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Time Series Regression with Mixtures of Integrated Processes


  • Chang, Yoosoon
  • Phillips, Peter C.B.


The paper develops a statistical theory for regressions with integrated regressors of unknown order and unknown cointegrating dimension. In practice, we are often unsure whether unit roots or cointegration is present in time series data, and we are also uncertain about the order of integration in some cases. This paper addresses issues of estimation and inference in cases of such uncertainty. Phillips (1995, Econometrica 63, 1023–1078) developed a theory for time series regressions with an unknown mixture of 1(0) and 1(1) variables and established that the method of fully modified ordinary least squares (FM-OLS) is applicable to models (including vector autoregressions) with some unit roots and unknown cointegrating rank. This paper extends these results to models that contain some I(0), I(1), and I(2) regressors. The theory and methods here are applicable to cointegrating regressions that include unknown numbers of I(0), I(1), and I(2) variables and an unknown degree of cointegration. Such models require a somewhat different approach than that of Phillips (1995). The paper proposes a residual-based fully modified ordinary least-squares (RBFMOLS) procedure, which employs residuals from a first-order autoregression of the first differences of the entire regressor set in the construction of the FMOLS estimator. The asymptotic theory for the RBFM-OLS estimator is developed and is shown to be normal for all the stationary coefficients and mixed normal for all the nonstationary coefficients. Under Gaussian assumptions, estimation of the cointegration space by RBFM-OLS is optimal even though the dimension of the space is unknown.

Suggested Citation

  • Chang, Yoosoon & Phillips, Peter C.B., 1995. "Time Series Regression with Mixtures of Integrated Processes," Econometric Theory, Cambridge University Press, vol. 11(5), pages 1033-1094, October.
  • Handle: RePEc:cup:etheor:v:11:y:1995:i:05:p:1033-1094_00

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    Cited by:

    1. Karabiyik, Hande & Reese, Simon & Westerlund, Joakim, 2017. "On the role of the rank condition in CCE estimation of factor-augmented panel regressions," Journal of Econometrics, Elsevier, vol. 197(1), pages 60-64.
    2. Mármol, Francesc & Dolado, Juan José, 1998. "FM-OLS estimation of cointegrating relationships among nonstationary fractionally integrated processes," DES - Working Papers. Statistics and Econometrics. WS 4672, Universidad Carlos III de Madrid. Departamento de Estadística.
    3. Syed Ammad Ali & Qazi Masood Ahmed & Lubna Naz, 2016. "Public spending on human capital formation and economic growth in Pakistan," Asia-Pacific Development Journal, United Nations Economic and Social Commission for Asia and the Pacific (ESCAP), vol. 23(1), pages 1-20, June.
    4. repec:eee:ecolet:v:174:y:2019:i:c:p:5-8 is not listed on IDEAS
    5. Baffes, John & Elbadawi, Ibrahim A. & O'Connell, Stephen A., 1997. "Single-equation estimation of the equilibrium real exchange rate," Policy Research Working Paper Series 1800, The World Bank.
    6. Dong Shin & Man-Suk Oh, 2003. "Tests for the order of integration against higher order integration," Statistical Papers, Springer, vol. 44(3), pages 383-396, July.

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