Definiton of information sets in rational expecations models I am struggling with the notion of 'information sets' in the context of rational expectation models in economics. I found interesting notes on the web (http://www2.econ.iastate.edu/tesfatsi/reintro.pdf) but I am not sure whether I am understanding the concept well. Let me explain my concerns along with the first example in the notes given in the link.
Consider a small model given in the following three equations:
$$ y_t =y_t^*+ap_{t-1}+b\mathbb{E}_{t-1}p_t\\
p_t =m_t+\varepsilon_t\\
\mathbb{E}_{t}p_{t+1}=\mathbb{E}(p_{t+1}\vert I_t)$$
where
$y_t^*$ notes the log of potential real GDP in period $t$
$y_t$ denotes the log of actual real GDP in period $t$
$\mathbb{E}_{t}p_{t+1}$ denotes the subjective forward-looking expectation of a representative agent in period $t$ regarding the price level in period $t+1$
$m_t$ denotes the log of the nominal money supply in period $t$
$\varepsilon_t$ is a stochastic shock at time $t$
$I_t$ denotes a period-t information set that is available to the representative agent at the end of the period $t$.
So my possibly stupid question is: What is $I_t$ or how is it defined?
To be more precise, let me outline what I think $I_t$ is.
First of all, because most economists apply the law of iterated expectations and other propositions that can be applied to conditional expectations, I suggest $I_t$ has to be a $\sigma$-Field because otherwise, one wouldn't be able to apply these propositions.
But how is this $\sigma$-Field defined?
Following the notes, Leigh Tesfatsion writes that the equations plus classification of variables and admissibility conditions together with the true variable values a,b and the deterministic exogenous process $(m_t)_{t \in \mathbb{N}}$ have to be part of the information set, as well as the properties of the probability distribution and properties of the stochastic shocks $(\varepsilon_t)_{t\in \mathbb{N}}$ and the values of past realizations of all variables.
Typically it is assumed that $\varepsilon=(\varepsilon_t)_{t\in \mathbb{N}}$ is a stochastic process defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$.
Thus I would say $I_{t}$ has to be a $\sigma$-Field over $\Omega$ and thus it has to be a system of subsets of $\Omega$, thus it cannot include specific equations, specific variable values nor variable classifications, or am I wrong?
Let $\mathbb{F}=(\mathcal{F}_t)_{t\in \mathbb{N}}$ be a filtration on $(\Omega,\mathcal{F},\mathbb{P})$, given as $\mathcal{F}_t=\sigma(\{\varepsilon_s:s\leq t\})$.
I thought $I_{t}$ to be the history of the stochastic process, i.e. $I_{t}=\mathcal{F}_{t}$, is this correct?
If not, could you provide me a (mathematical rigorous) definition of the information set $I_{t}$ or could you provide me some literature related to this issue?
Thanks in advance
Frank
Two notes.
A. &quot;Conditioning on information&quot; has always been applied in economics without much attention to probability theory-rigor, because it has (indeed) such a strong intuitive sense: &quot;based on the information I have (where &quot;information&quot; here means data, processing algorithms, psychological makeup, almost anything) I somehow form through a black-box process an expectation for the value of some variable&quot;.
B. Conditioning is always done with respect to a sigma-algebra. But in economics it is customary to write just the generator of the sigma-algebra, in our case $I_{t−1,i}$ and expect to be understood as $\sigma\left(I_{t−1,i}\right)$. So all you have to do is imagine $I_{t−1,i}$ as a set that can generate a sigma algebra. In that case, each element of the set can be anything really.
Note that the $\Omega$ in the probability space and the sigma-algebras that can come from it can literally be anything. We restrict $v$ to be a random variable, namely a function whose range is some numeric set, like the Reals or the Naturals, but the domain of $v$ can be, again, literally anything, and as multi-dimensional and as non-numeric as we like.