《計(jì)量經(jīng)濟(jì)學(xué)導(dǎo)論》ch課件

《《計(jì)量經(jīng)濟(jì)學(xué)導(dǎo)論》ch課件》由會(huì)員分享，可在線閱讀，更多相關(guān)《《計(jì)量經(jīng)濟(jì)學(xué)導(dǎo)論》ch課件（40頁珍藏版）》請(qǐng)?jiān)谘b配圖網(wǎng)上搜索。

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.Statistical inference in the regression modelHypothesis tests about population parametersConstruction of confidence intervals Sampling distributions of th

2、e OLS estimatorsThe OLS estimators are random variablesWe already know their expected values and their variancesHowever,for hypothesis tests we need to know their distributionIn order to derive their distribution we need additional assumptionsAssumption about distribution of errors:normal distributi

3、onMultiple RegressionAnalysis:Inference 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.Assumption MLR.6(Normality of error terms)independently ofIt is assumed that the unobservedfactors are normally dis

4、tributed around the population regression function.The form and the variance of the distribution does not depend onany of the explanatory variables.It follows that:Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a p

5、ublicly accessible website,in whole or in part.Discussion of the normality assumptionThe error term is the sum of many“different unobserved factorsSums of independent factors are normally distributed(CLT)Problems:How many different factors?Number large enough?Possibly very heterogenuous distribution

6、s of individual factorsHow independent are the different factors?The normality of the error term is an empirical questionAt least the error distribution should be close“to normalIn many cases,normality is questionable or impossible by definitionMultiple RegressionAnalysis:Inference 2013 Cengage Lear

7、ning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Discussion of the normality assumption(cont.)Examples where normality cannot hold:Wages(nonnegative;also:minimum wage)Number of arrests (takes on a small number of integer

8、values)Unemployment(indicator variable,takes on only 1 or 0)In some cases,normality can be achieved through transformations of the dependent variable(e.g.use log(wage)instead of wage)Under normality,OLS is the best(even nonlinear)unbiased estimatorImportant:For the purposes of statistical inference,

9、the assumption of normality can be replaced by a large sample sizeMultiple RegressionAnalysis:Inference 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.TerminologyTheorem 4.1(Normal sampling distribution

10、s)Under assumptions MLR.1 MLR.6:The estimators are normally distributed around the true parameters with the variance that was derived earlierThe standardized estimators follow a standard normal distributionGauss-Markov assumptions“Classical linear model(CLM)assumptions“Multiple RegressionAnalysis:In

11、ference 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.Testing hypotheses about a single population parameterTheorem 4.1(t-distribution for standardized estimators)Null hypothesis(for more general hypot

12、heses,see below)Under assumptions MLR.1 MLR.6:If the standardization is done using the estimated standard deviation(=standard error),the normal distribution is replaced by a t-distributionThe population parameter is equal to zero,i.e.after controlling for the other independent variables,there is no

13、effect of xj on y Note:The t-distribution is close to the standard normal distribution if n-k-1 is large.Multiple RegressionAnalysis:Inference 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.t-statistic(

14、or t-ratio)Distribution of the t-statistic if the null hypothesis is trueGoal:Define a rejection rule so that,if it is true,H0 is rejected only with a small probability(=significance level,e.g.5%)The t-statistic will be used to test the above null hypothesis.The farther the estimated coefficient is

15、away from zero,the less likely it is that the null hypothesis holds true.But what does far“away from zero mean?This depends on the variability of the estimated coefficient,i.e.its standard deviation.The t-statistic measures how many estimated standard deviations the estimated coefficient is away fro

16、m zero.Multiple RegressionAnalysis:Inference 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.Test against .Reject the null hypothesis in favour of the alternative hypothesis if the estimated coef-ficient

17、 is too large“(i.e.larger than a criti-cal value).Construct the critical value so that,if the null hypothesis is true,it is rejected in,for example,5%of the cases.In the given example,this is the point of the t-distribution with 28 degrees of freedom that is exceeded in 5%of the cases.!Reject if t-s

18、tatistic greater than 1.701Multiple RegressionAnalysis:InferenceTesting against one-sided alternatives(greater than zero)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:Wage equationTest whether,

19、after controlling for education and tenure,higher work experience leads to higher hourly wagesStandard errorsTest against .One would either expect a positive effect of experience on hourly wage or no effect at all.Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May no

20、t be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Wage equation(cont.)The effect of experience on hourly wage is statistically greater than zero at the 5%(and even at the 1%)significance level.“t-statisticDegrees of freedom;here the standard nor

