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time series data analysis.

Continuous Variables In the previous sections, we implicitly assumed that the conditioning variables are discrete. However, many conditioning variables are continuous. In this section, we take up this case and assume that the variables (y; x) are continuously distributed with a joint density function f(y; x): CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 19 As an example, take y = log(wage) and x = experience, the number of years of potential labor market experience9 . The contours of their joint density are plotted on the left side of Figure 2.6 for the population of white men with 12 years of education. Labor Market Experience (Years) Log Dollars per Hour 0 10 20 30 40 50 2.0 2.5 3.0 3.5 4.0 Conditional Mean (a) Joint Density of log(wage) and Experience and Conditional Mean Log Dollars per Hour Log Wage Conditional Density 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Exp=5 Exp=10 Exp=25 Exp=40 (b) Conditional Density Figure 2.6: White Men with High School Degree Given the joint density f(y; x) the variable x has the marginal density fx(x) = Z 1 1 f(y; x)dy: For any x such that fx(x) > 0 the conditional density of y given x is deÖned as fyjx (y j x) = f(y; x) fx(x) : (2.3) The conditional density is a (renormalized) slice of the joint density f(y; x) holding x Öxed. The slice is renormalized (divided by fx(x) so that it integrates to one and is thus a density.) We can visualize this by slicing the joint density function at a speciÖc value of x parallel with the y-axis. For example, take the density contours on the left side of Figure 2.6 and slice through the contour plot at a speciÖc value of experience, and then renormalize the slice so that it is a proper density. This gives us the conditional density of log(wage) for white men with 12 years of education and this level of experience. We do this for four levels of experience (5, 10, 25, and 40 years), and plot these densities on the right side of Figure 2.6. We can see that the distribution of wages shifts to the right and becomes more di§use as experience increases from 5 to 10 years, and from 10 to 25 years, but there is little change from 25 to 40 years experience. The CEF of y given x is the mean of the conditional density (2.3) m (x) = E (y j x) = Z 1 1 yfyjx (y j x)