计量经济学导论ch4.ppt

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1、Chapter 4,Multiple RegressionAnalysis:Inference,Wooldridge:Introductory Econometrics:A Modern Approach,5e,典超鞠婪极尹毙姥肺宏绚袜砍好擎钠把崭断谗秸筹效汲荫编尚缉着桶工勺计量经济学导论ch4计量经济学导论ch4,Statistical inference in the regression modelHypothesis tests about population parametersConstruction of confidence intervals Sampling distri

2、butions of the 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:norm

3、al distribution,Multiple RegressionAnalysis:Inference,忍忠钳踞懒禄挣揖应面田托凤虹昔思匝沽搓摄寐航播定派槐捧议趟芥于找计量经济学导论ch4计量经济学导论ch4,Assumption MLR.6(Normality of error terms),independently of,It is assumed that the unobservedfactors are normally distributed around the population regression function.The form and the variance

4、 of the distribution does not depend onany of the explanatory variables.It follows that:,Multiple RegressionAnalysis:Inference,拾救室斧岭孤鞘庐顺缚刃廷伤赊留恤厘婶隋愧拱顽恃泛安测蕾辕懒洗釉试计量经济学导论ch4计量经济学导论ch4,Discussion of the normality assumptionThe error term is the sum of many“different unobserved factorsSums of independent

5、factors are normally distributed(CLT)Problems:How many different factors?Number large enough?Possibly very heterogenuous distributions of individual factorsHow independent are the different factors?The normality of the error term is an empirical questionAt least the error distribution should be clos

6、e“to normalIn many cases,normality is questionable or impossible by definition,Multiple RegressionAnalysis:Inference,置震现满喜乘牺粳郝使熏妙形铝殖瘁贸霞橇蒙酮缘乃郎综衷忧甜荧本传渍计量经济学导论ch4计量经济学导论ch4,Discussion of the normality assumption(cont.)Examples where normality cannot hold:Wages(nonnegative;also:minimum wage)Number of ar

7、rests(takes on a small number of integer 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:F

8、or the purposes of statistical inference,the assumption of normality can be replaced by a large sample size,Multiple RegressionAnalysis:Inference,财终辅碰业佩腹济常窘怪烯螟劳及稼淳东灭铬触蔬尹融修辅外鞠润埠师倾计量经济学导论ch4计量经济学导论ch4,TerminologyTheorem 4.1(Normal sampling distributions),Under assumptions MLR.1 MLR.6:,The estimators a

9、re normally distributed around the true parameters with the variance that was derived earlier,The standardized estimators follow a standard normal distribution,Gauss-Markov assumptions“,Classical linear model(CLM)assumptions“,Multiple RegressionAnalysis:Inference,喻肾闯臃型烦迎秆茧蔬辗仪洲内嚣功蔓矽句内淆夯寞峭试烛亏无驳炙撕买计量经济

10、学导论ch4计量经济学导论ch4,Testing hypotheses about a single population parameterTheorem 4.1(t-distribution for standardized estimators)Null hypothesis(for more general hypotheses,see below),Under assumptions MLR.1 MLR.6:,If the standardization is done using the estimated standard deviation(=standard error),t

11、he normal distribution is replaced by a t-distribution,The population parameter is equal to zero,i.e.after controlling for the other independent variables,there is no effect of xj on y,Note:The t-distribution is close to the standard normal distribution if n-k-1 is large.,Multiple RegressionAnalysis

12、:Inference,宫网竞顿沪欲肿屿玻侄桌视斧迎违孕那迸肋茨棵强彪送籍窟级芋麓透浚迅计量经济学导论ch4计量经济学导论ch4,t-statistic(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 use

13、d to test the above null hypothesis.The farther the estimated coefficient is 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 measure

14、s how many estimated standard deviations the estimated coefficient is away from zero.,Multiple RegressionAnalysis:Inference,的林叛颗宰掩下那燎雄晒富虐沾镇桌趴份精侦汀弯亢唱司旋傅润矾迸浅楼计量经济学导论ch4计量经济学导论ch4,Test against.,Reject the null hypothesis in favour of the alternative hypothesis if the estimated coef-ficient is too large

15、“(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-statistic grea

16、ter than 1.701,Multiple RegressionAnalysis:Inference,Testing against one-sided alternatives(greater than zero),俏乘厄簿垄逸奈痈差嫩炳稻徒鲁苍刚漆钓份疹盒某晋锯弘艰祟柏借万狂棍计量经济学导论ch4计量经济学导论ch4,Example:Wage equationTest whether,after controlling for education and tenure,higher work experience leads to higher hourly wages,Standar

17、d errors,Test against.,One would either expect a positive effect of experience on hourly wage or no effect at all.,Multiple RegressionAnalysis:Inference,锌愁峻踩俞驹铭珠臻栈顾忆擞剖鸳疗贺浚矣胶王隐绿磁消咱怔肆撩殿累娘计量经济学导论ch4计量经济学导论ch4,Example:Wage equation(cont.),The effect of experience on hourly wage is statistically greater

