Probability and Statistics
Probability and Statistics ECON 209
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This 20 page Class Notes was uploaded by Aliya Schumm PhD on Wednesday October 28, 2015. The Class Notes belongs to ECON 209 at Vassar College taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/230541/econ-209-vassar-college in Economcs at Vassar College.
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Date Created: 10/28/15
eouomies 209 Regression Extensions Dummy Variables notquot New PM mil aw andD othetwise Example 1 G AEDAA1ALLZ VhexeMALFrlifthepexsonlsmaleand01f thepetson is male This genetatestwo equationsone otremales and one fclm femaleWAGE E EZED maleWAGE 3 AED4 lg a tap QED n 0 men and woman Example 11 VELOCITY 1 gINTEREsn 533 Whete Dis avanablewhlchtakus on the u i indicate that the velocity function shifted upwatd in 1975 Example 1 W GE 5 AED zc iJ WAGE A ZED WAGE 5 Em ZED WAGE 1 a ZED Y Dotherwlse r i h M u equations 0 all othets chtistians and Jewish Atest orthe hypothesis Hu 1 13 o is atest of 39 Why 7 o it other 1 ii Christian 2 irlewish Example IV I t ta 1 MALE Then weestimate the equanon WAGE A QED AMALE AEDM Ifthe thanto awoman s wage Thete ate two wage equations 1 WAGE ED l WAGE At 39 112 MED The tstisthewage equation rot women the second is the wage equation a mm The Halli l7 0 hypothesis is the hypothesis that thete is no dismmination against 01 in favoxofmm Example 5 De ne JMALE 7 I tames MALE WAGE 1 JIED AMALE M sJMALE petsohs and a are tespectavely the effect on wages of bemg male and the effect on wages ofbnng Iewash effects Acapmtes thas dfect Itasthe effect on wages ofbeahgboth Iewash ahdmale lf 115 as hohJewash petsohs D andENTEREST VELOCITY 1 leTEREST 1gb gum r m t hypothesas H a 0 usmg ah Ftest as atestofthehypothesasthatthete washo changeah 39 39 test 01 structutal moteahfozmatavewayto accomplash the same thng The test of H 113 a o as equavaleht to wage equataoh 01 men and awage equataoh 01 women Example v11 p l t m n otharvnse We then estamate the wage equataoh WAGE 1x 52E153E2 EE3 at a hagh school gaaduate wlthout acollege degzee wage 71x a and so forth It as ampoetaht when you are usahg a model thatyou ate cleat oh the meamhgofthe vanables If at axample E2 7 l amplaes thatEl 7l also the tesults wall bemtexpmtad daffetehtly thah af E2 7 l amplaes that El 7 0 Example vm Dummy vanables can also we use to model equataohs wath kmks and dascohtahuataes D7oatx lt xlzfx z N then estamate the equataohy A 2x 3DXX example a demand functaoa maght look lake thas SALES 13 ZPRICE 33131 5132 55133 whezeD1D2D3 ate dummaes fol spmg summet and fall Note thatohecategoeywhtet asleft out Eeon 2n9 Sample Smtis cs A Summary Random Mean Varianee Smndardized Variable Random Variable Popula on x u a Xu Sample mean f z 3 7 Sample sum S an no l Variables Popula on x 1391 11111 Sample mean P n nurn Propor on 7 Sample sum S M 1 Count quot11 n pup on mulnply the above vanance by the rollowmg zemcnm teem Economics 2n9 Probability and Sta s cs Handoutau Regression The Classical Linear Regression Model y li axo lxs a 2 3 12 n V number of obsemaaons k Lhe numbzr of coef clmls number of explanatory vanables 1 We make the followmg assumphons l mglx xh x u oi EO39JXanzi 1a 13 1711 l 53X 5131 2 E23XaXa Kb 62 3 cryDfarxs 4 E2x0furalllandj 1 5 n gt k 5 The explaname Vanables each have posthe mte Vanance 7 These 15 no pedi39ect multlcohneanty 8 In addition we frequently make the assumptlon 01131 the explanatory Vanables are non 50001135110 9 e N N052 We eshmate the model by oxdmary least squaes OLS 1 Vin zzy 21 0 I Z We choose 13 13k m orderlo minimize Ze s The eshmated coef ments are mdom vanables It can be molly shown that 503 13 aquot