I'm stuck with the following problem...
Assume that $ \mathbb{E}( y \vert x) = a + bx, $ where $a$ and $b$ are parameters. Define $ z :=\max(0, y)$. What are you able to conclude about the relationship between $ \mathbb{E}( y \vert x)$ and $\mathbb{E}( z \vert x)$ ?
I tried the following:
Attempt1: Decompose the positive and negative parts of y
$ y:= y^{+} - y^{-}$, where $ y^{+}=z, y^{-}=-\min(0, y)$ are positive and negative parts of y, respectively.
$\mathbb{E}( y \vert x) = \mathbb{E}( y^{+} \vert x) - \mathbb{E}( y^{-} \vert x)$
$ = \mathbb{E}( z \vert x) + \mathbb{E}( -y^{-} \vert x)$
$ = \mathbb{E}( z \vert x) + \mathbb{E}[ min(0, y) \vert x]$
For the 1st attempt, I tried to find distribution of $\min(0, y)$ with cdf(distribution fcn technique) but no assumption is given in the problem, so I failed....
Attempt2: Use CEF-Decomp property (Conditional Expectation Function)
According to Angrist-Pischke, MHE Thm 3.11,
$ Y = \mathbb{E}[ Y \vert X ] + \epsilon$, where $ \mathbb{E}[\epsilon \vert X] = 0$
$ \mathbb{E}( y \vert x) = a + bx \Rightarrow y = a + bx + \epsilon$ $ \mathbb{E}( z \vert x) = \mathbb{E}( z \vert x, y \geq 0)P(y \geq 0) + \mathbb{E}( z \vert x, y < 0)P(y < 0) $ $ = (a+bx)P(y \geq 0) $
since $\mathbb{E}( z \vert x, y < 0) = 0$
For the 2nd attempt, I failed to find a probability that Y is greater than or equal to zero... Someone told me that this problem is related to order statistics or median... Please help! Thanks in advance!!