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Inference in Time Series Regression When the Order of Integration of a Regressor is Unknown

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Graham Elliott
James H. Stock

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Abstract

It is well known that the distribution of statistics testing restrictions on the coefficients in time series regressions can depend on the order of integration of the regressors. In practice the order of integration is rarely blown. This paper examines two conventional approaches to this problem, finds them unsatisfactory, and proposes a new procedure. The two conventional approaches- simply to ignore unit root problems or to use unit root pretests to determine the critical values for second-stage inference - both often induce substantial size distortions. In the case of unit root pretests, this arises because type I and II pretest errors produce incorrect second-stage critical values and because, in many empirically plausible situations, the first stage test (the unit root test) and the second stage test (the exclusion restriction test) are dependent. Monte Carlo simulations reveal size distortions even if the regressor is stationary but has a large autoregressive root, a case that might arise for example in a regression of excess stock returns against the dividend yield. In the proposed alternative procedure, the second-stage test is conditional on a first-stage "unit root" statistic developed in Stock (1992); the second-stage critical values vary continuously with the value of the first-stage statistic. The procedure is shown to have the correct size asymptotically and to have good local asymptotic power against Granger-causality alternatives.

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Paper provided by National Bureau of Economic Research, Inc in its series NBER Technical Working Papers with number 0122.

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Date of creation: Jun 1992
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Handle: RePEc:nbr:nberte:0122

