IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2210.02599.html

The Local to Unity Dynamic Tobit Model

Author

Listed:
  • Anna Bykhovskaya
  • James A. Duffy

Abstract

This paper considers highly persistent time series that are subject to nonlinearities in the form of censoring or an occasionally binding constraint, such as are regularly encountered in macroeconomics. A tractable candidate model for such series is the dynamic Tobit with a root local to unity. We show that this model generates a process that converges weakly to a non-standard limiting process, that is constrained (regulated) to be positive. Surprisingly, despite the presence of censoring, the OLS estimators of the model parameters are consistent. We show that this allows OLS-based inferences to be drawn on the overall persistence of the process (as measured by the sum of the autoregressive coefficients), and for the null of a unit root to be tested in the presence of censoring. Our simulations illustrate that the conventional ADF test substantially over-rejects when the data is generated by a dynamic Tobit with a unit root, whereas our proposed test is correctly sized. We provide an application of our methods to testing for a unit root in the Swiss franc / euro exchange rate, during a period when this was subject to an occasionally binding lower bound.

Suggested Citation

  • Anna Bykhovskaya & James A. Duffy, 2022. "The Local to Unity Dynamic Tobit Model," Papers 2210.02599, arXiv.org, revised May 2024.
    • Handle: RePEc:arx:papers:2210.02599
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2210.02599
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Markus Hertrich, 2022. "Foreign exchange interventions under a minimum exchange rate regime and the Swiss franc," Review of International Economics, Wiley Blackwell, vol. 30(2), pages 450-489, May.
    2. Laura Liu & Hyungsik Roger Moon & Frank Schorfheide, 2023. "Forecasting with a panel Tobit model," Quantitative Economics, Econometric Society, vol. 14(1), pages 117-159, January.
    3. Brezigar-Masten, Arjana & Masten, Igor & Volk, Matjaž, 2021. "Modelin-g credit risk with a Tobit model of days past due," Journal of Banking & Finance, Elsevier, vol. 122(C).
    4. Liudas Giraitis & Peter C. B. Phillips, 2006. "Uniform Limit Theory for Stationary Autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(1), pages 51-60, January.
    5. Gao, Jiti, 2007. "Nonlinear time series: semiparametric and nonparametric methods," MPRA Paper 39563, University Library of Munich, Germany, revised 01 Sep 2007.
    6. Michel, Jon & de Jong, Robert M., 2018. "Mixing properties of the dynamic Tobit model with mixing errors," Economics Letters, Elsevier, vol. 162(C), pages 112-115.
    7. Anna Bykhovskaya & Peter C. B. Phillips, 2018. "Boundary Limit Theory for Functional Local to Unity Regression," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(4), pages 523-562, July.
    8. Peter C.B. Phillips, 1987. "Multiple Regression with Integrated Time Series," Cowles Foundation Discussion Papers 852, Cowles Foundation for Research in Economics, Yale University.
    9. Bruce E. Hansen, 1999. "The Grid Bootstrap And The Autoregressive Model," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 594-607, November.
    10. Robert Jong & Ana María Herrera, 2011. "Dynamic Censored Regression and the Open Market Desk Reaction Function," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(2), pages 228-237, April.
    11. Liang, Hanying & Phillips, Peter C.B. & Wang, Hanchao & Wang, Qiying, 2016. "Weak Convergence To Stochastic Integrals For Econometric Applications," Econometric Theory, Cambridge University Press, vol. 32(6), pages 1349-1375, December.
