計(jì)量經(jīng)濟(jì)學(xué)英文課件：Chapter 8 Multiple Regression Analysis Heteroscedasticity

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 8 Heteroscedasticity Wooldridge: Introductory Econometrics: A Modern Approach, 5e 2013 Cengage Learning. All Rights Reserved. May not be scan

2、ned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Chapter 8 Multiple Regression Analysis: Heteroscedasticity8.1 Consequences of Heteroskedasticity for OLS8.2 Heteroskedasticity-Robust Inference after OLS Estimation8.3 Testing for Heteroskedasticity8.4 Weighte

3、d Least Squares Estimation8.5 The Linear Probability Model RevisitedAssignments: Promblems 7 Computer Exercises C2, C5, C9, C11 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.Unaffected properties:

4、OLS still unbiased and consistent under heteroscedastictiy!Also, interpretation of R-squared is not changed.Affected properties:Heteroscedasticity invalidates variance formulas for OLS estimators.The usual F-tests and t-tests are not valid under heteroscedasticity. Under heteroscedasticity, OLS is n

5、o longer the best linear unbiased estimator (BLUE); there may be more efficient linear estimators. OLS is asymptotically efficient in the class of estimators. With relatively large sample sizes, it might no be so important to obtain an efficient estimator.Unconditional error variance is unaffected b

6、y heteroscedasticity (which refers to the conditional error variance) Chapter 8 Multiple Regression Analysis: Heteroscedasticity8.1 Consequences of Heteroskedasticity for OLSChapter 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessib

7、le website, in whole or in part.Formulas for OLS standard errors and related statistics have been developed that are robust to heteroscedasticity of unknown formAll formulas are only valid in large samplesFormula for heteroscedasticity-robust OLS standard errorUsing these formulas, the usual t-test

8、is valid asymptoticallyThe usual F-statistic does not work under heteroscedasticity, but we can obtain F (Wald) and LM statistics that are robust to heteroskedasticity of an unknown, arbitrary form. The heteroskedasticity-robust F statistic has no simple form, but it can be computed using certain st

9、atistical packages.Chapter 8 Multiple Regression Analysis: Heteroscedasticity8.2 Heteroskedasticity-Robust Inference after OLS Estimation (1/5)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

10、.Example: Hourly wage equationHeteroscedasticity robust standard errors may be larger or smaller than their nonrobust counterparts. The differences are often small in practice.F-statistics are also often not too different.If there is strong heteroscedasticity, differences may be larger. To be on the

11、 safe side, it is advisable to always compute robust standard errors.Chapter 8 Multiple Regression Analysis: Heteroscedasticity8.2 Heteroskedasticity-Robust Inference after OLS Estimation (2/5)Chapter 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to

12、a publicly accessible website, in whole or in part.Example 8.2 Heteroskedasticity-Robust F Statistic (gpa3.wf1)The heteroskedasticity-robust F statistic has no simple form, but it can be computed using certain statistical packages.smpl if spring=1ls(h) cumgpa c sat hsperc tothrs female black whiteCh

13、apter 8 Multiple Regression Analysis: Heteroscedasticity8.2 Heteroskedasticity-Robust Inference after OLS Estimation (3/5)Chapter0:0,0Perform the test in Eviews.blackwhiteH 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible websit

14、e, in whole or in part.Example 8.2 Heteroskedasticity-Robust F Statistic (gpa3.wf1)The heteroskedasticity-robust F statistic has no simple form, but it can be computed using certain statistical packages.smpl if spring=1ls(h) cumgpa c sat hsperc tothrs female black whiteChapter 8 Multiple Regression

15、Analysis: Heteroscedasticity8.2 Heteroskedasticity-Robust Inference after OLS Estimation (3/5)Chapter0:0,0Perform the test in Eviews.blackwhiteH 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.Chapt

16、er 8 Multiple Regression Analysis: Heteroscedasticity8.2 Heteroskedasticity-Robust Inference after OLS Estimation (4/5)ChapterA Heteroskedasticity-Robust LM Statistic011223344550451234123151232:0,01. Regressing on , gets residuals .2. Regressing on , gets residuals . Regressing on , gets residuals .

