Homework 2. is unbiased if. Y is consistent if. c. in real life you typically get to sample many times.

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1 Econ526 Mulile Choice. Homework 2 Choose he one ha bes comlees he saemen or answers he quesion. (1) An esimaor ˆ µ of he oulaion value µ is unbiased if a. ˆ µ = µ. b. has he smalles variance of all esimaors. c. µ. E ( µ ) µ d. ˆ (2) An esimaor ˆ µ of he oulaion value µ is consisen if a. ˆ µ µ. b. is mean square error is he smalles ossible. c. is normally disribued. d. 0. (3) The reason why esimaors have a samling disribuion is ha a. economics is no a recise science. b. individuals resond differenly o incenives. c. in real life you yically ge o samle many imes. d. he values of he exlanaory variable and he error erm differ across samles. (4) The -saisic is defined as follows: a. b. c. d µ σ n,0 2 µ,0 SE( ) ( µ ) 2,0 SE( )

2 (5) In he simle linear regression model, he regression sloe a. indicaes by how many ercen increases, given a one ercen increase in X. b. when mulilied wih he exlanaory variable will give you he rediced. c. indicaes by how many unis increases, given a one uni increase in X. d. reresens he elasiciy of on X. (6) The OLS esimaor is derived by a. connecing he i corresonding o he lowes X i observaion wih he i corresonding o he highes X i observaion. b. making sure ha he sandard error of he regression equals he sandard error of he sloe esimaor. c. minimizing he sum of absolue residuals. d. minimizing he sum of squared residuals. (7) Inerreing he inerce in a samle regression funcion is a. no reasonable because you never observe values of he exlanaory variables around he origin. b. reasonable because under cerain condiions he esimaor is BLUE. c. reasonable if your samle conains values of X i around he origin. d. no reasonable because economiss are ineresed in he effec of a change in X on he change in. (8) The OLS residuals a. can be calculaed using he errors from he regression funcion. b. can be calculaed by subracing he fied values from he acual values. c. are unknown since we do no know he oulaion regression funcion. d. should no be used in racice since hey indicae ha your regression does no run hrough all your observaions. = β + β X + u, (9) In he simle linear regression model i 0 1 i i a. he inerce is yically small and unimoran. b. β0 + β1x i reresens he oulaion regression funcion. c. he absolue value of he sloe is yically beween 0 and 1. d. β0 + β1x i reresens he samle regression funcion.

3 (10) To obain he sloe esimaor using he leas squares rincile, you divide he a. samle variance of X by he samle variance of. b. samle covariance of X and by he samle variance of. c. samle covariance of X and by he samle variance of X. d. samle variance of X by he samle covariance of X and. Shor Answers. The Firs Quesion (Q11) is a SAMPLE. (11) Sir Francis Galon, a cousin of James Darwin, examined he relaionshi beween he heigh of children and heir arens owards he end of he 19h cenury. I is from his sudy ha he name regression originaed. ou decide o udae his findings by collecing daa from 110 college sudens, and esimae he following relaionshi: Sudenh= Midarh, R 2 = 0.45, SER = 2.0 where Sudenh is he heigh of sudens in inches, and Midarh is he average of he arenal heighs. (Following Galon s mehodology, boh variables were adjused so ha he average female heigh was equal o he average male heigh.) (a)inerre he esimaed coefficiens. (b) Wha is he meaning of he regression R 2? (c) Wha is he redicion for he heigh of a child whose arens have an average heigh of inches? (d) Wha is he inerreaion of he SER here?. (e) Given he osiive inerce and he fac ha he sloe lies beween zero and one, wha can you say abou he heigh of sudens who have quie all arens? Who have quie shor arens? (f) Galon was concerned abou he heigh of he English arisocracy and referred o he above resul as regression owards mediocriy. Can you figure ou wha his concern was? Why do you hink ha we refer o his resul oday as Galon s Fallacy? Soluion : (a) For every one inch increase in he average heigh of heir arens, he suden s heigh increases by 0.73 of an inch. There is no reasonable inerreaion for he inerce. (b) The model exlains 45 ercen of he variaion in he heigh of sudens. (c) = (d) The SER is a measure of he sread of he observaions around he regression line. The magniude of he yical deviaion from he regression line or he yical regression error here is wo inches. (e) Tall arens will have, on average, all sudens, bu hey will no be as all as heir arens. Shor arens will have shor sudens, alhough on average, hey will be somewha aller han heir arens. (f) This is an examle of mean reversion. Since he arisocracy was, on average, aller, he was concerned ha heir

4 children would be shorer and resemble more he res of he oulaion. If his conclusion were rue, hen evenually everyone would be of he same heigh. However, we have no observed a decrease in he variance in heigh over ime. (12) The baseball eam neares o your home own is, once again, no doing well. Given ha your knowledge of wha i akes o win in baseball is vasly suerior o ha of managemen, you wan o find ou wha i akes o win in Major League Baseball (MLB). ou herefore collec he winning ercenage of all 30 baseball eams in MLB for 1999 and regress he winning ercenage on wha you consider he rimary deerminan for wins, which is qualiy iching (eam earned run average). ou find he following informaion on eam erformance: Summary of he Disribuion of Winning Percenage and Team Earned Run Average for MLB in 1999 Average Sandard deviaion Percenile 10% 25% 40% 50% 60% 75% 90% (median) Team ERA Winning Percenage (a) Wha is your execed sign for he regression sloe? Will i make sense o inerre he inerce? If no, should you omi i from your regression and force he regression line hrough he origin? (b) Wha is your execed sign for he regression sloe? Will i make sense o inerre he inerce? If no, should you omi i from your regression and force he regression line hrough he origin? (c) OLS esimaion of he relaionshi beween he winning ercenage and he eam ERA yields he following: winc = eamera, R 2 =0.49, SER = 0.06, where winc is measured as wins divided by games layed, so for examle a eam ha won half of is games would have Winc = Inerre your regression resuls. (d) I is yically sufficien o win 90 games o be in he layoffs and/or o win a division. Winning over 100 games a season is exceional: he Alana Braves had he mos wins in 1999 wih 103. Teams lay a oal

5 of 162 games a year. Given his informaion, do you consider he sloe coefficien o be large or small? (e) Wha would be he effec on he sloe, he inerce, and he regression R 2 if you measured Winc in ercenage oins, i.e., as (Wins/Games) 100? (f) Are you imressed wih he size of he regression R 2? Given ha here is 51% of unexlained variaion in he winning ercenage, wha migh some of hese facors be?