S a wllere A Is any other linear unbiased estimator of Thls last xesnlt says that 15 the Best Lmear Unblased Eshmatox BLUE of 1 Thls can be 2 quotJak be any hnear fmohon ofthe coeffments for stated 111016 genexally Let a 1 a2 whlch a bum unblased eshmatox emsts Then all 132 mtg3 15 the best lmear unbl sed estmlatox of al az z The Classical Normal Linear Regression CNLR model Ifln addmon to the above assumphons we assume 2 magi we get the olasnoal nomal lmeax legessmn model In thls oase Goodness of m SST lhe mm Vana on on Y Z 017 72 7 SSR lhexegessmnsum ofsquaxes me explmned momean Z 07 7 72 7 2 SSE sumofsquaxed moi vhexesodual Vexmm an Z 017 7 SST SSRSSE 2 SSR The Coef mml of Demmomnon R SST The gamma quuaxed E2 7 7 w 337 n 7 1 R 7 g 7 the eshmated unelauunbelwesn y and ysquared In Ihe bwanm xegxessmn model R2 7 52 To test joint hypotheses aboutseveral memcients we use cheF ado SSE quotiSSE whm SSE mdependenuesmuons omposed by lhe hypomesos Fox example voles lhe hypovhesos Ha z 33 o nos 1s equmlemmme o hypomesos um lhe uue R7squaxed 7 SSRk1N F39 SSEn7 k Extznsinn Nnnrlinzar qulaljnns Lima in paramztns 1 1 Rempmcal y A 7 r 2 Polynnmml y A f zx 53x2 kxlquot Ar 9 quot 9 Lngrlng 19y 1nA 1nx 1731an lny illnxz 1131 3 Notethatmdns formulannn 51 15mg 9195mm ananh xespectto x 4 Sem ng lny A 52 39 Note39 5 15 9999199 Lhepncmtage nhange mY 99599199911 mm aumt changem x Th1sf1mn Sanrlng y 11 132 1991 1131 93199 5 meumd 7 51 p22 EXANEPLES xeg y x Fxom Webster age 921 Number of ch 15 91 1 19 195 99 Model 7 191 911979 1 191 911979 Yxob gt 9 9 9999 919991 7 19 9992519 19 922259951 91999 9 9979 99 Risquaxed 9 9999 99991 7 171 799999 11 12 2999997 9999 Ms 99979 y Eoef 999 22 9 mm 959 Eonf Interval x 7 1 991917 9772919 19 995 9 999 9119959 1 219297 con 7 1 999251 9912925 1 125 9 991 2 211719 9 527799 xeg wage ed e27 Souxce 7 ss 99 Ms Number of ch 91 2 591 97 22 Model 7 2919 72115 2 1121 99299 Yxob gt 9 9 9999 919991 7 11291 7599 591 21 1529799 9799992 9 2929 99 Risquaxed 9 1999 99991 7 11975 1777 599 29 1999257 9999 Ms 1 5991 wage 7 Eoef 999 22 9 mm 959 Eonf Interval ed 7 9259199 9919995 11 975 9 999 7999121 1 995951 exy 7 1951291 9171997 9 119 9 999 9719192 19991 con 7 71 991997 1 219991 71 921 9 999 77 299991 72 599919 Econ 209 Fall 2002 Handout The Poisson Distribution Let 2 space mterval Let x be the humbex cf cccunehces The 1t can be shcwh that x has a P0155011 dlstnbuhch for 01 23 Pr 6 Example 1 Suppose that there are an average of 500 hlls p6 hour on a popularweb ills What 1 1s equaltc 10 low m Pgtc 20 7 001866 20 Example 11 Suppcse a swnchboard can handle up tc 2 calls pet mmute Suppcse that the rmnllle the swnchboard ls ove oaded A 12 0311 p6 rmnllle 1302 17Pgtc 0 Ptx 1 lots 2 1 12 equot 2 12 2 12 1 986 014 quot 0 z 2x quot quot quot39 W l T I n IV m 00 000 then are 10 cases ofthe dlsease In a ally OfSUUUUU I 5 m is 5 2 0181 10 Pg 10 the Ah lnlexeshng feature ofthe Pmssch dlsmbuhan us that Em A and a A L6H dmsny mchon on 15 gwenby mm MN The cumulative density function 15 F 2 2 Vanable 151655 than 01 equal a gmequ Value For example m m s 2 What 15 the pmbabmty that them 15 a ve minute we Interval wnh no can A 12 Plgt5 17m 2 Econ 209 End of the semester notes Review Problems 1 Given the variables the interest rate r the stock of money m and gross domestic product y the following is a regression on the transformed variables g lnr lnr g lnm lnm g 10Y 10W reg lnr lnm lny Source 1 SS df MS Number of obs 32 1 F 2 29 831 Model 1 161417643 2 807088215 Prob gt F 00014 Residual 1 281646021 29 097119318 R squared 03643 1 Adj R squared 03205 Total 1 443063664 31 142923763 Root MSE 31164 lnr 1 Coef Std Err t Pgt1t1 95 Conf Interval lnm 1 1960497 5225924 3751 0001 3029318 8916753 