<p>Two notes.</p>
<p><strong>A.</strong> &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;.</p>
<p><strong>B.</strong> 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 <span class="math-container">$I_{t−1,i}$</span> and expect to be understood as <span class="math-container">$\sigma\left(I_{t−1,i}\right)$</span>. So all you have to do is imagine <span class="math-container">$I_{t−1,i}$</span> as a set that can generate a sigma algebra. In that case, each element of the set can be anything really.</p>
<p>Note that the <span class="math-container">$\Omega$</span> in the probability space and the sigma-algebras that can come from it can <em>literally be anything</em>. We restrict <span class="math-container">$v$</span> to be a random variable, namely a function whose range is some numeric set, like the Reals or the Naturals, but the <em>domain</em> of <span class="math-container">$v$</span> can be, again, literally anything, and as multi-dimensional and as non-numeric as we like.</p>

CC BY-SA 4.0
2020-12-03T06:54:40.277Z
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?
Frank 
CC BY-SA 4.0
2020-11-25T14:42:00.840Z
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?
Frank 
CC BY-SA 4.0
2020-11-25T16:12:18.493Z
@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.
Alecos Papadopoulos
CC BY-SA 4.0
2020-11-23T16:14:06.160Z
2020-11-23T16:14:06.160Z
Alecos Papadopoulos
<p>The &quot;answer&quot; 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:</p>
<p>Leigh Tesfatsion (Prof. of Econ, Iowa State University, Ames, IA 50011-1054)
Last Updated: 29 September 2019
&quot;Introductory Notes to Rational Expectations&quot;
<a href="http://www2.econ.iastate.edu/tesfatsi/reintro.pdf" rel="nofollow noreferrer">http://www2.econ.iastate.edu/tesfatsi/reintro.pdf</a></p>
<p>From page 2:  &quot;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.&quot;</p>

CC BY-SA 4.0
2022-01-20T16:35:06.997Z
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.
Frank 
CC BY-SA 4.0
2022-01-10T17:14:33.637Z
2022-01-10T17:14:33.637Z
Leigh Tesfatsion
<p>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 (<a href="http://www2.econ.iastate.edu/tesfatsi/reintro.pdf" rel="noreferrer">http://www2.econ.iastate.edu/tesfatsi/reintro.pdf</a>) 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.</p>
<p>Consider a small model given in the following three equations:
<span class="math-container">$$ y_t =y_t^*+ap_{t-1}+b\mathbb{E}_{t-1}p_t\\
p_t =m_t+\varepsilon_t\\
\mathbb{E}_{t}p_{t+1}=\mathbb{E}(p_{t+1}\vert I_t)$$</span></p>
<p>where</p>
<p><span class="math-container">$y_t^*$</span> notes the log of potential real GDP in period <span class="math-container">$t$</span></p>
<p><span class="math-container">$y_t$</span> denotes the log of actual real GDP in period <span class="math-container">$t$</span></p>
<p><span class="math-container">$\mathbb{E}_{t}p_{t+1}$</span> denotes the subjective forward-looking expectation of a representative agent in period <span class="math-container">$t$</span> regarding the price level in period <span class="math-container">$t+1$</span></p>
<p><span class="math-container">$m_t$</span> denotes the log of the nominal money supply in period <span class="math-container">$t$</span></p>
<p><span class="math-container">$\varepsilon_t$</span> is a stochastic shock at time <span class="math-container">$t$</span></p>
<p><span class="math-container">$I_t$</span> denotes a period-t information set that is available to the representative agent at the end of the period <span class="math-container">$t$</span>.</p>
<p><strong>So my possibly stupid question is:</strong> What is <span class="math-container">$I_t$</span> or how is it defined?</p>
<p>To be more precise, let me outline what I think <span class="math-container">$I_t$</span> is.</p>
<p>First of all, because most economists apply the law of iterated expectations and other propositions that can be applied to conditional expectations, I suggest <span class="math-container">$I_t$</span> has to be a <span class="math-container">$\sigma$</span>-Field because otherwise, one wouldn't be able to apply these propositions.</p>
<p>But how is this <span class="math-container">$\sigma$</span>-Field defined?</p>
<p>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 <span class="math-container">$(m_t)_{t \in \mathbb{N}}$</span> have to be part of the information set, as well as the properties of the probability distribution and properties of the stochastic shocks <span class="math-container">$(\varepsilon_t)_{t\in \mathbb{N}}$</span> and the values of past realizations of all variables.</p>
<p>Typically it is assumed that <span class="math-container">$\varepsilon=(\varepsilon_t)_{t\in \mathbb{N}}$</span> is a stochastic process defined on the probability space <span class="math-container">$(\Omega,\mathcal{F},\mathbb{P})$</span>.
Thus I would say <span class="math-container">$I_{t}$</span> has to be a <span class="math-container">$\sigma$</span>-Field over <span class="math-container">$\Omega$</span> and thus it has to be a system of subsets of <span class="math-container">$\Omega$</span>, thus it cannot include specific equations, specific variable values nor variable classifications, or am I wrong?</p>
<p>Let <span class="math-container">$\mathbb{F}=(\mathcal{F}_t)_{t\in \mathbb{N}}$</span> be a filtration on <span class="math-container">$(\Omega,\mathcal{F},\mathbb{P})$</span>, given as <span class="math-container">$\mathcal{F}_t=\sigma(\{\varepsilon_s:s\leq t\})$</span>.</p>
<p>I thought <span class="math-container">$I_{t}$</span> to be the history of the stochastic process, i.e. <span class="math-container">$I_{t}=\mathcal{F}_{t}$</span>, is this correct?</p>
<p>If not, could you provide me a (mathematical rigorous) definition of the information set <span class="math-container">$I_{t}$</span> or could you provide me some literature related to this issue?</p>
<p>Thanks in advance</p>
<p>Frank</p>

