# 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?

Frank

• Great question and nice thought process.
– user28226
Commented Jun 26, 2020 at 14:44
• 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. Commented Jun 26, 2020 at 14:46
• @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. Commented Jun 27, 2020 at 20:09
• @ Michael Greinecker: Thanks for adding the definition of the $\sigma$-operator. Commented Jun 29, 2020 at 6:24
• @ 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). Commented Jun 29, 2020 at 6:32

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.

• 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? Commented Nov 25, 2020 at 14:42
• @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. Commented Nov 25, 2020 at 16:12
• 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? Commented Dec 3, 2020 at 6:54

The "answer" to the original question in this thread is briefly given on page 2 of the following notes cited in this question, and more carefully explained in the appendix appearing at the end of these notes:

Leigh Tesfatsion (Prof. of Econ, Iowa State University, Ames, IA 50011-1054) Last Updated: 29 September 2019 "Introductory Notes to Rational Expectations" http://www2.econ.iastate.edu/tesfatsi/reintro.pdf

From page 2: "Moreover, as discussed more carefully in Appendix A.6, the conditioning information set It−1 must be a collection of assertions that are true for a (possibly empty) subset A of possible worlds to which an objectively true probability P(A) can be assigned."

• Thanks for pointing to the enlightening appendices. Yes, we can define conditional expectation with respect to certain elements of a sigma field (as in A.6) or with respect to a whole sigma field (as in A4). In the latter case, it is a random variable. On p. 24 you state the law of iterated expectations that is often applied in economic models and in those contexts it is said that the object, the expectations are conditioned for, is the information set. So I conclude that the information set in those contexts is a sigma field. This viewpoint seems different to A.6 of your REintro notes. Commented Jan 20, 2022 at 16:35

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.

• I'm not Frank, sorry
– user28226
Commented Jun 26, 2020 at 17:17
• @_the_rainbox. I moved the comment. my apologies. Commented Jun 27, 2020 at 20:10
• @ 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 $I_t=\sigma(\{p_s,m_s,y_s,y_s^*,\varepsilon_s : s\leq t\})$? Commented Jun 29, 2020 at 6:43
• 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.
– user28226
Commented Jun 29, 2020 at 7:47
• 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 $\Omega = \{x_t,x_{t-1},..., y_t,y_{t-1}, ..., u_t, u_{t-1}, ...\}$ ? Commented Jun 29, 2020 at 9:50