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1、Chapter 2,The Simple Regression Model,Wooldridge:Introductory Econometrics:A Modern Approach,5eInstructed by professor Yuan,Huiping,Chapter 2 The Simple Regression Model,2.1 Definition of the Simple Regression Model,2.2 Deriving the Ordinary Least Squares Estimates,2.3 Algebraic Properties of OLS on
2、 Any Sample of Data,2.4 Units of Measurement and Functional Form,The End,2.5 Expected Values and Variances of the OLS Estimators,2.6 Regression through the Origin and Regression on a Constant,Introduction to Eviews,Assignments:Problems 6-10,Computer Exercises C2,C4,C6,Dependent variable,explained va
3、riable,response variable,Independent variable,explanatory variable,regressor,Error term,disturbance,unobservables,Intercept,Slope parameter,Explains variable in terms of variable“,Chapter 2 The Simple Regression Model,Chapter,End,2.1 Definition of the Simple Regression Model(1/6),Interpretation of t
4、he simple linear regression modelThe simple linear regression model is rarely applicable in prac-tice but its discussion is useful for pedagogical reasons,Studies how varies with changes in:“,as long as,By how much does the dependent variable change if the independent variable is increased by one un
5、it?,Interpretation only correct if all otherthings remain equal when the indepen-dent variable is increased by one unit,Chapter 2 The Simple Regression Model,Chapter,End,2.1 Definition of the Simple Regression Model(2/6),Example:Soybean yield and fertilizerExample:A simple wage equation,Measures the
6、 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 fixed,Labor force experience,tenure with current employer,work ethic,intelligence,Chapter 2 The Simpl
7、e Regression Model,Chapter,End,2.1 Definition of the Simple Regression Model(3/6),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 unobserved factors,The cond
8、itional mean independence assumption is unlikely to hold becauseindividuals 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),Population regression function(PFR)The conditional mean indep
9、endence assumption implies thatThis means that the average value of the dependent variable can be expressed as a linear function of the explanatory variable,Chapter 2 The Simple Regression Model,Chapter,End,2.1 Definition of the Simple Regression Model(5/6),Chapter 2 The Simple Regression Model,Chap
10、ter,End,2.1 Definition of the Simple Regression Model(6/6),In order to estimate the regression model one needs dataA random sample of observations,First observation,Second observation,Third observation,n-th observation,Value of the expla-natory variable of the i-th observation,Value of the dependent
11、variable of the i-th ob-servation,Chapter 2 The Simple Regression Model,Chapter,End,2.2 Deriving the Ordinary Least Squares Estimates(1/10),Fit as good as possible a regression line through the data points:,Fitted regression line,For example,the i-th data point,Chapter 2 The Simple Regression Model,
12、Chapter,End,2.2 Deriving the Ordinary Least Squares Estimates(2/10),What does as good as possible“mean?Regression residualsMinimize sum of squared regression residualsOrdinary Least Squares(OLS)estimates,Chapter 2 The Simple Regression Model,Chapter,End,2.2 Deriving the Ordinary Least Squares Estima
13、tes(3/10),Supplementary materials,Chapter 2 The Simple Regression Model,Chapter,End,2.2 Deriving the Ordinary Least Squares Estimates(4/10),The summation operator：,See A.1 The Summation Operator and Descriptive Statistics at p703-705.,Sample:,Population:,CEO Salary and return on equityFitted regress
14、ionCausal interpretation?,Salary in thousands of dollars,Return on equity of the CEOs firm,Intercept,If the return on equity increases by 1 percent,then salary is predicted to change by 18,501$,Chapter 2 The Simple Regression Model,Chapter,End,2.2 Deriving the Ordinary Least Squares Estimates(5/10),
15、Fitted regression line(depends on sample),Unknown population regression line,Chapter 2 The Simple Regression Model,Chapter,End,2.2 Deriving the Ordinary Least Squares Estimates(6/10),Wage and educationFitted regressionCausal interpretation?,Hourly wage in dollars,Years of education,Intercept,In the
