# What is the expectation of the error term conditional on Y?

In an OLS model, we usually assume $(Y_i, X_i)$ are i.i.d., and $U|X$ has mean $0$ and variance $\sigma^2$, and then run a regression of $Y$ on $X$.

But when we are running a regression of $X$ on $Y$ instead, what will be $U|Y$ like (we assume the iid and $U|X$ assumptions still hold)？ Will it still have zero mean?

Thanks!

• The answer depends on the distribution of the $X_i$.
– whuber
Feb 5, 2017 at 21:44

## 1 Answer

The model you specified is, in fact, partial in the sense that you only have conditional expectation/variance of $Y$ given $X$. In order to answer your question, you need to know the value $E[X|Y]$. This is possible when you have marginal distribution of $X$. So as whuber pointed out in the comment, it depends on the marginal distribution of $X$. If your model is $Y=\beta X+U$, the answer is yes if and only if $E[X|Y] = \frac{1}{\beta}Y$

Some special cases when $U$ is normal. If $X$ is normal, then your answer is YES as you have independence. If $X$ follows a Bernoulli with $p=\frac{1}{2}$ and $\beta=1$, you can easily see that $E[X|Y] \neq Y$ is never true.