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I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process: \begin{equation*} d\mathbf{S}_t = \mathbf{\kappa}(\mathbf{\theta} - \mathbf{S}_t)dt + \mathbf{\sigma}d\mathbf{W}_t, \end{equation*} using the vector autoregression: \begin{equation*} \mathbf{S}_t = \mathbf{A} + \mathbf{BS}_{t-1} + \mathbf{\epsilon}_t. \end{equation*}

Recall that the ideal co-integration factor corresponds to the eigenvector of $\kappa = \mathbf{I-B}$$\kappa = \mathbf{I_2-B}$ with the largest corresponding eigenvalue. However, I am encountering an empirical situation in which $\kappa$ is not diagonalizable over $\mathbb{R}$ and has only complex$\mathbb{C}$-valued eigenvalues and eigenvectors.

This situation does not seem to be treated in any of the literature, even though there appears to be no reason, a priori, that it should not arise. What are best practices for handling this situation, i.e., computing the co-integration factor when $\kappa$ is not diagonalizable?

I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process: \begin{equation*} d\mathbf{S}_t = \mathbf{\kappa}(\mathbf{\theta} - \mathbf{S}_t)dt + \mathbf{\sigma}d\mathbf{W}_t, \end{equation*} using the vector autoregression: \begin{equation*} \mathbf{S}_t = \mathbf{A} + \mathbf{BS}_{t-1} + \mathbf{\epsilon}_t. \end{equation*}

Recall that the ideal co-integration factor corresponds to the eigenvector of $\kappa = \mathbf{I-B}$ with the largest corresponding eigenvalue. However, I am encountering an empirical situation in which $\kappa$ is not diagonalizable over $\mathbb{R}$ and has only complex eigenvalues and eigenvectors.

This situation does not seem to be treated in any of the literature, even though there appears to be no reason, a priori, that it should not arise. What are best practices for handling this situation, i.e., computing the co-integration factor when $\kappa$ is not diagonalizable?

I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process: \begin{equation*} d\mathbf{S}_t = \mathbf{\kappa}(\mathbf{\theta} - \mathbf{S}_t)dt + \mathbf{\sigma}d\mathbf{W}_t, \end{equation*} using the vector autoregression: \begin{equation*} \mathbf{S}_t = \mathbf{A} + \mathbf{BS}_{t-1} + \mathbf{\epsilon}_t. \end{equation*}

Recall that the ideal co-integration factor corresponds to the eigenvector of $\kappa = \mathbf{I_2-B}$ with the largest corresponding eigenvalue. However, I am encountering an empirical situation in which $\kappa$ is not diagonalizable over $\mathbb{R}$ and has only $\mathbb{C}$-valued eigenvalues and eigenvectors.

This situation does not seem to be treated in any of the literature, even though there appears to be no reason, a priori, that it should not arise. What are best practices for handling this situation, i.e., computing the co-integration factor when $\kappa$ is not diagonalizable?

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  • 11
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Non-diagonalizable matrix in a discretized Ornstein-Uhlenbeck process

I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process: \begin{equation*} d\mathbf{S}_t = \mathbf{\kappa}(\mathbf{\theta} - \mathbf{S}_t)dt + \mathbf{\sigma}d\mathbf{W}_t, \end{equation*} using the vector autoregression: \begin{equation*} \mathbf{S}_t = \mathbf{A} + \mathbf{BS}_{t-1} + \mathbf{\epsilon}_t. \end{equation*}

Recall that the ideal co-integration factor corresponds to the eigenvector of $\kappa = \mathbf{I-B}$ with the largest corresponding eigenvalue. However, I am encountering an empirical situation in which $\kappa$ is not diagonalizable over $\mathbb{R}$ and has only complex eigenvalues and eigenvectors.

This situation does not seem to be treated in any of the literature, even though there appears to be no reason, a priori, that it should not arise. What are best practices for handling this situation, i.e., computing the co-integration factor when $\kappa$ is not diagonalizable?