<p>I am trying to replicate the results of <a href="https://faculty.chicagobooth.edu/john.cochrane/research/papers/macroann.pdf" rel="noreferrer">Cochrane, 1998</a>. Most of the paper is just describing the theory behind The Fiscal Theory of the Price Level. But from p. 42 he begins the econometrics aspect.</p>

<p>Cochrane uses a frictionless model so the government's flow budget constraint can be written as:</p>

<p>$v_{t} = \frac{1}{r_{t+1}}(s_{t+1} + v_{t+1})$ (1)</p>

<p>where $v_{t} = \frac{B_{t-1}(t)}{p_{t}}$ i.e, the real value of a bond issued in $t-1$ and matures in $t$. $s_{t}$ is the real primary surplus and $r_{t}$ is the real rate of return on government bonds.</p>

<p>For theoretical reasons he deflates the variables by consumption (as opposed to output) and writes the flow constraint as:</p>

<p>$\frac{v_{t}}{c_{t}} = \frac{1}{r_{t+1}}\frac{c_{t+1}}{c_{t}} (\frac{s_{t+1}}{c_{t+1}} + \frac{v_{t+1}}{c_{t+1}})$. If we define $\beta = \frac{1}{r_{t+1}} \frac{c_{t+1}}{c_{t}}$, the flow constraint can be iterated forward and expressed as:</p>

<p>$vc_{t} = E_{t} \sum_{j=1}^{\infty} \beta^{j} sc_{t+j}$ (2)</p>

<p>Note that $vc_{t}$ and $sc_{t}$
 and just $vc_{t} = \frac{v_{t}}{c_{t}} - E(\frac{v_{t}}{c_{t}})$  and $sc_{t} = \frac{s_{t}}{c_{t}} - E(\frac{s_{t}}{c_{t}})$. Here is my first source of confusion: Why does he subtract the mean from each series? I understand that this is "deviations from the mean" but surely the flow constraint could be iterated forward even without subtracting the means?</p>

<p>A key aspect of the fiscal theory is that primary surpluses evolve exogenously. Thus, Cochrane defines the following:</p>

<p>$sc_{t} = a_{t} + z_{t}$ (3) where</p>

<p>$a_{t} = \phi_{a}a_{t-1} + \epsilon_{at}$</p>

<p>$z_{t} = \phi_{z} z_{t-1} + \epsilon_{zt}$</p>

<p>note that $a_{t}$ and $z_{t}$ are unobserved variables.</p>

<p>Substituting this into equation (2) we find that:</p>

<p>$vc_{t} = \frac{\beta \phi_{a}}{1 - \beta \phi_{a}} a_{t} + \frac{\beta \phi_{z}}{1 - \beta \phi_{z}} z_{t} $ (4)</p>

<p>So equation (3) and (4) describe the observed variables (sc,vc) in terms of the unobserved ones (a,z). We can put this in state-space form:</p>

<p>Let $Y_{t} = (sc_{t}, vc_{t})'$ be the vector of observed variables and let $X_{t} = (a_{t},z_{t})'$ be the vector of unobserved variables. Then the system can be represented as:</p>

<p>$Y_{t} = BX_{t}$ and $X_{t} = AX_{t-1} + \epsilon_{t}$ where</p>

<p>$ B =\begin{bmatrix}
1 &amp; 1 \\ 
 \frac{\beta \phi_{a}}{1 - \beta \phi_{a}}&amp;  \frac{\beta \phi_{z}}{1 - \beta \phi_{z}} 
\end{bmatrix}$, $A = \begin{bmatrix}
\phi_{a} &amp; 0 \\ 
 0&amp; \phi_{z} 
\end{bmatrix}$,  $\epsilon_{t} = (\epsilon_{at}, \epsilon_{zt})'$</p>

<p>This can be expressed as </p>

<p>$Y_{t} = B(AX_{t-1} + \epsilon_{t}) =BA \underbrace{B^{-1}Y_{t-1}}_{\text{$X_{t-1}$}} + \underbrace{e_{t}}_{\text{$B\epsilon_{t}$}}$ (5)</p>

<p>So (5) is a reduced form equation which can be estimated. I have estimated the model and obtained $\phi_{a}$ and $\phi_{z}$ as well as the reduced form covariance matrix from which I was able to infer the "structural" covariance matrix. You can find Cochrane's results on p.43.</p>

