计量经济学基础第6章.pptx

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1、Principles of Econometrics,4th EditionPage 1Chapter 6:Further Inference in the Multiple Regression ModelChapter 6Further Inference in the Multiple Regression ModelWalter R.Paczkowski Rutgers UniversityPrinciples of Econometrics,4th EditionPage 2Chapter 6:Further Inference in the Multiple Regression

2、Model6.1 Joint Hypothesis Testing6.2 The Use of Nonsample Information6.3 Model Specification6.4 Poor Data,Collinearity,and Insignificance6.5 PredictionChapter ContentsPrinciples of Econometrics,4th EditionPage 3Chapter 6:Further Inference in the Multiple Regression ModelEconomists develop and evalua

3、te theories about economic behaviorHypothesis testing procedures are used to test these theoriesThe theories that economists develop sometimes provide nonsample information that can be used along with the information in a sample of data to estimate the parameters of a regression modelA procedure tha

4、t combines these two types of information is called restricted least squaresPrinciples of Econometrics,4th EditionPage 4Chapter 6:Further Inference in the Multiple Regression Model6.1 Joint Hypothesis TestingPrinciples of Econometrics,4th EditionPage 5Chapter 6:Further Inference in the Multiple Regr

5、ession ModelA null hypothesis with multiple conjectures,expressed with more than one equal sign,is called a joint hypothesis1.Example:Should a group of explanatory variables should be included in a particular model?2.Example:Does the quantity demanded of a product depend on the prices of substitute

6、goods,or only on its own price?6.1Joint Hypothesis Testing6.1Testing Joint HypothesesPrinciples of Econometrics,4th EditionPage 6Chapter 6:Further Inference in the Multiple Regression ModelBoth examples are of the form:The joint null hypothesis in Eq.6.1 contains three conjectures (three equal signs

7、):4=0,5=0,and 6=0A test of H0 is a joint test for whether all three conjectures hold simultaneouslyEq.6.16.1Joint Hypothesis Testing6.1Testing Joint HypothesesPrinciples of Econometrics,4th EditionPage 7Chapter 6:Further Inference in the Multiple Regression ModelConsider the model:Test whether or no

8、t advertising has an effect on sales but advertising is in the model as two variables6.1.1Testing the Effect of Advertising:The F-TestEq.6.26.1Joint Hypothesis TestingPrinciples of Econometrics,4th EditionPage 8Chapter 6:Further Inference in the Multiple Regression ModelAdvertising will have no effe

9、ct on sales if 3=0 and 4=0 Advertising will have an effect if 3 0 or 4 0 or if both 3 and 4 are nonzeroThe null hypotheses are:6.1Joint Hypothesis Testing6.1.1Testing the Effect of Advertising:The F-TestPrinciples of Econometrics,4th EditionPage 9Chapter 6:Further Inference in the Multiple Regressio

10、n ModelRelative to the null hypothesis H0:3=0,4=0 the model in Eq.6.2 is called the unrestricted modelThe restrictions in the null hypothesis have not been imposed on the modelIt contrasts with the restricted model,which is obtained by assuming the parameter restrictions in H0 are true6.1Joint Hypot

11、hesis Testing6.1.1Testing the Effect of Advertising:The F-TestPrinciples of Econometrics,4th EditionPage 10Chapter 6:Further Inference in the Multiple Regression ModelWhen H0 is true,3=0 and 4=0,and ADVERT and ADVERT2 drop out of the modelThe F-test for the hypothesis H0:3=0,4=0 is based on a compar

12、ison of the sums of squared errors(sums of squared least squares residuals)from the unrestricted model in Eq.6.2 and the restricted model in Eq.6.3 Shorthand notation for these two quantities is SSEU and SSER,respectivelyEq.6.36.1Joint Hypothesis Testing6.1.1Testing the Effect of Advertising:The F-T

