数学本科论文外文翻译

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：  李  山            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：  包东娥 （讲师）    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), R2= 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 portion Two important issues in applied economics are (1) understanding how changing theunits of measurement of the dependent and/or independent 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 OLS Statistics In 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 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 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: sala?rdol = 963,191 +18,501 roe. We obtain the intercept and slope in (2.40) simply by multiplying the intercept and the slope 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 sala?rdol = 963,191, so the predicted salary is $963,191 [the same value we obtained from equation (2.39)]. Furthermore, if roe increases by one, then the predicted salary increases by $18,501; again, this is what we concluded 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 of measurement. If the dependent variable is multiplied by the constant c—which means each value in the sample is multiplied byc—then 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 moving from salary to salardol. Chapter 2 The Simple Regression Model We 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 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 on roedec, we obtain    sal?ary =963.191 + 1850.1 roedec. The coefficient on roedec is 100 times the coefficient on roe in (2.39). This is as it should be. Changing roe by one percentage point is equivalent to Δroedec = 0.01. From (2.41), if Δroedec = 0.01, then Δsal?ary = 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 independent variable 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. The 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 regression. 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 on the units of measurement of our variables. For example, the amount 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 that R2 is, in fact, invariant to changes in the units of y or x. Incorporating Nonlinearities in Simple Regression So 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 economic 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 sciences, 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 additional 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 given 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) =β0 +β1educ + u,(2.42) where log(.) denotes the natural logarithm. (See Appendix A for a review of logarithms.) In particular, if Δu =0, then %Δwage = (100*β1) Δeduc.(2.43) Notice how we multiply β1 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 wage 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(β0+β1educ + u). This equation is graphed in Figure 2.6, with u = 0. Estimating a model such as (2.42) is straightForward when 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 β?0 andβ?1 from 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, R2 =0.186. The coefficient on educ 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 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, 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,”  then the twelfth year of education—graduation from high school—could 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 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(salary =β0 +β1log(sales) +u, (2.45) where β1 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 1 Regression Analysis with Cross-Sectional Data log(sal?ary) = 4.822 ?+0.257 log(sales) (2.46) n =209, R2= 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 percent—the 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 applied work. The interpretation of such models will not be much different in the multiple regression case. It is also useful to note what happens to the intercept and slope estimates if we change the units of measurement of the dependent variable when it appears in logarithmic Form. Because the change to logarithmic Form approximates a proportionate change, it makes sense that nothing happens to the slope. We can see this by writing the rescaled variable as c1yi for each observation i. The original equation is log(yi) =β0 +β1xi +ui. If we add log(c1) to both sides, we get log(c1) + log(yi) + [log(c1) β0] +β1xi + ui, or log(c1yi) ? [log(c1) +β0] +β1xi +ui.(Remember that the sum of the logs is equal to the log of their product as shown in Appendix A.) Therefore, the slope is still ?1, but the intercept is now log(c1) ? ?0. Similarly, if the independent variable is log(x), and we change the units of measurement of x before taking the log, the slope remains the same but the intercept does not change. You will be asked to verify these claims in Problem 2.9. We end this subsection by summarizing four combinations of functional forms available from using either the original variable or its natural log. In Table 2.3, x and y stand for the variables in their original form. The model with y as the dependent variable and x as the independent variable is called the level-level model, because each variable appears in its level form. The model with log(y) as the dependent variable and x as the independent variable is called the log-level model. We will not explicitly discuss the level-log model here, because it arises less often in practice. In any case, we will see examples of this model in later chapters. Chapter 2 The Simple Regression Model Table 2.3 The last column in Table 2.3 gives the interpretation of β1. In the log-level model,  100*β1 is sometimes called the semi-elasticity of y with respect to x. As we mentioned in Example 2.11, in the log-log model, β1 is the elasticity of y with respect to x. Table 2.3 warrants careful study, as we will refer to it often in the remainder of the text. The Meaning of “Linear” Regression The simple regression model that we have studied in this chapter is also called the simple linear regression model. Yet, as we have just seen, the general model also allows For certain nonlinear relationships. So what does “linear” mean here? You can see by looking at equation (2.1) that y =β0 +β1x + u. The key is that this equation is linear in the parameters, β0 and β1. There are no restrictions on how y and x relate to the original explained and explanatory variables of interest. As we saw in Examples 2.7 and 2.8, y and x can be natural logs of variables, and this is quite common in applications. But we need not stop there. For example, nothing prevents us from using simple regression to estimate a model such as cons =β0 +β1√inc+u, where cons is annual consumption and inc is annual income. While the mechanics of simple regression do not depend on how y and x are defined, the interpretation of the coefficients does depend on their definitions. For successful empirical work, it is much more important to become proficient at interpreting coefficients than to become efficient at computing formulas such as (2.19). We will get much more practice with interpreting the estimates in OLS regression lines when we study multiple regression. There are plenty of models that cannot be cast as a linear regression model because they are not linear in their parameters; an example is cons = 1/(β0 +β1inc) + u. Estimation of such models takes us into the realm of the nonlinear regression model, which is beyond the scope of this text. For most applications, choosing a model that can be put into the linear regression framework is sufficient. EXPECTED VALUES AND VARIANCES OF THE OLS ESTIMATORS In Section 2.1, we defined the population model y =β0 +β1x +u, and we claimed that the key assumption For simple regression analysis to be useful is that the expected value of u given any value of x is zero. In Sections 2.2, 2.3, and 2.4, we discussed the algebraic properties of OLS estimation. We now return to the population model and study the statistical properties of OLS. In other words, we now view β?0 and β?1 as estimators for the parameters ?0 and ?1 that appear in the population model. This means that we will study properties of the distributions of ??0 and ??1 over different random samples from the population. (Appendix C contains definitions of estimators and reviews some of their important properties.) Unbiasedness of OLS We begin by establishing the unbiasedness of OLS under a simple set of assumptions. For future reference, it is useful to number these assumptions using the prefix “SLR” for simple linear regression. The first assumption defines the population model. 测量单位和函数形式 在投票结果方程（2.28）中，R2= 0.505。因此，竞选支出的份额解释只是在这样的选举结果的变化百分之50。这是一个相当大的部分 在应用经济学的两个重要的问是：（1）了解如何改变单位的依赖性和/或独立变量的OLS估计和测量的影响（2）知道如何将经济学中流行的功能形式为回归。一个完整的理解的功能形式问题所需要的数学在附录A了 测量单位的OLS变化的影响统计 在例2.3中，我们选择了测量数千美元的年薪，和净资产收益率是衡量百分之一（而不是一个小数点）。关键是要知道薪水和净资产收益率是衡量在这个例子中，为了使方程估计的意义（2.39）。我们还必须知道，OLS估计预期的方式改变时，完全的依赖和独立变量的测量单位的变化。在example2.3，假设，而不是千美元衡量的工资，我们衡量的美元。让salardol工资以美元（salardol = 845761会被解释为845761美元。）。当然，salardol有数千美元的测量一个简单的关系：salardol工资？1000？工资。我们不需要实际运行的salardol对罗伊的回归知道估计方程为：萨拉?RDOL = 963191 + 18501净资产收益率。 我们得到的截距和斜率在（2.40）简单地乘以拦截和 边坡（2.39）1000。这给出了方程（2.39）和（2.40）相同的解释。看（2.40），如果净资产收益率= 0，然后萨拉?RDOL = 963191，所以预测的工资963191美元[我们得到的方程相同的值（2.39）]。此外，如果净资产收益率增加了一个，然后预测的工资增加18501美元；再次，这是我们从我们前面的分析方程（2.39）。