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 6 Multiple RegressionAnalysis: Further IssuesWooldridge: Introductory Econometrics: A Modern Approach, 5e羽盆肮忘底艺大掘驾僵魔渴馒墒紊帚贰靡光馏锅窗昂聊扦枉抢骑漏花苫糕计量经济学导论ch6计量经济学导论ch6 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.More on Functional FormMore on using logarithmic functional formsConvenient percentage/elasticity interpretationSlope coefficients of logged variables are invariant to rescalingsTaking logs often eliminates/mitigates problems with outliersTaking logs often helps to secure normality and homoscedasticityVariables measured in units such as years should not be loggedVariables measured in percentage points should also not be loggedLogs must not be used if variables take on zero or negative valuesIt is hard to reverse the log-operation when constructing predictionsMultiple Regression Analysis: Further Issues券开里少柑邑侄脸酶焰摔达螺伯缆国唐篷恿族凯态命叶湿悄镣柏庇设蜗葛计量经济学导论ch6计量经济学导论ch6 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.Using quadratic functional formsExample: Wage equationMarginal effect of experienceThe first year of experience increases the wage by some .30$, the second year by .298-2(.0061)(1) = .29$ etc.Concave experience profileMultiple Regression Analysis: Further Issues铬艺酥籍花秸件就橡媒悟母窥崖说蚌阵载淳浓鲜绰邓海钒遵遵猜扣舵旺力计量经济学导论ch6计量经济学导论ch6 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.Wage maximum with respect to work experienceDoes this mean the return to experience becomes negative after 24.4 years?Not necessarily. It depends on how many observations in the sample lie right of theturnaround point.In the given example, these are about 28% of the observations. There may be a speci-fication problem (e.g. omitted variables). Multiple Regression Analysis: Further Issues采墒宠咎记目委殆遂兑躬裕性肝奠斤哎烩蓄丈涎秦挎堑玫铲霄悬内稍鹏佳计量经济学导论ch6计量经济学导论ch6 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: Effects of pollution on housing pricesDoes this mean that, at a low number of rooms, more rooms are associated with lower prices?Nitrogen oxide in air, distance from em-ployment centers, student/teacher ratioMultiple Regression Analysis: Further Issues侧间胰答益湾目搔筑左推博箍兜纺遇法你莉建车菠珐鲁肿窃悠线确迷滚氏计量经济学导论ch6计量经济学导论ch6 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.Calculation of the turnaround pointThis area can be ignored as it concerns only 1% of the observations.Increase rooms from 5 to 6:Increase rooms from 6 to 7:Turnaround point:Multiple Regression Analysis: Further Issues翼湃绪嫂述蜒舰援尹官侠半捡巢侩巡谱访钮昆障育乒节搅嘶师艘乡殆阉恬计量经济学导论ch6计量经济学导论ch6 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.Other possibilitiesHigher polynomialsMultiple Regression Analysis: Further Issues耻昼言琼瞥始沥弗磐遥吃男愚齐账赶悉御蛔侮昔掏科宠纽缆轻斡此也阻耀计量经济学导论ch6计量经济学导论ch6 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.Models with interaction termsInteraction effects complicate interpretation of parametersInteraction termThe effect of the number of bedrooms depends on the level of square footageEffect of number of bedrooms, but for a square footage of zeroMultiple Regression Analysis: Further Issues庄钠岛晰台淮靴赐旗后瘪冤幻酗套销柏梅俞裙纪市涝帅酒刹拧超荷撮脉宫计量经济学导论ch6计量经济学导论ch6 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.Population means; may be replaced by sample meansReparametrization of interaction effectsAdvantages of reparametrizationEasy interpretation of all parametersStandard errors for partial effects at the mean values availableIf necessary, interaction may be centered at other interesting valuesEffect of x2 if all variables take on their mean valuesMultiple Regression Analysis: Further Issues豢犯乞求贰浦菲隋淑媒众愧拿咨起信赣辅舅奈戌坝裂振团淬六多圆筒迎祭计量经济学导论ch6计量经济学导论ch6 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.More on goodness-of-fit and selection of regressorsGeneral remarks on R-squaredA high R-squared does not imply that there is a causal interpretationA low R-squared does not preclude precise estimation of partial effectsAdjusted R-squaredWhat is the ordinary R-squared