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1、Chapter 2 The Simple Regression ModelWooldridge:Introductory Econometrics:A Modern Approach,5eDefinition of the simple linear regression modelDependent variable,explained variable,response variable,Independent variable,explanatory variable,regressor,Error term,disturbance,unobservables,InterceptSlop
2、e parameterExplains variable in terms of variable “The Simple Regression ModelInterpretation 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
3、 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 increased by one unitThe Simple Regression ModelExample:Soybean yield and fertilizerExample:A simple wage eq
4、uationMeasures 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,intelligence Th
5、e Simple Regression ModelWhen is there a causal 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 b
6、ecauseindividuals with more education will also be more intelligent on average.The Simple Regression ModelPopulation 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 function of t
7、he explanatory variableThe Simple Regression Model Population regression functionFor individuals with ,the average value of isThe Simple Regression ModelIn order to estimate the regression model one needs dataA random sample of observationsFirst observationSecond observationThird observationn-th obs
8、ervationValue of the expla-natory variable of the i-th observationValue of the dependentvariable of the i-th ob-servationThe Simple Regression ModelFit as good as possible a regression line through the data points:Fitted regression lineFor example,the i-th data pointThe Simple Regression ModelWhat d
9、oes as good as possible“mean?Regression residualsMinimize sum of squared regression residualsOrdinary Least Squares(OLS)estimatesThe Simple Regression ModelCEO Salary and return on equityFitted regressionCausal interpretation?Salary in thousands of dollarsReturn on equity of the CEOs firmInterceptIf
10、 the return on equity increases by 1 percent,then salary is predicted to change by 18,501$The Simple Regression Model Fitted regression line(depends on sample)Unknown population regression lineThe Simple Regression ModelWage and educationFitted regressionCausal interpretation?Hourly wage in dollarsY
11、ears of educationInterceptIn the sample,one more year of education wasassociated with an increase in hourly wage by 0.54$The Simple Regression ModelVoting outcomes and campaign expenditures(two parties)Fitted regressionCausal interpretation?Percentage of vote for candidate APercentage of campaign ex
12、penditures 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 ModelProperties of OLS on any sample of dataFitted values and residualsAlgebraic properties of OLS regressionFitte
13、d or predicted valuesDeviations from regression line(=residuals)Deviations from regression line sum up to zeroCorrelation between deviations and regressors is zeroSample averages of y and x lie on regression lineThe Simple Regression Model For example,CEO number 12s salary was526,023$lower than pred
14、icted using thethe information on his firms return on equity The Simple Regression ModelGoodness-of-FitMeasures 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
15、 variation explained by regressionResidual sum of squares,represents variation notexplained by regressionThe Simple Regression ModelDecomposition of total variationGoodness-of-fit measure(R-squared)Total variationExplained partUnexplained partR-squared measures the fraction of the total variation th
16、at is explained by the regressionThe Simple Regression ModelCEO 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 only 1.3%of the total variation in salariesThe regression explains 85.6%of the total variation in election outcomesThe Simple Regression ModelIncorporating nonlinearities:Semi-logarithmic formRegression of log wages on years of eductionThis chan
