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Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

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  • Søren Johansen
  • Morten Ørregaard Nielsen

Abstract

We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model based on the conditional Gaussian likelihood. The model allows the process X(t) to be fractional of order d and cofractional of order d-b; that is, there exist vectors β for which β′X(t) is fractional of order d-b. The parameters d and b satisfy either d≥b≥1/2, d=b≥1/2, or d=d₀≥b≥1/2. Our main technical contribution is the proof of consistency of the maximum likelihood estimators on the set 1/2≤b≤d≤d₁ for any d₁≥d₀. To this end, we consider the conditional likelihood as a stochastic process in the parameters, and prove that it converges in distribution when errors are i.i.d. with suitable moment conditions and initial values are bounded. We then prove that the estimator of β is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. We also find the asymptotic distribution of the likelihood ratio test for cointegration rank, which is a functional of fractional Brownian motion of type II.
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Suggested Citation

  • Søren Johansen & Morten Ørregaard Nielsen, 2012. "Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model," Econometrica, Econometric Society, vol. 80(6), pages 2667-2732, November.
  • Handle: RePEc:ecm:emetrp:v:80:y:2012:i:6:p:2667-2732
    DOI: ECTA9299
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    File URL: http://hdl.handle.net/10.3982/ECTA9299
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    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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