supposed to measure?is an estimate forPopulation R-squaredMultiple Regression Analysis: Further Issues繁竭焚慷哈拇润恰连贪酮只吭栋汤冗拎礁豆水绑律龙玻儒顽廉签抚卉链啮计量经济学导论ch6计量经济学导论ch6 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.Adjusted R-squared (cont.)A better estimate taking into account degrees of freedom would beThe adjusted R-squared imposes a penalty for adding new regressorsThe adjusted R-squared increases if, and only if, the t-statistic of a newly added regressor is greater than one in absolute valueRelationship between R-squared and adjusted R-squaredCorrect degrees of freedom of nominator and denominatorThe adjusted R-squared may even get negativeMultiple Regression Analysis: Further Issues匡嚣趴驯林扼捎焉卯掐邱够掣豹抓天菊辟冬煤又简走锣格导裳加刃壹绘放计量经济学导论ch6计量经济学导论ch6 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.Using adjusted R-squared to choose between nonnested modelsModels are nonnested if neither model is a special case of the otherA comparison between the R-squared of both models would be unfair to the first model because the first model contains fewer parametersIn the given example, even after adjusting for the difference in degrees of freedom, the quadratic model is preferredMultiple Regression Analysis: Further Issues伊抹酶好始储拍荒下荧匀恶瓜总撼桩诀拜改伏顷闪谰椎竿螟舀蒙忿选嘿乘计量经济学导论ch6计量经济学导论ch6 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.Comparing models with different dependent variablesR-squared or adjusted R-squared must not be used to compare models which differ in their definition of the dependent variableExample: CEO compensation and firm performanceThere is muchless variationin log(salary)that needs tobe explained than in salaryMultiple Regression Analysis: Further Issues践绍据非扶沫项柳蔫码衰技霖略疟烃瞬蓖拭艺哗矽坡擅个梦到魏闪烽奋隐计量经济学导论ch6计量经济学导论ch6 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.Controlling for too many factors in regression analysisIn some cases, certain variables should not be held fixedIn a regression of traffic fatalities on state beer taxes (and other factors) one should not directly control for beer consumptionIn a regression of family health expenditures on pesticide usage among farmers one should not control for doctor visitsDifferent regressions may serve different purposesIn a regression of house prices on house characteristics, one would only include price assessments if the purpose of the regression is to study their validity; otherwise one would not include themMultiple Regression Analysis: Further Issues芬临消寒郁内幻通诈杰硕夷凌芽订具硅亭预歉柿枪挟趾全血欢见失叮跌阻计量经济学导论ch6计量经济学导论ch6 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.Adding regressors to reduce the error varianceAdding regressors may excarcerbate multicollinearity problemsOn the other hand, adding regressors reduces the error variance Variables that are uncorrelated with other regressors should be added because they reduce error variance without increasing multicollinearityHowever, such uncorrelated variables may be hard to findExample: Individual beer consumption and beer pricesIncluding individual characteristics in a regression of beer consumption on beer prices leads to more precise estimates of the price elasticityMultiple Regression Analysis: Further Issues娘坑柔扶抹卓少腔险苔汉峻岭撅以翠娟望紊束征堑裹姬敏棺棍鸥屡筹剂丙计量经济学导论ch6计量经济学导论ch6 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.Predicting y when log(y) is the dependent variableUnder the additional assumption that is independent of :Prediction for yMultiple Regression Analysis: Further Issues趾竹亿怕蚤戳睡瓮录烬加盟刮媚筋蓉榆裹八阁倔怂铣凭邵荆爪墒涎饶疮皑计量经济学导论ch6计量经济学导论ch6 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.Comparing R-squared of a logged and an unlogged specificationThese are the R-squareds for the predictions of the unlogged salary variable (although the second regression is originally for logged salaries). Both R-squareds can now be directly compared.Multiple Regression Analysis: Further Issues队楼躁垄惩墒遥师圆弯伦琼阻废佯未屹还桃鲤读找链氰艺蛮澜茸辙勃赛斗计量经济学导论ch6计量经济学导论ch6