21、mal approximation appliesCritical values for the 5%and the 1%significance level(these are conventional significance levels).The null hypothesis is rejected because the t-statistic exceeds the critical value.Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be sc

22、anned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Test against .Reject the null hypothesis in favour of the alternative hypothesis if the estimated coef-ficient is too small“(i.e.smaller than a criti-cal value).Construct the critical value so that,if the null

23、hypothesis is true,it is rejected in,for example,5%of the cases.In the given example,this is the point of the t-distribution with 18 degrees of freedom so that 5%of the cases are below the point.!Reject if t-statistic less than-1.734Multiple RegressionAnalysis:InferenceTesting against one-sided alte

24、rnatives(less than zero)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:Student performance and school sizeTest whether smaller school size leads to better student performanceTest against .Do lar

25、ger schools hamper student performance or is there no such effect?Percentage of studentspassing maths testAverage annual tea-cher compensationStaff per one thou-sand studentsSchool enrollment(=school size)Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be scan

26、ned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Student performance and school size(cont.)One cannot reject the hypothesis that there is no effect of school size on student performance(not even for a lax significance level of 15%).t-statisticDegrees of

27、 freedom;here the standard normal approximation appliesCritical values for the 5%and the 15%significance level.The null hypothesis is not rejected because the t-statistic is not smaller than the critical value.Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be

28、 scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Student performance and school size(cont.)Alternative specification of functional form:Test against .R-squared slightly higher Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights R

29、eserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Student performance and school size(cont.)The hypothesis that there is no effect of school size on student performance can be rejected in favor of the hypothesis that the effect is

30、negative.t-statisticCritical value for the 5%significance level!reject null hypothesisHow large is the effect?+10%enrollment !-0.129 percentage points students pass test(small effect)Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicat

31、ed,or posted to a publicly accessible website,in whole or in part.Testing against two-sided alternativesTest against .Reject the null hypothesis in favour of the alternative hypothesis if the absolute value of the estimated coefficient is too large.Construct the critical value so that,if the null hy

32、pothesis is true,it is rejected in,for example,5%of the cases.In the given example,these are the points of the t-distribution so that 5%of the caseslie in the two tails.!Reject if absolute value of t-statistic is less than-2.06 or greater than 2.06Multiple RegressionAnalysis:Inference 2013 Cengage L

33、earning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Example:Determinants of college GPALectures missed per weekThe effects of hsGPA and skipped are significantly different from zero at the 1%significance level.The effect

34、of ACT is not significantly different from zero,not even at the 10%significance level.For critical values,use standard normal distributionMultiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website

35、,in whole or in part.Statistically significant“variables in a regressionIf a regression coefficient is different from zero in a two-sided test,the corresponding variable is said to be statistically significant“If the number of degrees of freedom is large enough so that the nor-mal approximation appl

36、ies,the following rules of thumb apply:statistically significant at 10%level“statistically significant at 5%level“statistically significant at 1%level“Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly acces

37、sible website,in whole or in part.Guidelines for discussing economic and statistical significanceIf a variable is statistically significant,discuss the magnitude of the coefficient to get an idea of its economic or practical importanceThe fact that a coefficient is statistically significant does not

38、 necessa-rily mean it is economically or practically significant!If a variable is statistically and economically important but has the wrong“sign,the regression model might be misspecified If a variable is statistically insignificant at the usual levels(10%,5%,1%),one may think of dropping it from t

39、he regressionIf the sample size is small,effects might be imprecisely estimated so that the case for dropping insignificant variables is less strongMultiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessib

40、le website,in whole or in part.Testing more general hypotheses about a regression coefficientNull hypothesist-statisticThe test works exactly as before,except that the hypothesized value is substracted from the estimate when forming the statisticHypothesized value of the coefficientMultiple Regressi

41、onAnalysis:Inference 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:Campus crime and enrollmentAn interesting hypothesis is whether crime increases by one percent if enrollment is increased by o

42、ne percentThe hypothesis is rejected at the 5%levelEstimate is different from one but is this difference statistically significant?Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in who

43、le or in part.Computing p-values for t-testsIf the significance level is made smaller and smaller,there will be a point where the null hypothesis cannot be rejected anymoreThe reason is that,by lowering the significance level,one wants to avoid more and more to make the error of rejecting a correct

44、H0The smallest significance level at which the null hypothesis is still rejected,is called the p-value of the hypothesis testA small p-value is evidence against the null hypothesis because one would reject the null hypothesis even at small significance levelsA large p-value is evidence in favor of t

