<p>I am now learning <em>Recursive Macroeconomics Theory</em> by Lars Ljungqvist and Thomas J. Sargent and have a question about the spectral density of a covariance stationary process.</p>
<blockquote>
<p><strong>(<strong>Section 2.4 in LS</strong>)</strong> <span class="math-container">$$x_{t+1}=Ax_{t}+C\omega_{t+1}\tag{2.4.1}$$</span>
for <span class="math-container">$t=0,1,2,\dots,$</span> where <span class="math-container">$x_{t}$</span> is an <span class="math-container">$n\times 1$</span> state variable, <span class="math-container">$x_{0}$</span> is the initial condition drawn from a probability distribution with mean <span class="math-container">$\mu_{0}$</span> and covariance matrix <span class="math-container">$\Sigma_{0}$</span>. <span class="math-container">$A$</span> is an <span class="math-container">$n\times n$</span> matrix and <span class="math-container">$C$</span> is an <span class="math-container">$n\times m$</span> matrix and <span class="math-container">$\omega_{t+1}$</span> is an <span class="math-container">$m\times 1$</span> vector which is i.i.d and <span class="math-container">$\omega_{t+1}\sim N(0,I)$</span>.</p>
<p><strong>(Section 2.11 in LS)</strong> The spectral density matrix of the covariance stationary distribution <span class="math-container">$S_{x}(\omega)$</span> is defined to be the Fourier transformation of the covariogram of <span class="math-container">$x$</span>, <span class="math-container">$$S_{x}(\omega)=\sum_{\tau=-\infty}^{\infty}C_{x}(\tau)e^{-i\omega \tau}$$</span></p>
<p>For the system <span class="math-container">$(2.4.1)$</span>, the spectral density of the stationary distribution is <span class="math-container">$$S_{x}(\omega)=[I-Ae^{-i\omega}]^{-1}CC'[I-A'e^{+i\omega}]^{-1}\tag{2.11.2}$$</span></p>
</blockquote>
<p>My question is that, how can we derive equation <span class="math-container">$(2.11.2)$</span>?</p>
<p><strong>My attempt:</strong>  by equation <span class="math-container">$(2.4.1)$</span>, we know <span class="math-container">$$C_{x}(\tau)=Cov(x_{t},x_{t+\tau})=Cov\left(A^{\tau}x_{t}+\sum_{j=0}^{\tau}A^{j}C\omega_{t+\tau-j},x_{t}\right)=Cov(A^{\tau}x_{t},x_{t})=A^{\tau}CC'$$</span></p>
<p>The third equality follows from the independence between <span class="math-container">$w_{j}$</span> and <span class="math-container">$x_{t}$</span> for <span class="math-container">$j\geq t$</span> and the last equality follows from the fact that <span class="math-container">$$\mathbb E(x_{t+1}-Ax_{t})(x_{t+1}-Ax_{t})'=CC'$$</span></p>
<p>Then we can get <span class="math-container">$$S_{x}(\omega)=\sum_{\tau=-\infty}^{\infty}A^{\tau}CC'e^{-i\omega\tau}$$</span>
Because <span class="math-container">$C_{x}(\tau)=C_{x}(-\tau)$</span>, we can simplify the equation above and finally get <span class="math-container">$$S_{x}(\omega)=CC'[I-Ae^{-i\omega}]^{-1}+CC'A^{-1}e^{i\omega}[I-A^{-1}e^{+i\omega}]^{-1}$$</span>Which is different from <span class="math-container">$(2.11.2)$</span></p>
<p>Can someone help me with this? Thanks in advance!</p>

CC BY-SA 4.0
2022-05-24T03:16:45.887Z
Hi: I ddn't go through all of the details to be 100 percent sure so this is only a vague reference. if you go to page 18 at the link below and define $\psi(B) = (I - A)^{-1}$ you can write the AR(p) as a linear proccess composed of $\epsilon_t$ terms. Then, once you have it written that way, I'm pretty sure the derivation there will give you the expression that you are seeking. https://www.stat.berkeley.edu/~bartlett/courses/153-fall2010/lectures/16.pdf
mark leeds
CC BY-SA 4.0
2022-05-24T03:17:38.417Z
I think the way you are doing it is interesting but you are trying to work with the AR piece and the MA piece simultaneously which makes things more complicated. Good luck and I hope it helps.
mark leeds
CC BY-SA 4.0
2022-05-24T05:43:07.483Z
Just one more thing: I think that the problem with your approach is that there is covariance between $\omega_i$ and $x_{j}$ when $i < j$ because past $\omega_{i}$ are all entering into the $x_{j}$ for $i < j$. This is why it's better to re-write the response as a function of only past $\omega_{i}$ and nothing else.
mark leeds
<econometrics>
Spectral density of a covariance stationary process
John
# Spectral density of a covariance stationary process

