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– Threshold and smooth transition models.

CEF Error The CEF error e is deÖned as the di§erence between y and the CEF evaluated at the random vector x: e = y m(x): By construction, this yields the formula y = m(x) + e: (2.7) In (2.7) it is useful to understand that the error e is derived from the joint distribution of (y; x); and so its properties are derived from this construction. Many authors in econometrics denote the CEF error using the Greek letter ” (epsilon). I do not follow this convention since the error e is a random variable similar to y and x, and typically use Latin characters for random variables. A key property of the CEF error is that it has a conditional mean of zero. To see this, by the linearity of expectations, the deÖnition m(x) = E (y j x) and the Conditioning Theorem E (e j x) = E ((y m(x)) j x) = E (y j x) E (m(x) j x) = m(x) m(x) = 0: This fact can be combined with the law of iterated expectations to show that the unconditional mean is also zero. E (e) = E (E (e j x)) = E (0) = 0: We state this and some other results forma CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 22 Theorem 2.4 Properties of the CEF error If E jyj < 1 then 1. E (e j x) = 0: 2. E (e) = 0: 3. If E jyj r < 1 for r  1 then E jej r < 1: 4. For any function h (x) such that E jh (x) ej < 1 then E (h (x) e) = 0: The proof of the third result is deferred to Section 2.36: The fourth result, whose proof is left to Exercise 2.3, implies that e is uncorrelated with any function of the regressors. The equations y = m(x) + e E (e j x) = 0 together imply that m(x) is the CEF of y given x. It is important to understand that this is not a restriction. These equations hold true by deÖnition. The condition E (e j x) = 0 is implied by the deÖnition of e as the di§erence between y and the CEF m (x): The equation E (e j x) = 0 is sometimes called a conditional mean restriction, since the conditional mean of the error e is restricted to equal zero. The property is also sometimes called mean independence, for the conditional mean of e is 0 and thus independent of x. However, it does not imply that the distribution of e is independent of x: Sometimes the assumption ìe is independent of xîis added as a convenient simpliÖcation, but it is not generic feature of the conditional mean. Typically and generally, e and x are jointly dependent, even though the conditional mean of e is zero. As an example, the contours of the joint density of e and experience are plotted in Figure 2.7 for the same population as Figure 2.6. Notice that the shape of the conditional distribution varies with the level of experience. As a simple example of a case where x and e are mean independent yet dependent, let e = x” where x and ” are independent N(0; 1): Then conditional on x; the error e has the distribution N(0; x2 ): Thus E (e j x) = 0 and e is mean independent of x; yet e is not fully independent of x: Mean independence does not imply full independence.