Conditional expectation; regression

Question

Assume that

E[$y_{it}|c_i, x_{it}$] = $c_i$ + $x'_{it}\beta$

Eliminate $c_i$ by taking the expectation with respect to $c_i$, leading to

E[$y_{it}|x_{it}$] = E[$c_i|x_{it}$] + $x'_{it}\beta$

Can anybody explain this step?

Answer 1

Sure: For every random variables $x$, $y$, $z$, with $y$ integrable, one has $E(E(y\mid x,z)\mid x)=E(y\mid x)$. This is called the tower property and is mentioned in every decent introductory chapter on conditional expectations.

In your case, $E(y\mid x,z)=z+x'\beta$ hence $E(y\mid x)=E(z+x'\beta\mid x)=E(z\mid x)+x'\beta$.