《計(jì)量經(jīng)濟(jì)學(xué)》ch-02-wooldridg

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1、 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 2 The Simple Regression ModelWooldridge: Introductory Econometrics: A Modern Approach, 5eInstructed by professor Yuan, Huiping 2013 Cengage

2、Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 2 The Simple Regression Model2.1 Definition of the Simple Regression Model 2.2 Deriving the Ordinary Least Squares Estimates 2.3 Algebraic Properties of O

3、LS on Any Sample of Data2.4 Units of Measurement and Functional Form The End2.5 Expected Values and Variances of the OLS Estimators 2.6 Regression through the Origin and Regression on a Constant Introduction to EviewsAssignments: Problems 6-10, Computer Exercises C2, C4, C6 2013 Cengage Learning. Al

4、l Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Dependent variable,explained variable,response variable, Independent variable,explanatory variable,regressor, Error term,disturbance,unobservables, Intercept Slope parameterEx

5、plains variable in terms of variable “Chapter 2 The Simple Regression ModelChapter End2.1 Definition of the Simple Regression Model (1/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interpretat

6、ion of the simple linear regression modelThe simple linear regression model is rarely applicable in prac-tice but its discussion is useful for pedagogical reasonsStudies how varies with changes in :“as long asBy how much does the dependent variable change if the independent variable is increased by

7、one unit? Interpretation only correct if all otherthings remain equal when the indepen-dent variable is increased by one unitChapter 2 The Simple Regression Model Chapter End 2.1 Definition of the Simple Regression Model (2/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or

8、 duplicated, or posted to a publicly accessible website, in whole or in part. Example: Soybean yield and fertilizerExample: A simple wage equationMeasures the effect of fertilizer on yield, holding all other factors fixed Rainfall,land quality, presence of parasites, Measures the change in hourly wa

9、ge given another year of education, holding all other factors fixed Labor force experience,tenure with current employer, work ethic, intelligence Chapter 2 The Simple Regression Model Chapter End 2.1 Definition of the Simple Regression Model (3/6) 2013 Cengage Learning. All Rights Reserved. May not

10、be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. When is there a causal interpretation?Conditional mean independence assumptionExample: wage equation e.g. intelligence The explanatory variable must notcontain information about the meanof the unobserv

11、ed factors The conditional mean independence assumption is unlikely to hold because individuals with more education will also be more intelligent on average. Chapter 2 The Simple Regression Model Chapter End 2.1 Definition of the Simple Regression Model (4/6) 2013 Cengage Learning. All Rights Reserv

12、ed. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Population regression function (PFR)The conditional mean independence assumption implies thatThis means that the average value of the dependent variable can be expressed as a linear functio

13、n of the explanatory variableChapter 2 The Simple Regression Model Chapter End 2.1 Definition of the Simple Regression Model (5/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 2 The Simp

14、le Regression Model Chapter End 2.1 Definition of the Simple Regression Model (6/6)f(y|x) xx 1 x2 x3E(y|x2)E(y|x1) E(y|x3) Population regression function:E(y|x)=b0+b1xy, E(y|x) 0 1 0 10y x uE u xE y x xb bb b= + += + 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicate

15、d, or posted to a publicly accessible website, in whole or in part. In order to estimate the regression model one needs dataA random sample of observationsFirst observationSecond observationThird observationn-th observation Value of the expla-natory variable of the i-th observation Value of the depe

16、ndentvariable of the i-th ob-servationChapter 2 The Simple Regression Model Chapter End 2.2 Deriving the Ordinary Least Squares Estimates (1/10) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Fit

17、as good as possible a regression line through the data points:Fitted regression lineFor example, the i-th data pointChapter 2 The Simple Regression Model Chapter End 2.2 Deriving the Ordinary Least Squares Estimates (2/10) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or dup

18、licated, or posted to a publicly accessible website, in whole or in part. What does as good as possible“ mean?Regression residualsMinimize sum of squared regression residualsOrdinary Least Squares (OLS) estimatesChapter 2 The Simple Regression Model Chapter End 2.2 Deriving the Ordinary Least Square