CC BY-SA 4.0
2020-06-26T14:46:01.370Z
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.
Michael Greinecker
CC BY-SA 4.0
2020-06-27T20:09:40.617Z
@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.
mark leeds
CC BY-SA 4.0
2020-06-29T06:24:10.470Z
@ Michael Greinecker: Thanks for adding the definition of the $\sigma$-operator.
Frank 
CC BY-SA 4.0
2020-06-29T06:32:22.163Z
@ 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).
Frank 
CC BY-SA 4.0
2020-06-29T06:38:18.103Z
@ Mark Leeds: Thanks for the literature recommendation, I'll check it immediately.
Frank 
CC BY-SA 4.0
2020-07-02T13:39:41.247Z
This is a case where there is not really any gain to pin down a rigorous definition as an economist. Many macroeconomists can't tell the difference between different sigma-algebras except in trivial cases or even state the definition of a sigma-algebra (as the answer below shows)---it's also not clear why they should care to. "Information set" as used here originates from an economic context (RE), not mathematics.  Taken to data, this might even mean OLS regression. "Pretend" it's a sigma algebra when convenient---e.g. when a LIE is implicitly invoked.
Michael
CC BY-SA 4.0
2020-07-06T14:28:48.553Z
@ Michael, yes you may be right when you are saying that there is no gain to pin down a rigorous definition as an economist (although I would say giving and working with precise definitions is the core of science). However, I think it's unsatisfactory when the same thing is treated differently in various contexts. When I am reading Paul Klein's Paper on solving a RE model, he just assumes that the exogenous shock process has to be adapted to a filtration (his information sets). It is quite different from saying that initial values or variable classifications are contained in information sets.
Frank 
CC BY-SA 4.0
2020-07-26T15:28:29.433Z
@Frank: Only seeing this now. Also, I see your point as far as the problem with not defining a sigma field. But, as far as that goes, can't one just say that the expectations and conditional expectations are in the information set at time $t$. RE says that the agent has the same exact information as the econometrician at time $t$. I see how you want to be rigorous ( and it's appreciated. I have found the RE literature to be difficult to understand. they often don't define much )  but I'm  not clear on the problem, if one defines the information set at time $t$ in the way I just defined it.
mark leeds
CC BY-SA 4.0
2020-07-28T10:59:44.327Z
@ Mark Leeds: Thanks again for your response and for the discussion. Yes, I would agree with your sentence 'RE says that the agent has the same exact information as the econometrician at time t'. I talked to Leigh Tasfatsion directly and she highlighted that the RE concept depends not on a specific random theory (we typically impose to one introduced by Kolomorov in 1933). So one implication is that RE are in general independent of sigma fields and probability spaces. (Part 1/2)
Frank 
CC BY-SA 4.0
2020-07-28T11:14:27.327Z
However, when we build a model, we have to define the underlying probability and information concept. Doing so, one typically applies already existing mathematical theories. One possible way to do this is to assume that information at time t is represented by a sigma field, offering nice properties like the law of iterated expectations and others. If we, one the other hand, define the information set to be part of a sigma field (as Leigh did), i.e. as an event, it's also fine but then we cannot use the law of iterated expectations (or am I wrong?).  (Part 2/2)
Frank 
CC BY-SA 4.0
2020-07-28T13:49:38.333Z
@Frank: yeah, it sounds like the best way to think about it is to think of the information at time t as being represented by a sigma field. This way, you can use all the standard machinery such as conditional expectation etc.You sound like you're really into RE so, if you know of or find a great textbook ( my favorite is pesaran ) that goes into RE and doesn't assume that one has the brain and knowledge  of sargent or lucas, it's appreciated.  Lucas's book,  business cycle theory contains most of his papers and helped me a lot also. He's the clearest writer IMHO when it comes to RE papers.
mark leeds
<definition><information><rational-expectations>
Definiton of information sets in rational expecations models
Frank 
# Definiton of information sets in rational expecations models

