# Replicating a state-space model

I am trying to replicate the results of Cochrane, 1998. 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.

• 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) – BenBernke Mar 3 '18 at 12:41
• This would essentially lead to producing three AR(1) processes. However, I am still not sure how Cochrane produced the simulations in fig. 11. – BenBernke Mar 3 '18 at 12:43