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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

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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Publication status: published as Economic Theory, vol 10, (1994) pp 672-700.
Handle: RePEc:nbr:nberte:0122

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  1. Denis Kwiatkowski & Peter C.B. Phillips & Peter Schmidt, 1991. "Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root: How Sure Are We That Economic Time Series Have a Unit Root?," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 979, Cowles Foundation for Research in Economics, Yale University.
  2. Matthew D. Shapiro & N. Gregory Mankiw, 1984. "Trends, Random Walks, and Tests of the Permanent Income Hypothesis," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 725, Cowles Foundation for Research in Economics, Yale University.
  3. Sargan, John Denis & Bhargava, Alok, 1983. "Testing Residuals from Least Squares Regression for Being Generated by the Gaussian Random Walk," Econometrica, Econometric Society, Econometric Society, vol. 51(1), pages 153-74, January.
  4. Fama, Eugene F, 1991. " Efficient Capital Markets: II," Journal of Finance, American Finance Association, American Finance Association, vol. 46(5), pages 1575-617, December.
  5. Campbell, John Y, 1991. "A Variance Decomposition for Stock Returns," Economic Journal, Royal Economic Society, Royal Economic Society, vol. 101(405), pages 157-79, March.
  6. 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.
  7. Phillips, P C B, 1991. "Optimal Inference in Cointegrated Systems," Econometrica, Econometric Society, Econometric Society, vol. 59(2), pages 283-306, March.
  8. 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, Econometric Society, vol. 58(1), pages 145-63, January.
  9. Tanaka, Katsuto, 1990. "Testing for a Moving Average Unit Root," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 6(04), pages 433-444, December.
  10. Sims, Christopher A & Stock, James H & Watson, Mark W, 1990. "Inference in Linear Time Series Models with Some Unit Roots," Econometrica, Econometric Society, Econometric Society, vol. 58(1), pages 113-44, January.
  11. Peter C.B. Phillips, 1988. "Spectral Regression for Cointegrated Time Series," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 872, Cowles Foundation for Research in Economics, Yale University.
  12. Peter C.B. Phillips & Joon Y. Park, 1986. "Statistical Inference in Regressions with Integrated Processes: Part 2," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 819R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1987.
  13. Flavin, Marjorie A, 1981. "The Adjustment of Consumption to Changing Expectations about Future Income," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 89(5), pages 974-1009, October.
  14. Hiro Y. Toda & Peter C.B. Phillips, 1991. "Vector Autoregression and Causality: A Theoretical Overview and Simulation Study," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1001, Cowles Foundation for Research in Economics, Yale University.
  15. Fama, Eugene F. & French, Kenneth R., 1989. "Business conditions and expected returns on stocks and bonds," Journal of Financial Economics, Elsevier, Elsevier, vol. 25(1), pages 23-49, November.
  16. Perron, P., 1991. "A Test for Changes in a Polynomial Trend Functions for a Dynamioc Time Series," Papers, Princeton, Department of Economics - Econometric Research Program 363, Princeton, Department of Economics - Econometric Research Program.
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  1. Some thoughts on the Reinhart and Rogoff debate
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