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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
2、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 Asymptotics5.1 Consistency5.2 Asymptotic Normality and Large Sample Inference5.3 Asymptotic Efficiency of OLS5.0 IntroductionAssignments: Com
3、puter Exercises C2 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.So far we focused on properties of OLS that hold for any sampleProperties of OLS that hold for any sample/sample sizeExpected value
4、s/unbiasedness 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.5Without assuming nor-malit
5、y of the error term! Chapter 5 Multiple RegressionAnalysis: OLS Asymptotics5.0 IntroductionChapter 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.ConsistencyInterpretation:Consistency means that th
6、e probability that the estimate is arbitrari- ly close to the true population value can be made arbitrarily 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:Th
7、e estimate converges in proba-bility to the true population valueChapter 5 Multiple RegressionAnalysis: OLS Asymptotics5.1 Consistency (1/3)Chapter 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.Ch
8、apter 5 Multiple RegressionAnalysis: OLS Asymptotics5.1 Consistency (2/3)ChapterTheorem 5.1 (Consistency of OLS)Special case of simple regression modelAssumption MLR.4One can see that the slope estimate is consistent if the explanatory variable is exogenous, i.e. un-correlated with the error term.Al
9、l explanatory variables must be uncorrelated with the error term. This assumption is weaker than the zero conditional mean assumption MLR.4. 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
10、 Multiple RegressionAnalysis: OLS Asymptotics5.1 Consistency (3/3)ChapterFor consistency of OLS, only the weaker MLR.4 is neededOLS turns out to be biased under Assumption MLR.4Asymptotic analog of omitted variable biasTrue modelThere is no omitted variable bias if the omitted variable is irrelevant
11、 or uncorrelated with the included variableBiasMisspecified model 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 assumption
12、 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
13、.5:also In large samples, the standardized estimates are normally distributedChapter 5 Multiple RegressionAnalysis: OLS Asymptotics5.2 Asymptotic Normality and Large Sample Inference (1/5)Chapter 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a pub
14、licly accessible website, in whole or in part.Converges toConverges toPractical consequencesIn 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
15、MLR.5 are still necessary, esp. homoscedasticityAsymptotic analysis of the OLS sampling errorsConverges to a fixed numberChapter 5 Multiple RegressionAnalysis: OLS Asymptotics5.2 Asymptotic Normality and Large Sample Inference (2/5)Chapter 2013 Cengage Learning. All Rights Reserved. May not be scann
16、ed, 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 o
17、f observationsChapter 5 Multiple RegressionAnalysis: OLS Asymptotics5.2 Asymptotic Normality and Large Sample Inference (3/5)Chapter 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 Multipl
18、e RegressionAnalysis: OLS Asymptotics5.2 Asymptotic Normality and Large Sample Inference (4/5)ChapterThe Lagrange Multiplier Statistic: test multiple exclusion restrictionsThe LM statistic relies on the Gauss-Markov assumptions without the normality assumption.The procedure for testing H0:(i) Regres
19、s the restricted model,(ii) Auxiliary regression: (iii) Compute(iv) If LMc, the null hypothesis is rejected. or If p-valuea, the null hypothesis is rejected. What would happen to if we regressed y on all regressors in step (i)? 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied o
20、r duplicated, or posted to a publicly accessible website, in whole or in part.Chapter 5 Multiple RegressionAnalysis: OLS Asymptotics5.2 Asymptotic Normality and Large Sample Inference (5/5)ChapterExample 5.3 Economic Model of Crimecrime1.wf1ls narr86 c pcnv avgsen tottime ptime86 qemp86We find that
21、both avgsen and tottime are insignificant. So we want to know if they are jointly insignificant.(i) Regress the restricted model.ls narr86 c pcnv ptime86 qemp86series u=resid(ii) Regress the auxiliary regression.ls u c pcnv avgsen tottime ptime86 qemp86(iii) Compute(iv) Given a=5%, c=5.99. So LMc, we cannot reject that both avgsen and tottime are joint insignificant.
