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Impulse response and forecast error variance asymptotics in nonstationary VARs

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  • Phillips, Peter C. B.

Abstract

Impulse response and forecast error variance matrix asymptotics are developed for VAR models with some roots at or near unity and some cointegration. For such models, it is shown that impulse responses that are estimated from an unrestricted VAR are inconsistent at long horizons and tend to random variables rather than the true impulse responses in the limit. The asymmetric distribution of the limit variates helps to explain the asymmetry of the finite sample distributions of the estimated impulse responses that is often found in simulations. VAR regressions also give inconsistent estimates of the forecast error variance of the optimal predictor at long horizons, and have a tendency to understate this variance. Moreover, predictions from an unrestricted nonstationary VAR are not optimal in the sense that they do not converge to the optimal predictors, at least for long horizons. In these respects, the asymptotic theory of prediction and policy analysis for nonstationary VAR's is very different from that which applies in stationary VAR's. By contrast, in a reduced rank regression the impulse response and forecast error variance matrix estimates are consistent and predictions from the fitted RRR model are asymptotically optimal, all provided the cointegrating rank is correctly specified or consistently estimated. Some simulations are reported which show these findings to be relevant in finite samples, and which assess the sensitivity of forecasting performance and policy analysis to certain design features of models in the VAR class.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Econometrics.

Volume (Year): 83 (1998)
Issue (Month): 1-2 ()
Pages: 21-56

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Handle: RePEc:eee:econom:v:83:y:1998:i:1-2:p:21-56

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Web page: http://www.elsevier.com/locate/jeconom

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  1. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
  2. Peter C.B. Phillips, 1987. "Multiple Regression with Integrated Time Series," Cowles Foundation Discussion Papers 852, Cowles Foundation for Research in Economics, Yale University.
  3. Peter C.B. Phillips, 1994. "Model Determination and Macroeconomic Activity," Cowles Foundation Discussion Papers 1083, Cowles Foundation for Research in Economics, Yale University.
  4. Olivier Jean Blanchard & Danny Quah, 1990. "The Dynamic Effects of Aggregate Demand and Supply Disturbances," NBER Working Papers 2737, National Bureau of Economic Research, Inc.
  5. Phillips, Peter C. B., 1995. "Bayesian model selection and prediction with empirical applications," Journal of Econometrics, Elsevier, vol. 69(1), pages 289-331, September.
  6. Christopher A. Sims & Tao Zha, 1994. "Error Bands for Impulse Responses," Cowles Foundation Discussion Papers 1085, Cowles Foundation for Research in Economics, Yale University.
  7. 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.
  8. Peter C.B. Phillips & Joon Y. Park, 1986. "Statistical Inference in Regressions with Integrated Processes: Part 1," Cowles Foundation Discussion Papers 811R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1987.
  9. Peter C.B. Phillips, 1988. "Optimal Inference in Cointegrated Systems," Cowles Foundation Discussion Papers 866R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1989.
  10. Peter C.B. Phillips, 1992. "Bayes Methods for Trending Multiple Time Series with an Empirical Application to the US Economy," Cowles Foundation Discussion Papers 1025, Cowles Foundation for Research in Economics, Yale University.
  11. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-80, November.
  12. Peter C.B. Phillips, 1986. "Regression Theory for Near-Integrated Time Series," Cowles Foundation Discussion Papers 781R, Cowles Foundation for Research in Economics, Yale University, revised Jan 1987.
  13. Litterman, Robert B, 1986. "Forecasting with Bayesian Vector Autoregressions-Five Years of Experience," Journal of Business & Economic Statistics, American Statistical Association, vol. 4(1), pages 25-38, January.
  14. Peter C.B. Phillips, 1992. "Bayes Models and Forecasts of Australian Macroeconomic Time Series," Cowles Foundation Discussion Papers 1024, Cowles Foundation for Research in Economics, Yale University.
  15. Spencer, David E, 1989. "Does Money Matter? The Robustness of Evidence from Vector Autoregressions," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 21(4), pages 442-54, November.
  16. Bowden,Roger J. & Turkington,Darrell A., 1990. "Instrumental Variables," Cambridge Books, Cambridge University Press, number 9780521385824, November.
  17. Park, Joon Y. & Phillips, Peter C.B., 1989. "Statistical Inference in Regressions with Integrated Processes: Part 2," Econometric Theory, Cambridge University Press, vol. 5(01), pages 95-131, April.
  18. Peter C.B. Phillips & Bruce E. Hansen, 1988. "Statistical Inference in Instrumental Variables," Cowles Foundation Discussion Papers 869R, Cowles Foundation for Research in Economics, Yale University, revised Apr 1989.
  19. Richard M. Todd, 1990. "Vector autoregression evidence on monetarism: another look at the robustness debate," Quarterly Review, Federal Reserve Bank of Minneapolis, issue Spr, pages 19-37.
  20. Peter C.B. Phillips, 1993. "Fully Modified Least Squares and Vector Autoregression," Cowles Foundation Discussion Papers 1047, Cowles Foundation for Research in Economics, Yale University.
  21. Chao, John C. & Phillips, Peter C. B., 1999. "Model selection in partially nonstationary vector autoregressive processes with reduced rank structure," Journal of Econometrics, Elsevier, vol. 91(2), pages 227-271, August.
  22. Christ, Carl F, 1975. "Judging the Performance of Econometric Models of the U.S. Economy," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 16(1), pages 54-74, February.
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