16、sample,one more year of education wasassociated with an increase in hourly wage by 0.54$,Chapter 2 The Simple Regression Model,Chapter,End,2.2 Deriving the Ordinary Least Squares Estimates(7/10),Voting outcomes and campaign expenditures(two parties)Fitted regressionCausal interpretation?,Percentage
17、of vote for candidate A,Percentage of campaign expenditures candidate A,Intercept,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 Sq
18、uares Estimates(8/10),Chapter 2 The Simple Regression Model,Chapter,End,2.2 Deriving the Ordinary Least Squares Estimates(9/10),Chapter 2 The Simple Regression Model,Chapter,End,2.2 Deriving the Ordinary Least Squares Estimates(10/10),Fitted values and residualsAlgebraic properties of OLS regression
19、,Fitted or predicted values,Deviations from regression line(=residuals),Chapter 2 The Simple Regression Model,Chapter,End,2.3 Algebraic Properties of OLS on Any Sample of Data(1/6),Deviations from regression line sum up to zero,Correlation between deviations and regressors is zero,Sample averages of
20、 y and x lie on regression line,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 Model,Chapter,End,2.3 Algebraic Properties of OLS on Any Sample of Data(2/6),Goodness-of-FitMeasures of Variation,H
21、ow 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 regression,Residual sum of squares,represents variation notexplained by regression,Chapter 2 The Sim
22、ple Regression Model,Chapter,End,2.3 Algebraic Properties of OLS on Any Sample of Data(3/6),Decomposition of total variationGoodness-of-fit measure(R-squared),Total variation,Explained part,Unexplained part,R-squared measures the fraction of the total variation that is explained by the regression,Ch
23、apter 2 The Simple Regression Model,Chapter,End,2.3 Algebraic Properties of OLS on Any Sample of Data(4/6),Chapter 2 The Simple Regression Model,Chapter,End,2.3 Algebraic Properties of OLS on Any Sample of Data(5/6),CEO Salary and return on equityVoting outcomes and campaign expendituresCaution:A hi
24、gh R-squared does not necessarily mean that the regression has a causal interpretation!,The regression explains 85.6%of the total variation in election outcomes,Chapter 2 The Simple Regression Model,Chapter,End,The regression explains only 1.3%of the total variation in salaries,ceosal1.wf1ls salary
25、c roe,2.3 Algebraic Properties of OLS on Any Sample of Data(6/6),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,Incorporating nonlinearities:Semi-logarithmic formRegression of log wage
26、s 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 Simple Regression Model,Chapter,End,2.4 Units of Measurement and Functional Form(2/6),Fitted regressi
27、on,The 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 education,Chapter 2 The Simple Regression Model,Chapter,End,wage1.wf1ls log(wage)c educ,2.4 Units of Measurement and Functional Form(3/6),Chapter 2 The Simple
28、 Regression Model,Chapter,End,2.4 Units of Measurement and Functional Form(4/6),Supplementary materials,The proportionate change in x in moving from x0 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.,Incorporating nonli
29、nearities: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 logarithm of his/her firms sales,Logarithmic changes are always percentage changes,Chapter
30、 2 The Simple Regression Model,Chapter,End,2.4 Units of Measurement and Functional Form(5/6),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
31、in the variablesThe interpretation of the coefficients depends 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),ceosal1.wf1ls log(salary)c log(sales),+1%sales!+0.257%salary,The estimated regression coefficients are random
32、 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 is,Data is random and depends on particular sample that has been drawn,Chapter 2 The Simple Regression Model,Chapter,End,2.5 Exp
33、ected Values and Variances of the OLS Estimators(1/17),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 linear,The data is a random sample drawn from the population,Each d
