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Panel smooth transition regression models.

Notice the way the law is applied. The inner expectation conditions on x1 and x2, while the outer expectation conditions only on x1: The iterated expectation yields the simple answer E (y j x1); the expectation conditional on x1 alone. Sometimes we phrase this as: ìThe smaller information set wins.î CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 21 As an example E (log(wage) j sex = man; race = white) P (race = whitejsex = man) + E (log(wage) j sex = man; race = black) P (race = blackjsex = man) + E (log(wage) j sex = man; race = other) P (race = otherjsex = man) = E (log(wage) j sex = man) or numerically 3:07 0:84 + 2:86 0:08 + 3:03 0:08 = 3:05: A property of conditional expectations is that when you condition on a random vector x you can e§ectively treat it as if it is constant. For example, E (x j x) = x and E (g (x) j x) = g (x) for any function g(): The general property is known as the Conditioning Theorem. Theorem 2.3 Conditioning Theorem If E jyj < 1 then E (g (x) y j x) = g (x) E (y j x): (2.5) If in addition E jg (x) yj < 1 then E (g (x) y) = E (g (x) E (y j x)): (2.6) The proofs of Theorems 2.1, 2.2 and 2.3 are given in Section 2.36.