19、s Estimates (3/10) 11 0 121 , n i ii n iix x y y y xx xb b b= = = = 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Supplementary materialsChapter 2 The Simple Regression Model Chapter End 2.2 Deri

20、ving the Ordinary Least Squares Estimates (4/10) 11 21 n i ii n iix x y yx xb = = = i i i ii ii ix x y y x y yx x yx y nxy = = = The summation operator： 0ix x =See A.1 The Summation Operator and Descriptive Statistics at p703-705. 2013 Cengage Learning. All Rights Reserved. May not be scanned, copie

21、d or duplicated, or posted to a publicly accessible website, in whole or in part.Sample: Population:CEO Salary and return on equityFitted regressionCausal interpretation?Salary in thousands of dollars Return on equity of the CEOs firmIntercept If the return on equity increases by 1 percent,then sala

22、ry is predicted to change by 18,501 $Chapter 2 The Simple Regression ModelChapter End2.2 Deriving the Ordinary Least Squares Estimates (5/10) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Fitted

23、regression line(depends on sample)Unknown population regression lineChapter 2 The Simple Regression Model Chapter End 2.2 Deriving the Ordinary Least Squares Estimates (6/10) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible webs

24、ite, in whole or in part. Wage and educationFitted regressionCausal interpretation?Hourly wage in dollars Years of educationIntercept In the sample, one more year of education wasassociated with an increase in hourly wage by 0.54 $Chapter 2 The Simple Regression Model Chapter End 2.2 Deriving the Or

25、dinary Least Squares Estimates (7/10) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Voting outcomes and campaign expenditures (two parties)Fitted regressionCausal interpretation?Percentage of vot

26、e for candidate A Percentage of campaign expenditures candidate AIntercept If candidate As share of spending increases by onepercentage point, he or she receives 0.464 percen- tage points more of the total vote Chapter 2 The Simple Regression Model Chapter End 2.2 Deriving the Ordinary Least Squares

27、 Estimates (8/10) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 2 The Simple Regression Model Chapter End 2.2 Deriving the Ordinary Least Squares Estimates (9/10) : : 0 0cov , cov , 000 c

28、ov , 0 x E u x E uE u xE u E E uE uE u x xu x E u xu x x = = = = = =mean independencezero conditional mean 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 2 The Simple Regression Model Chap

29、ter End 2.2 Deriving the Ordinary Least Squares Estimates (10/10) 0 1 0 1 0 10 1 11 and : and 0 00cov , 0 0 a0 0 cond 0cov , var cov , r 0va v ,y x u E u x E y xE u x u E xu E x y xy x u E u xy x xE u x E u uy xx xb b b bb bb b bb = + + = = = = = = + + = = = =method of momentsanother method : 2013 C

30、engage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Fitted values and residualsAlgebraic properties of OLS regressionFitted or predicted values Deviations from regression line (= residuals)Chapter 2 The Simp

31、le Regression Model Chapter End 2.3 Algebraic Properties of OLS on Any Sample of Data (1/6) 0 1 y xb b= +1 0n ii u= = 1 0n i ii u x= = 1 0n i ii u y= =Deviations from regression line sum up to zero Correlation between deviations and regressors is zero Sample averages of y and x lie on regression lin

32、e y y= 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. For example, CEO number 12s salary was 526,023 $ lower than predicted using thethe information on his firms return on equity Chapter 2 The Sim

33、ple Regression Model Chapter End 2.3 Algebraic Properties of OLS on Any Sample of Data (2/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Goodness-of-FitMeasures of VariationHow well does the ex

34、planatory variable explain the dependent variable?“Total sum of squares,represents total variation in dependent variable Explained sum of squares,represents variation explained by regression Residual sum of squares,represents variation notexplained by regression Chapter 2 The Simple Regression Model

