# What does expectation of $\varepsilon_{c,t}$ conditional on $_c$ mean?

From a discussion, I recieve a mathematic answer, 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 $$c$$"

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.

• 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 $\epsilon$ Jun 5, 2021 at 21:59
• @Knowledge-chaser yes I omitted subscripts for brevity sake but by $\epsilon$ I meant $\epsilon_{c,t}$
– 1muflon1
Jun 5, 2021 at 22:03
• @Knowledge-chaser oh I noticed I was using epsilon instead of varepsilon I will edit that probably that was bit confusing
– 1muflon1
Jun 5, 2021 at 22:07
• Yes, it is exactly what I mean, thank you , 1muflon1 Jun 6, 2021 at 6:24