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# Time series factor analysis.

One beneÖt of focusing on conditional means is that they reduce complicated distributions to a single summary measure, and thereby facilitate comparisons across groups. Because of this simplifying property, conditional means are the primary interest of regression analysis and are a major focus in econometrics. Table 2.1 allows us to easily calculate average wage di§erences between groups. For example, we can see that the wage gap between men and women continues after disaggregation by race, as the average gap between white men and white women is 25%, and that between black men and black women is 13%. We also can see that there is a race gap, as the average wages of blacks are substantially less than the other race categories. In particular, the average wage gap between white men and black men is 21%, and that between white women and black women is 9%. 2.4 Log Di§erences* A useful approximation for the natural logarithm for small x is log (1 + x)  x: (2.1) This can be derived from the inÖnite series expansion of log (1 + x) : log (1 + x) = x x 2 2 + x 3 3 x 4 4 +    = x + O(x 2 ): The symbol O(x 2 ) means that the remainder is bounded by Ax2 as x ! 0 for some A < 1: A plot of log (1 + x) and the linear approximation x is shown in Figure 2.4. We can see that log (1 + x) and the linear approximation x are very close for jxj  0:1, and reasonably close for jxj  0:2, but the di§erence increases with jxj. Now, if y is c% greater than y; then y = (1 + c=100)y: Taking natural logarithms, log y = log y + log(1 + c=100) or log y log y = log(1 + c=100)  c 100 where the approximation is (2.1). This shows that 100 multiplied by the di§erence in logarithms is approximately the percentage di§erence between y and y , and this approximation is quite good for jcj  CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 17 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 log(1+x) x Figure 2.4: log(1 + x)