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. James H. Stock & Kenneth D. West, 1988. "Integrated Regressors and Tests of the Permanent Income Hypothesis," NBER Working Papers 2359, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  2. Kwiatkowski, D. & Phillips, P.C.B. & Schmidt, P., 1990. "Testing the Null Hypothesis of Stationarity Against the Alternative of Unit Root : How Sure are we that Economic Time Series have a Unit Root?," Papers 8905, Michigan State - Econometrics and Economic Theory.
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  3. Peter C.B. Phillips & Joon Y. Park, 1986. "Statistical Inference in Regressions with Integrated Processes: Part 2," Cowles Foundation Discussion Papers 819R, Cowles Foundation, Yale University, revised Feb 1987. [Downloadable!]
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  4. Nabeya, Seiji & Tanaka, Katsuto, 1990. "A General Approach to the Limiting Distribution for Estimators in Time Series Regression with Nonstable Autoregressive Errors," Econometrica, Econometric Society, vol. 58(1), pages 145-63, January. [Downloadable!] (restricted)
  5. Peter C.B. Phillips, 1988. "Spectral Regression for Cointegrated Time Series," Cowles Foundation Discussion Papers 872, Cowles Foundation, Yale University. [Downloadable!]
  6. Hiro Y. Toda & Peter C.B. Phillips, 1991. "Vector Autoregression and Causality: A Theoretical Overview and Simulation Study," Cowles Foundation Discussion Papers 1001, Cowles Foundation, Yale University. [Downloadable!]
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  7. Matthew D. Shapiro & N. Gregory Mankiw, 1984. "Trends, Random Walks, and Tests of the Permanent Income Hypothesis," Cowles Foundation Discussion Papers 725, Cowles Foundation, Yale University. [Downloadable!]
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  8. Campbell, John Y, 1991. "A Variance Decomposition for Stock Returns," Economic Journal, Royal Economic Society, vol. 101(405), pages 157-79, March. [Downloadable!] (restricted)
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  9. Sargan, John Denis & Bhargava, Alok, 1983. "Testing Residuals from Least Squares Regression for Being Generated by the Gaussian Random Walk," Econometrica, Econometric Society, vol. 51(1), pages 153-74, January. [Downloadable!] (restricted)
  10. Sims, Christopher A & Stock, James H & Watson, Mark W, 1990. "Inference in Linear Time Series Models with Some Unit Roots," Econometrica, Econometric Society, vol. 58(1), pages 113-44, January. [Downloadable!] (restricted)
  11. Fama, Eugene F. & French, Kenneth R., 1989. "Business conditions and expected returns on stocks and bonds," Journal of Financial Economics, Elsevier, vol. 25(1), pages 23-49, November. [Downloadable!] (restricted)
  12. Fama, Eugene F, 1991. " Efficient Capital Markets: II," Journal of Finance, American Finance Association, vol. 46(5), pages 1575-617, December. [Downloadable!] (restricted)
  13. Flavin, Marjorie A, 1981. "The Adjustment of Consumption to Changing Expectations about Future Income," Journal of Political Economy, University of Chicago Press, vol. 89(5), pages 974-1009, October. [Downloadable!] (restricted)
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Simon van Norden & Huntley Schaller & ), 1995. "Regime Switching in Stock Market Returns," Econometrics 9502002, EconWPA. [Downloadable!]
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  2. Ai Deng, 2005. "Understanding Spurious Regression in Financial Economics," Boston University - Department of Economics - Working Papers Series WP2005-048, Boston University - Department of Economics. [Downloadable!]
  3. Rossen Valkanov, 1999. "Equity Premium and Dividend Yield regressions: A lot of noise, little information, confusing results," University of California at Los Angeles, Anderson Graduate School of Management 1103, Anderson Graduate School of Management, UCLA. [Downloadable!]
  4. Robert F. Stambaugh, 1999. "Predictive Regressions," NBER Technical Working Papers 0240, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  5. Fève, Patrick & Hénin, Pierre-Yves, 1998. "Assessing effective sustainability of fiscal policy within the G-7," CEPREMAP Working Papers (Couverture Orange) 9815, CEPREMAP. [Downloadable!]
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  6. Efstathios Paparoditis & Dimitris Politis, 2000. "Large-sample inference in the general AR(1) model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 9(2), pages 487-509, December. [Downloadable!] (restricted)
  7. Jeeman Jung & Robert J. Shiller, 2002. "One Simple Test of Samuelson's Dictum for the Stock Market," Cowles Foundation Discussion Papers 1386, Cowles Foundation, Yale University. [Downloadable!]
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  8. Alex Maynard & Katsumi Shimotsu, 2007. "Covariance-based orthogonality tests for regressors with unknown persistence," Working Papers 1122, Queen's University, Department of Economics. [Downloadable!]
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  9. John Y. Campbell & Luis Viceira, 2005. "The Term Structure of the Risk-Return Tradeoff," NBER Working Papers 11119, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  10. Geert Bekaert & Robert J. Hodrick & David A. Marshall, 1996. "On Biases in Tests of the Expecations Hypothesis of the Term Structure Of Interest Rates," NBER Technical Working Papers 0191, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  11. Cosme Vodounou, 1998. "Inférence fondée sur les statistiques des rendements de long terme," CIRANO Working Papers 98s-20, CIRANO. [Downloadable!]
  12. Lubos Pastor & Robert F. Stambaugh, 2007. "Predictive Systems: Living with Imperfect Predictors," NBER Working Papers 12814, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  13. Malcolm P. Baker & Ryan Taliaferro & Jeffrey Wurgler, 2004. "Pseudo Market Timing and Predictive Regressions," NBER Working Papers 10823, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
  14. Walter Torous & Shu Yan, 2000. "Predictive Regressions Revisited," University of California at Los Angeles, Anderson Graduate School of Management 1028, Anderson Graduate School of Management, UCLA. [Downloadable!]
  15. Greg Tkacz, 2002. "Inflation Changes, Yield Spreads, and Threshold Effects," Working Papers 02-40, Bank of Canada. [Downloadable!]
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  16. In Choi & Timothy K. Chue, 2006. "Subsampling-Based Tests of Stock-Return Predictability," Hi-Stat Discussion Paper Series d06-178, Institute of Economic Research, Hitotsubashi University. [Downloadable!]
  17. Ulrich Müller & Mark W. Watson, 2009. "Low-Frequency Robust Cointegration Testing," NBER Working Papers 15292, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
  18. John Y. Campbell & Motohiro Yogo, 2003. "Efficient Tests of Stock Return Predictability," NBER Working Papers 10026, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  19. Stanislav Anatolyev & Nikolay Gospodinov, 2007. "Modeling Financial Return Dynamics by Decomposition," Working Papers w0095, Center for Economic and Financial Research (CEFIR). [Downloadable!]
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