13、estPrinciples of Econometrics,4th EditionPage 11Chapter 6:Further Inference in the Multiple Regression ModelThe F-statistic determines what constitutes a large reduction or a small reduction in the sum of squared errorswhere J is the number of restrictions,N is the number of observations and K is th

14、e number of coefficients in the unrestricted modelEq.6.46.1Joint Hypothesis Testing6.1.1Testing the Effect of Advertising:The F-TestPrinciples of Econometrics,4th EditionPage 12Chapter 6:Further Inference in the Multiple Regression ModelIf the null hypothesis is true,then the statistic F has what is

15、 called an F-distribution with J numerator degrees of freedom and N-K denominator degrees of freedomIf the null hypothesis is not true,then the difference between SSER and SSEU becomes largeThe restrictions placed on the model by the null hypothesis significantly reduce the ability of the model to f

16、it the data6.1Joint Hypothesis Testing6.1.1Testing the Effect of Advertising:The F-TestPrinciples of Econometrics,4th EditionPage 13Chapter 6:Further Inference in the Multiple Regression ModelThe F-test for our sales problem is:1.Specify the null and alternative hypotheses:The joint null hypothesis

17、is H0:3=0,4=0.The alternative hypothesis is H0:3 0 or 4 0 both are nonzero2.Specify the test statistic and its distribution if the null hypothesis is true:Having two restrictions in H0 means J=2 Also,recall that N=75:6.1Joint Hypothesis Testing6.1.1Testing the Effect of Advertising:The F-TestPrincip

18、les of Econometrics,4th EditionPage 14Chapter 6:Further Inference in the Multiple Regression ModelThe F-test for our sales problem is(Continued):3.Set the significance level and determine the rejection region4.Calculate the sample value of the test statistic and,if desired,the p-valueThe correspondi

19、ng p-value is p=P(F(2,71)8.44)=0.00056.1Joint Hypothesis Testing6.1.1Testing the Effect of Advertising:The F-TestPrinciples of Econometrics,4th EditionPage 15Chapter 6:Further Inference in the Multiple Regression ModelThe F-test for our sales problem is(Continued):5.State your conclusionSince F=8.44

20、 Fc=3.126,we reject the null hypothesis that both 3=0 and 4=0,and conclude that at least one of them is not zeroAdvertising does have a significant effect upon sales revenue6.1Joint Hypothesis Testing6.1.1Testing the Effect of Advertising:The F-TestPrinciples of Econometrics,4th EditionPage 16Chapte

21、r 6:Further Inference in the Multiple Regression ModelConsider again the general multiple regression model with(K-1)explanatory variables and K unknown coefficientsTo examine whether we have a viable explanatory model,we set up the following null and alternative hypotheses:6.1.2Testing the Significa

22、nce of the ModelEq.6.5Eq.6.66.1Joint Hypothesis TestingPrinciples of Econometrics,4th EditionPage 17Chapter 6:Further Inference in the Multiple Regression ModelSince we are testing whether or not we have a viable explanatory model,the test for Eq.6.6 is sometimes referred to as a test of the overall

23、 significance of the regression model.Given that the t-distribution can only be used to test a single null hypothesis,we use the F-test for testing the joint null hypothesis in Eq.6.6 6.1Joint Hypothesis Testing6.1.2Testing the Significance of the ModelPrinciples of Econometrics,4th EditionPage 18Ch

24、apter 6:Further Inference in the Multiple Regression ModelThe unrestricted model is that given in Eq.6.5The restricted model,assuming the null hypothesis is true,becomes:Eq.6.76.1Joint Hypothesis Testing6.1.2Testing the Significance of the ModelPrinciples of Econometrics,4th EditionPage 19Chapter 6:

25、Further Inference in the Multiple Regression ModelThe least squares estimator of 1 in this restricted model is:The restricted sum of squared errors from the hypothesis Eq.6.6 is:6.1Joint Hypothesis Testing6.1.2Testing the Significance of the ModelPrinciples of Econometrics,4th EditionPage 20Chapter

