<p>Conditional expectation just means expectation conditional on some additional variable. I assume you already understand that expected value is is the average value or mean of a random variable and that the observed data can be viewed as the values of a collection of independent identically distributed random variables. Consequently, sample mean is defined as the expectation of the data with respect to the empirical distribution for the observed data so for a random sample <span class="math-container">$E[x]=\mu_x$</span>.</p>
<p>Conditional expectation just means we are conditioning the expectation on some other set of variables. For example, suppose that at a university expected grade (X) of a sample of students is 8 (I am using Dutch grading system that is on scale 1-10), then let us suppose that expected grade of females in that sample is 9 and expected grade of males in that sample is 7. If we denote gender as a dummy where <span class="math-container">$D=1$</span> signifies females and <span class="math-container">$D=0$</span> signifies males, then in this case:</p>
<p><em>unconditional expected grade is</em> <span class="math-container">$E[X]=8$</span></p>
<p><em>conditional expected grade, conditional on being female is</em> <span class="math-container">$E[X|D=1]=9$</span></p>
<p><em>conditional expected grade, conditional on being male is</em> <span class="math-container">$E[X|D=0]=7$</span></p>
<p>Thus saying that &quot;<em><span class="math-container">$δ_c$</span> to be the expectation of <span class="math-container">$ε_{c,t}$</span> conditional on <span class="math-container">$c$</span></em>&quot; is just saying that <span class="math-container">$\delta_c$</span> is the expected value <span class="math-container">$\varepsilon$</span> conditional on what values <span class="math-container">$c$</span> take. For example, if <span class="math-container">$c$</span> is number of children you have <span class="math-container">$\delta_c$</span> would be expected value <span class="math-container">$\varepsilon$</span> given the number of children you happen to have rather than an expectation for anyone with any number of children.</p>

CC BY-SA 4.0
2021-06-05T21:59:49.827Z
I think this part ("Thus saying that "*$δ_c$ to be the expectation of $ε_{c,t}$ conditional on $c$*" is just saying that $\delta_c$ is the expected value **$\epsilon$** conditional on what values $c$ take") should be ("Thus saying that "*$δ_c$ to be the expectation of $ε_{c,t}$ conditional on $c$*" is just saying that $\delta_c$ is the expected value  of **$ε_{c,t}$** conditional on what values $c$ take"). I mean $ε_{c,t}$ rather than $\epsilon$
Phil Nguyen
CC BY-SA 4.0
2021-06-05T22:03:43.000Z
@Knowledge-chaser yes I omitted subscripts for brevity sake but by $\epsilon$ I meant $\epsilon_{c,t}$
1muflon1
CC BY-SA 4.0
2021-06-05T22:07:50.997Z
@Knowledge-chaser oh I noticed I was using epsilon instead of varepsilon I will edit that probably that was bit confusing
1muflon1
CC BY-SA 4.0
2021-06-06T06:24:07.483Z
Yes, it is exactly what I mean, thank you , 1muflon1
Phil Nguyen
CC BY-SA 4.0
2021-06-05T11:58:21.973Z
2021-06-05T22:08:15.010Z
2021-06-05T22:08:15.010Z
1muflon1
<p>From a discussion, I <a href="https://economics.stackexchange.com/a/45333/35009">recieve a mathematic answer</a>, I understand until one point</p>
<blockquote>
<p>This decomposition can always be made by setting <strong><span class="math-container">$\delta_c$</span> to be the
expectation of <span class="math-container">$\varepsilon_{c,t}$</span> conditional on <span class="math-container">$c$</span></strong> and <span class="math-container">$\gamma_t$</span>
to be the expectation of <span class="math-container">$\varepsilon_{c,t}$</span> conditional on <span class="math-container">$t$</span></p>
</blockquote>
<p>Can I ask what does this mean when saying &quot;<strong><span class="math-container">$\delta_c$</span> to be the
expectation of <span class="math-container">$\varepsilon_{c,t}$</span> conditional on <span class="math-container">$c$</span></strong>&quot;</p>

<mathematical-economics>
What does expectation of $\varepsilon_{c,t}$ conditional on $_c$ mean?
Phil Nguyen
# What does expectation of $\varepsilon_{c,t}$ conditional on $_c$ mean?

(Asked by: username_0 on 2021-06-05)

From a discussion, I [recieve a mathematic answer](https://economics.stackexchange.com/a/45333/35009), I understand until one point

> This decomposition can always be made by setting **$\delta\_c$ to be the
> expectation of $\varepsilon\_{c,t}$ conditional on $c$** and $\gamma\_t$
> to be the expectation of $\varepsilon\_{c,t}$ conditional on $t$

Can I ask what does this mean when saying "**$\delta\_c$ to be the
expectation of $\varepsilon\_{c,t}$ conditional on $c$**"

**Tags:** mathematical-economics


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## 1 answers

### Answer by username_1 on 2021-06-05. Score: 3 | Accepted answer

Conditional expectation just means expectation conditional on some additional variable. I assume you already understand that expected value is is the average value or mean of a random variable and that the observed data can be viewed as the values of a collection of independent identically distributed random variables. Consequently, sample mean is defined as the expectation of the data with respect to the empirical distribution for the observed data so for a random sample $E[x]=\mu\_x$.

Conditional expectation just means we are conditioning the expectation on some other set of variables. For example, suppose that at a university expected grade (X) of a sample of students is 8 (I am using Dutch grading system that is on scale 1-10), then let us suppose that expected grade of females in that sample is 9 and expected grade of males in that sample is 7. If we denote gender as a dummy where $D=1$ signifies females and $D=0$ signifies males, then in this case:

*unconditional expected grade is* $E[X]=8$

*conditional expected grade, conditional on being female is* $E[X|D=1]=9$

*conditional expected grade, conditional on being male is* $E[X|D=0]=7$

Thus saying that "*$δ\_c$ to be the expectation of $ε\_{c,t}$ conditional on $c$*" is just saying that $\delta\_c$ is the expected value $\varepsilon$ conditional on what values $c$ take. For example, if $c$ is number of children you have $\delta\_c$ would be expected value $\varepsilon$ given the number of children you happen to have rather than an expectation for anyone with any number of children.

> **Comments:**
> **username_0**: I think this part ("Thus saying that "*$δ_c$ to be the expectation of $ε_{c,t}$ conditional on $c$*" is just saying that $\delta_c$ is the expected value **$\epsilon$** conditional on what values $c$ take") should be ("Thus saying that "*$δ_c$ to be the expectation of $ε_{c,t}$ conditional on $c$*" is just saying that $\delta_c$ is the expected value  of **$ε_{c,t}$** conditional on what values $c$ take"). I mean $ε_{c,t}$ rather than $\epsilon$  
> **username_1**: @Knowledge-chaser yes I omitted subscripts for brevity sake but by $\epsilon$ I meant $\epsilon_{c,t}$  
> **username_1**: @Knowledge-chaser oh I noticed I was using epsilon instead of varepsilon I will edit that probably that was bit confusing  
> **username_0**: Yes, it is exactly what I mean, thank you , username_1  

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