(Asked by: username_0 on 2020-06-26)

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, username_5 username_5 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

username_0

**Tags:** definition, information, rational-expectations

> **Comments:**
> **username_1**: 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.  
> **username_2**: @username_0: 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.  
> **username_0**: @ username_1: Thanks for adding the definition of the $\sigma$-operator.  
> **username_0**: @ Mark username_2: 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).  
> **username_0**: @ Mark username_2: Thanks for the literature recommendation, I'll check it immediately.  
> **username_3**: This is a case where there is not really any gain to pin down a rigorous definition as an economist. Many macroeconomists can't tell the difference between different sigma-algebras except in trivial cases or even state the definition of a sigma-algebra (as the answer below shows)---it's also not clear why they should care to. "Information set" as used here originates from an economic context (RE), not mathematics.  Taken to data, this might even mean OLS regression. "Pretend" it's a sigma algebra when convenient---e.g. when a LIE is implicitly invoked.  
> **username_0**: @ username_1, yes you may be right when you are saying that there is no gain to pin down a rigorous definition as an economist (although I would say giving and working with precise definitions is the core of science). However, I think it's unsatisfactory when the same thing is treated differently in various contexts. When I am reading Paul Klein's Paper on solving a RE model, he just assumes that the exogenous shock process has to be adapted to a filtration (his information sets). It is quite different from saying that initial values or variable classifications are contained in information sets.  
> **username_2**: @username_0: Only seeing this now. Also, I see your point as far as the problem with not defining a sigma field. But, as far as that goes, can't one just say that the expectations and conditional expectations are in the information set at time $t$. RE says that the agent has the same exact information as the econometrician at time $t$. I see how you want to be rigorous ( and it's appreciated. I have found the RE literature to be difficult to understand. they often don't define much )  but I'm  not clear on the problem, if one defines the information set at time $t$ in the way I just defined it.  
> **username_0**: @ Mark username_2: Thanks again for your response and for the discussion. Yes, I would agree with your sentence 'RE says that the agent has the same exact information as the econometrician at time t'. I talked to username_5 Tasfatsion directly and she highlighted that the RE concept depends not on a specific random theory (we typically impose to one introduced by Kolomorov in 1933). So one implication is that RE are in general independent of sigma fields and probability spaces. (Part 1/2)  
> **username_0**: However, when we build a model, we have to define the underlying probability and information concept. Doing so, one typically applies already existing mathematical theories. One possible way to do this is to assume that information at time t is represented by a sigma field, offering nice properties like the law of iterated expectations and others. If we, one the other hand, define the information set to be part of a sigma field (as username_5 did), i.e. as an event, it's also fine but then we cannot use the law of iterated expectations (or am I wrong?).  (Part 2/2)  
> **username_2**: @username_0: yeah, it sounds like the best way to think about it is to think of the information at time t as being represented by a sigma field. This way, you can use all the standard machinery such as conditional expectation etc.You sound like you're really into RE so, if you know of or find a great textbook ( my favorite is pesaran ) that goes into RE and doesn't assume that one has the brain and knowledge  of sargent or lucas, it's appreciated.  Lucas's book,  business cycle theory contains most of his papers and helped me a lot also. He's the clearest writer IMHO when it comes to RE papers.  

---

## 2 answers

### Answer by username_4 on 2020-11-23. Score: 5

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.

> **Comments:**
> **username_0**: 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?  
> **username_4**: @username_0Don'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.  
> **username_0**: 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?  

---

### Answer by username_5 on 2022-01-10. Score: 2

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:

username_5 username_5 (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."

> **Comments:**
> **username_0**: 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.  

---

