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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

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  • $\begingroup$ Great question and nice thought process. $\endgroup$ – S. Iason Koutsoulis Jun 26 '20 at 14:44
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    $\begingroup$ If $(\Omega,\Sigma,\mu)$ is a probability space and $g:\Omega\to\mathbb{R}$ is a random variable, the information of $g$ is usually modeled by the $\sigma$-algebra generated by $g$; the coarsest $\sigma$-algebra on $\Omega$ under which $g$ is still measurable. Conditionion on $g$ is then the same thing as conditioning on this $\sigma$-algebra. $\endgroup$ – Michael Greinecker Jun 26 '20 at 14:46
  • $\begingroup$ @Frank: I'm not certain of the exact technical defintion but forcing $I_t$ to be a sigma field is too restrictive. My best guess would be that it is $I_t$ + the equations + any any known probability distributions + any relevant data from time t and earlier. $I_t$ is supposed to be everything and any information that the econometric modeller has at his disposal at time $t$. Pesaran's book ( The limits of RE) must discuss this but I can't remember if he provides the exact technical definition. Still it's a book worth checking out if you haven't already. It's the best text I know of for RE. $\endgroup$ – mark leeds Jun 27 '20 at 20:09
  • $\begingroup$ @ Michael Greinecker: Thanks for adding the definition of the $\sigma$-operator. $\endgroup$ – Frank Jun 29 '20 at 6:24
  • $\begingroup$ @ Mark Leeds: Thanks for your suggestion, but I have a problem with this 'definition'. Conditional expectations are typically (in mathematics) defined regarding a $\sigma$-Field, thus all the 'nice properties' like the law of iterated expectations are proved to hold. If you say, $I_t$ is not a $\sigma$-Field, I am not sure how to define the conditional expectation nor why it should be allowed to apply a law of iterated expectations (what is typically done by economists). $\endgroup$ – Frank Jun 29 '20 at 6:32
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Two notes.

A. "Conditioning on information" has always been applied in economics without much attention to probability theory-rigor, because it has (indeed) such a strong intuitive sense: "based on the information I have (where "information" 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".

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.

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  • $\begingroup$ Thanks for your notes. I agree with A. I understand that conditioning on information is independent of prob. theory. My point was that if mathematical propositions are applied to these conditional expectations there is a need for explaining the mathematical structure. As far as I understand you right, you confirm my point of view. I totally agree with B. This confirms my view of information sets. And you are right $\Omega$ is arbitrary but when it comes to $\mathcal{I}$ we need at least some measurability conditions.Regarding B what would you say is the generator of the example in my question? $\endgroup$ – Frank Nov 25 '20 at 14:42
  • $\begingroup$ @Frank Don't fret about it. "All relevant information" will do, even in a subjective sense. Imagine $\Omega$ as a set where each element is a vector containing all possible combinations of bits and pieces of information that are deemed subjectively relevant to agent $i$. The assumption is, exactly, that this subjective modelling of $v$ still is able to lead agent $i$ "close" to the objective conditional expectation, plus a zero-mean error. $\endgroup$ – Alecos Papadopoulos Nov 25 '20 at 16:12
  • $\begingroup$ Thanks for your patience but I still don't get it.Consider a model with various variables. Of course, I can say, that all these variables build the generator of the information set - In the model context, this is all relevant information. The question is whether this is necessary - Is there a smaller generator that leads to the same sigma field? This would imply that the agent may not need the whole information of the model.As I suggested in the question, isn't it enough to consider the information set as the sigma filed generated by the stochastic variables in the model? $\endgroup$ – Frank Dec 3 '20 at 6:54
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In McCallum's, "Monetary Economics", it is implied that $ I_t $ is a set that contains all information $ \{x_t,x_{t-1},..., y_t,y_{t-1}, ..., u_t, u_{t-1}, ...\} $, where $x_t$ is the value of variable $x$ for time $t$.

This means that an information set contains all the known variable prices, up to period $t$ and prior, including knowledge over stochastic variables ($u_t$) - and therefore stochastic trends.

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  • $\begingroup$ I'm not Frank, sorry $\endgroup$ – S. Iason Koutsoulis Jun 26 '20 at 17:17
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    $\begingroup$ @_the_rainbox. I moved the comment. my apologies. $\endgroup$ – mark leeds Jun 27 '20 at 20:10
  • $\begingroup$ @ the_rainbox: So you would say, that $I_t$ is the $\sigma$-Field generated by all model variables/processes? To come back to the example above, you suggest that $I_t=\sigma(\{p_s,m_s,y_s,y_s^*,\varepsilon_s : s\leq t\}) $? $\endgroup$ – Frank Jun 29 '20 at 6:43
  • $\begingroup$ the sigma-field is not needed - it usually represents a subset of a full set. Like, when sets A, B are included within Ω, you don't want the σ({Α,Β}) set, instead you want the Ω set. There's no need for sampling parts of Ω, you already know the entirety of Ω. $\endgroup$ – S. Iason Koutsoulis Jun 29 '20 at 7:47
  • $\begingroup$ I don't get your point. You say that the information set ist not a $\sigma$-Field, right? What I wanted to say with my last comment is, that $I_t$ is the smallest $\sigma$-Field on $\Omega$ under which all model processes are measureable, (See the comment of Michael Greinecker under the question). I further do not understand this part of your comment 'you already know the entirety of $\Omega$' Do you mean that $\Omega = \{x_t,x_{t-1},..., y_t,y_{t-1}, ..., u_t, u_{t-1}, ...\}$ ? $\endgroup$ – Frank Jun 29 '20 at 9:50

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