计量经济学英文课件：Chapter 2 The Simple Regression Model

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 Model 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly a

2、ccessible 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 OLS on Any Sample of Data2.4 Units of Measurement and Functional Form The End2.5 Expected Values and

3、 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. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible web

4、site, in whole or in part.Dependent variable,explained variable,response variable,Independent variable,explanatory variable,regressor,Error term,disturbance,unobservables,InterceptSlope parameterExplains variable in terms of variable “Chapter 2 The Simple Regression ModelChapterEnd2.1 Definition of

5、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.Interpretation of the simple linear regression modelThe simple linear regression model is rarely applicable in prac-

6、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 one unit?Interpretation only correct if all otherthings remain equal when the indepen-dent variable is in

7、creased by one unitChapter 2 The Simple Regression ModelChapterEnd2.1 Definition of the Simple Regression Model (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.Example: Soybean yield and ferti

8、lizerExample: 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 wagegiven another year of education, holding all other factors fixedLabor force experience,tenure with current e

9、mployer, work ethic, intelligence Chapter 2 The Simple Regression ModelChapterEnd2.1 Definition of the Simple Regression Model (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.When is there a c

10、ausal interpretation?Conditional mean independence assumptionExample: wage equatione.g. intelligence The explanatory variable must notcontain information about the meanof the unobserved factors The conditional mean independence assumption is unlikely to hold becauseindividuals with more education wi

11、ll also be more intelligent on average.Chapter 2 The Simple Regression ModelChapterEnd2.1 Definition of the Simple Regression Model (4/6) 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.Population r

12、egression function (PFR)The conditional mean independence assumption implies thatThis means that the average value of the dependent variable can be expressed as a linear function of the explanatory variableChapter 2 The Simple Regression ModelChapterEnd2.1 Definition of the Simple Regression Model (

13、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 Simple Regression ModelChapterEnd2.1 Definition of the Simple Regression Model (6/6)f(y|x)xx1x2x3E(y|x2)E(y|x1)E(y|x3)Population re

14、gression function:E(y|x)=b0+b1xy, E(y|x)01010yxuE u xE y xxbbbb=+=+ 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.In order to estimate the regression model one needs dataA random sample of observa

15、tionsFirst observationSecond observationThird observationn-th observationValue of the expla-natory variable of the i-th observationValue of the dependentvariable of the i-th ob-servationChapter 2 The Simple Regression ModelChapterEnd2.2 Deriving the Ordinary Least Squares Estimates (1/10) 2013 Cenga

16、ge Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Fit as good as possible a regression line through the data points:Fitted regression lineFor example, the i-th data pointChapter 2 The Simple Regression ModelCha

17、pterEnd2.2 Deriving the Ordinary Least Squares Estimates (2/10) 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.What does as good as possible“ mean?Regression residualsMinimize sum of squared regres

18、sion residualsOrdinary Least Squares (OLS) estimatesChapter 2 The Simple Regression ModelChapterEnd 2.2 Deriving the Ordinary Least Squares Estimates (3/10)110121, niiiniixxyyyxxxbbb= 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly access

19、ible website, in whole or in part.Supplementary materialsChapter 2 The Simple Regression ModelChapterEnd 2.2 Deriving the Ordinary Least Squares Estimates (4/10)1121niiiniixxyyxxb=iiiiiiiixxyyxyyxx yx ynxy=The summation operator：0ixx=See A.1 The Summation Operator and Descriptive Statistics at p703-

20、705. 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.Sample:Population:CEO Salary and return on equityFitted regressionCausal interpretation?Salary in thousands of dollarsReturn on equity of the CEO

21、s firmInterceptIf the return on equity increases by 1 percent,then salary is predicted to change by 18,501 $Chapter 2 The Simple Regression ModelChapterEnd2.2 Deriving the Ordinary Least Squares Estimates (5/10) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or

22、 posted to a publicly accessible website, in whole or in part. Fitted regression line(depends on sample)Unknown population regression lineChapter 2 The Simple Regression ModelChapterEnd2.2 Deriving the Ordinary Least Squares Estimates (6/10) 2013 Cengage Learning. All Rights Reserved. May not be sca

23、nned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Wage and educationFitted regressionCausal interpretation?Hourly wage in dollarsYears of educationInterceptIn the sample, one more year of education wasassociated with an increase in hourly wage by 0.54 $Chapt

24、er 2 The Simple Regression ModelChapterEnd2.2 Deriving the Ordinary 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 part

25、ies)Fitted regressionCausal interpretation?Percentage of vote for candidate APercentage of campaign expenditures candidate AInterceptIf candidate As share of spending increases by onepercentage point, he or she receives 0.464 percen-tage points more of the total voteChapter 2 The Simple Regression M

26、odelChapterEnd2.2 Deriving the Ordinary Least Squares 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 ModelChapterEnd2.2 Deriving the Ordinary Least

27、Squares Estimates (9/10) : : 0 0cov,cov,000cov,0 xE u xE uE u xE uEE uE uE u xxu xEu xu xx=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

28、 Regression ModelChapterEnd2.2 Deriving the Ordinary Least Squares Estimates (10/10) 0101010111 and : and 000cov,00 a00cond 0cov,varcov,r0vav,yxuE u xE yxE ux uE xuE x yxyxuE u xy xxE u xE uuyxxxbbbbbbbbbb=+=+=method of momentsanother method : 2013 Cengage Learning. All Rights Reserved. May not be s

