多元回归分析假设检验.ppt

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1、多元回归分析假设检验1现在学习的是第1页，共42页Assumptions of the Classical Linear Model(CLM)nSo far,we know that given the Gauss-Markov assumptions,OLS is BLUE,nIn order to do classical hypothesis testing,we need to add another assumption(beyond the Gauss-Markov assumptions)nAssume that u is independent of x1,x2,xk and

2、u is normally distributed with zero mean and variance s s2:u Normal(0,s2)2现在学习的是第2页，共42页CLM Assumptions(cont.)nUnder CLM,OLS is not only BLUE,but is the minimum variance unbiased estimator,which means that OLS has the smallest variance among unbiased estimators;we no longer have to restrict our comp

3、arison to estimators that are linear in yi.nWe can summarize the population assumptions of CLM as followsny|x Normal(b0+b1x1+bkxk,s2)nWhile for now we just assume normality,clear that sometimes not the casenLarge samples will let us drop normality3现在学习的是第3页，共42页.x1x2The homoskedastic normal distribu

4、tion with a single explanatory variableE(y|x)=b0+b1xyf(y|x)Normaldistributions4现在学习的是第4页，共42页Normal Sampling Distributions 2222j1,1errors theofn combinatiolinear a isit becausenormally ddistribute is 0,1Normal thatso,Normal st variableindependen theof valuessample theon lconditiona s,assumption CLM

5、Under thejjjjjjjjjjjjRSSTsdRSSTVarsdVarsbsbbbbbbbb5现在学习的是第5页，共42页The t Test 1,11:freedom of degrees theNoteby estimate tohave webecausenormal)(vson distributi a is thisNote sassumption CLM Under the2222221knuRSSTseknttseijjjknjjjssbssbbb6现在学习的是第6页，共42页The t Test(cont)nKnowing the sampling distributi

6、on for the standardized estimator allows us to carry out hypothesis testsnStart with a null hypothesisnFor example,H0:bj=0nIf accept null,then accept that xj has no effect on y,controlling for other xs7现在学习的是第7页，共42页The t Test(cont)0jH,hypothesis null accept theo whether tdetermine torulerejection a

7、 withalong statistic our use then willWe:for statistic theform toneedfirst eour test w perform Totsettjjjbbbb8现在学习的是第8页，共42页t Test:One-Sided Alternativesn Besides our null,H0,we need an alternative hypothesis,H1,and a significance leveln H1 may be one-sided,or two-sidedn H1:bj 0 and H1:bj 0One-Sided

8、 Alternatives(cont)0ca1 aFail to rejectreject abbabbbb1orcsePctPcsePctPjjjjjj11现在学习的是第11页，共42页An Example:Hourly Wage EquationnWage determination:(wooldridge,p123)nlog(wge)=0.284+0.092educ+0.0041exper+0.022tenuren (0.104)(0.007)(0.0017)(0.003)n n=526 R2=0.316nWhether the return to exper,controlling f

9、or educ and tenure,is zero in the population,against the alternative that it is positive.nH0:bexper=0 vs.H1:bexper 0nThe t statistic is t=0.0041/0.00172.41nThe degree of freedom:df=n-k-1=526-3-1=522nThe critical value of 5%is 1.645nAnd the t statistic is larger than the critical value,ie.,2.411.645n

10、That is,we will reject the null hypothesis and bexper is really positive.01.6451 aFail to reject5%reject12现在学习的是第12页，共42页Another example:Student Performance and School SizenWhether the school size has effect on student performance?qmath10,math test scores,reveal the student performanceqtotcomp,avera

11、ge annual teacher compensationqstaff,the number of staff per one thousand studentsqenroll,student enrollment,reveal the school size.nThe Model Equationqmath10=b0+b1totcomp+b2staff+b3enrollqH0:benroll=0,H1:benroll-1.645,so we cant reject the null hypothesis.-1.645reject-09113现在学习的是第13页，共42页One-sided

12、vs Two-sidednBecause the t distribution is symmetric,testing H1:bj 0 is straightforward.The critical value is just the negative of beforenWe can reject the null if the t statistic than c then we fail to reject the nullnFor a two-sided test,we set the critical value based on a/2 and reject H1:bj 0 if

13、 the absolute value of the t statistic c14现在学习的是第14页，共42页yi =b0 +b1Xi1 +bkXik+uiH0:bj=0 H1:bj 0c0a/21 a-ca/2Two-Sided Alternativesrejectrejectfail to reject abbabbbb1orcsePctPcsePctPjjjjjj15现在学习的是第15页，共42页Summary for H0:bj=0nUnless otherwise stated,the alternative is assumed to be two-sidednIf we re

14、ject the null,we typically say“xj is statistically significant at the a%level”nIf we fail to reject the null,we typically say“xj is statistically insignificant at the a%level”16现在学习的是第16页，共42页An Example:Determinants of College GPA(wooldridge,p128)nVariables:qcolGPA,college GPAqskipped,the average nu

