测量单位和函数形式外文翻译.docx

1、测量单位和函数形式外文翻译河南科技学院2014届本科毕业论文（设计）英文文献及翻译UNITS OF MEASUREMENT AND FUNCTIONAL FORM学生姓名： 所在院系： 数学科学学院 所学专业： 数学与应用数学 导师姓名： 完成时间： 2013年12月25日 UNITS OF MEASUREMENT AND FUNCTIONAL FORM ( V o t i n g O u t c o m e s a n d C a m p a i g n E x p e n d i t u r e s )In the voting outcome equation in (2.28), R=

2、 0.505. Thus, the share of campaign expenditures explains just over 50 percent of the variation in the election outcomes for this sample. This is a fairly sizable portionTwo important issues in applied economics are (1) understanding how changing theunits of measurement of the dependent and/or indep

3、endent variables affects OLS estimates and (2) knowing how to incorporate popular functional forms used in economics into regression analysis. The mathematics needed for a full understanding of functional form issues is reviewed in Appendix A.The Effects of Changing Units of Measurement on OLSStatis

4、ticsIn Example 2.3, we chose to measure annual salary in thousands of dollars, and the return on equity was measured as a percent (rather than as a decimal). It is crucial to know how salary and roe are measured in this example in order to make sense of the estimates in equation (2.39). We must also

5、 know that OLS estimates change in entirely expected ways when the units of measurement of the dependent and independent variables change. In Example2.3, suppose that, rather than measuring salary in thousands of dollars, we measure it in dollars. Let salardol be salary in dollars (salardol =845,761

6、 would be interpreted as $845,761.). Of course, salardol has a simple relationship to the salary measured in thousands of dollars: salardol ? 1,000?salary. We do not need to actually run the regression of salardol on roe to know that the estimated equation is: salardol = 963,191 +18,501 roe. We obta

7、in the intercept and slope in (2.40) simply by multiplying the intercept and theslope in (2.39) by 1,000. This gives equations (2.39) and (2.40) the same interpretation.Looking at (2.40), if roe = 0, then salardol = 963,191, so the predicted salary is$963,191 the same value we obtained from equation

8、 (2.39). Furthermore, if roeincreases by one, then the predicted salary increases by $18,501; again, this is what weconcluded from our earlier analysis of equation (2.39).Generally, it is easy to figure out what happens to the intercept and slope estimates when the dependent variable changes units o

9、f measurement. If the dependent variable is multiplied by the constant cwhich means each value in the sample is multiplied bycthen the OLS intercept and slope estimates are also multiplied by c. (This assumes nothing has changed about the independent variable.) In the CEO salary example, c ?1,000 in

10、 moving from salary to salardol.Chapter 2The Simple Regression ModelWe can also use the CEO salary example to see what happens when we change the units of measurement of the independent variable. Define roedec =roe/100 to be the decimal equivalent of roe; thus, roedec =0.23 means a return on equity

11、of23 percent. To focus on changing the unitsof measurement of the independent variable, we return to our original dependent variable, salary, which is measured in thousands of dollars. When we regress salary onroedec, we obtain salary =963.191 + 1850.1 roedec.The coefficient on roedec is 100 times t

12、he coefficient on roe in (2.39). This is as it should be. Changing roe by one percentage point is equivalent to roedec = 0.01. From (2.41), if roedec = 0.01, then salary = 1850.1(0.01) = 18.501, which is what is obtained by using (2.39). Note that, in moving from (2.39) to (2.41), the independentvar

13、iable was divided by 100, and so the OLS slope estimate was multiplied by 100, preserving the interpretation of the equation. Generally, if the independent variable is divided or multiplied by some nonzero constant, c, then the OLS slope coefficient is also multiplied or divided by c respectively.Th

14、e intercept has not changed in (2.41) because roedec =0 still corresponds to a zero return on equity. In general, changing the units of measurement of only the independent variable does not affect the intercept.In the previous section, we defined R-squared as a goodness-of-fit measure for OLS regres

15、sion. We can also ask what happens to R2 when the unit of measurement of either the independent or the dependent variable changes. Without doing any algebra, we should know the result: the goodness-of-fit of the model should not depend onthe units of measurement of our variables. For example, the am

16、ount of variation in salary, explained by the return on equity, should not depend on whether salary is measured in dollars or in thousands of dollars or on whether return on equity is a percent or a decimal. This intuition can be verified mathematically: using the definition of R2, it can be shown t

