Conditional Expectation
A conditional expectation is the expected value of a conditioned probability distribution.
Description
The conditional expectation of Y given X can be expressed as:
Note that while an expected value is a scalar moment, a conditional expectation is a function of X. This is sometimes referred to as the conditional expectation function (CEF).
Properties
By the law of total probability, the expected value of a conditional expectation is the unconditioned expectation: E[ E[Y|X] ] = E[Y]. This is often called the tower probability.
Estimation by Regression
More generally, what is meant by a conditional expectation is the expected value of Y while controlling for X, as in E[Y|X]. This is the case for regression, which estimates the conditional expectation: E[Y|X] ≈ Xβ.
In the case of a pure linear regression like yi = β0 + β1x1i + β2x2i + ϵi, there is a straightforward interpretation of the regression coefficients as E[Y|X1, X2] = β0 + β1X1 + β2X2.
In the case of a transformed linear regression, backtransformation must be performed to interpret the coefficients.
For example, ln(yi) = β0 + β1x1i + β2x2i + ϵi backtransforms to yi = exp(β0 + β1x1i + β2x2i + ϵi) = exp(β0) * exp(β1x1i) * exp(β2x2i) * exp(ϵi). It follows that E[log(Y)|X]] is a geometric mean. Furthermore, while E[ϵ] = 0 follows from the normality assumption and ordinarily enables the error term to vanish from the expectation, the log-normal expectation is E[exp(ϵ)] = exp(σ2/2). It follows that the regression fitting this model will always underestimate the true conditional expectation. If the errors are exactly normally distributed, then multiplying by this factor is sufficient. Otherwise consider Duan's smearing factor.
As another example, the log-log transformation: ln(yi) = β0 + β1ln(x1i) + β2ln(x2i) + ϵi. This backtransforms to:
