Sorry for the long title, but my problem is quite specific and hard to explain in one title.
I am currently learning about the Roy Model (treatment effect analysis).
There is one derivation step at my slides, which I do not understand.
We calculate the expected outcome with treatment in the tretment group (dummy D is treatment or not treatment). This is written as
\begin{align} E[Y_1|D=1] \end{align}
since $Y_1=\mu_1 + U_1$ this can be rewritten as \begin{align} E[Y_1|D=1] &= E[\mu_1+U_1|D=1]\\ &=\mu_1+ E[U_1|D=1] \end{align} before we also said, that $D=1$ if $Y_1>Y_0$ so it follows:
$Y_1-Y_0>0$
$\mu_1+U_1-(\mu_0-U_0)>0$
$(\mu_1+U_1)/\sigma-(\mu_0-U_0)/ \sigma >0$
$Z-\epsilon>0$
so $D=1$ if $\epsilon<Z$
Therefore it holds, that \begin{align} E[Y_1|D=1] &=\mu_1 + E[U_1|\epsilon<Z] \end{align}
It is further known, that \begin{align} \begin{bmatrix} U_1 \\ U_0 \\ \epsilon \end{bmatrix}=N\left( \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} \sigma_1^2 & \sigma_{10} & \sigma_{1\epsilon} \\ \sigma_{10} & \sigma_{0}^2 & \sigma_{0\epsilon} \\ \sigma_{1\epsilon} & \sigma_{0\epsilon} & \sigma_{\epsilon}^2 \end{bmatrix}\right) \end{align}
therefore it follows: $P(D=1)=P(\epsilon<Z)=\Phi(Z)$
So now comes my question, the slides say, that \begin{align} \mu_1 - E[U_1|\epsilon<Z] =\mu_1 - \sigma_{1\epsilon} \frac{\phi(Z)}{\Phi(Z)} \end{align} And I do not understand why?
I know, that if two random variables follow a standard bivariate normal distribution: $E[u_1|u_2)=\rho u_2$
so $E[u_1|u_2>c)=E[\rho u_2|u_2>c]=\rho E[u_2|u_2>c)=\rho\frac{\phi(c)}{1-\Phi(c)}$
Therefore I would have expected a "plus" and not a minus sign? Also why do we use the covariance $\sigma_{1\epsilon}$ and not the correlation $\rho$? So I would have expected something like
\begin{align} \mu_1 - E[U_1|\epsilon<Z] =\mu_1 + \rho \frac{\phi(Z)}{\Phi(Z)} \end{align}
I am aware of the fact, that if I do the truncation from above the $1-\Phi(c)$ becomes a $\Phi(c)$.