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I am trying to understand proof of BK conditions given in the appendix to their paper. In their paper authors have original model:

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There $X_{t + 1} \in \mathbb{R}^{n}, P_{t + 1} \in \mathbb{R}^{m}$ This model is transformed into

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by using the fact that $A = C^{-1} \Lambda C$. $\Lambda$ also is the matrix of eigenvalues of A, which are sorted in increasing order by modulus: $|\lambda_{11}| < |\lambda_{22}| < ... < |\lambda_{(m+n)(m+n)}|$. After constuction of $\Lambda$ we can decompose it into two submatrixes:

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There $\Lambda_1$ consist of eigenvalues that are inside unit circle in $\mathbb{C}$. And $\Lambda_2$ consist of eigenvalues that are outside unit circle. By definition $\Lambda_1$ has $\bar{n}$ rows and columns. And $\Lambda_2$ has $\bar{m}$ rows and columns. It folows then that vector $Y_t$ may not have same amount of elements as $X_t$ because

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and $B_{11} \in \mathbb{R}^{n \times \bar{n}}$. So $Y_t$ has $\bar{n}$ elements. We also know that $n + m = \bar{n} + \bar{m}$.

So there are my question: Appendix of paper Blanchard Kahn (1980) says that we can divide our original system of equations into two subsystems (yes, we can). Second subsystem will be $E_t[Q_{t+1}] = \Lambda_2 Q_t + (C_{21} \gamma_1 + C_{22} \gamma_2) Z_t$. It is also stated that system will explode unless:

$Q_t = -\sum_{i=0}^{\infty} \Lambda_2^{-i-1}(C_{21}\gamma_1 + C{22} \gamma_2)E_t[Z_{t+i}] \quad(1)$

However it does not seem to be obvious since we obtain (1) by just iterating formula $E_t[Q_{t+1+i}] = \Lambda_2 Q_{t+i} + (C_{21} \gamma_1 + C_{22} \gamma_2) Z_{t + i}$ for $i=0,...,\infty$. So how does non-explosiveness follows from (1)?

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Expression (1) only follows from $\mathbb{E}_t[Q_{t+1}]=\Lambda_2Q_t+(C_{21} \gamma_1 +C_{22}\gamma_2)Z_t$ if we make an assumption about how fast $Q_t$ grows relative to $\Lambda_2$.

To see this, notice you can write the expression $\mathbb{E}_t[Q_{t+1}]=\Lambda_2 Q_t +(C_{21}\gamma_1+C_{22}\gamma_2)Z_t$ as $$Q_t=-\Lambda_2^{-1}(C_{21}\gamma_1+C_{22}\gamma_2)Z_t+\Lambda_2^{-1}\mathbb{E}_t[Q_{t+1}]$$ This defines a recurrence relation of the form $Q_t=\Lambda_2^{-1}B_t+\Lambda_2^{-1}\mathbb{E}_t[Q_{t+1}]$ where $B_t=-(C_{21}\gamma_1+C_{22}\gamma_2)Z_t$. Iterating this expression forwards $$\begin{align*} Q_t &= \Lambda_2^{-1}B_t+\Lambda_2^{-1}\mathbb{E}_t[Q_{t+1}] \\ &=\Lambda_2^{-1}B_t +\Lambda_2^{-1}\mathbb{E}_t[\Lambda_2^{-1}B_{t+1}+\Lambda_2^{-1}\mathbb{E}_{t+1}[Q_{t+1}]] \\ &= \Lambda_2^{-1}B_t +\Lambda_2^{-2}\mathbb{E}_t[B_{t+1}]+\Lambda_2^{-2}\underbrace{\mathbb{E}_t[\mathbb{E}_{t+1}[Q_{t+1}]]}_{=\mathbb{E}_t[Q_{t+1}]} \\ &=\Lambda_2^{-1}B_t +\Lambda_2^{-2}\mathbb{E}_t[B_{t+1}]+\Lambda_2^{-2}\mathbb{E}_t[Q_{t+1}] \\ & \ \ \vdots \\ &=\sum_{i=0}^{\infty}\Lambda_2^{-i-1}\mathbb{E}_t[B_{t+i}]+\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}] \\ &=\sum_{i=0}^{\infty}\Lambda_2^{-i-1}\mathbb{E}_t[-(C_{21}\gamma_1+C_{22}\gamma_2)Z_{t+i}]+\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}] \\ &=-\sum_{i=0}^{\infty}\Lambda_2^{-i-1}(C_{21}\gamma_1+C_{22}\gamma_2)\mathbb{E}_t[Z_{t+i}]+\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}] \end{align*}$$

Therefore, non-explosiveness corresponds to $\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}]=0$. As $\Lambda_2$ contains the eigenvalues outside the unit circle, $\Lambda_2^{-\tau-1}$ tends to the 0 matrix. Therefore, this condition is saying that for non-explosive behaviour we need $\mathbb{E}_t[Q_{t+\tau}]$ to grow at a rate slower than $\Lambda_2^{\tau}$. This limit condition is similar to the transversality condition which also precludes explosive paths.

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  • $\begingroup$ This was really helpful and enlightening. Thank you. $\endgroup$
    – mark leeds
    Commented May 10 at 22:38

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