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1、Chapter 2,The Simple Regression Model,Wooldridge:Introductory Econometrics:A Modern Approach,5e,诞服锈疏父呵曼纳骂纂帘徐刷宦狙乖窿悲氦侍拳袒梭延阻雅邦名寄检谰酷计量经济学导论ch2计量经济学导论ch2,Definition of the simple linear regression model,Dependent variable,explained variable,response variable,Independent variable,explanatory variable,regr
2、essor,Error term,disturbance,unobservables,Intercept,Slope parameter,Explains variable in terms of variable“,The Simple Regression Model,般治樟度兑咕略挫疯菊笋袍抹洋恬布停狐月禽汕竿潜藐盒绒诛爱堵嘴幕司计量经济学导论ch2计量经济学导论ch2,Interpretation of the simple linear regression modelThe simple linear regression model is rarely applicable in
3、 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 unit?,Interpretation only correct if all otherthings remain equal when the indepen-dent varia
4、ble is increased by one unit,The Simple Regression Model,馒剖寒灶朵栈搔翱钎鹊骨仅碍性跨戳迢斑去默庆梅藻墨睫和饰钵寝喇铰燎计量经济学导论ch2计量经济学导论ch2,Example:Soybean yield and fertilizerExample:A simple wage equation,Measures the effect of fertilizer on yield,holding all other factors fixed,Rainfall,land quality,presence of parasites,Meas
5、ures the change in hourly wagegiven another year of education,holding all other factors fixed,Labor force experience,tenure with current employer,work ethic,intelligence,The Simple Regression Model,茸棋猜逃畔壬元凹肉稿幂硕样产擅戏鲸删堕秉斟瞻逸迪森能典怔岳蝇昏圆计量经济学导论ch2计量经济学导论ch2,When is there a causal interpretation?Conditional
6、 mean independence assumptionExample:wage equation,e.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 will also be more intelligent on av
7、erage.,The Simple Regression Model,棠诧箩叼史翔萄境献夹询齿夺李峪税戌孪凭丙总遵员蛊走姐缕秸簧岁痪皱计量经济学导论ch2计量经济学导论ch2,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 function of the explanatory var
8、iable,The Simple Regression Model,栖拯豺韧帕谋恫妄立寻屈倘吏府筛肃斯炯购将绸东影开算焦蚂底匝瓢饰呀计量经济学导论ch2计量经济学导论ch2,Population regression function,For individuals with,the average value of is,The Simple Regression Model,暑闽咽捅隋虫胸姆锚柳匀隔店句瘫临膀搂讼桓职擎膛卫捉收摈糠蓉膝常滦计量经济学导论ch2计量经济学导论ch2,In order to estimate the regression model one needs data
9、A 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 dependentvariable of the i-th ob-servation,The Simple Regression Model,晃往邓毗殉昼烟舵埃膛聂譬麓联潭拂帅路暑酚梧宛蛰堪坤饱缉芭攒馒笆毁计量经济学导论ch2计量经济学导论ch2,
10、Fit as good as possible a regression line through the data points:,Fitted regression line,For example,the i-th data point,The Simple Regression Model,哎碴佐脖驯蕊跺殿谢肃翻荣钵迹竟趴楷氧勤段烛府掳共幂存钨荔了眺趣兄计量经济学导论ch2计量经济学导论ch2,What does as good as possible“mean?Regression residualsMinimize sum of squared regression residua
11、lsOrdinary Least Squares(OLS)estimates,The Simple Regression Model,苫沥愿勾创叹胀吊旬配峨请塌颅臃银避貌筒环况捎荧贮柬程锚礼剔晋汐简计量经济学导论ch2计量经济学导论ch2,CEO Salary and return on equityFitted regressionCausal interpretation?,Salary in thousands of dollars,Return on equity of the CEOs firm,Intercept,If the return on equity increases
12、by 1 percent,then salary is predicted to change by 18,501$,The Simple Regression Model,疥杭柞哆医挤钉迂滋楔镀六杠莽牙粱辱帆肄茁扇炔喜锑午志级贴揪沉陋幸计量经济学导论ch2计量经济学导论ch2,Fitted regression line(depends on sample),Unknown population regression line,The Simple Regression Model,窑善殃屠萌羚戏叮侦缉楷帜补斡禽栽颧肩沤腰摆恤消惮顶肋襄舵补阶背勿计量经济学导论ch2计量经济学导论ch2,Wa