    12. Barbara Rossi, 2013. "Exchange Rate Predictability," Journal of Economic Literature, American Economic Association, vol. 51(4), pages 1063-1119, December.
    13. Steven Wei, 1999. "A bayesian approach to dynamic tobit models," Econometric Reviews, Taylor & Francis Journals, vol. 18(4), pages 417-439.
    14. Terasvirta, Timo & Tjostheim, Dag & Granger, Clive W. J., 2010. "Modelling Nonlinear Economic Time Series," OUP Catalogue, Oxford University Press, number 9780199587155.
    15. Hong, Seung Hyun & Phillips, Peter C. B., 2010. "Testing Linearity in Cointegrating Relations With an Application to Purchasing Power Parity," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(1), pages 96-114.
    16. Diansheng Dong & Todd M. Schmit & Harry Kaiser, 2012. "Modelling household purchasing behaviour to analyse beneficial marketing strategies," Applied Economics, Taylor & Francis Journals, vol. 44(6), pages 717-725, February.
    17. Andrews, Donald W K & Chen, Hong-Yuan, 1994. "Approximately Median-Unbiased Estimation of Autoregressive Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(2), pages 187-204, April.
    18. Sophocles Mavroeidis, 2021. "Identification at the Zero Lower Bound," Econometrica, Econometric Society, vol. 89(6), pages 2855-2885, November.
    19. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    20. Hahn, Jinyong & Kuersteiner, Guido, 2010. "Stationarity and mixing properties of the dynamic Tobit model," Economics Letters, Elsevier, vol. 107(2), pages 105-111, May.
    21. Gao, Jiti & Tjøstheim, Dag & Yin, Jiying, 2013. "Estimation in threshold autoregressive models with a stationary and a unit root regime," Journal of Econometrics, Elsevier, vol. 172(1), pages 1-13.
    22. Gregor von Schweinitz & Lena Tonzer & Manuel Buchholz, 2021. "Monetary policy through exchange rate pegs: The removal of the Swiss franc‐Euro floor and stock price reactions," International Review of Finance, International Review of Finance Ltd., vol. 21(4), pages 1382-1406, December.
    23. Cavaliere, Giuseppe & Xu, Fang, 2014. "Testing for unit roots in bounded time series," Journal of Econometrics, Elsevier, vol. 178(P2), pages 259-272.
    24. Selva Demiralp & Òscar Jordà, 2002. "The announcement effect: evidence from open market desk data," Economic Policy Review, Federal Reserve Bank of New York, vol. 8(May), pages 29-48.
    25. Anna Mikusheva, 2007. "Uniform Inference in Autoregressive Models," Econometrica, Econometric Society, vol. 75(5), pages 1411-1452, September.
    26. Saikkonen, Pentti, 2008. "Stability Of Regime Switching Error Correction Models Under Linear Cointegration," Econometric Theory, Cambridge University Press, vol. 24(1), pages 294-318, February.
    27. Sarno, Lucio & Valente, Giorgio, 2006. "Deviations from purchasing power parity under different exchange rate regimes: Do they revert and, if so, how?," Journal of Banking & Finance, Elsevier, vol. 30(11), pages 3147-3169, November.
    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. Bykhovskaya, Anna & Duffy, James A., 2024. "The local to unity dynamic Tobit model," Journal of Econometrics, Elsevier, vol. 241(2).
    2. Duffy, James A. & Mavroeidis, Sophocles & Wycherley, Sam, 2025. "Cointegration with occasionally binding constraints," Journal of Econometrics, Elsevier, vol. 252(PA).
    3. Sabzikar, Farzad & Wang, Qiying & Phillips, Peter C.B., 2020. "Asymptotic theory for near integrated processes driven by tempered linear processes," Journal of Econometrics, Elsevier, vol. 216(1), pages 192-202.
    4. Andrews, Donald W.K. & Guggenberger, Patrik, 2012. "Asymptotics for LS, GLS, and feasible GLS statistics in an AR(1) model with conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 169(2), pages 196-210.