35、 Chapter End 2.3 Algebraic Properties of OLS on Any Sample of Data (3/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Decomposition of total variationGoodness-of-fit measure (R-squared)Total var

36、iation Explained part Unexplained partR-squared measures the fraction of the total variation that is explained by the regressionChapter 2 The Simple Regression Model Chapter End 2 220 1 ,RR cor y y = 2.3 Algebraic Properties of OLS on Any Sample of Data (4/6) 2013 Cengage Learning. All Rights Reserv

37、ed. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 2 The Simple Regression Model Chapter End 22 2 22 1 12 2 22 1 11 1 21 1 1 22 1 2 Prove , .Proof: , , n ni i i ii in nn n i ii i i ii in n ni i i i i i i i in ii iiR cor y y y y y y

38、y y y ycor y y y y y yy y y yy y y y y y u y y y yy ycor y y y y= = = = = = = = + = = 21n R= = 2.3 Algebraic Properties of OLS on Any Sample of Data (5/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in

39、part. CEO Salary and return on equityVoting outcomes and campaign expendituresCaution: A high R-squared does not necessarily mean that the regression has a causal interpretation! The regression explains 85.6 % of the total variation in election outcomesChapter 2 The Simple Regression Model Chapter E

40、ndThe regression explains only 1.3 %of the total variation in salaries ceosal1.wf1ls salary c roe2.3 Algebraic Properties of OLS on Any Sample of Data (6/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or i

41、n part. Chapter 2 The Simple Regression Model Chapter End 2.4 Units of Measurement and Functional Form (1/6)The Effects of Changing Units of Measurement on OLS Statistics 0 11 0 1 10 1 21 0 11 21 2963.191 18.501963191 18501 , where 1000 .963.191 1850.1 , whe? ? re 100.salary roesalarydol roe salardo

42、l salarysalary roedec roedec roesalc arydol roedecy xc c xy cc cc cy xc yb bb bb bb b= += + = = + = += += += += + 212Does R depend on the units of measurement of variables?c x 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible web

43、site, in whole or in part. Incorporating nonlinearities: Semi-logarithmic formRegression of log wages on years of eductionThis changes the interpretation of the regression coefficient:Natural logarithm of wage Percentage change of wage if years of education are increased by one year Chapter 2 The Si

44、mple Regression Model Chapter End 2.4 Units of Measurement and Functional Form (2/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Fitted regressionThe wage increases by 8.3 % for every additiona

45、l year of education(= return to education)For example: Growth rate of wage is 8.3 %per year of education Chapter 2 The Simple Regression Model Chapter End wage1.wf1ls log(wage) c educ2.4 Units of Measurement and Functional Form (3/6) 1 0.083 8.3%wage wageeduc b = = = 2013 Cengage Learning. All Right

46、s Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 2 The Simple Regression Model Chapter End 2.4 Units of Measurement and Functional Form (4/6)Supplementary materialsThe proportionate change in x in moving from x0 to x1:The

47、percentage change in x in moving from x0 to x1 :See A.1 The Summation Operator and Descriptive Statistics at p707-709. 0 1 0 0 x x x x x = 0% 100 x x x = 11 0.083 8.3% 1 0 .30 8wage wageeducwageeduc bb = = = = = 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or

48、 posted to a publicly accessible website, in whole or in part. Incorporating nonlinearities: Log-logarithmic formCEO salary and firm salesThis changes the interpretation of the regression coefficient:Natural logarithm of CEO salary Percentage change of salary if sales increase by 1 % Natural logarit

49、hm of his/her firms sales Logarithmic changes are always percentage changes Chapter 2 The Simple Regression Model Chapter End 2.4 Units of Measurement and Functional Form (5/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible we

50、bsite, in whole or in part. CEO salary and firm sales: fitted regressionFor example:The log-log form postulates a constant elasticity model, whereas the semi-log form assumes a semi-elasticity modellinear in the parameters and nonlinear in the variablesThe interpretation of the coefficients depends

