<p>Expression (1) only follows from <span class="math-container">$\mathbb{E}_t[Q_{t+1}]=\Lambda_2Q_t+(C_{21} \gamma_1 +C_{22}\gamma_2)Z_t$</span> if we make an assumption about how fast <span class="math-container">$Q_t$</span> grows relative to <span class="math-container">$\Lambda_2$</span>.</p>
<p>To see this, notice you can write the expression <span class="math-container">$\mathbb{E}_t[Q_{t+1}]=\Lambda_2 Q_t +(C_{21}\gamma_1+C_{22}\gamma_2)Z_t$</span> as <span class="math-container">$$Q_t=-\Lambda_2^{-1}(C_{21}\gamma_1+C_{22}\gamma_2)Z_t+\Lambda_2^{-1}\mathbb{E}_t[Q_{t+1}]$$</span>
This defines a recurrence relation of the form <span class="math-container">$Q_t=\Lambda_2^{-1}B_t+\Lambda_2^{-1}\mathbb{E}_t[Q_{t+1}]$</span> where <span class="math-container">$B_t=-(C_{21}\gamma_1+C_{22}\gamma_2)Z_t$</span>. Iterating this expression forwards <span class="math-container">$$\begin{align*} Q_t &amp;= \Lambda_2^{-1}B_t+\Lambda_2^{-1}\mathbb{E}_t[Q_{t+1}] \\
&amp;=\Lambda_2^{-1}B_t +\Lambda_2^{-1}\mathbb{E}_t[\Lambda_2^{-1}B_{t+1}+\Lambda_2^{-1}\mathbb{E}_{t+1}[Q_{t+1}]] \\
&amp;= \Lambda_2^{-1}B_t +\Lambda_2^{-2}\mathbb{E}_t[B_{t+1}]+\Lambda_2^{-2}\underbrace{\mathbb{E}_t[\mathbb{E}_{t+1}[Q_{t+1}]]}_{=\mathbb{E}_t[Q_{t+1}]} \\
&amp;=\Lambda_2^{-1}B_t +\Lambda_2^{-2}\mathbb{E}_t[B_{t+1}]+\Lambda_2^{-2}\mathbb{E}_t[Q_{t+1}] \\
&amp; \ \ \vdots \\
&amp;=\sum_{i=0}^{\infty}\Lambda_2^{-i-1}\mathbb{E}_t[B_{t+i}]+\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}] \\
&amp;=\sum_{i=0}^{\infty}\Lambda_2^{-i-1}\mathbb{E}_t[-(C_{21}\gamma_1+C_{22}\gamma_2)Z_{t+i}]+\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}] \\
&amp;=-\sum_{i=0}^{\infty}\Lambda_2^{-i-1}(C_{21}\gamma_1+C_{22}\gamma_2)\mathbb{E}_t[Z_{t+i}]+\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}]
\end{align*}$$</span></p>
<p>Therefore, non-explosiveness corresponds to <span class="math-container">$\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}]=0$</span>. As <span class="math-container">$\Lambda_2$</span> contains the eigenvalues outside the unit circle, <span class="math-container">$\Lambda_2^{-\tau-1}$</span> tends to the 0 matrix. Therefore, this condition is saying that for non-explosive behaviour we need <span class="math-container">$\mathbb{E}_t[Q_{t+\tau}]$</span> to grow at a rate slower than <span class="math-container">$\Lambda_2^{\tau}$</span>. This limit condition is similar to the transversality condition which also precludes explosive paths.</p>