<p>The next part is where I am very confused, he says:</p>

<blockquote>
  <p>The government chooses debt so that inflation is a function of the two state variables z, a</p>
</blockquote>

<p>$dp_{t} = \Delta ln(p_{t}) - E(\Delta ln(p_{t})) = [\alpha_{a} \quad \alpha_{z}] [a_{t} \quad z_{t}]'$ (6)</p>

<p>This is the structural model. Cochrane then says</p>

<blockquote>
  <p>I choose the parameters $\alpha$ so that </p>
</blockquote>

<p>$dp_{t} = [1 \quad -0.21] [sc_{t} \quad vc_{t}]'$</p>

<p>And lastly, he goes on to simulate this model. I am able to follow up until he introduces the price level ($dp_{t}$). My two questions are:</p>

<ol>
<li><p>Why do I need to subtract the means from both series? I.e why even introduce $vc_{t}$ and $sc_{t}$ and not just stick with $v_{t}$ and $s_{t}$. And how do I actually calculate these values? Should I simply subtract the sample mean from each observation? </p></li>
<li><p>I don't understand how to incorporate the price level in to the model. I somewhat understand the structural model, equation (6). But how do I use this to make simulations? I was able to find the AR(1) coefficients and the structural covariance matrix. But how do I use these to simulate the model as Cochrane did.</p></li>
</ol>

<p>I apologise for the long post, if you want further information I recommend p.42-46 from the paper.</p>

<p>Any feedback would be greatly appreciated.</p>

CC BY-SA 3.0
2018-03-03T12:41:14.127Z
I am not familiar with the Cochrane's empirical approach but I have read the theory part of his paper. So don't take this comment as a definite answer but here are my thoughts: Since you have surpluses and real debt value in terms of the two exogenous series and you know that inflation is a function of these processes as well -  Could you not infer $\alpha_{a}$ and $\alpha_{z}$ from the observed inflation equation? Then you would have the three variables in terms of 2 exogenous series and you know the AR coefficients. Then it should be quite simple to simulate the model (this would be fig.10)
user11767
CC BY-SA 3.0
2018-03-03T12:43:50.357Z
This would essentially lead to producing three AR(1) processes. However, I am still not sure how  Cochrane produced the simulations in fig. 11.
user11767
<macroeconomics><time-series><simulations>
Replicating a state-space model
WoodfordJr
# Replicating a state-space model

(Asked by: username_0 on 2018-02-28)

I am trying to replicate the results of [Cochrane, 1998](https://faculty.chicagobooth.edu/john.cochrane/research/papers/macroann.pdf). Most of the paper is just describing the theory behind The Fiscal Theory of the Price Level. But from p. 42 he begins the econometrics aspect.

Cochrane uses a frictionless model so the government's flow budget constraint can be written as:

$v\_{t} = \frac{1}{r\_{t+1}}(s\_{t+1} + v\_{t+1})$ (1)

where $v\_{t} = \frac{B\_{t-1}(t)}{p\_{t}}$ i.e, the real value of a bond issued in $t-1$ and matures in $t$. $s\_{t}$ is the real primary surplus and $r\_{t}$ is the real rate of return on government bonds.

For theoretical reasons he deflates the variables by consumption (as opposed to output) and writes the flow constraint as:

$\frac{v\_{t}}{c\_{t}} = \frac{1}{r\_{t+1}}\frac{c\_{t+1}}{c\_{t}} (\frac{s\_{t+1}}{c\_{t+1}} + \frac{v\_{t+1}}{c\_{t+1}})$. If we define $\beta = \frac{1}{r\_{t+1}} \frac{c\_{t+1}}{c\_{t}}$, the flow constraint can be iterated forward and expressed as:

$vc\_{t} = E\_{t} \sum\_{j=1}^{\infty} \beta^{j} sc\_{t+j}$ (2)

Note that $vc\_{t}$ and $sc\_{t}$
and just $vc\_{t} = \frac{v\_{t}}{c\_{t}} - E(\frac{v\_{t}}{c\_{t}})$ and $sc\_{t} = \frac{s\_{t}}{c\_{t}} - E(\frac{s\_{t}}{c\_{t}})$. Here is my first source of confusion: Why does he subtract the mean from each series? I understand that this is "deviations from the mean" but surely the flow constraint could be iterated forward even without subtracting the means?

