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Traditionally, if $x_t$ and $y_t$ are both $I(1)$, they are cointegrated when there exists some linear combination $z_t=y_t-$$\gamma$ $x_t$ such that $z_t$ is stationary or $I(0)$.

My question is if I can claim that $x_t$ and $y_t$ are cointegrated (perhaps a special case) when $z_t=y_t-$$\gamma_t$$x_t$ where $z_t$~$I(0)$? Here, I allow the cointegrating vector (coefficient) $\gamma_t$ to vary over time.

In this situation, $\gamma_t$ adapts to covariance structure between the $x_t$ and $y_t$ if said cov structure evolves over time. Since $z_t$ is still stationary, is there some term that is appropriate I can use in place of traditional cointegration? I was thinking 'time-varying cointegration' or something along those lines.

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Yes, the concept of non-linear cointegration has been introduced in the litterature, and there are various authors who adressed the case of time varying cointegration. Look for example at:

  • Bierens, Martins, Time Varying Cointegration, Econometric Theory, 2010, Page 1 of 38
  • Park, Hahn, Cointegrating Regressions With Time Varying Coefficients, Econometric Theory, 15, 1999,

There is also a strand of papers looking at very flexible (time) varying models in a non-parametric way, so-called functional cointegration, see among others:

  • Xiao, Functional-coefficient cointegration models, Journal of Econometrics 152 (2009) 81–92
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