《計量經(jīng)濟(jì)學(xué)導(dǎo)論》ch(7)

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, 5e 2013 Cengage Learning. All Rights Reserved. May not

2、 be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Definition of the simple linear regression modelDependent variable,explained variable,response variable,Independent variable,explanatory variable,regressor,Error term,disturbance,unobservables,Intercep

3、tSlope parameterExplains variable in terms of variable “The Simple Regression Model 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 lin

4、ear 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 one unit?Interpretation only correct if all otherthings

5、 remain equal when the indepen-dent variable is increased by one unitThe Simple Regression Model 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 fertilizerExample: A simpl

6、e 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 employer, work ethic,

7、intelligence The Simple Regression Model 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 causal interpretation?Conditional mean independence assumptionExample: wage equatione.g. inte

8、lligence 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 will also be more intelligent on average.The Simple Regression Model 2013 Cengage Learning. Al

9、l Rights Reserved. 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

10、linear function of the explanatory variableThe Simple Regression Model 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 regression functionFor individuals with , the average value of isTh

11、e Simple Regression Model 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 observationsFirst observationSecond observationTh

12、ird observationn-th observationValue of the expla-natory variable of the i-th observationValue of the dependentvariable of the i-th ob-servationThe Simple Regression Model 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website

13、, 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 pointThe Simple Regression Model 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website,

14、in whole or in part.What does as good as possible“ mean?Regression residualsMinimize sum of squared regression residualsOrdinary Least Squares (OLS) estimatesThe Simple Regression Model 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly acce

15、ssible website, in whole or in part.CEO Salary and return on equityFitted regressionCausal interpretation?Salary in thousands of dollarsReturn on equity of the CEOs firmInterceptIf the return on equity increases by 1 percent,then salary is predicted to change by 18,501 $The Simple Regression Model 2

16、013 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 regression line(depends on sample)Unknown population regression lineThe Simple Regression Model 2013 Cengage Learning. All Rights Reserved. May

17、 not be scanned, 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

18、0.54 $The Simple Regression Model 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 vote for

19、 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 voteThe Simple Regression Model 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied

20、 or duplicated, or posted to a publicly accessible website, in whole or in part.Properties of OLS on any sample of dataFitted values and residualsAlgebraic properties of OLS regressionFitted or predicted valuesDeviations from regression line (= residuals)Deviations from regression line sum up to zer

21、oCorrelation between deviations and regressors is zeroSample averages of y and x lie on regression lineThe Simple Regression Model 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 n

22、umber 12s salary was526,023 $ lower than predicted using thethe information on his firms return on equity The Simple Regression Model 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-FitM

23、easures of VariationHow well does the explanatory variable explain the dependent variable?“Total sum of squares,represents total variation in dependent variable Explained sum of squares,represents variation explained by regressionResidual sum of squares,represents variation notexplained by regressio

24、nThe Simple Regression Model 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 variationExplained partUnexplained partR-squared

25、 measures the fraction of the total variation that is explained by the regressionThe Simple Regression Model 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.CEO Salary and return on equityVoting out

26、comes and campaign expendituresCaution: A high R-squared does not necessarily mean that the regression has a causal interpretation!The regression explains only 1.3 %of the total variation in salariesThe regression explains 85.6 % of the total variation in election outcomesThe Simple Regression Model

27、 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.Incorporating nonlinearities: Semi-logarithmic formRegression of log wages on years of eductionThis changes the interpretation of the regression coef

28、ficient:Natural logarithm of wagePercentage change of wage if years of education are increased by one yearThe Simple Regression Model 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 regressio

29、nThe wage increases by 8.3 % for every additional year of education(= return to education)For example:Growth rate of wage is 8.3 %per year of educationThe Simple Regression Model 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible

30、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 firms salesLogarithmi

31、c changes are always percentage changesThe Simple Regression Model 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.CEO salary and firm sales: fitted regressionFor example:The log-log form postulates

32、 a constant elasticity model, whereas the semi-log form assumes a semi-elasticity model+ 1 % sales ! + 0.257 % salaryThe Simple Regression Model 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.Expec

33、ted values and variances of the OLS estimatorsThe 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 depends on pa

34、rticular sample that has been drawnThe Simple Regression Model 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.Standard assumptions for the linear regression modelAssumption SLR.1 (Linear in paramet

35、ers)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 equationThe Simple Regression Model 2013 Cengage Learning. All Rights Reserved. May not be scanned,

36、 copied or duplicated, or posted to a publicly accessible website, in whole or in part.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 holdsD

37、raw completely randomly a worker from the populationThe wage and the years of education 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 u

38、sed to estimate the linear relationship between wages and educationThe Simple Regression Model 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 deviati

39、onfrom the populationrelationship for the i-th worker:The Simple Regression Model 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 regression model (cont.)Assumption SLR.3

40、(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 the dependent variable)The value of t

41、he explanatory variable must contain no information about the mean of the unobserved factorsThe Simple Regression Model 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 O

42、LS)Interpretation of unbiasednessThe estimated coefficients may be smaller or 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 sampli

43、ng was repeated, i.e. if drawing the random sample und doing the estimation was repeated many timesIn a given sample, estimates may differ considerably from true valuesThe Simple Regression Model 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a pub

44、licly accessible website, in whole or in part.Variances of the OLS estimatorsDepending on the sample, the estimates will be nearer or farther away from the true population valuesHow far can we expect our estimates to be away from the true population values on average (= sampling variability)?Samplin

45、g variability is measured by the estimators variancesAssumption SLR.5 (Homoskedasticity)The value of the explanatory variable must contain no information about the variability of the unobserved factorsThe Simple Regression Model 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied

46、or duplicated, or posted to a publicly accessible website, in whole or in part.Graphical illustration of homoskedasticityThe variability of the unobservedinfluences does not dependent on the value of the explanatory variableThe Simple Regression Model 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.An example for heteroskedasticity: Wage and educationThe variance of the unobserved determinants of wages increaseswith the level of educa