A key aspect of the fiscal theory is that primary surpluses evolve exogenously. Thus, Cochrane defines the following:

$sc\_{t} = a\_{t} + z\_{t}$ (3) where

$a\_{t} = \phi\_{a}a\_{t-1} + \epsilon\_{at}$

$z\_{t} = \phi\_{z} z\_{t-1} + \epsilon\_{zt}$

note that $a\_{t}$ and $z\_{t}$ are unobserved variables.

Substituting this into equation (2) we find that:

$vc\_{t} = \frac{\beta \phi\_{a}}{1 - \beta \phi\_{a}} a\_{t} + \frac{\beta \phi\_{z}}{1 - \beta \phi\_{z}} z\_{t} $ (4)

So equation (3) and (4) describe the observed variables (sc,vc) in terms of the unobserved ones (a,z). We can put this in state-space form:

Let $Y\_{t} = (sc\_{t}, vc\_{t})'$ be the vector of observed variables and let $X\_{t} = (a\_{t},z\_{t})'$ be the vector of unobserved variables. Then the system can be represented as:

$Y\_{t} = BX\_{t}$ and $X\_{t} = AX\_{t-1} + \epsilon\_{t}$ where

$ B =\begin{bmatrix}
1 & 1 \\
\frac{\beta \phi\_{a}}{1 - \beta \phi\_{a}}& \frac{\beta \phi\_{z}}{1 - \beta \phi\_{z}}
\end{bmatrix}$, $A = \begin{bmatrix}
\phi\_{a} & 0 \\
0& \phi\_{z}
\end{bmatrix}$, $\epsilon\_{t} = (\epsilon\_{at}, \epsilon\_{zt})'$

This can be expressed as

$Y\_{t} = B(AX\_{t-1} + \epsilon\_{t}) =BA \underbrace{B^{-1}Y\_{t-1}}\_{\text{$X\_{t-1}$}} + \underbrace{e\_{t}}\_{\text{$B\epsilon\_{t}$}}$ (5)

So (5) is a reduced form equation which can be estimated. I have estimated the model and obtained $\phi\_{a}$ and $\phi\_{z}$ as well as the reduced form covariance matrix from which I was able to infer the "structural" covariance matrix. You can find Cochrane's results on p.43.

The next part is where I am very confused, he says:

> The government chooses debt so that inflation is a function of the two state variables z, a

$dp\_{t} = \Delta ln(p\_{t}) - E(\Delta ln(p\_{t})) = [\alpha\_{a} \quad \alpha\_{z}] [a\_{t} \quad z\_{t}]'$ (6)

This is the structural model. Cochrane then says

> I choose the parameters $\alpha$ so that

$dp\_{t} = [1 \quad -0.21] [sc\_{t} \quad vc\_{t}]'$

And lastly, he goes on to simulate this model. I am able to follow up until he introduces the price level ($dp\_{t}$). My two questions are:

1. Why do I need to subtract the means from both series? I.e why even introduce $vc\_{t}$ and $sc\_{t}$ and not just stick with $v\_{t}$ and $s\_{t}$. And how do I actually calculate these values? Should I simply subtract the sample mean from each observation?
2. I don't understand how to incorporate the price level in to the model. I somewhat understand the structural model, equation (6). But how do I use this to make simulations? I was able to find the AR(1) coefficients and the structural covariance matrix. But how do I use these to simulate the model as Cochrane did.

I apologise for the long post, if you want further information I recommend p.42-46 from the paper.

Any feedback would be greatly appreciated.

**Tags:** macroeconomics, time-series, simulations
> **Comments:**
> **username_1**: I am not familiar with the Cochrane's empirical approach but I have read the theory part of his paper. So don't take this comment as a definite answer but here are my thoughts: Since you have surpluses and real debt value in terms of the two exogenous series and you know that inflation is a function of these processes as well -  Could you not infer $\alpha_{a}$ and $\alpha_{z}$ from the observed inflation equation? Then you would have the three variables in terms of 2 exogenous series and you know the AR coefficients. Then it should be quite simple to simulate the model (this would be fig.10)  
> **username_1**: This would essentially lead to producing three AR(1) processes. However, I am still not sure how  Cochrane produced the simulations in fig. 11.  

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## 0 answers

