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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 5 Multiple RegressionAnalysis:OLS AsymptoticsWooldridge:Introductory Econometrics:A Modern Approach,5e 2013 Cengage Learning.All Rights Reserved.M
2、ay not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.So far we focused on properties of OLS that hold for any sampleProperties of OLS that hold for any sample/sample sizeExpected values/unbiasedness under MLR.1 MLR.4Variance formulas under MLR.1 MLR.5
3、Gauss-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.5Without assuming nor-mality of the error term!Multiple RegressionAnalysis:OLS Asymptotics 201
4、3 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.ConsistencyInterpretation:Consistency means that the probability that the estimate is arbitrari-ly close to the true population value can be made arbitrarily
5、high by increasing the sample sizeConsistency is a minimum requirement for sensible estimatorsAn estimator is consistent for a population parameter iffor arbitrary and .Alternative notation:The estimate converges in proba-bility to the true population valueMultiple RegressionAnalysis:OLS Asymptotics
6、 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 5.1(Consistency of OLS)Special case of simple regression modelAssumption MLR.4One can see that the slope estimate is consistent if the explanatory
7、 variable is exogenous,i.e.un-correlated with the error term.All explanatory variables must be uncorrelated with the error term.This assumption is weaker than the zero conditional mean assumption MLR.4.Multiple RegressionAnalysis:OLS Asymptotics 2013 Cengage Learning.All Rights Reserved.May not be s
8、canned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.For consistency of OLS,only the weaker MLR.4 is neededAsymptotic analog of omitted variable biasTrue modelMisspecified modelThere is no omitted variable bias if the omitted variable is irrelevant or uncorrelat
9、ed with the included variableBiasMultiple RegressionAnalysis:OLS Asymptotics 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.Asymptotic normality and large sample inferenceIn practice,the normality assum
10、ption 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 MLR.1 MLR.
11、5:also In large samples,the standardized estimates are normally distributedMultiple RegressionAnalysis:OLS Asymptotics 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.Converges toConverges toPractical co
12、nsequencesIn large samples,the t-distribution is close to the N(0,1)distributionAs 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
13、errorsConverges to a fixed numberMultiple RegressionAnalysis:OLS Asymptotics 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.Asymptotic analysis of the OLS sampling errors(cont.)This is why large samples are betterExample:Standard errors in a birth weight equationshrinks with the rate shrinks with the rate Use only the first half of observationsMultiple RegressionAnalysis:OLS Asymptotics