17、3. Run the regyxxxxxuHyx xxuxx xxrxx xxr12212ression of 1 on , . aru r uLMnSSR 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 8 Multiple Regression Analysis: Heteroscedasticity8.2 Heteroske

18、dasticity-Robust Inference after OLS Estimation (5/5)ChapterExample 8.3 Heteroskedasticity-Robust LM Statistic (crime1.wf1)equation eq_r.ls narr86 c pcnv ptime86 qemp86 inc86 black hispanseries u=residequation eq_1.ls avgsen c pcnv ptime86 qemp86 inc86 black hispanseries r1=residequation eq_2.ls avg

19、sen2 c pcnv ptime86 qemp86 inc86 black hispanseries r2=residequation eq_a.ls 1 r1*u r2*uscalar lm=obssmpl-eq_a.ssr LM=4.00scalar p=chisq(lm,2) p=0.135520:0,0avgsenavgsenH 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website,

20、 in whole or in part.It may still be interesting whether there is heteroscedasticity because then OLS may not be the most efficient linear estimator anymoreBreusch-Pagan test for heteroscedasticityUnder MLR.4The mean of u2 must not vary with x1, x2, , xkChapter 8 Multiple Regression Analysis: Hetero

21、scedasticity8.3 Testing for Heteroskedasticity (1/7)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.Breusch-Pagan test for heteroscedasticity (cont.)Regress squared residuals on all expla-na

22、tory variables and test whether this regression has explanatory power.A large test statistic (= a high R-squared) is evidence against the null hypothesis.Alternative test statistic (= Lagrange multiplier statistic, LM). Again, high values of the test statistic (= high R-squared) lead to rejection of

23、 the null hypothesis that the expected value of u2 is unrelated to the explanatory variables.Chapter 8 Multiple Regression Analysis: Heteroscedasticity8.3 Testing for Heteroskedasticity (2/7)Chapter 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a

24、publicly accessible website, in whole or in part.Example: Heteroscedasticity in housing price equationsIn the logarithmic specification, homoscedasticity cannot be rejectedHeteroscedasticityChapter 8 Multiple Regression Analysis: Heteroscedasticity8.3 Testing for Heteroskedasticity (3/7)Chapter 2013

25、 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 8 Multiple Regression Analysis: Heteroscedasticity8.3 Testing for Heteroskedasticity (4/7)ChapterExample 8.4 Heteroskedasticity in Housing Price E

26、quations (hprice1.wf1)equation eq_y.ls price c lotsize sqrft bdrmsseries u2=resid2equation eq_u2.ls u2 c lotsize sqrft bdrmsscalar f=eq_u2.f f=5.34scalar pf=fdist(f,3,84) pf=0.002scalar lm=obssmpl*eq_u2.r2lm=14.09scalar plm=chisq(lm,3)plm=0.0028equation eq_lny.ls log(price) c log(lotsize) log(sqrft)

27、 bdrmsseries u2=resid2equation eq_u2.ls u2 c log(lotsize) log(sqrft) bdrmsscalar ff=eq_u2.fff=1.41scalar pff=fdist(ff,3,84)pff=0.245scalar lmm=obssmpl*eq_u2.r2 lmm=4.22scalar plmm=chisq(lmm,3) plmm=0.238Reject H0.Fail to reject H0. 2013 Cengage Learning. All Rights Reserved. May not be scanned, copi

28、ed or duplicated, or posted to a publicly accessible website, in whole or in part.White test for heteroscedasticityDisadvantage of this form of the White testIncluding all squares and interactions leads to a large number of esti-mated parameters (e.g. k=6 leads to 27 parameters to be estimated)Regre

29、ss squared residuals on all expla-natory variables, their squares, and in-teractions (here: example for k=3)The White test detects more general deviations from heteroscedasticity than the Breusch-Pagan testChapter 8 Multiple Regression Analysis: Heteroscedasticity8.3 Testing for Heteroskedasticity (