lny 1 5319644 1341171 3966 0000 2576642 8062646 cons 1 2995116 7923452 3780 0001 4615644 1374589 a How would you interpret the coefficient of lny of lnm b A 10 increase in the money stock would lead to what predicted change in r 2 The following is output from stata reg wage ed R squared 01459 wage 1 Coef Std Err t Pgt1t1 95 Conf Interval ed 1 7504489 0787299 953 0000 5957892 9051086 cons 1 7459421 1045404 071 0476 2799568 1307684 a How would you interpret the coefficient of ed b What would you predict for the wage for a person with 14 years of ed 3 The following is output from stata reg lnwage ed R squared 01447 lnwage 1 Coef Std Err t Pgt1t1 95 Conf Interval ed 1 0767605 0080906 949 0000 0608671 0926539 cons 1 1059858 1074298 987 0000 8488189 1270896 a How would you interpret the coefficient of ed b What would you predict for the wage for a person with 14 years of ed 4 The follo g marfe mar reg lhwage ed wing is output from stata riedfemale married female marfe Source SS df MS Number of obs 534 7 F 4 529 3939 Model 340674714 4 851686785 Prob gt F 00000 Residual 114374424 529 21620874 R squared 02295 Adj R squared 02237 Total 148441895 533 278502617 Root MSE 46498 lhwage Coef Std Err t Pgtt 95 Cohf Interval ed 0768516 007714 996 0000 0616978 0920054 married 2868351 0573656 500 0000 1741428 3995273 female 063109 0689404 092 0360 1985395 0723216 marfe 2601734 0851219 306 0002 4273917 092955 cohs 9785543 1103712 887 0000 7617347 1195374 test female married F 2 529 2179 Prob gt F 00000 test female married marfe F 3 529 1940 Prob gt F 00000 test female marfe F 2 529 2144 Prob gt F 00000 test married marfe F 2 529 1259 Prob gt F 00000 test married marfe 0 F 1 529 018 Prob gt F 06717 test female marfe 0 F 1 529 4204 Prob gt F 00000 test female married marfe 0 F 1 529 038 Prob gt F 05367 test female married 0 F 1 529 406 Prob gt F 00444 test female married F 1 529 3256 Prob gt F 00000 a Write the predicted equation for a single female b What would you predict as a wage for a single female with 12 years of ed c How would you interpret the coefficient of marfe d Oh the basis of this output would you say that there was evidence of discrimination against single men relative to married e Oh the basis of this output would you say that there was evidence of discrimination against single men relative to married women f Oh the basis of this output would you say that there was evidence of discrimination single women relative to married men g Oh the basis of this output would you say that there was evidence of discrimination single women relative to married women h Oh the basis of this output would you say that discrimination against sihgle persons is worse for women than for men 5 How would you interpret the sigh of the coefficient of exp2 in the following output g exp2 exp92 lhwage ed exp exp2 lhwage Coef Std Err t Pgtt 95 Cohf Ihterval ed 0897588 0083203 10788 0000 0734139 1061036 exp 0349377 0056491 6 185 0000 0238403 046035 exp2 0005362 0001245 4307 0000 0007808 0002916 cohs 5202983 1236135 4 209 0000 2774659 7631308 Sketch the relationship between lhwage and experience You are the president ofa firm which manufactures hard drives for computers You are considering coming out with a new model ofhard drive The model which you are now producing fails on the average 1 of the time in the first mo nth Your chief engineer assures you that this model will fail less frequently that The following test is designed 1000 hard drives are produced and tested How would the following information affect your choice ofa significance level Answer separately for each case A The chiefengineer is willing to stake his reputation on the fact that the new drive is superior and you have reason