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1、Chapter 5,Multiple RegressionAnalysis:OLS Asymptotics,Wooldridge:Introductory Econometrics:A Modern Approach,5e,僻敛盼芝摆尧挨彭息樟汕肠池铆检久偶悼加歇淹钝曹强贺勒辐漂置躇帮痛计量经济学导论ch5计量经济学导论ch5,So far we focused on properties of OLS that hold for any sampleProperties of OLS that hold for any sample/sample sizeExpected values/un
2、biasedness under MLR.1 MLR.4Variance formulas under MLR.1 MLR.5Gauss-Markov Theorem under MLR.1 MLR.5Exact sampling distributions/tests under MLR.1 MLR.6Properties of OLS that hold in large samplesConsistency under MLR.1 MLR.4Asymptotic normality/tests under MLR.1 MLR.5,Without assuming nor-mality o
3、f the error term!,Multiple RegressionAnalysis:OLS Asymptotics,盒勉拥酶佛陌雇兹庄义侩焚威菌馏础忘菊掉隶朝纠疗韵据震帧型剪尊以薪计量经济学导论ch5计量经济学导论ch5,ConsistencyInterpretation:Consistency means that the probability that the estimate is arbitrari-ly close to the true population value can be made arbitrarily high by increasing the samp
4、le sizeConsistency is a minimum requirement for sensible estimators,An estimator is consistent for a population parameter if,for arbitrary and.,Alternative notation:,The estimate converges in proba-bility to the true population value,Multiple RegressionAnalysis:OLS Asymptotics,拖拦词猾晚筛鲤独瞅庆展辨鄂构杂吩涧区综盔斤疮
5、菊餐梯玄症疚锣车袱凡计量经济学导论ch5计量经济学导论ch5,Theorem 5.1(Consistency of OLS)Special case of simple regression modelAssumption MLR.4,One can see that the slope estimate is consistent if the explanatory variable is exogenous,i.e.un-correlated with the error term.,All explanatory variables must be uncorrelated with
6、the error term.This assumption is weaker than the zero conditional mean assumption MLR.4.,Multiple RegressionAnalysis:OLS Asymptotics,扫粤悍饺幂陋刁沮验藕筒抢霜戮瓢途渗募殊竿慌目坪阶晃旧明滤米石悉铱计量经济学导论ch5计量经济学导论ch5,For consistency of OLS,only the weaker MLR.4 is neededAsymptotic analog of omitted variable bias,True model,Missp
7、ecified model,There is no omitted variable bias if the omitted variable is irrelevant or uncorrelated with the included variable,Bias,Multiple RegressionAnalysis:OLS Asymptotics,援碌鲸之僻拙使紊误哀霞袖娄熄宝斯某廊鞋复哦胰脂戏械旷半袍兼图霍承计量经济学导论ch5计量经济学导论ch5,Asymptotic normality and large sample inferenceIn practice,the normal
8、ity assumption MLR.6 is often questionableIf MLR.6 does not hold,the results of t-or F-tests may be wrongFortunately,F-and t-tests still work if the sample size is large enoughAlso,OLS estimates are normal in large samples even without MLR.6Theorem 5.2(Asymptotic normality of OLS),Under assumptions
9、MLR.1 MLR.5:,also,In large samples,the standardized estimates are normally distributed,Multiple RegressionAnalysis:OLS Asymptotics,夸型衣容熟耪榔瘟辉侯孰瑰书厂盔暂的狈季克擎沟狐各酌扮裙炎猫楔赫哄计量经济学导论ch5计量经济学导论ch5,Converges to,Converges to,Practical consequencesIn large samples,the t-distribution is close to the N(0,1)distributi
10、onAs a consequence,t-tests are valid in large samples without MLR.6The same is true for confidence intervals and F-testsImportant:MLR.1 MLR.5 are still necessary,esp.homoscedasticityAsymptotic analysis of the OLS sampling errors,Converges to a fixed number,Multiple RegressionAnalysis:OLS Asymptotics
11、,郁昆疽丙俄郭疲箩瓢除员斌潘味慕筏卵矫瓢剖晶请濒彼痊末禄袖矣孽戊硫计量经济学导论ch5计量经济学导论ch5,Asymptotic analysis of the OLS sampling errors(cont.)This is why large samples are betterExample:Standard errors in a birth weight equation,shrinks with the rate,shrinks with the rate,Use only the first half of observations,Multiple RegressionAnalysis:OLS Asymptotics,瘸陇霜雌容绅欠艳谰扰汐票粟模蔬收茹凤馈涡矮蔑数篡损锰乏逞杉建签网计量经济学导论ch5计量经济学导论ch5,
