I'm currently taking a course in time series and have been struggling with understanding the Johansen trace test. Specifically, the calculation of the eigenvalues for the Likelihood ratio statistic. Suppose we have an m-dimensional VAR(p) model $$Y_t = \Pi_1 Y_{t-1} +\Pi_2 Y_{t-1} +\dots + \Pi_p Y_{t-p} +\varepsilon_t$$ Then the VECM corresponding to this model is $$\Delta Y_t = \Pi Y_{t-1} +\sum_{j=1}^{p-1} \Gamma_j \Delta Y_{t-j} +\varepsilon_t$$ Where $$\Pi = -\Pi(1)=\Pi_1 +\Pi_2 +\dots + \Pi_p -I_m$$ From what I understand, the LR-statistic for the trace test comparing $$H_0:\text{rank}(\Pi)=r$$ vs $$H_1 : \text{rank}(\Pi) \geq r+1$$ is calculated as $$LR(r)=\frac{n-p}{2}\sum_{j=1}^r \log (1-\hat{\lambda}_j)$$ Where $1\geq \hat{\lambda}_1 \geq \hat{\lambda}_2 \geq \dots \geq \hat{\lambda}_m \geq 0$ are the eigenvalues of $\Pi$. But I'm a little confused how the eigenvalues lie between 0 and 1. I have heard that they are the square of the canonical correlations between $\Delta Y_t$ and $Y_{t-1}$. However, if we have all eigenvalues of $\Pi_1 + \Pi_2 +\dots +\Pi_p$ inside, or on, the unit circle then how can $\Pi$ have eigenvalues greater than 0?

Any help would be greatly appreciated.



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