26、6:Further Inference in the Multiple Regression ModelThus,to test the overall significance of a model,but not in general,the F-test statistic can be modified and written as:Eq.6.86.1Joint Hypothesis Testing6.1.2Testing the Significance of the ModelPrinciples of Econometrics,4th EditionPage 21Chapter

27、6:Further Inference in the Multiple Regression ModelFor our problem,note:1.We are testing:2.If H0 is true:6.1Joint Hypothesis Testing6.1.2Testing the Significance of the ModelPrinciples of Econometrics,4th EditionPage 22Chapter 6:Further Inference in the Multiple Regression ModelFor our problem,note

28、(Continued):3.Using a 5%significance level,we find the critical value for the F-statistic with(3,71)degrees of freedom is Fc=2.734.Thus,we reject H0 if F 2.734.4.The required sums of squares are SST=3115.482 and SSE=1532.084 which give an F-value of:p-value =P(F 24.459)=0.00006.1Joint Hypothesis Tes

29、ting6.1.2Testing the Significance of the ModelPrinciples of Econometrics,4th EditionPage 23Chapter 6:Further Inference in the Multiple Regression ModelFor our problem,note(Continued):5.Since 24.459 2.734,we reject H0 and conclude that the estimated relationship is a significant oneNote that this con

30、clusion is consistent with conclusions that would be reached using separate t-tests for the significance of each of the coefficients6.1Joint Hypothesis Testing6.1.2Testing the Significance of the ModelPrinciples of Econometrics,4th EditionPage 24Chapter 6:Further Inference in the Multiple Regression

31、 ModelWe used the F-test to test whether 3=0 and 4=0 in:Suppose we want to test if PRICE affects SALES6.1.3Relationship Between the t-and F-TestsEq.6.9Eq.6.106.1Joint Hypothesis TestingPrinciples of Econometrics,4th EditionPage 25Chapter 6:Further Inference in the Multiple Regression ModelThe F-valu

32、e for the restricted model is:The 5%critical value is Fc=F(0.95,1,71)=3.976 We reject H0:2=06.1Joint Hypothesis Testing6.1.3Relationship Between the t-and F-TestsPrinciples of Econometrics,4th EditionPage 26Chapter 6:Further Inference in the Multiple Regression ModelUsing the t-test:The t-value for

33、testing H0:2=0 against H1:2 0 is t=7.640/1.045939=7.30444Its square is t=(7.30444)2=53.355,identical to the F-value6.1Joint Hypothesis Testing6.1.3Relationship Between the t-and F-TestsPrinciples of Econometrics,4th EditionPage 27Chapter 6:Further Inference in the Multiple Regression ModelThe elemen

34、ts of an F-test1.The null hypothesis H0 consists of one or more equality restrictions on the model parameters k 2.The alternative hypothesis states that one or more of the equalities in the null hypothesis is not true3.The test statistic is the F-statistic in(6.4)4.If the null hypothesis is true,F h

35、as the F-distribution with J numerator degrees of freedom and N-K denominator degrees of freedom5.When testing a single equality null hypothesis,it is perfectly correct to use either the t-or F-test procedure:they are equivalent6.1Joint Hypothesis Testing6.1.3Relationship Between the t-and F-TestsPr

36、inciples of Econometrics,4th EditionPage 28Chapter 6:Further Inference in the Multiple Regression ModelThe conjectures made in the null hypothesis were that particular coefficients are equal to zero The F-test can also be used for much more general hypotheses Any number of conjectures(K)involving li

37、near hypotheses with equal signs can be tested6.1.4More General F-Tests6.1Joint Hypothesis TestingPrinciples of Econometrics,4th EditionPage 29Chapter 6:Further Inference in the Multiple Regression ModelConsider the issue of testing:If ADVERT0=$1,900 per month,then:orEq.6.116.1Joint Hypothesis Testi

