<p>Let the $n$ states of the finite-state markov chain be denoted
by $\{x_1,...,x_n\}$ and let $\vec e = [e(x_1), ..., e(x_n)]'$.
Now,
first note that because $X_{t+1} \mid X_t$ is independent of $W_{t+1}$, we can write
\begin{align*}
\exp(\eta) e(x) &amp;= E[\exp(D'x + x' F W_{t+1})] E[ e(X_{t+1}) \mid X_t = x] \\
  &amp;= \exp(D'x + x' F F' x)\, E[ e(X_{t+1}) \mid X_t = x] .
\end{align*}
Then, because this must hold for all values $x= x_1, ..., x_n$, we
can assemble the following vector equation,
\begin{align*}
\vec e \, \exp(\eta)
  &amp;= 
\begin{bmatrix}
\exp(D'x_1 + x_1' F F' x_1)\, E[ e(X_{t+1}) \mid X_t = x_1] \\
... \\
\exp(D'x_n + x_n' F F' x_n)\, E[ e(X_{t+1}) \mid X_t = x_n]  \\
\end{bmatrix}\\
  &amp;= 
\text{diag} \begin{bmatrix}
\exp(D'x_1 + x_1' F F' x_1) \\
... \\
\exp(D'x_n + x_n' F F' x_n) \\
\end{bmatrix}
\, \mathbb P \, \vec e \\
  &amp;= 
\text{diag}(f)\, \mathbb P \, \vec e \\
\end{align*}
where $f = [ \exp(D'x_1 + x_1' F F' x_1),  ..., \exp(D'x_n + x_n' F F' x_n) ]'$
and $\text{diag}$ is the operator that takes a vector and places
it on the diagonal of a matrix where the off-diagonal elements are zero.
So, we can express the equation as a linear operator $\mathbb M$ on $\vec e$
with
\begin{align*}
\text{diag}(f)\, \mathbb P \, \vec e
  &amp;= \vec e \, \exp(\eta) \\
\mathbb M \vec e &amp;= \vec e \, \exp(\eta).
\end{align*}</p>

CC BY-SA 3.0
2015-01-19T19:24:49.613Z
2015-01-19T19:24:49.613Z
jmbejara
<p><strong>NOTE:</strong> This question is related to the econometric method explored in the following two questions:</p>

<ul>
<li><a href="https://economics.stackexchange.com/questions/3000/multiplicative-factorization-of-stochastic-growth-time-series-solving-for-an-ei">Multiplicative factorization of stochastic growth time series--solving for an eigenfunction/eigenvector</a></li>
<li><a href="https://economics.stackexchange.com/questions/3009/example-of-the-change-of-measure-proposed-in-hansen-2012?lq=1">Example of the change of measure proposed in Hansen (2012)</a></li>
</ul>

<p><strong>QUESTION:</strong> Suppose that $X_t$ is an $n$-state Markov chain with transition probability matrix $\mathbb P$ and realized values given by $n$-dimensional coordinate vectors. Suppose that $\{W_{t+1} \}$ is an iid sequence of multivariate normally distributed random vectors. How would I represent an equation of the form
$$
E[\exp(D'X_t + X_t' F W_{t+1}) e(X_{t+1}) \mid X_t = x] = \exp(\eta) e(x)
$$
as an eigenvector problem for a matrix $\mathbb M$? How can I represent $\mathbb M$ in terms of the primitives of the problem?</p>

<econometrics><time-series><eigenvalue><markov-chain><linear-algebra>
How can I write a conditional expectation of finite state markov process in matrix notation
jmbejara
# How can I write a conditional expectation of finite state markov process in matrix notation

(Asked by: username_0 on 2015-01-19)

**NOTE:** This question is related to the econometric method explored in the following two questions:

* [Multiplicative factorization of stochastic growth time series--solving for an eigenfunction/eigenvector](https://economics.stackexchange.com/questions/3000/multiplicative-factorization-of-stochastic-growth-time-series-solving-for-an-ei)
* [Example of the change of measure proposed in Hansen (2012)](https://economics.stackexchange.com/questions/3009/example-of-the-change-of-measure-proposed-in-hansen-2012?lq=1)

**QUESTION:** Suppose that $X\_t$ is an $n$-state Markov chain with transition probability matrix $\mathbb P$ and realized values given by $n$-dimensional coordinate vectors. Suppose that $\{W\_{t+1} \}$ is an iid sequence of multivariate normally distributed random vectors. How would I represent an equation of the form
$$
E[\exp(D'X\_t + X\_t' F W\_{t+1}) e(X\_{t+1}) \mid X\_t = x] = \exp(\eta) e(x)
$$
as an eigenvector problem for a matrix $\mathbb M$? How can I represent $\mathbb M$ in terms of the primitives of the problem?

**Tags:** econometrics, time-series, eigenvalue, markov-chain, linear-algebra

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## 1 answers

### Answer by username_0 on 2015-01-19. Score: 2

Let the $n$ states of the finite-state markov chain be denoted
by $\{x\_1,...,x\_n\}$ and let $\vec e = [e(x\_1), ..., e(x\_n)]'$.
Now,
first note that because $X\_{t+1} \mid X\_t$ is independent of $W\_{t+1}$, we can write
\begin{align\*}
\exp(\eta) e(x) &= E[\exp(D'x + x' F W\_{t+1})] E[ e(X\_{t+1}) \mid X\_t = x] \\
&= \exp(D'x + x' F F' x)\, E[ e(X\_{t+1}) \mid X\_t = x] .
\end{align\*}
Then, because this must hold for all values $x= x\_1, ..., x\_n$, we
can assemble the following vector equation,
\begin{align\*}
\vec e \, \exp(\eta)
&=
\begin{bmatrix}
\exp(D'x\_1 + x\_1' F F' x\_1)\, E[ e(X\_{t+1}) \mid X\_t = x\_1] \\
... \\
\exp(D'x\_n + x\_n' F F' x\_n)\, E[ e(X\_{t+1}) \mid X\_t = x\_n] \\
\end{bmatrix}\\
&=
\text{diag} \begin{bmatrix}
\exp(D'x\_1 + x\_1' F F' x\_1) \\
... \\
\exp(D'x\_n + x\_n' F F' x\_n) \\
\end{bmatrix}
\, \mathbb P \, \vec e \\
&=
\text{diag}(f)\, \mathbb P \, \vec e \\
\end{align\*}
where $f = [ \exp(D'x\_1 + x\_1' F F' x\_1), ..., \exp(D'x\_n + x\_n' F F' x\_n) ]'$
and $\text{diag}$ is the operator that takes a vector and places
it on the diagonal of a matrix where the off-diagonal elements are zero.
So, we can express the equation as a linear operator $\mathbb M$ on $\vec e$
with
\begin{align\*}
\text{diag}(f)\, \mathbb P \, \vec e
&= \vec e \, \exp(\eta) \\
\mathbb M \vec e &= \vec e \, \exp(\eta).
\end{align\*}


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