<p>Let <span class="math-container">$z_t =\rho z_{t-1}+\epsilon_t$</span>, where <span class="math-container">$|\rho|&lt;1$</span> and <span class="math-container">$E_t e_{t+1}=0$</span></p>
<p>I need to solve the stochastic difference equation <span class="math-container">$E_t[y_{t+2}-2ay_{t+1}+by_t]=z_t$</span> for
<span class="math-container">$y_t$</span>, <span class="math-container">$t\geq0$</span>, where <span class="math-container">$0&lt;b&lt;1$</span> and <span class="math-container">$2a&gt;b+1$</span></p>
<p>In class, my professor said that when you face this kind of SDE, you can define a new variable and use Blanchard-Kahn method to get a solution. He even gave me a hint that the solution <span class="math-container">$y_t$</span> should look like <span class="math-container">$y_t = \delta y_{t-1}+\gamma z_t$</span>, where <span class="math-container">$\delta, \gamma$</span> are some functions of the parameters.</p>
<p>First, using the hint, I tried to change <span class="math-container">$y_{t+2}=\delta y_{t+1}+\gamma z_{t+2} = \delta y_{t+1}+\gamma\rho z_{t+1}+\epsilon_{t+1}$</span>. So, I changed the SDE to this one.</p>
<blockquote>
<p><span class="math-container">$E_t[\delta y_{t+1}+\gamma\rho z_{t+1}+\epsilon_{t+1}-2ay_{t+1}+by_t]=(\delta-2a)E_ty_{t+1} +by_t+\gamma \rho E_t z_{t+1} = z_t$</span></p>
</blockquote>
<p>With this and the given equation on <span class="math-container">$z$</span>, <span class="math-container">$E_tz_{t+1} =\rho z_{t}$</span>, I tried to solve it using Blanchard-Kaehn method.</p>
<p>However, I couldn't get the solution since I don't know exact the range of <span class="math-container">$\delta$</span> (I'm not sure even if my approach was correct)</p>
<p>Could you please tell me how to solve this problem? I don't know even how to start solving this one..</p>

CC BY-SA 4.0
2022-09-27T20:26:46.567Z
Hi: I can't answer your question or help you but I can say that I reached into this literature a few years back and never found it very clear. In fact, I found it extremely difficult because it seemed like you always had to understand the whole literature in order to understand one piece of it. But  here's something that I didn't find in my past investigations that seems more  readable than average. pages 50-70  look particularly relevant to your problem. https://www.mv.helsinki.fi/home/ajripatt/opetus/metrics/slides.pdf
mark leeds
CC BY-SA 4.0
2022-09-27T20:33:52.933Z
This looks possibly helpful also.  http://web.sgh.waw.pl/~jg23234/do/EMiDO%20-%20wyk03%20-%20Blanchard%20Kahn.pdf
mark leeds
CC BY-SA 4.0
2022-09-27T20:42:24.910Z
One more for the road. http://www.sef.hku.hk/~yluo/teaching/econ6012/2013topic5ii.pdf
mark leeds
CC BY-SA 4.0
2022-09-28T05:01:24.060Z
@markleeds thank you! But I actually reviewed the materials you gave before asking the question, and still did not understand..
OhRoBo
CC BY-SA 4.0
2022-09-28T20:10:28.107Z
I wish that I could help you. Maybe ask your professor if he knows of a good reference for this topic. I never really found  a good one that illustrated the various methods ( blanchard-kahn, uhlig, sargent, sims ) with examples.
mark leeds
CC BY-SA 4.0
2022-09-28T20:21:38.563Z
I hope that you didn't find this one earlier because there are some good and hopefully useful examples in it. https://www.academia.edu/30393697/How_to_get_the_Blanchard_Kahn_form_from_a_general_linear_rational_expectations_model
mark leeds
CC BY-SA 4.0
2022-09-29T07:17:01.380Z
@markleeds thanks a lot! Your resource is really helpful
OhRoBo
CC BY-SA 4.0
2022-09-30T15:50:43.093Z
I hope it helps. Like  I said, I've dealt with the "rational expectations" monster before and he's a handful !!!!!!!!  As an aside, Pesaran's "limits to rational expectations" is , IMHO, one of the best books as far as explaining what's going on with RE. But I don't know if it has a blanchard-khan example in it  ?.
mark leeds
<macroeconomics><difference-equations><rbc>
How to derive a solution to stochastic difference equation using Blanchard-Kahn method?
OhRoBo
# How to derive a solution to stochastic difference equation using Blanchard-Kahn method?

