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# Asymmetric price transmission models.

In this context it is useful to transform the data by taking the natural logarithm5 . Figure 2.2 shows the density of log hourly wages log(wage) for the same population, with its mean 2.95 drawn in with the arrow. The density of log wages is much less skewed and fat-tailed than the density of the level of wages, so its mean E (log(wage)) = 2:95 is a much better (more robust) measure6 of central tendency of the distribution. For this reason, wage regressions typically use log wages as a dependent variable rather than the level of wages. Another useful way to summarize the probability distribution F(u) is in terms of its quantiles. For any 2 (0; 1); the th quantile of the continuous7 distribution F is the real number q which satisÖes F (q ) = : The quantile function q ; viewed as a function of ; is the inverse of the distribution function F: The most commonly used quantile is the median, that is, q0:5 = m: We sometimes refer to quantiles by the percentile representation of ; and in this case they are often called percentiles, e.g. the median is the 50th percentile. 2.3 Conditional Expectation We saw in Figure 2.2 the density of log wages. Is this distribution the same for all workers, or does the wage distribution vary across subpopulations? To answer this question, we can compare wage distributions for di§erent groups ñ for example, men and women. The plot on the left in Figure 2.3 displays the densities of log wages for U.S. men and women with their means (3.05 and 5Throughout the text, we will use log(y) or log y to denote the natural logarithm of y: 6More precisely, the geometric mean exp (E (log w)) = \$19:11 is a robust measure of central tendency. 7 If F is not continuous the deÖnition is q = inffu : F(u)  g CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 15 2.81) indicated by the arrows. We can see that the two wage densities take similar shapes but the density for men is somewhat shifted to the right with a higher mean. Log Dollars per Hour Log Wage Density 0 1 2 3 4 5 6 Women Men (a) Women and Men Log Dollars per Hour Log Wage Density 1.8 3.2 4.6 white men white women black men black women (b) By Sex and Race Figure 2.3: Log Wage Density by Sex and Race The values 3.05 and 2.81 are the mean log wages in the subpopulations of men and women workers. They are called the conditional means (or conditional expectations) of log wages given sex. We can write their speciÖc values as E (log(wage) j sex = man) = 3:05 E (log(wage) j sex = woman) = 2:81