to trust his judgement B The changover to the production of the new model is very costly C The market for hard drives is very competitive and you chief rival has just announced a hard drive with a lower failure rate You are hired as a statistical consultant to the New York state gaming commission to test the fairness of the roulette wheels used in aparticular casino You are concerned that the probability ofthe outcomes 0 or 00 which favor the casino might be greater than 119 You are designing a hypo thesis test where the null hypothesis is that the probability of0 or 00 is 119 How would the following information affect your choice ofa significance level Why A The casino owners are reputed to be gangsters B Ifthe casino is wrongly charged with ch eating the gaming commission will face a major lawsuit The Hypergeomeme Distriblm on Miscellaneous Notes slzeN Lelx fllnchon ofx ls r N7 r c c x n e a PM Mm m lc N n r N7 n Lent 7 ThenllcanbeshownllmEx mg ml 02 mtl lr 7 N N7 1 dlsmbmlon Example Suppose that there are 25 pencils u a class 15 uses ml 10 womeu Suppose that four 91 x Fmd m 3 ml nd he mean ml vmuoe ofx Solunou 4lt152524 Px3 if IHSZ 102 84 a 2Hlt 5gtlt 5 Economics 209 Notes on the Poisson Distribution and the Binomial Consider the following problem On the average cars arrive at a toll booth at the rate of it per unit of time For example cars might arrive at a toll booth at rate 2 per minute What is the probability that in a given minute x cars arrive Suppose we tried to solve this using the Binomial distribution We might divide the unit of time into n sub intervals Then the average number of cars arriving in the sub interval If this number is suf ciently small we might treat it as the probability that a car would arrive in the sub interval In our example if 2 cars per minute arrive then 130 of a car arrives on the average per second We might treat this as the probability that a car arrives in a given interval of one second Then we could use the Binomial probability Pltxgtltgtxlt1 gt In our example ifn 60 andy 2 Px 5 0348 This is not quite accurate because there is a possibility that more than one car would arrive in a one second interval So our calculation slightly understates the true probability that x 5 We can get a better answer if we break a minute into tenths of a second Then n 600 and the probability 1300 that a car arrives in a given interval Px 5 0360 If we continue this process of breaking up the time interval into smaller and smaller subinterval we get the Poisson probability as a limit For this case Px50361 Formally we can show that m quot 1 7 x n n x my 339 hem quot1 9 Yl g k medial weMK39TWJlt TJEEJ W J MK V A A 3 l w E v A H 3 v 3 A 3 Econ 209 End of the semester notes Review Problems Solutions 1 a The coefficient of lny is the elasticity of r with respect to y The coefficient of lnm is the elasticity of r with respect to m b A 10 increase in the money stock would lead to a 196 decrease in r 2 The following is output from stata reg wage ed R squared 01459 wage Coef Std Err t Pgtt 95 Conf Interval ed 7504489 0787299 953 0000 5957892 9051086 cons 7459421 1045404 071 0476 2799568 1307684 a The coefficient of ed says that an additional year of schooling increase the wage by 75 75 centm b wage 7459421 7504489l4 976 3 The following is output from stata a The coefficient of ed says that an additional year of schooling increases the wage 77 b lnwage 1059858 076760514 2134505 wage exp2134505 84528613 4 The following is output from stata test female married F 2 529 2179 Prob gt F 00000 test female married marfe F 3 529 1940 Prob gt F 00000 test female marfe F 2 529 2144 Prob gt F 00000 test married marfe F 2 529 1259 Prob gt F 00000 test married