18、than zero at the 5%(and even at the 1%)significance level.“,t-statistic,Degrees of freedom;here the standard normal approximation applies,Critical values for the 5%and the 1%significance level(these are conventional significance levels).The null hypothesis is rejected because the t-statistic exceeds

19、 the critical value.,Multiple RegressionAnalysis:Inference,果货弃郧吝评求旱啦竖奠陆鹰窍恳勋浚托返酗毖饭缆椭戴糠终耸谬孜救磅计量经济学导论ch4计量经济学导论ch4,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 valu

20、e 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 18 degrees of freedom so that 5%of the cases are below the point.!Reject if t-statistic less than-1.734,Multiple RegressionAnalysis:Inference,Testin

21、g against one-sided alternatives(less than zero),惺罩电贝绢咱却杂虎咎第讹浇俺妇萝慕聋辣撩硬赢骚押乘该痹未猾癸惋抓计量经济学导论ch4计量经济学导论ch4,Example:Student performance and school sizeTest whether smaller school size leads to better student performance,Test against.,Do larger schools hamper student performance or is there no such effect?

22、,Percentage of studentspassing maths test,Average annual tea-cher compensation,Staff per one thou-sand students,School enrollment(=school size),Multiple RegressionAnalysis:Inference,竞淳矽惮恰蛆廷坊络磋畅曲姆始仲丧凤蚊找沫姜悉诛煤怔军潭趾吮弦猴洱计量经济学导论ch4计量经济学导论ch4,Example:Student performance and school size(cont.),One cannot rej

23、ect the hypothesis that there is no effect of school size on student performance(not even for a lax significance level of 15%).,t-statistic,Degrees of freedom;here the standard normal approximation applies,Critical values for the 5%and the 15%significance level.The null hypothesis is not rejected be

24、cause the t-statistic is not smaller than the critical value.,Multiple RegressionAnalysis:Inference,居王鬼晋哈窿荚邹纹尽惯禁有碎忘蚁侦嗜匙爆惹湾消汁珐悟掳豌戈板抵腹计量经济学导论ch4计量经济学导论ch4,Example:Student performance and school size(cont.)Alternative specification of functional form:,Test against.,R-squared slightly higher,Multiple Re

25、gressionAnalysis:Inference,爹惭又缺痉款再恩圃禽羹伴阐附平抓吠邱邮冠觅亩析馅泉纷子箔栈柳窘圈计量经济学导论ch4计量经济学导论ch4,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 negative.,t-statistic,Critical

26、 value for the 5%significance level!reject null hypothesis,How large is the effect?,+10%enrollment!-0.129 percentage points students pass test,(small effect),Multiple RegressionAnalysis:Inference,毫伤敖蔼积顿秒隘坟怂辞曹捞遮捉共兹浊焕染敞箕捞森唤伍劝震湖痹厩袋计量经济学导论ch4计量经济学导论ch4,Testing against two-sided alternatives,Test against

27、.,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 hypothesis is true,it is rejected in,for example,5%of the cases.In the given example,these are the points of the t-dis

28、tribution 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 2.06,Multiple RegressionAnalysis:Inference,望去凶茵琢茎掸跟须解仆祁铆痘沪锗纸禾奄笺瘤细硝檀熄妥城拦侩培雷砒计量经济学导论ch4计量经济学导论ch4,Example:Determinants of college GPA,Lectures missed per week,The effects of

29、hsGPA and skipped are significantly different from zero at the 1%significance level.The effect of ACT is not significantly different from zero,not even at the 10%significance level.,For critical values,use standard normal distribution,Multiple RegressionAnalysis:Inference,棱愁戌创论哼丝辟历么济窃浴播草火攫血恤郡忠刺舵僻矽和沉

30、圈骋造帅额计量经济学导论ch4计量经济学导论ch4,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

31、 applies,the following rules of thumb apply:,statistically significant at 10%level“,statistically significant at 5%level“,statistically significant at 1%level“,Multiple RegressionAnalysis:Inference,角硷螟弦扑闺涵蜡痢华宽微归凹仲觉净存铃骑舵攒搭宣甸距夕烈苫儒晕汽计量经济学导论ch4计量经济学导论ch4,Guidelines for discussing economic and statistica

32、l 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 necessa-rily mean it is economically or practically significant!If a variable is s

33、tatistically 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 the regressionIf the sample size is small,effects might be imprecisely estimated so

34、that the case for dropping insignificant variables is less strong,Multiple RegressionAnalysis:Inference,忱笼斑题潦蛰迎云芝冻灌淳奄朝答脏氟砰李仗侧毫挫妊敬误悸腿斧等钦谱计量经济学导论ch4计量经济学导论ch4,Testing more general hypotheses about a regression coefficientNull hypothesist-statisticThe test works exactly as before,except that the hypoth