45、he null hypothesisP-values are more informative than tests at fixed significance levelsMultiple RegressionAnalysis:Inference 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.How the p-value is computed(he

46、re:two-sided test)The p-value is the significance level at which one is indifferent between rejecting and not rejecting the null hypothesis.In the two-sided case,the p-value is thus the probability that the t-distributed variable takes on a larger absolute value than the realized value of the test s

47、tatistic,e.g.:From this,it is clear that a null hypothesis is rejected if and only if the corresponding p-value is smaller than the significance level.For example,for a significance level of 5%the t-statistic would not lie in the rejection region.value of test statisticThese would be the critical va

48、lues for a 5%significance levelMultiple RegressionAnalysis:Inference 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.Critical value oftwo-sided testConfidence intervalsSimple manipulation of the result i

49、n Theorem 4.2 implies thatInterpretation of the confidence intervalThe bounds of the interval are randomIn repeated samples,the interval that is constructed in the above way will cover the population regression coefficient in 95%of the cases Lower bound of the Confidence intervalUpper bound of the C

50、onfidence intervalConfidence levelMultiple RegressionAnalysis:Inference 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.Confidence intervals for typical confidence levelsRelationship between confidence i

51、ntervals and hypotheses tests reject in favor of Use rules of thumbMultiple RegressionAnalysis:Inference 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:Model of firms R&D expendituresSpending on

52、 R&DAnnual salesProfits as percentage of salesThe effect of sales on R&D is relatively precisely estimated as the interval is narrow.Moreover,the effect is significantly different from zero because zero is outside the interval.This effect is imprecisely estimated as the in-terval is very wide.It is

53、not even statisticallysignificant because zero lies in the interval.Multiple RegressionAnalysis:Inference 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.Testing hypotheses about a linear combination of

54、parametersExample:Return to education at 2 year vs.at 4 year collegesYears of education at 2 year collegesYears of education at 4 year collegesTest against .A possible test statistic would be:The difference between the estimates is normalized by the estimated standard deviation of the difference.The

55、 null hypothesis would have to be rejected if the statistic is too negative“to believe that the true difference between the parameters is equal to zero.Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly acce

56、ssible website,in whole or in part.Insert into original regressionImpossible to compute with standard regression output becauseAlternative methodUsually not available in regression outputDefine and test against .a new regressor(=total years of college)Multiple RegressionAnalysis:Inference 2013 Cenga

57、ge Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Estimation resultsThis method works always for single linear hypothesesTotal years of collegeHypothesis is rejected at 10%level but not at 5%levelMultiple Regression

58、Analysis:Inference 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.Testing multiple linear restrictions:The F-testTesting exclusion restrictions Years in the leagueAverage number of games per yearSalary

59、of major lea-gue base ball playerBatting averageHome runs per yearRuns batted in per yearagainstTest whether performance measures have no effect/can be exluded from regression.Multiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or p

60、osted to a publicly accessible website,in whole or in part.Estimation of the unrestricted modelNone of these variabels is statistically significant when tested individuallyIdea:How would the model fit be if these variables were dropped from the regression?Multiple RegressionAnalysis:Inference 2013 C

61、engage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible website,in whole or in part.Estimation of the restricted modelTest statisticThe sum of squared residuals necessarily increases,but is the increase statistically significant?The relative inc

62、rease of the sum of squared residuals when going fromH1 to H0 follows a F-distribution(ifthe null hypothesis H0 is correct)Number of restrictionsMultiple RegressionAnalysis:Inference 2013 Cengage Learning.All Rights Reserved.May not be scanned,copied or duplicated,or posted to a publicly accessible

63、website,in whole or in part.Rejection rule(Figure 4.7)A F-distributed variable only takes on positive values.This corresponds to the fact that the sum of squared residuals can only increase if one moves from H1 to H0.Choose the critical value so that the null hypo-thesis is rejected in,for example,5

64、%of the cases,although it is true.Multiple RegressionAnalysis:Inference 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.Test decision in exampleDiscussionThe three variables are jointly significant“They

65、were not significant when tested individuallyThe likely reason is multicollinearity between them Number of restrictions to be tested Degrees of freedom inthe unrestricted modelThe null hypothesis is overwhel-mingly rejected(even at very small significance levels).Multiple RegressionAnalysis:Inferenc

66、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.Test of overall significance of a regressionThe test of overall significance is reported in most regression packages;the null hypothesis is usually overwhelmingly rejectedThe null hypothesis states that the explanatory variables are not useful at all in explaining the dependent variableRestricted model(regression on constant)Multiple RegressionAnalysi