    5. Lin, Yingqian & Tu, Yundong, 2020. "Robust inference for spurious regressions and cointegrations involving processes moderately deviated from a unit root," Journal of Econometrics, Elsevier, vol. 219(1), pages 52-65.
    6. Donald W. K. Andrews & Patrik Guggenberger, 2014. "A Conditional-Heteroskedasticity-Robust Confidence Interval for the Autoregressive Parameter," The Review of Economics and Statistics, MIT Press, vol. 96(2), pages 376-381, May.
    7. Marcet, Albert & Jarociński, Marek, 2010. "Autoregressions in small samples, priors about observables and initial conditions," Working Paper Series 1263, European Central Bank.
    8. Petropoulos, Fotios & Apiletti, Daniele & Assimakopoulos, Vassilios & Babai, Mohamed Zied & Barrow, Devon K. & Ben Taieb, Souhaib & Bergmeir, Christoph & Bessa, Ricardo J. & Bijak, Jakub & Boylan, Joh, 2022. "Forecasting: theory and practice," International Journal of Forecasting, Elsevier, vol. 38(3), pages 705-871.
      • Fotios Petropoulos & Daniele Apiletti & Vassilios Assimakopoulos & Mohamed Zied Babai & Devon K. Barrow & Souhaib Ben Taieb & Christoph Bergmeir & Ricardo J. Bessa & Jakub Bijak & John E. Boylan & Jet, 2020. "Forecasting: theory and practice," Papers 2012.03854, arXiv.org, revised Jan 2022.
    9. Lieberman, Offer & Phillips, Peter C.B., 2020. "Hybrid stochastic local unit roots," Journal of Econometrics, Elsevier, vol. 215(1), pages 257-285.
    10. Christis Katsouris, 2023. "Limit Theory under Network Dependence and Nonstationarity," Papers 2308.01418, arXiv.org, revised Aug 2023.
    11. Carlos A. Medel & Pablo M. Pincheira, 2016. "The out-of-sample performance of an exact median-unbiased estimator for the near-unity AR(1) model," Applied Economics Letters, Taylor & Francis Journals, vol. 23(2), pages 126-131, February.
    12. Sofiane H. Sekioua, 2004. "Real interest parity (RIP) over the 20th century: New evidence based on confidence intervals for the dominant root and half-lives of shocks," Money Macro and Finance (MMF) Research Group Conference 2004 91, Money Macro and Finance Research Group.
    13. Josep Lluís Carrion-i-Silvestre & María Dolores Gadea & Antonio Montañés, 2017. "“Unbiased estimation of autoregressive models forbounded stochastic processes," AQR Working Papers 201710, University of Barcelona, Regional Quantitative Analysis Group, revised Dec 2017.
    14. Ordoñez-Callamand, Daniel & Hernandez-Leal, Juan D. & Villamizar-Villegas, Mauricio, 2018. "When multiple objectives meet multiple instruments: Identifying simultaneous monetary shocks," International Review of Economics & Finance, Elsevier, vol. 58(C), pages 78-101.
    15. Christopher J. Neely & David E. Rapach, 2008. "Real interest rate persistence: evidence and implications," Review, Federal Reserve Bank of St. Louis, vol. 90(Nov), pages 609-642.
    16. Yiu Lim Lui & Weilin Xiao & Jun Yu, 2022. "The Grid Bootstrap for Continuous Time Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(3), pages 1390-1402, June.
    17. Inoue, Atsushi & Kilian, Lutz, 2020. "The uniform validity of impulse response inference in autoregressions," Journal of Econometrics, Elsevier, vol. 215(2), pages 450-472.
    18. Andrews, Donald W.K. & Cheng, Xu & Guggenberger, Patrik, 2020. "Generic results for establishing the asymptotic size of confidence sets and tests," Journal of Econometrics, Elsevier, vol. 218(2), pages 496-531.
    19. Fallahi, Firouz, 2012. "The stationarity of consumption–income ratios: Evidence from bootstrapping confidence intervals," Economics Letters, Elsevier, vol. 115(1), pages 137-140.
    20. Christis Katsouris, 2022. "Asymptotic Theory for Unit Root Moderate Deviations in Quantile Autoregressions and Predictive Regressions," Papers 2204.02073, arXiv.org, revised Aug 2023.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:arx:papers:2210.02599. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.