(Asked by: username_0 on 2022-05-23)

I am now learning *Recursive Macroeconomics Theory* by Lars Ljungqvist and Thomas J. Sargent and have a question about the spectral density of a covariance stationary process.

> **(**Section 2.4 in LS**)** $$x\_{t+1}=Ax\_{t}+C\omega\_{t+1}\tag{2.4.1}$$
> for $t=0,1,2,\dots,$ where $x\_{t}$ is an $n\times 1$ state variable, $x\_{0}$ is the initial condition drawn from a probability distribution with mean $\mu\_{0}$ and covariance matrix $\Sigma\_{0}$. $A$ is an $n\times n$ matrix and $C$ is an $n\times m$ matrix and $\omega\_{t+1}$ is an $m\times 1$ vector which is i.i.d and $\omega\_{t+1}\sim N(0,I)$.
>
> **(Section 2.11 in LS)** The spectral density matrix of the covariance stationary distribution $S\_{x}(\omega)$ is defined to be the Fourier transformation of the covariogram of $x$, $$S\_{x}(\omega)=\sum\_{\tau=-\infty}^{\infty}C\_{x}(\tau)e^{-i\omega \tau}$$
>
> For the system $(2.4.1)$, the spectral density of the stationary distribution is $$S\_{x}(\omega)=[I-Ae^{-i\omega}]^{-1}CC'[I-A'e^{+i\omega}]^{-1}\tag{2.11.2}$$

My question is that, how can we derive equation $(2.11.2)$?

**My attempt:** by equation $(2.4.1)$, we know $$C\_{x}(\tau)=Cov(x\_{t},x\_{t+\tau})=Cov\left(A^{\tau}x\_{t}+\sum\_{j=0}^{\tau}A^{j}C\omega\_{t+\tau-j},x\_{t}\right)=Cov(A^{\tau}x\_{t},x\_{t})=A^{\tau}CC'$$

The third equality follows from the independence between $w\_{j}$ and $x\_{t}$ for $j\geq t$ and the last equality follows from the fact that $$\mathbb E(x\_{t+1}-Ax\_{t})(x\_{t+1}-Ax\_{t})'=CC'$$

Then we can get $$S\_{x}(\omega)=\sum\_{\tau=-\infty}^{\infty}A^{\tau}CC'e^{-i\omega\tau}$$
Because $C\_{x}(\tau)=C\_{x}(-\tau)$, we can simplify the equation above and finally get $$S\_{x}(\omega)=CC'[I-Ae^{-i\omega}]^{-1}+CC'A^{-1}e^{i\omega}[I-A^{-1}e^{+i\omega}]^{-1}$$Which is different from $(2.11.2)$

Can someone help me with this? Thanks in advance!

**Tags:** econometrics

> **Comments:**
> **username_1**: Hi: I ddn't go through all of the details to be 100 percent sure so this is only a vague reference. if you go to page 18 at the link below and define $\psi(B) = (I - A)^{-1}$ you can write the AR(p) as a linear proccess composed of $\epsilon_t$ terms. Then, once you have it written that way, I'm pretty sure the derivation there will give you the expression that you are seeking. https://www.stat.berkeley.edu/~bartlett/courses/153-fall2010/lectures/16.pdf  
> **username_1**: I think the way you are doing it is interesting but you are trying to work with the AR piece and the MA piece simultaneously which makes things more complicated. Good luck and I hope it helps.  
> **username_1**: Just one more thing: I think that the problem with your approach is that there is covariance between $\omega_i$ and $x_{j}$ when $i < j$ because past $\omega_{i}$ are all entering into the $x_{j}$ for $i < j$. This is why it's better to re-write the response as a function of only past $\omega_{i}$ and nothing else.  

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