30、5/7)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.Alternative form of the White testExample: Heteroscedasticity in (log) housing price equationsThis regression indirectly tests the depende

31、nce of the squared residuals on the explanatory variables, their squares, and interactions, because the predicted value of y and its square implicitly contain all of these terms. Chapter 8 Multiple Regression Analysis: Heteroscedasticity8.3 Testing for Heteroskedasticity (6/7)Chapter222,32 or nuFFLM

32、n R 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 8 Multiple Regression Analysis: Heteroscedasticity8.3 Testing for Heteroskedasticity (7/7)ChapterExample 8.5 Special Form of the White Tes

33、t in the Log Housing Price Equationequation eq_lny.ls log(price) c log(lotsize) log(sqrft) bdrmsseries u2=resid2series lnyf=log(price)-residequation eq_u2.ls u2 c lnyf lnyf2scalar lm=obssmpl*eq_u2.r2 lm=3.45scalar plm=chisq(lm,2) plm=0.178We can view tests for heteroskedasticity as general misspecif

34、ication tests. But it is better to test for functional form first, and then test for heteroskedasticity. 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 8 Multiple Regression Analysis: Heter

35、oscedasticity8.4 Weighted Least Squares EstimationChapter8.4.1 The Heteroskedasticity Is Known up to a Multiplicative Constant8.4.2 The Heteroskedasticity Function Must Be Estimated: Feasible GLS8.4.3 What If the Assumed Heteroskedasticity Function Is Wrong?8.4.4 Point Prediction and Prediction Inte

36、rvals with Heteroskedasticity 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.Transformed modelThe functional form of the heteroscedasticity is knownChapter 8 Multiple Regression Analysis: Heterosce

37、dasticity8.4.1 The Heteroskedasticity Is Known up to a Multiplicative Constant (1/6)The EndChapter 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.Example: Savings and incomeThe transformed model is

38、 homoscedasticIf the other Gauss-Markov assumptions hold as well, OLS applied to the transformed model is the best linear unbiased estimator!Note that this regression model has no interceptChapter 8 Multiple Regression Analysis: HeteroscedasticityThe EndChapter8.4.1 The Heteroskedasticity Is Known u

39、p to a Multiplicative Constant (2/6) 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.OLS in the transformed model is weighted least squares (WLS)WLS is a special case of generalized least squares (G

40、LS)Chapter 8 Multiple Regression Analysis: HeteroscedasticityThe EndChapter8.4.1 The Heteroskedasticity Is Known up to a Multiplicative Constant (3/6)Observations with a large variance get a smaller weight in the optimization problem 2013 Cengage Learning. All Rights Reserved. May not be scanned, co

41、pied or duplicated, or posted to a publicly accessible website, in whole or in part.Example 8.6: Financial wealth equation (401ksubs.wf1)WLS estimates have considerably smaller standard errors (which is line with the expectation that they are more efficient).Assumed form of heteroscedasticity:Net fi

42、nancial wealthParticipation in 401K pension plan Chapter 8 Multiple Regression Analysis: HeteroscedasticityThe EndChapter8.4.1 The Heteroskedasticity Is Known up to a Multiplicative Constant (4/6) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a pu

43、blicly accessible website, in whole or in part.Chapter 8 Multiple Regression Analysis: HeteroscedasticityThe EndChapter8.4.1 The Heteroskedasticity Is Known up to a Multiplicative Constant (5/6)Example 8.6 Financial Wealth Equation (cont.)401ksubs.wf1smpl if fsize=1ls(h) nettfa inc cls(w=1/sqrt(inc)

44、 nettfa inc cls(h) nettfa inc (age-25)2 male e401k cls(w=1/sqrt(inc) nettfa inc (age-25)2 male e401k cSome WLS estimates are substantially below OLS estimates and suggests a misspecification of the functional form. 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated,

45、 or posted to a publicly accessible website, in whole or in part.Important special case of heteroscedasticityIf the observations are reported as averages at the city/county/state/-country/firm level, they should be weighted by the size of the unitAverage contribution to pension plan in firm iAverage