29、canned, 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 valuesDeviations from regression line (= residuals)Chapter 2 The Simple Regression ModelChapterEnd2.3 Algebraic Propertie

30、s of OLS on Any Sample of Data (1/6)01yxbb=+10niiu=10niiiu x=1 0niiiu y=Deviations from regression line sum up to zeroCorrelation between deviations and regressors is zeroSample averages of y and x lie on regression lineyy= 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or du

31、plicated, or posted to a publicly accessible website, in whole or in part. For example, CEO number 12s salary was526,023 $ lower than predicted using thethe information on his firms return on equity Chapter 2 The Simple Regression ModelChapterEnd2.3 Algebraic Properties of OLS on Any Sample of Data

32、(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 explanatory variable explain the dependent variable?“Total sum of squares,represents total

33、 variation in dependent variable Explained sum of squares,represents variation explained by regressionResidual sum of squares,represents variation notexplained by regressionChapter 2 The Simple Regression ModelChapterEnd2.3 Algebraic Properties of OLS on Any Sample of Data (3/6) 2013 Cengage Learnin

34、g. 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 variationExplained partUnexplained partR-squared measures the fraction of the total variation tha

35、t is explained by the regressionChapter 2 The Simple Regression ModelChapterEnd22201,RRcor y y= 2.3 Algebraic Properties of OLS on Any Sample of Data (4/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in

36、 part.Chapter 2 The Simple Regression ModelChapterEnd22222112222111121112212Prove ,.Proof:,nniiiiiinnnniiiiiiiinnniiiiiiiiiniiiiRcor y yyyyyyyyycor y yyyyyyyyyyyyyyyuyyyyyycor y yyy= =+=21nR=2.3 Algebraic Properties of OLS on Any Sample of Data (5/6) 2013 Cengage Learning. All Rights Reserved. May n

37、ot be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in 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.

38、6 % of the total variation in election outcomesChapter 2 The Simple Regression ModelChapterEndThe regression explains only 1.3 %of the total variation in salariesceosal1.wf1ls salary c roe2.3 Algebraic Properties of OLS on Any Sample of Data (6/6) 2013 Cengage Learning. All Rights Reserved. May not

39、be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Chapter 2 The Simple Regression ModelChapterEnd2.4 Units of Measurement and Functional Form (1/6)The Effects of Changing Units of Measurement on OLS Statistics0110110121011212963.19118.50196319118501, w

40、here 1000.963.1911850.1, whe? ?re 100.salaryroesalarydolroesalardolsalarysalaryroedecroedecroesalcarydolroedecyxccxycccccyxc ybbbbbbbb=+=+=+= +=+=+=+=+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, o

41、r posted to a publicly accessible website, 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 wagePercentage change of wage if years of education are inc

42、reased by one yearChapter 2 The Simple Regression ModelChapterEnd2.4 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 increase

43、s by 8.3 % for every additional year of education(= return to education)For example:Growth rate of wage is 8.3 %per year of educationChapter 2 The Simple Regression ModelChapterEndwage1.wf1ls log(wage) c educ2.4 Units of Measurement and Functional Form (3/6)10.0838.3%wage wageeducb= 2013 Cengage Lea

44、rning. 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 ModelChapterEnd2.4 Units of Measurement and Functional Form (4/6)Supplementary materialsThe proportionate change in x in moving from x

45、0 to x1:The percentage change in x in moving from x0 to x1 :See A.1 The Summation Operator and Descriptive Statistics at p707-709.0100 x xxxx=0%100 xx x =110.0838.3%1 0.308wage wageeducwageeducbb= 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a pu

46、blicly 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 salaryPercentage change of salary if sales increase by 1 % Natural logarithm of his/her fir

47、ms salesLogarithmic changes are always percentage changesChapter 2 The Simple Regression ModelChapterEnd2.4 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 website, in whole or in

48、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 on how y and x are defi

49、ned. Chapter 2 The Simple Regression ModelChapterEnd2.4 Units of Measurement and Functional Form (6/6)10.257salary salarysales salesb=ceosal1.wf1ls log(salary) c log(sales)+ 1 % sales ! + 0.257 % salary 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted t

50、o 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 samples isData is random and depend

51、s on particular sample that has been drawnChapter 2 The Simple Regression ModelChapterEnd2.5 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 accessible website, in whole or in par

52、t.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 therefore followsthe population equationCh

53、apter 2 The Simple Regression ModelChapterEnd2.5 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.Discussion of random sampling: Wage and ed

54、ucationThe 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 education of the worker drawn are random because on

55、e 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 Simple Regression ModelChapterEnd2.5 Expected Va

56、lues 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 deviationfrom the populationrelationship for the i-th

57、worker:Chapter 2 The Simple Regression ModelChapterEnd2.5 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 whole or in part.Assumptions for the linear regres

58、sion 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 explanatory variablelead to different values of th

59、e dependent variable)The value of the explanatory variable must contain no information about the mean of the unobserved factorsChapter 2 The Simple Regression ModelChapterEndFurther, E(ui|x1, xn) = 0 and E(ui uj|x1, xn) = 0 .2.5 Expected Values and Variances of the OLS Estimators (5/17) 2013 Cengage

60、 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 larger, depending on the sample that is the result of