(Asked by: username_0 on 2022-09-27)

Let $z\_t =\rho z\_{t-1}+\epsilon\_t$, where $|\rho|<1$ and $E\_t e\_{t+1}=0$

I need to solve the stochastic difference equation $E\_t[y\_{t+2}-2ay\_{t+1}+by\_t]=z\_t$ for
$y\_t$, $t\geq0$, where $0<b<1$ and $2a>b+1$

In class, my professor said that when you face this kind of SDE, you can define a new variable and use Blanchard-Kahn method to get a solution. He even gave me a hint that the solution $y\_t$ should look like $y\_t = \delta y\_{t-1}+\gamma z\_t$, where $\delta, \gamma$ are some functions of the parameters.

First, using the hint, I tried to change $y\_{t+2}=\delta y\_{t+1}+\gamma z\_{t+2} = \delta y\_{t+1}+\gamma\rho z\_{t+1}+\epsilon\_{t+1}$. So, I changed the SDE to this one.

> $E\_t[\delta y\_{t+1}+\gamma\rho z\_{t+1}+\epsilon\_{t+1}-2ay\_{t+1}+by\_t]=(\delta-2a)E\_ty\_{t+1} +by\_t+\gamma \rho E\_t z\_{t+1} = z\_t$

With this and the given equation on $z$, $E\_tz\_{t+1} =\rho z\_{t}$, I tried to solve it using Blanchard-Kaehn method.

However, I couldn't get the solution since I don't know exact the range of $\delta$ (I'm not sure even if my approach was correct)

Could you please tell me how to solve this problem? I don't know even how to start solving this one..

**Tags:** macroeconomics, difference-equations, rbc

> **Comments:**
> **username_1**: Hi: I can't answer your question or help you but I can say that I reached into this literature a few years back and never found it very clear. In fact, I found it extremely difficult because it seemed like you always had to understand the whole literature in order to understand one piece of it. But  here's something that I didn't find in my past investigations that seems more  readable than average. pages 50-70  look particularly relevant to your problem. https://www.mv.helsinki.fi/home/ajripatt/opetus/metrics/slides.pdf  
> **username_1**: This looks possibly helpful also.  http://web.sgh.waw.pl/~jg23234/do/EMiDO%20-%20wyk03%20-%20Blanchard%20Kahn.pdf  
> **username_1**: One more for the road. http://www.sef.hku.hk/~yluo/teaching/econ6012/2013topic5ii.pdf  
> **username_0**: @username_1 thank you! But I actually reviewed the materials you gave before asking the question, and still did not understand..  
> **username_1**: I wish that I could help you. Maybe ask your professor if he knows of a good reference for this topic. I never really found  a good one that illustrated the various methods ( blanchard-kahn, uhlig, sargent, sims ) with examples.  
> **username_1**: I hope that you didn't find this one earlier because there are some good and hopefully useful examples in it. https://www.academia.edu/30393697/How_to_get_the_Blanchard_Kahn_form_from_a_general_linear_rational_expectations_model  
> **username_0**: @username_1 thanks a lot! Your resource is really helpful  
> **username_1**: I hope it helps. Like  I said, I've dealt with the "rational expectations" monster before and he's a handful !!!!!!!!  As an aside, Pesaran's "limits to rational expectations" is , IMHO, one of the best books as far as explaining what's going on with RE. But I don't know if it has a blanchard-khan example in it  ?.  

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## 0 answers