46、 earnings and age in firm iPercentage firm contributes to planHeteroscedastic error termError variance if errors are homoscedastic at the employee levelIf errors are homoscedastic at the employee level, WLS with weights equal to firm size mi should be used. If the assumption of homoscedasticity at t

47、he employee level is not exactly right, one can calculate robust standard errors after WLS (i.e. for the transformed model).Chapter 8 Multiple Regression Analysis: HeteroscedasticityThe EndChapter8.4.1 The Heteroskedasticity Is Known up to a Multiplicative Constant (6/6)Individual-level:Firm-level:

48、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.Unknown heteroscedasticity function (feasible GLS)A particular, fairly flexible approachFeasible GLS is consistent and asymptotically more efficient t

49、han OLS.Multiplicative error (assumption: independent of the explanatory variables)Use inverse values of the estimated heteroscedasticity funtion as weights in WLSChapter 8 Multiple Regression Analysis: Heteroscedasticity8.4.2 The Heteroskedasticity Function Must Be Estimated: Feasible GLS (1/4)The

50、EndChapterAssumed general form of heteroscedasticity; exp-function is used to ensure positivity 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.Example 8.7: Demand for cigarettesEstimation by OLSCig

51、arettes smoked per dayLogged income and cigarette priceReject homo-scedasticitySmoking restrictions in restaurantsChapter 8 Multiple Regression Analysis: HeteroscedasticityThe EndChapter8.4.2 The Heteroskedasticity Function Must Be Estimated: Feasible GLS (2/4) 2013 Cengage Learning. All Rights Rese

52、rved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Estimation by FGLSDiscussionThe income elasticity is now statistically significant; other coefficients are also more precisely estimated (without changing qualit. results)Now statistically

53、 significantChapter 8 Multiple Regression Analysis: HeteroscedasticityThe EndChapter8.4.2 The Heteroskedasticity Function Must Be Estimated: Feasible GLS (3/4) 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole o

54、r in part.Chapter 8 Multiple Regression Analysis: HeteroscedasticityThe EndChapter8.4.2 The Heteroskedasticity Function Must Be Estimated: Feasible GLS (4/4)Example 8.7 Demand for Cigarettes (cont.) (smoke.wf1)equation eq_y.ls cigs c log(income) log(cigpric) educ age age2 restaurntest for heterosked

55、asticityseries u2=resid2equation eq_u2.ls u2 c log(income) log(cigpric) educ age age2 restaurnscalar lm=807*eq_u2.r2 lm=32scalar p=chisq(lm,6) p=0.000015 feasible GLS procedureequation eq_g.ls log(u2) c log(income) log(cigpric) educ age age2 restaurnseries ghat=log(u2)-residseries hhat=exp(ghat)equa

56、tion eq_fgls.ls(w=1/sqrt(hhat) cigs c log(income) log(cigpric) educ age age2 restaurn 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.What if the assumed heteroscedasticity function is wrong?If the

57、heteroscedasticity function is misspecified, WLS is still consistent under MLR.1 MLR.4, but robust standard errors should be computedWLS is consistent under MLR.4 but not necessarily under MLR.4If OLS and WLS produce very different estimates, this typically indi-cates that some other assumptions (e.

58、g. MLR.4) are wrongIf there is strong heteroscedasticity, it is still often better to use a wrong form of heteroscedasticity in order to increase efficiencyChapter 8 Multiple Regression Analysis: Heteroscedasticity8.4.3 What If the Assumed Heteroskedasticity Function Is Wrong? (1/3)The EndChapter 20

59、13 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 8 Multiple Regression Analysis: Heteroscedasticity8.4.3 What If the Assumed Heteroskedasticity Function Is Wrong? (2/3)The EndChapterOLS with ro

60、bust S.E.WLS withNonrobust S.E.WLS withRobust S.E. 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 8 Multiple Regression Analysis: Heteroscedasticity8.4.3 What If the Assumed Heteroskedastic