34、ata point therefore followsthe population equation,Chapter 2 The Simple Regression Model,Chapter,End,2.5 Expected Values and Variances of the OLS Estimators(2/17),Discussion of random sampling:Wage and educationThe population consists,for example,of all workers of country AIn the population,a linear
35、 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 one does not know beforehand which worker is drawnThrow back worker into population and repeat random d
36、raw timesThe wages and years of education of the sampled workers are used to estimate the linear relationship between wages and education,Chapter 2 The Simple Regression Model,Chapter,End,2.5 Expected Values and Variances of the OLS Estimators(3/17),The values drawnfor the i-th worker,The implied de
37、viationfrom 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),Assumptions for the linear regression model(cont.)Assumption SLR.3(Sample variation in explanatory variable)Assumption SLR.4(Zer
38、o 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 the explanatory variable must contain no information about the
39、mean of the unobserved factors,Chapter 2 The Simple Regression Model,Chapter,End,Further,E(ui|x1,xn)=0 and E(ui uj|x1,xn)=0.,2.5 Expected Values and Variances of the OLS Estimators(5/17),Theorem 2.1(Unbiasedness of OLS)Interpretation of unbiasednessThe estimated coefficients may be smaller or larger
40、,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 was repeated m
41、any timesIn a given sample,estimates may differ considerably from true values,Chapter 2 The Simple Regression Model,Chapter,End,2.5 Expected Values and Variances of the OLS Estimators(6/17),Chapter 2 The Simple Regression Model,Chapter,End,PROOF of Theorem 2.1(Unbiasedness of OLS):It is easy to show
42、,2.5 Expected Values and Variances of the OLS Estimators(7/17),Chapter 2 The Simple Regression Model,Chapter,End,Example 2.12(p48,meap93.wf1)ls math10 c lnchprgNote the interpretation on the negative slope.,2.5 Expected Values and Variances of the OLS Estimators(8/17),Variances of the OLS estimators
43、Depending 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)?Sampling variability is measured by the estimators variancesAssumption SLR.5(Homoskedas
44、ticity),The value of the explanatory variable must contain no information about the variability of the unobserved factors,Chapter 2 The Simple Regression Model,Chapter,End,2.5 Expected Values and Variances of the OLS Estimators(9/17),Graphical illustration of homoskedasticity,The variability of the
45、unobservedinfluences does not dependent on the value of the explanatory variable,Chapter 2 The Simple Regression Model,Chapter,End,2.5 Expected Values and Variances of the OLS Estimators(10/17),An example for heteroskedasticity:Wage and education,The variance of the unobserved determinants of wages
46、increaseswith the level of education,Chapter 2 The Simple Regression Model,Chapter,End,2.5 Expected Values and Variances of the OLS Estimators(11/17),Chapter 2 The Simple Regression Model,Chapter,End,By assumptions SLR.4 and SLR.5,we have,2.5 Expected Values and Variances of the OLS Estimators(12/17
47、),Theorem 2.2(Variances of OLS estimators)Conclusion:The sampling variability of the estimated regression coefficients will be the higher the larger the variability of the unobserved factors,and the lower,the higher the variation in the explanatory variable,Under assumptions SLR.1 SLR.5:where these
48、are conditional on the sample values x1,xn.,Chapter 2 The Simple Regression Model,Chapter,End,2.5 Expected Values and Variances of the OLS Estimators(13/17),Proof of Theorem 2.2(Variances of OLS estimators),Chapter 2 The Simple Regression Model,Chapter,End,2.5 Expected Values and Variances of the OL
49、S Estimators(14/17),Estimating the error variance,The variance of u does not depend on x,i.e.is equal to the unconditional variance,One could estimate the variance of theerrors by calculating the variance of the residuals in the sample;unfortunately this estimate would be biased,An unbiased estimate
50、 of the error variance can be obtained by substracting the number of estimated regression coefficients from the number of observations,Chapter 2 The Simple Regression Model,Chapter,End,2.5 Expected Values and Variances of the OLS Estimators(15/17),Theorem 2.3(Unbiasedness of the error variance)Proof