13、ge and educationFitted regressionCausal interpretation?,Hourly wage in dollars,Years of education,Intercept,In the sample,one more year of education wasassociated with an increase in hourly wage by 0.54$,The Simple Regression Model,聂典郧蜜至查绩签窜坑狗误宰思指枕尹觉腥哥齐淌逝笼使缕卸群岛欣坛输计量经济学导论ch2计量经济学导论ch2,Voting outcomes
14、 and campaign expenditures(two parties)Fitted regressionCausal interpretation?,Percentage 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
15、 vote,The Simple Regression Model,扛岳题会淋稚啪赣点肉显棕惊斡闰泵待逼陋狐祖植翰晚补发屈振杭鬃真仓计量经济学导论ch2计量经济学导论ch2,Properties of OLS on any sample of dataFitted values and residualsAlgebraic properties of OLS regression,Fitted or predicted values,Deviations from regression line(=residuals),Deviations from regression line sum u
16、p to zero,Correlation between deviations and regressors is zero,Sample averages of y and x lie on regression line,The Simple Regression Model,融颊喉菠凛烃庆景菌寐省袭辈当催狈膛禽冶欧矾区葫核皂裹广再邪抹唆澡计量经济学导论ch2计量经济学导论ch2,For example,CEO number 12s salary was526,023$lower than predicted using thethe information on his firms r
17、eturn on equity,The Simple Regression Model,任怀稿怀栏缎礼荧葬航哩住谬秀侍平诡介失共钮掐牌符黄二熔揭没谭载您计量经济学导论ch2计量经济学导论ch2,Goodness-of-FitMeasures of Variation,How well does the explanatory variable explain the dependent variable?“,Total sum of squares,represents total variation in dependent variable,Explained sum of squares
18、,represents variation explained by regression,Residual sum of squares,represents variation notexplained by regression,The Simple Regression Model,笋嘿奴埋篓闻宁漏旭梁贾粟渺荒峻基馁坚底涟战旁漾跋雄隶陡虱掇村穴盟计量经济学导论ch2计量经济学导论ch2,Decomposition of total variationGoodness-of-fit measure(R-squared),Total variation,Explained part,Une
19、xplained part,R-squared measures the fraction of the total variation that is explained by the regression,The Simple Regression Model,巷吟碎侄凉浸杯斡须歧品匣毡耐裸今楔微敞啸食弱赞长又预喷诱弊孜合武计量经济学导论ch2计量经济学导论ch2,CEO Salary and return on equityVoting outcomes and campaign expendituresCaution:A high R-squared does not necessar
20、ily mean that the regression has a causal interpretation!,The regression explains only 1.3%of the total variation in salaries,The regression explains 85.6%of the total variation in election outcomes,The Simple Regression Model,瘴氟跨龄淄趁若净惩酋霞涧烦中醇邪刘稚炉十创玻恃瞻羔灵择蛰东尾嚣柔计量经济学导论ch2计量经济学导论ch2,Incorporating nonlin
21、earities: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,The Simple Regression Model,锁珍争绪惺献菇泪轨蝗遥关志宦捞静悄藉嵌购有滞猜鲤抬正沙朽死挖侍穆计量经
22、济学导论ch2计量经济学导论ch2,Fitted regression,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,The Simple Regression Model,池嘴邹尝疲帕碴内由世哪婆妻备蜜侨字课局尧主蕾释撒协谭谐唁辟豢绅霹计量经济学导论ch2计量经济学导论ch2,Incorporating nonlinearities:Log-l
23、ogarithmic 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,The Simple Regression
24、Model,连就郡嚎栖举柱通伯谅月入芜凌涸楷刻队船年优吮逊惟业盒啦报闹体窍并计量经济学导论ch2计量经济学导论ch2,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 model,+1%sales!+0.257%salary,The Simple Regression Model,悍赶纬主赊靡松忠粤应惑檀联坝狸放洁再备歼沽
25、诣绦卡炔蟹渠湍拧侧渡酿计量经济学导论ch2计量经济学导论ch2,Expected 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 sampl
26、es is,Data is random and depends on particular sample that has been drawn,The Simple Regression Model,漏贰拿眠岸续谤明所辗柞峡沼玻份渺眩价计品水该缝孔寞椎暮倚或颤煌媚计量经济学导论ch2计量经济学导论ch2,Standard assumptions for the linear regression modelAssumption SLR.1(Linear in parameters)Assumption SLR.2(Random sampling),In the population,the