51、on how y and x are defined. Chapter 2 The Simple Regression Model Chapter End 2.4 Units of Measurement and Functional Form (6/6)1 0.257salary salarysales sales b = = ceosal1.wf1ls log(salary) c log(sales)+ 1 % sales ! + 0.257 % salary 2013 Cengage Learning. All Rights Reserved. May not be scanned, c

52、opied or duplicated, or posted to a publicly accessible website, in whole or in part. The estimated regression coefficients are random variables because they are calculated from a random sampleThe question is what the estimators will estimate on average and how large their variability in repeated sa

53、mples isData is random and depends on particular sample that has been drawnChapter 2 The Simple Regression Model Chapter End 2.5 Expected Values and Variances of the OLS Estimators (1/17) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly ac

54、cessible website, in whole or in part. Standard assumptions for the linear regression modelAssumption SLR.1 (Linear in parameters)Assumption SLR.2 (Random sampling)In the population, the relationship between y and x is linearThe data is a random sample drawn from the population Each data point there

55、fore follows the population equation Chapter 2 The Simple Regression Model Chapter End 2.5 Expected Values and Variances of the OLS Estimators (2/17) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

56、 Discussion of random sampling: Wage and educationThe population consists, for example, of all workers of country AIn the population, a linear relationship between wages (or log wages) and years of education holdsDraw completely randomly a worker from the populationThe wage and the years of educatio

57、n of the worker drawn are random because one does not know beforehand which worker is drawnThrow back worker into population and repeat random draw timesThe wages and years of education of the sampled workers are used to estimate the linear relationship between wages and educationChapter 2 The Simpl

58、e Regression Model Chapter End 2.5 Expected Values and Variances of the OLS Estimators (3/17) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The values drawnfor the i-th workerThe implied deviatio

59、nfrom the populationrelationship for the i-th worker:Chapter 2 The Simple Regression Model Chapter End 2.5 Expected Values and Variances of the OLS Estimators (4/17) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in w

60、hole or in part. Assumptions for the linear regression model (cont.)Assumption SLR.3 (Sample variation in explanatory variable)Assumption SLR.4 (Zero conditional mean)The values of the explanatory variables are not all the same (otherwise it would be impossible to stu-dy how different values of the

61、explanatory variablelead to different values of the dependent variable)The value of the explanatory variable must contain no information about the mean of the unobserved factorsChapter 2 The Simple Regression Model Chapter EndFurther, E(ui|x1, xn) = 0 and E(ui uj|x1, xn) = 0 . 2.5 Expected Values an

62、d Variances of the OLS Estimators (5/17) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Theorem 2.1 (Unbiasedness of OLS)Interpretation of unbiasednessThe estimated coefficients may be smaller or

63、larger, depending on the sample that is the result of a random drawHowever, on average, they will be equal to the values that charac-terize the true relationship between y and x in the populationOn average“ means if sampling was repeated, i.e. if drawing the random sample and doing the estimation wa

64、s repeated many timesIn a given sample, estimates may differ considerably from true valuesChapter 2 The Simple Regression Model Chapter End 2.5 Expected Values and Variances of the OLS Estimators (6/17) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted t

65、o a publicly accessible website, in whole or in part. Chapter 2 The Simple Regression Model Chapter End 11 1 21 1 1 11 1 1,.,0 11 1 0 1 11 0000 1 ,where and .1 ,., ,. ,., ,., ., # n x i ii i x ix i i nnx xnnE x SST d ud x x SST x xSST d E u x xE E xxEE x x xx x uE ybbb b b bb bb b bbb b bbb= += = =

66、+ = = = = = + += PROOF of Theorem 2.1 (Unbiasedness of OLS):It is easy to show2.5 Expected Values and Variances of the OLS Estimators (7/17) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 2 The Simple Regression Model Chapter EndExample 2.12 (p48, meap93.wf1) ls math10 c lnchprgNote the interpretation on the negative slope. 20.31932.14408, 0.171.math10 lnchprgn R= = 2.5 Expected Values