35、esized value is substracted from the estimate when forming the statistic,Hypothesized value of the coefficient,Multiple RegressionAnalysis:Inference,墓蛮钱爱淬嫡灵罚种削长卷斧苹恃酉尺疡洁远港馅副咨袁痢无奈赞涛卿伦计量经济学导论ch4计量经济学导论ch4,Example:Campus crime and enrollmentAn interesting hypothesis is whether crime increases by one per

36、cent if enrollment is increased by one percent,The hypothesis is rejected at the 5%level,Estimate is different from one but is this difference statistically significant?,Multiple RegressionAnalysis:Inference,修妈绽思收烫烘去椽菊搀淌读粥稚上党险潭呢蜡酱栅赴侧鞠诀涅闸秦竣蕉计量经济学导论ch4计量经济学导论ch4,Computing p-values for t-testsIf the si

37、gnificance 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 H0The smallest significance level at which the null h

38、ypothesis 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 the null hypothesisP-values are more informative than

39、tests at fixed significance levels,Multiple RegressionAnalysis:Inference,盏姬榨剪情钓叛默惦燃伯刀笋茹慎钙熏讳摊浮酵幼腺廊奶仑郡蛾敞咯邻灌计量经济学导论ch4计量经济学导论ch4,How the p-value is computed(here:two-sided test),The p-value is the significance level at which one is indifferent between rejecting and not rejecting the null hypothesis.In

40、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 statistic,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 significanc

41、e level.For example,for a significance level of 5%the t-statistic would not lie in the rejection region.,value of test statistic,These would be the critical values for a 5%significance level,Multiple RegressionAnalysis:Inference,奇城闺傈勤肇拐漏鲁副纶瓣窒绍维擎苟挨宁蜂参瓣凑乓杉啥困耕徒铃堕杆计量经济学导论ch4计量经济学导论ch4,Critical value oft

42、wo-sided test,Confidence intervalsSimple manipulation of the result in 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

43、 the cases,Lower bound of the Confidence interval,Upper bound of the Confidence interval,Confidence level,Multiple RegressionAnalysis:Inference,背劲魄撬姿贷鹊饲肢巴昧集哎即铺萍辉王芬值滩锅指狙龋膝捂轴哟靶疵佑计量经济学导论ch4计量经济学导论ch4,Confidence intervals for typical confidence levelsRelationship between confidence intervals and hypothe

44、ses tests,reject in favor of,Use rules of thumb,Multiple RegressionAnalysis:Inference,灶衰玛干惦谓雌哥岁怯搜庞铅褒奶戳烩碍平憋辛渣总莆案锌狭移划虑味脏计量经济学导论ch4计量经济学导论ch4,Example:Model of firms R&D expenditures,Spending on R&D,Annual sales,Profits as percentage of sales,The effect of sales on R&D is relatively precisely estimated

45、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 not even statisticallysignificant because zero lies in the interval.,Multiple RegressionAnalysis:Inference

46、,饲叭咖放顽屠日器糜赘赎弃掩讶衍匙练徽十殆踩扦妊襄钦缘恨磅姚氰归偏计量经济学导论ch4计量经济学导论ch4,Testing hypotheses about a linear combination of parametersExample:Return to education at 2 year vs.at 4 year colleges,Years of education at 2 year colleges,Years of education at 4 year colleges,Test against.,A possible test statistic would be:,T

47、he difference between the estimates is normalized by the estimated standard deviation of the difference.The 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,石阅

48、骸郝沥贩敢嚼血鸦舔缄概藤屏探姨咎惹伦颊系玛司谈宅曰漓熏扼晦烯计量经济学导论ch4计量经济学导论ch4,Insert into original regression,Impossible to compute with standard regression output becauseAlternative method,Usually not available in regression output,Define and test against.,a new regressor(=total years of college),Multiple RegressionAnalysis:

49、Inference,减挎扼倘晒桩唱绞唐泳镭孔仓任贺晦吾诱窥娥叛棉桓体袖鄂挝随撵速悍顶计量经济学导论ch4计量经济学导论ch4,Estimation resultsThis method works always for single linear hypotheses,Total years of college,Hypothesis is rejected at 10%level but not at 5%level,Multiple RegressionAnalysis:Inference,拙科搁辩工萧翠混音擞琅踊鞍必赫腔旧它揖扭巳叔屉伸捏兔欺韶联昆退冕计量经济学导论ch4计量经济学导论c

50、h4,Testing multiple linear restrictions:The F-testTesting exclusion restrictions,Years in the league,Average number of games per year,Salary of major lea-gue base ball player,Batting average,Home runs per year,Runs batted in per year,against,Test whether performance measures have no effect/can be ex