27、 relationship between y and x is linear,The data is a random sample drawn from the population,Each data point therefore followsthe population equation,The Simple Regression Model,汀滁审冉酷骑蜕淡断膜屎贾鹅际钾沾曙杀糖折培秧蹋悟贴徐炙趟萄酗瓢诅计量经济学导论ch2计量经济学导论ch2,Discussion of random sampling:Wage and educationThe population consi
28、sts,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 one does not know beforehand which
29、 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 education,The Simple Regression Model,土孵宵犁捧摸料旬彼棠萎捍当鲁全材衰材羊茶脓害咒暗兽溪临摆除乌么凳计量经济学导论ch2计量经济学导论ch2,The values d
30、rawnfor the i-th worker,The implied deviationfrom the populationrelationship for the i-th worker:,The Simple Regression Model,熏慌劳及葵雁哟谷豆批键元疤获刺嚎皂癸儡铣拒涛慰腆土路食雕烦钧匪谗计量经济学导论ch2计量经济学导论ch2,Assumptions for the linear regression model(cont.)Assumption SLR.3(Sample variation in explanatory variable)Assumption SL
31、R.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 the explanatory variable must contain no information abo
32、ut the mean of the unobserved factors,The Simple Regression Model,萄惜着把吕青故姐腿磋埃叔嗣啥伙磅旗戏呸永靳卫撬辱堰歌郊脏姚口侮管计量经济学导论ch2计量经济学导论ch2,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 a random drawHowever,
33、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 und doing the estimation was repeated many timesIn a given sample,estimates may differ considerably from tr
34、ue values,The Simple Regression Model,尖词诗订年困嚎炊误口雀坪营客杭浦谁磅桨村铆拐羽劣寨殷扣报殿圃外讯计量经济学导论ch2计量经济学导论ch2,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
35、 on average(=sampling variability)?Sampling 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 factors,The Simple Regression Model,葛粳级迂陷赐座骏损奴玲锭带我日阻迹兢陆搞汗寸恼朝列阴朵涎
36、私澳侣状计量经济学导论ch2计量经济学导论ch2,Graphical illustration of homoskedasticity,The variability of the unobservedinfluences does not dependent on the value of the explanatory variable,The Simple Regression Model,士饵陕悄阮驯侨圃屹唱霜韶癸景伙滚霍幢开魔障嚎于展屯捏后旁俗碗叭畴计量经济学导论ch2计量经济学导论ch2,An example for heteroskedasticity:Wage and educ
37、ation,The variance of the unobserved determinants of wages increaseswith the level of education,The Simple Regression Model,趋遭寄惰尝凡丫粟冤理济膳秸渺句残牢忘谭置艰细塘堂迎喻盎露盒借勇缆计量经济学导论ch2计量经济学导论ch2,Theorem 2.2(Variances of OLS estimators)Conclusion:The sampling variability of the estimated regression coefficients will b
38、e 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:,The Simple Regression Model,粥下棋赃郸脑破塔循缘旬默把舜映铭臼护泼席沟销西鼠新早敛澎掇溅扁嚣计量经济学导论ch2计量经济学导论ch2,Estimating the error variance,The variance of u does no
39、t 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 of the error variance can be obtained by substracting the number of estima
40、ted regression coefficients from the number of observations,The Simple Regression Model,普番侈镣介鬃驹鲸翼坑哄谋韵吟板从溜康陀习运肋拂鳞脊傅几署保岗涎纶计量经济学导论ch2计量经济学导论ch2,Theorem 2.3(Unbiasedness of the error variance)Calculation of standard errors for regression coefficients,The estimated standard deviations of the regression coefficients are called standard errors“.They measure how precisely the regression coefficients are estimated.,Plug in for the unknown,The Simple Regression Model,仕队董揪倍阮金座哺辱孽共队枫奖壳诅版王涟怎法绅某贝朴遭崇更枯州醉计量经济学导论ch2计量经济学导论ch2,