marfe 0 F 1 529 018 Prob gt F 06717 test female marfe 0 F 1 529 4204 Prob gt F 00000 test female married marfe 0 F 1 529 038 Prob gt F 05367 test female married 0 F 1 529 406 Prob gt F 00444 test female married F 1 529 3256 Prob gt F 00000 a lnwage 9785543 063109 0768516ed b lnwage 9785543 063109 076851612 18376645 wage exp18376645 62818499 c The coefficient of marfe is the advantage or disadvantage of being both married and female in addition to the advantages or disadvantages of being married and being female d Yes the coefficient of married is positive and significant e No we fail to reject the hypothesis that the coefficients of femalemarriedmarfe f Yes reject the hypothesis that female married married gt female No we fail to reject the hypothesis that married marfe 0 No the coefficient of marfe is negative The coefficient of epo is negative which means that the relationship is concave from below You are the president ofa firm which manufactures hard drives for computers You are considering coming out with a new model ofhard drive The model which you are now producing fails on the average 1 of the time in the first mo nth Your chief engineer assures you that this model will fail less frequently that The following test is designed 1000 hard drives are produced and tested How would the following information affect your choice ofa significanc e level Answer separately for each case A The chiefengineer is willing to stake his reputation on the fact that the new drive is superior and you have reason to trust his judgement That the chief engineer in whom you have faith is staking his reputation on the fact that the null hypothesis is false Raise the significance level B The changover to the production of the new model is very costly The cost ofa type I error is very high lower the signicance level C The market for hard drives is very competitive and you chiefrival has just announced a hard drive with a lower failure rate The cost ofa type II error is very large Raise the significance level You are hired as a statistical consultant to the New York state gaming commission to test the fairness ofthe roulette wheels used in aparticular casino You are concerned that the probability of the outcomes 0 or 00 which favor the casino might be greater than 119 You are designing a hypothesis test where the null hypothesis is that the probability of 0 or 00 is 119 How would the following information affect your choice ofa significance level Why A The casino owners are reputed to be gangsters The probability that the house is cheating is high ie the prob ability that the null hypothesis is true is low Raise the significance level B If the casino is wrongly charged with ch eating the gaming commission will face a major lawsuit The cost ofa type I error is high Lower the significance level Econ 209 Estimators Paramemr Estimamr Mean of Variance of Esn mated esn mamd Estimamr Es mamr Variance 2 S X 7 l1 l1 n n 70 7 II p1 p P quot T Companng two populnnons Independent Samples M 2 X1 X M 2 quot1 53 Fl P 75 Which dismbm nn m Use Normal or t Rules of Thumb a known use normal C3581 Po 1311011 Normal P 7 unknown usel 5 known use normal if cm applies Case 11 P0 ula on Nonenolm l P 5 unknown use t J CLT applies use normal if CLT applies Case 111 Pmpomons awemlse use non pammetric methods mvolvmg diffnences In means that as two disarm eases to wander 1 Vananees unknown but assumed equal S S2 2 1 t 1 x 7 H quot2 n 1 n 1s degrees of freedom 7 n n2 7 2 t 1 1 2 Sn 1 S2 wheze quot mnsz 2 Vananees unknown but nnt assumed equal vnth degzees or rzeedom given by the ronowmg expzessmn39 2 2 2 S S I2 n H degees orrzeedom 15 equal to n n r 2
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