15、mber of lectures missed per weekqACT,achievement test scoreqhsGPA,high school GPAnThe estimated modelqolGPA=1.39+0.412 hsGPA+0.015 ACT 0.083 skippedq (0.33)(0.094)(0.011)(0.026)q n=141,R2=0.234nH0:bskipped=0,H1:bskipped 0nfd:n-k-1=137,the critical value t137=1.96nThe t statistic is|-0.083/0.026|=3.1

16、9 t137=1.96,so we will reject the null hypothesis and the bskipped is signanificantly beyond zero.-1.96reject-3.191.96reject17现在学习的是第17页，共42页Testing other hypothesesnA more general form of the t statistic recognizes that we may want to test something like H0:bj=aj nIn this case,the appropriate t sta

17、tistic is teststandard for the 0 where,jjjjaseatbb18现在学习的是第18页，共42页An Example:Campus Crime and Enrollment(wooldridge,p129)lVariableslcrime,the annual number of crimes on college campuseslenroll,student enrollment,reveal the size of college.lThe regression modelllog(crime)=b0+b1log(enroll)+ulWhether

18、b1=1,that is H0:b1=1,H1:b1 1llog(crime)=6.63+1.27 log(enroll)l (1.03)(0.11)n=97 R2=0.585ldf:n-k-1=95,the critical value at 5%is t95=1.645lThe t-statistic is(1.27-1)/0.112.45t95=1.645lSo we reject the null hypothesis and the evidence prove that b1 1.19现在学习的是第19页，共42页An Example:House Prices and Air Po

19、llution(wooldridge,p131)nVariablesqprice,median housing price;qnox,the amount of nitrogen oxide in the air,in parts per million;qdist,a weighted distance of the community from five employment centers,in miles;qrooms,the average number of rooms in houses in the communityqStratio,the average student-t

20、eacher ratio of schools in the community.nThe estimated modelqlog(prie)=11.08-0.954log(nox)-0.134log(dis)+0.255rooms-0.052stratioq (0.32)(0.117)(0.043)(0.019)(0.006)q n=506 R2=581nThe null hypothesis H0:blog(nox)=-1,H1:blog(nox)-1nThe t statistic is(-0.954-(-1)/0.117=0.393,and the critical value t=1

21、.96.0.393|t|”and“95%Conf.Interval”,respectively23现在学习的是第23页，共42页Testing a Linear Combinationn Suppose instead of testing whether b1 is equal to a constant,you want to test if it is equal to another parameter,that is H0:b1=b2,or b1-b2=0n Use same basic procedure for forming a t statistic2121bbbbset24

22、现在学习的是第24页，共42页Testing Linear Combination(cont)211221122221212121212121,of estimatean is where2,2 then,SincebbbbbbbbbbbbbbbbCovssseseseCovVarVarVarVarse25现在学习的是第25页，共42页Testing a Linear Combo(cont)nSo,to use formula,need s12,which standard output does not havenMany packages will have an option to ge

23、t it,or will just perform the test for younIn Stata,after reg y x1 x2 xk you would type test x1=x2 to get a p-value for the testnMore generally,you can always restate the problem to get the test you want26现在学习的是第26页，共42页Example:nSuppose you are interested in the effect of campaign expenditures on ou

24、tcomesnModel is voteA=b0+b1log(expendA)+b2log(expendB)+b3prtystrA+unH0:b1=-b2,or H0:q1=b1+b2=0nb1=q1 b2,so substitute in and rearrange voteA=b0+q1log(expendA)+b2log(expendB-expendA)+b3prtystrA+u27现在学习的是第27页，共42页Example(cont):nThis is the same model as originally,but now you get a standard error for

25、b1 b2=q1 directly from the basic regressionnAny linear combination of parameters could be tested in a similar mannernOther examples of hypotheses about a single linear combination of parameters:qb1=1+b2;b1=5b2;b1=-1/2b2;etc 28现在学习的是第28页，共42页Multiple Linear RestrictionsnEverything weve done so far ha

26、s involved testing a single linear restriction,(e.g.b1=0 or b1=b2)nHowever,we may want to jointly test multiple hypotheses about our parametersnA typical example is testing“exclusion restrictions”we want to know if a group of parameters are all equal to zero29现在学习的是第29页，共42页Testing Exclusion Restric

27、tionsnNow the null hypothesis might be something like H0:bk-q+1=0,.,bk=0nThe alternative is just H1:H0 is not truenCant just check each t statistic separately,because we want to know if the q parameters are jointly significant at a given level it is possible for none to be individually significant a

28、t that level30现在学习的是第30页，共42页Exclusion Restrictions(cont)nTo do the test we need to estimate the“restricted model”without xk-q+1,xk included,as well as the“unrestricted model”with all xs includednIntuitively,we want to know if the change in SSR is big enough to warrant inclusion of xk-q+1,xk edunres

29、trict isur and restricted isr where,1knSSRqSSRSSRFururr31现在学习的是第31页，共42页The F statisticnThe F statistic is always positive,since the SSR from the restricted model cant be less than the SSR from the unrestrictednEssentially the F statistic is measuring the relative increase in SSR when moving from th