38、ng6.1.4More General F-TestsPrinciples of Econometrics,4th EditionPage 30Chapter 6:Further Inference in the Multiple Regression ModelNote that when H0 is true,3=1 3.84 so that:orEq.6.126.1Joint Hypothesis Testing6.1.4More General F-TestsPrinciples of Econometrics,4th EditionPage 31Chapter 6:Further I

39、nference in the Multiple Regression ModelThe calculated value of the F-statistic is:For =0.05,the critical value is Fc=3.976 Since F=0.9362 Fc=3.976,we do not reject H0 We conclude that an advertising expenditure of$1,900 per month is optimal is compatible with the data6.1Joint Hypothesis Testing6.1

40、.4More General F-TestsPrinciples of Econometrics,4th EditionPage 32Chapter 6:Further Inference in the Multiple Regression ModelThe t-value is t=0.9676F=0.9362 is equal to t2=(0.9676)2The p-values are identical:6.1Joint Hypothesis Testing6.1.4More General F-TestsPrinciples of Econometrics,4th Edition

41、Page 33Chapter 6:Further Inference in the Multiple Regression ModelSuppose we have:In this case,we can no longer use the F-testBecause F=t2,the F-test cannot distinguish between the left and right tails as is needed for a one-tail testWe restrict ourselves to the t-distribution when considering alte

42、rnative hypotheses that have inequality signs such as 6.1.4aOne-tail TestEq.6.136.1Joint Hypothesis TestingPrinciples of Econometrics,4th EditionPage 34Chapter 6:Further Inference in the Multiple Regression ModelMost software packages have commands that will automatically compute t-and F-values and

43、their corresponding p-values when provided with a null hypothesisThese tests belong to a class of tests called Wald tests6.1.5Using Computer Software6.1Joint Hypothesis TestingPrinciples of Econometrics,4th EditionPage 35Chapter 6:Further Inference in the Multiple Regression ModelSuppose we conjectu

44、re that:We formulate the joint null hypothesis:Because there are J=2 restrictions to test jointly,we use an F-test A t-test is not suitable6.1Joint Hypothesis Testing6.1.5Using Computer SoftwarePrinciples of Econometrics,4th EditionPage 36Chapter 6:Further Inference in the Multiple Regression Model6

45、.2 The Use of Nonsample InformationPrinciples of Econometrics,4th EditionPage 37Chapter 6:Further Inference in the Multiple Regression ModelIn many estimation problems we have information over and above the information contained in the sample observations This nonsample information may come from man

46、y places,such as economic principles or experience When it is available,it seems intuitive that we should find a way to use it6.2The Use of Nonsample InformationPrinciples of Econometrics,4th EditionPage 38Chapter 6:Further Inference in the Multiple Regression ModelConsider the log-log functional fo

47、rm for a demand model for beer:This model is a convenient one because it precludes infeasible negative prices,quantities,and income,and because the coefficients 2,3,4,and 5 are elasticitiesEq.6.146.2The Use of Nonsample InformationPrinciples of Econometrics,4th EditionPage 39Chapter 6:Further Infere

48、nce in the Multiple Regression ModelA relevant piece of nonsample information can be derived by noting that if all prices and income go up by the same proportion,we would expect there to be no change in quantity demanded For example,a doubling of all prices and income should not change the quantity

49、of beer consumed This assumption is that economic agents do not suffer from money illusion6.2The Use of Nonsample InformationPrinciples of Econometrics,4th EditionPage 40Chapter 6:Further Inference in the Multiple Regression ModelHaving all prices and income change by the same proportion is equivale

50、nt to multiplying each price and income by a constant,say:Eq.6.156.2The Use of Nonsample InformationPrinciples of Econometrics,4th EditionPage 41Chapter 6:Further Inference in the Multiple Regression ModelTo have no change in ln(Q)when all prices and income go up by the same proportion,it must be tr