CC BY-SA 4.0
2024-05-10T22:38:02.620Z
This was really helpful and enlightening. Thank you.
mark leeds
CC BY-SA 4.0
2024-05-09T09:16:05.610Z
2024-05-09T09:16:05.610Z
Joseph Basford
<p>I am trying to understand proof of BK conditions given in the appendix to their paper.
In their paper authors have original model:</p>
<p><a href="https://i.sstatic.net/fz5xq3K6.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/fz5xq3K6.png" alt="enter image description here" /></a></p>
<p>There <span class="math-container">$X_{t + 1} \in \mathbb{R}^{n}, P_{t + 1} \in \mathbb{R}^{m}$</span> This model is transformed into</p>
<p><a href="https://i.sstatic.net/EDuSLYmZ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EDuSLYmZ.png" alt="enter image description here" /></a></p>
<p>by using the fact that <span class="math-container">$A = C^{-1} \Lambda C$</span>. <span class="math-container">$\Lambda$</span> also is the matrix of eigenvalues of A, which are sorted in increasing order by modulus: <span class="math-container">$|\lambda_{11}| &lt; |\lambda_{22}| &lt; ... &lt; |\lambda_{(m+n)(m+n)}|$</span>.
After constuction of <span class="math-container">$\Lambda$</span> we can decompose it into two submatrixes:</p>
<p><a href="https://i.sstatic.net/Cc9CWrkA.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Cc9CWrkA.png" alt="enter image description here" /></a></p>
<p>There <span class="math-container">$\Lambda_1$</span> consist of eigenvalues that are inside unit circle in <span class="math-container">$\mathbb{C}$</span>. And <span class="math-container">$\Lambda_2$</span> consist of eigenvalues that are outside unit circle. By definition <span class="math-container">$\Lambda_1$</span> has <span class="math-container">$\bar{n}$</span> rows and columns. And <span class="math-container">$\Lambda_2$</span> has <span class="math-container">$\bar{m}$</span> rows and columns. It folows then that vector <span class="math-container">$Y_t$</span> may not have same amount of elements as <span class="math-container">$X_t$</span> because</p>
<p><a href="https://i.sstatic.net/M69r5aep.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/M69r5aep.png" alt="enter image description here" /></a></p>
<p>and <span class="math-container">$B_{11} \in \mathbb{R}^{n \times \bar{n}}$</span>. So <span class="math-container">$Y_t$</span> has <span class="math-container">$\bar{n}$</span> elements. We also know that <span class="math-container">$n + m = \bar{n} + \bar{m}$</span>.</p>
<p>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 <span class="math-container">$E_t[Q_{t+1}] = \Lambda_2 Q_t + (C_{21} \gamma_1 + C_{22} \gamma_2) Z_t$</span>. It is also stated that system will explode unless:</p>
<p><span class="math-container">$Q_t = -\sum_{i=0}^{\infty} \Lambda_2^{-i-1}(C_{21}\gamma_1 + C{22} \gamma_2)E_t[Z_{t+i}]    \quad(1)$</span></p>
<p>However it does not seem to be obvious since we obtain (1) by just iterating formula <span class="math-container">$E_t[Q_{t+1+i}] = \Lambda_2 Q_{t+i} + (C_{21} \gamma_1 + C_{22} \gamma_2) Z_{t + i}$</span> for <span class="math-container">$i=0,...,\infty$</span>. So how does non-explosiveness follows from (1)?</p>

<macroeconomics>
Proof of Blanchard Kahn conditions
Maksim Tarasenko
# Proof of Blanchard Kahn conditions

(Asked by: username_0 on 2024-05-09)

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

[![enter image description here](https://i.sstatic.net/fz5xq3K6.png)](https://i.sstatic.net/fz5xq3K6.png)

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

[![enter image description here](https://i.sstatic.net/EDuSLYmZ.png)](https://i.sstatic.net/EDuSLYmZ.png)

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:

[![enter image description here](https://i.sstatic.net/Cc9CWrkA.png)](https://i.sstatic.net/Cc9CWrkA.png)

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

[![enter image description here](https://i.sstatic.net/M69r5aep.png)](https://i.sstatic.net/M69r5aep.png)

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)?

**Tags:** macroeconomics


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## 1 answers

### Answer by username_1 on 2024-05-09. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_2**: This was really helpful and enlightening. Thank you.  

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