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Define the conditional expectation function (CEF) by $E[Y_i|X_i]$.

In Mostly Harmless Econometrics, Angrist and Pischke (2009, p. 30) write:

Because $X_i$ is random, the CEF is random

I do not understand the above statement.

Say $Y_i$ is real-valued. Then $E[Y_i|X_i]$ gives us, for each specific $X_i$, a real number. In what way is this random?

mdewey
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user20311
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1 Answers1

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Say $Y_i$ is real-valued. Then $E[Y_i|X_i]$ gives us, for each specific $X_i$, a real number. In what way is this random?

Your first two sentences are entirely correct, so think about the implication of this. This means that the conditional expectation is a function mapping each possible outcome of $X_i$ to a real number. Thus, if we let $\mathscr{X}$ denote the range of possible values of $X_i$ then there is some function $h:\mathscr{X} \rightarrow \mathbb{R}$ such that:

$$\mathbb{E}(Y_i|X_i) = h(X_i) \quad \quad \quad \quad \quad \mathbb{E}(Y_i|X_i=x) = h(x).$$

So, we have established that the conditional expectation is a function of the conditioning variable. In the first of the two cases shown in the above equations, the conditioning variable is treated as a random variable, so our function of that variable is also random.

Ben
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