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model of the log variance.

Conditional Expectation Function An important determinant of wage levels is education. In many empirical studies economists measure educational attainment by the number of years8 of schooling, and we will write this variable as education. The conditional mean of log wages given sex, race, and education is a single number for each category. For example E (log(wage) j sex = man; race = white; education = 12) = 2:84: We display in Figure 2.5 the conditional means of log(wage) for white men and white women as a function of education. The plot is quite revealing. We see that the conditional mean is increasing in years of education, but at a di§erent rate for schooling levels above and below nine years. Another striking feature of Figure 2.5 is that the gap between men and women is roughly constant for all education levels. As the variables are measured in logs this implies a constant average percentage gap between men and women regardless of educational attainment. In many cases it is convenient to simplify the notation by writing variables using single characters, typically y; x and/or z. It is conventional in econometrics to denote the dependent variable (e.g. log(wage)) by the letter y; a conditioning variable (such as sex ) by the letter x; and multiple conditioning variables (such as race, education and sex ) by the subscripted letters x1; x2; :::; xk. Conditional expectations can be written with the generic notation E (y j x1; x2; :::; xk) = m(x1; x2; :::; xk): 8Here, education is deÖned as years of schooling beyond kindergarten. A high school graduate has education=12, a college graduate has education=16, a Masterís degree has education=18, and a professional degree (medical, law or PhD) has education=20. CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 18 ● ● ● ● ● ● ● ● ● ● ● ● 2.0 2.5 3.0 3.5 4.0 Years of Education Log Dollars per Hour 4 6 8 10 12 14 16 18 20 ● white men white women Figure 2.5: Mean Log Wage as a Function of Years of Education We call this the conditional expectation function (CEF). The CEF is a function of (x1; x2; :::; xk) as it varies with the variables. For example, the conditional expectation of y = log(wage) given (x1; x2) = (sex ; race) is given by the six entries of Table 2.1. The CEF is a function of (sex ; race) as it varies across the entries. For greater compactness, we will typically write the conditioning variables as a vector in R k : x = 0 BBB@ x1 x2 . . . xk 1 CCCA : (2.2) Here we follow the convention of using lower case bold italics x to denote a vector. Given this notation, the CEF can be compactly written as E (y j x) = m (x): The CEF E (y j x) is a random variable as it is a function of the random variable x. It is also sometimes useful to view the CEF as a function of x. In this case we can write m (u) = E (y j x = u), which is a function of the argument u. The expression E (y j x = u) is the conditional expectation of y; given that we know that the random variable x equals the speciÖc value u. However, sometimes in econometrics we take a notational shortcut and use E (y j x) to refer to this function. Hopefully, the use of E (y j x) should be apparent from the context.