30、e unrestricted to restricted modeln q=number of restrictions,or dfr dfurn n k 1=dfur32现在学习的是第32页，共42页The F statistic(cont)nTo decide if the increase in SSR when we move to a restricted model is“big enough”to reject the exclusions,we need to know about the sampling distribution of our F statnNot surp

31、risingly,F Fq,n-k-1,where q is referred to as the numerator degrees of freedom and n k 1 as the denominator degrees of freedom 33现在学习的是第33页，共42页0c1 af(F)FThe F statistic(cont)rejectfail to rejectReject H0 at a significance level if F ca34现在学习的是第34页，共42页Example:the determinations of league baseball p

32、layers salaries(wooldridge,p143)nThe regression modelqlog(salary)=b0+b1year+b2gamesyr+b3bavg+b4hrunsyr+b5rbisyr+uqsalary,the 1993 total salaryqyears,years in the leagueqgamesyr,average games played per yearqbavg,career batting averageqhrunsyr,home runs per yearqrbisyr,runs battled in per yearnThe nu

33、ll hypothesis is H0:b30,b40,b50,which is called multiple hypotheses test or joint hypotheses test.The alternative hypothesis is H1:H0 is not true.The unrestricted model:qlog(salary)=11.19+0.0689year+0.0126gamesyr+0.00098bavg+0.0144hrunsyr+0.0108rbisyrq (0.29)(0.0689)(0.0026)(0.00110)(0.0161)(0.0072)

34、q n=353,SSR=183.186,R2=0.6278nThe restricted modelqlog(salary)=11.22+0.0713year+0.0202gamesyrq (0.11)(0.0125)(0.0013)q n=353,SSR=198.311,R2=0.597135现在学习的是第35页，共42页Example:the determinations of league baseball players salaries,cont.nThe restricted number and the degree of the freedom of restricted mo

35、del is q=3;nThe degree of freedom of unrestricted model is 353-5-1=347;nThen the F statistic ishypothesis null reject thecan we53.855.9 and.53.8is 5%at 347 and 3 freedom of valuecritical theand55.9347/186.1863/186.186-198.3111347,3FknSSRqSSRSSRFururr36现在学习的是第36页，共42页The R2 form of the F statisticnBe

36、cause the SSRs may be large and unwieldy,an alternative form of the formula is usefulnWe use the fact that SSR=SST(1 R2)for any regression,so can substitute in for SSRu and SSRuredunrestrict isur and restricted isr again where,11222knRqRRFurrur37现在学习的是第37页，共42页Example:Parents Education in a Birth We

37、ight Equation(wooldridge,p150)nVariablesqbwght,birth weight in pounds;qcigs,average number of cigarettes the mother smoked per day during pregnancy;qparity,the birth order of this child;qfaminc,annual family income;qmotheduc,years of schooling for the mother;qfatheduc,years of schooling for the fath

38、er.nModel:bwght=b0+b1cigs+b2parity+b3faminc+b4motheduc+b5fatheduc+unWhether the parents education has any effect on birth weight?This is stated as H0:b4=0,b5=0,so q=2.qbwght=114.5240.596cigs+1.788parity+0.056faminc-0.370motheduc+0.472fatheducq (3.728)(0.110)(0.659)(0.037)(0.320)(0.283)q n=1191 R2=0.

39、0387qbwght=115.4700.598cigs+1.832parity+0.067famincq (1.656)(0.109)(0.658)(0.032)q n=1191 R2=0.0364nF Statistic is F=(0.0387-0.0364)/2/(1-0.0387)/(1191-5-1)=1.42 F2,1185=19.5qWe fail to reject H0.In other words,motheduc and fatheduc are jointly insignificant in the birth weight equation38现在学习的是第38页，

40、共42页Overall SignificancenA special case of exclusion restrictions is to test H0:b1=b2=bk=0nSince the R2 from a model with only an intercept will be zero,the F statistic is simply1122knRkRF39现在学习的是第39页，共42页General Linear Restrictionsn The basic form of the F statistic will work for any set of linear

41、restrictionsn First estimate the unrestricted model and then estimate the restricted modeln In each case,make note of the SSRn Imposing the restrictions can be tricky will likely have to redefine variables again40现在学习的是第40页，共42页Example:nUse same voting model as beforenModel is voteA=b0+b1log(expendA

42、)+b2log(expendB)+b3prtystrA+unnow null is H0:b1=1,b3=0nSubstituting in the restrictions:voteA=b0+log(expendA)+b2log(expendB)+u,sonUse voteA-log(expendA)=b0+b2log(expendB)+u as restricted model41现在学习的是第41页，共42页F Statistic SummarynJust as with t statistics,p-values can be calculated by looking up the percentile in the appropriate F distributionnStata will do this by entering:display fprob(q,n k 1,F),where the appropriate values of F,q,and n k 1 are usednIf only one exclusion is being tested,then F=t2,and the p-values will be the same42现在学习的是第42页，共42页