments," NBER Technical Working Papers 0313, National Bureau of Economic Research, Inc.
    14. Firmin Doko Tchatoka & Wenjie Wang, 2020. "Uniform Inference after Pretesting for Exogeneity," School of Economics and Public Policy Working Papers 2020-05, University of Adelaide, School of Economics and Public Policy.
    15. Surach Tanboon, 2005. "On the Validity and Refinement of the Use of Rainfall as Instrument for Transitory Income," Working Papers 2005-10, Monetary Policy Group, Bank of Thailand.
    16. Gomes, Fábio Augusto Reis & Ribeiro, Priscila Fernandes, 2015. "Estimating the elasticity of intertemporal substitution taking into account the precautionary savings motive," Journal of Macroeconomics, Elsevier, vol. 45(C), pages 108-123.
    17. Firmin Doko Tchatoka & Jean‐Marie Dufour, 2014. "Identification‐robust inference for endogeneity parameters in linear structural models," Econometrics Journal, Royal Economic Society, vol. 17(1), pages 165-187, February.
    18. Doko Tchatoka, Firmin, 2011. "Testing for partial exogeneity with weak identification," MPRA Paper 39504, University Library of Munich, Germany, revised Mar 2012.
    19. Murray Michael P., 2017. "Linear Model IV Estimation When Instruments Are Many or Weak," Journal of Econometric Methods, De Gruyter, vol. 6(1), pages 1-22, January.
    20. Jondeau, Eric & Le Bihan, Hervé, 2008. "Examining bias in estimators of linear rational expectations models under misspecification," Journal of Econometrics, Elsevier, vol. 143(2), pages 375-395, April.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:13:y:2025:i:11:p:1849-:d:1670290. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.