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# Distributed-lag linear structural equation models.

Here we have used the common and convenient convention of using the single character y to denote a random variable, rather than the more cumbersome label wage. A general deÖnition of the mean is presented in Section 2.31. The mean U.S. wage (\$23.90) is indicated in the right panel of Figure 2.1 by the arrow. We sometimes use the notation Ey instead of E (y) when the variable whose expectation is being taken is clear from the context. There is no distinction in meaning. The mean is a convenient measure of central tendency because it is a linear operator and arises naturally in many economic models. A disadvantage of the mean is that it is not robust4 especially in the presence of substantial skewness or thick tails, which are both features of the wage distribution as can be seen easily in the right panel of Figure 2.1. Another way of viewing this is that 64% of workers earn less that the mean wage of \$23.90, suggesting that it is incorrect to describe the mean as a ìtypicalîwage rate. 1The distribution and density are estimated nonparametrically from the sample of 50,742 full-time non-military wage-earners reported in the March 2009 Current Population Survey. The wage rate is constructed as annual individual wage and salary earnings divided by hours worked. 2 If F is not continuous the deÖnition is m = inffu : F(u)  1 2 g 3The median is not sensitive to pertubations in the tails of the distribution. 4The mean is sensitive to pertubations in the tails of the distribution CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 14 Log Dollars per Hour Log Wage Density 0 1 2 3 4 5 6 Figure 2.2: Log Wage Density 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 ) = :