AppendixA - xd 7/14/99 8:51 PM Page 643 A p p e n d i x A...

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T his appendix covers some basic mathematics that are used in econometric analy- sis. We summarize various properties of the summation operator, study properties of linear and certain nonlinear equations, and review proportions and percents. We also present some special functions that often arise in applied econometrics, includ- ing quadratic functions and the natural logarithm. The first four sections require only basic algebra skills. Section A.5 contains a brief review of differential calculus; while a knowledge of calculus is not necessary to understand most of the text, it is used in some end-of-chapter appendices and in several of the more advanced chapters in Part III. A.1 THE SUMMATION OPERATOR AND DESCRIPTIVE STATISTICS The summation operator is a useful shorthand for manipulating expressions involving the sums of many numbers, and it plays a key role in statistics and econometric analy- sis. If { x i : i ± 1,…, n } denotes a sequence of n numbers, then we write the sum of these numbers as ± n i ± 1 x i ² x 1 ² x 2 ² ² x n . (A.1) With this definition, the summation operator is easily shown to have the following prop- erties: PROPERTY SUM. 1: For any constant c , ± n i ± 1 c ± nc . (A.2) PROPERTY SUM. 2: For any constant c , ± n i ± 1 cx i ± c ± n i ± 1 x i . (A.3) 643 Appendix A Basic Mathematical Tools xd 7/14/99 8:51 PM Page 643
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PROPERTY SUM. 3: If {( x i , y i ): i ± 1,2,…, n } is a set of n pairs of numbers, and a and b are constants, then ± n i ± 1 ( ax i ² by i ) ± a ± n i ± 1 x i ² b ± n i ± 1 y i . (A.4) It is also important to be aware of some things that cannot be done with the sum- mation operator. Let {( x i , y i ): i ± n } again be a set of n pairs of numbers with y i ³ 0 for each i . Then, ± n i ± 1 ( x i / y i ) ³ ² ± n i ± 1 x i ³´² ± n i ± 1 y i ³ . In other words, the sum of the ratios is not the ratio of the sums. In the n ± 2 case, the application of familiar elementary algebra also reveals this lack of equality: x 1 / y 1 ² x 2 / y 2 ³ ( x 1 ² x 2 )/( y 1 ² y 2 ). Similarly, the sum of the squares is not the square of the sum: ± n i ± 1 x 2 i ³ ² ± n i ± 1 x i ³ 2 , except in special cases. That these two quantities are not gener- ally equal is easiest to see when n ± 2: x 2 1 ² x 2 2 ³ ( x 1 ² x 2 ) 2 ± x 2 1 ² 2 x 1 x 2 ² x 2 2 . Given n numbers { x i : i ± 1,…, n }, we compute their average or mean by adding them up and dividing by n : x ¯ ± (1/ n ± n i ± 1 x i . (A.5) When the x i are a sample of data on a particular variable (such as years of education), we often call this the sample average (or sample mean ) to emphasize that it is com- puted from a particular set of data. The sample average is an example of a descriptive statistic ; in this case, the statistic describes the central tendency of the set of points x i .
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AppendixA - xd 7/14/99 8:51 PM Page 643 A p p e n d i x A...

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