17、hat R2 is, in fact, invariant to changes in the units of y or x. Incorporating Nonlinearities in Simple RegressionSo far we have focused on linear relationships between the dependent and independent variables. As we mentioned in Chapter 1, linear relationships are not nearly general enough for all e

18、conomic applications. Fortunately, it is rather easy to incorporate many nonlinearities into simple regression analysis by appropriately defining the dependent and independent variables. Here we will cover two possibilities that often appear in applied work.In reading applied work in the social scie

19、nces, you will often encounter regression equations where the dependent variable appears in logarithmic form. Why is this done? Recall the wage-education example, where we regressed hourly wage on years of education. We obtained a slope estimate of 0.54 see equation (2.27), which means that each add

20、itional year of education is predicted to increase hourly wage by 54 cents.Because of the linear nature of (2.27), 54 cents is the increase for either the first year of education or the twentieth year; this may not be reasonable.Suppose, instead, that the percentage increase in wage is the same give

21、n one more year of education. Model (2.27) does not imply a constant percentage increase: the percentage increases depends on the initial wage. A model that gives (approximately) a constant percentage effect is log(wage) =0 +1educ + u,(2.42) where log(.) denotes the natural logarithm. (See Appendix

22、A for a review of logarithms.) In particular, if u =0, then %wage = (100*1) educ.(2.43) Notice how we multiply 1 by 100 to get the percentage change in wage given one additional year of education. Since the percentage change in wage is the same for each additional year of education, the change in wa

23、ge for an extra year of education increases aseducation increases; in other words, (2.42) implies an increasing return to education.By exponenttiating (2.42), we can write wage =exp(0+1educ + u). This equationis graphed in Figure 2.6, with u = 0.Estimating a model such as (2.42) is straightforward w

24、hen using simple regression. Just define the dependent variable, y, to be y = log(wage). The independent variable is represented by x = educ. The mechanics of OLS are the same as before: the intercept and slope estimates are given by the formulas (2.17) and (2.19). In other words, we obtain 0 and1 f

25、rom the OLS regression of log(wage) on educ. E X A M P L E 2 . 1 0( A L o g W a g e E q u a t i o n )Using the same data as in Example 2.4, but using log(wage) as the dependent variable, we obtain the following relationship: log(wage) =0.584 +0.083 educ(2.44) n = 526, R =0.186.The coefficient on edu

26、c has a percentage interpretation when it is multiplied by 100: wage increases by 8.3 percent for every additional year of education. This is what economists mean when they refer to the “return to another year of education.” It is important to remember that the main reason for using the log of wage

27、in (2.42) is to impose a constant percentage effect of education on wage. Once equation (2.42) is obtained, the natural log of wage is rarely mentioned. In particular, it is not correct to say that another year of education increases log(wage) by 8.3%.The intercept in (2.42) is not very meaningful,

28、as it gives the predicted log(wage), when educ =0. The R-squared shows that educ explains about 18.6 percent of the variation in log(wage) (not wage). Finally, equation (2.44) might not capture all of the non-linearity in the relationship between wage and schooling. If there are “diploma effects,” t

29、hen the twelfth year of educationgraduation from high schoolcould be worth much more than the eleventh year. We will learn how to allow for this kind of nonlinearity in Chapter 7. Another important use of the natural log is in obtaining a constant elasticity model.E X A M P L E 2 . 1 1( C E O S a l

30、a r y a n d F i r m S a l e s )We can estimate a constant elasticity model relating CEO salary to firm sales. The data set is the same one used in Example 2.3, except we now relate salary to sales. Let sales be annual firm sales, measured in millions of dollars. A constant elasticity model is log(sa

31、lary =0 +1log(sales) +u, (2.45) where 1 is the elasticity of salary with respect to sales. This model falls under the simple regression model by defining the dependent variable to be y = log(salary) and the independent variable to be x = log(sales). Estimating this equation by OLS gives Part 1Regres

32、sion Analysis with Cross-Sectional Datalog(salary) = 4.822 ?+0.257 log(sales)(2.46)n =209, R= 0.211.The coefficient of log(sales) is the estimated elasticity of salary with respect to sales. It implies that a 1 percent increase in firm sales increases CEO salary by about 0.257 percentthe usual interpretation of an elasticity.The two functional forms covered in this section will often arise in the remainder of this text. We have covered models containing natural logarithms here because they appear so frequently in