UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

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1 UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed: Ope book exam, where all wrtte ad prted resources as well as calculator - s allowed The grades gve: A-F, wth A as the best ad E as the weakest passg grade F s fal Problem A Let (X,Y) be two radom varables (rv s) such that X s ormally dstrbuted wth expectato, EX ( ) 0, ad varace, var( X ) (wrtte short, X ~ N (0,) ), ad the codtoal dstrbuto of Y gve that X x s fxed, s ormal wth codtoal expectato, E( Y x) x, ad codtoal varace, var( Y x) (wrtte short Y x ~ N( x,) [ A remder you may eed below: The geeral ormal probablty desty fucto (pdf) forv ~ N(, ) s gve by ( ) v fu ( v ;, ) e, v ad the momet geeratg fucto (mgf) by t t M ( t) e, t ] V Wrte up the formula for the pdf, f ( y x ), of Y gve that X x Show that E( Y ) 0 ad var( Y ) [Ht Use, eg, the law of total expectato] Show that the correlato coeffcet betwee X ad Y s [Ht Remember that E( XY ) E X E Y X ]

2 B Let X, X,, X be depedet ad detcally dstrbuted (d) where X ~ N(0, ),,,, (ad, hece, wth E( X ) 0, var( X ) 0, ad stadard devato, SD( X ) ) Show that the maxmum lkelhood estmator (mle) of s ˆ X Show that ˆ s also the momet method estmator (mme) of C Let X be as defed secto B Expla why X X s ch-square dstrbuted wth degree of freedom ( short: ~ ),,,, [Ht You may take as a fact (that you do ot eed to prove), that f Z ~ N (0, ), the Z s ch-square dstrbuted wth degree of freedom ( Z ~ ) ] Show that the mle, ˆ, s ubased ad has varace, ˆ var( ) Show that the mle, ˆ s effcet the sese that t has smallest possble varace amog all ubased estmators for that are based o X, X,, X D Expla why ˆ (short: ~ ) ˆ s ch-square dstrbuted wth degrees of freedom [Ht You may beeft from the ht C() ad other relevat establshed propertes of the ch-square dstrbuto ] Data: We have 5 observatos, x, x,, x, where 5 x 98 Assumg the model secto B, develop ad calculate a (exact) 95% cofdece terval (c), based o ˆ, for the stadard devato of X, e, for [Ht Fd frst a 95% c for derved from the probablty ˆ statemet, P c c 095, where c, c are sutable percetles the relevat ch-square dstrbuto The trasform ths terval to a 95% c for ] Why s ˆ a cosstet estmator of? Is ˆ ubased for? reasos for your aswers Gve

3 E Retur to the stuato secto A Show that the margal dstrbuto of Y s ormal (N(0, )) [Ht There are several ways to show ths If you do ot have a way of your ow, you may try oe of the followg two approaches: ty Approach : Wrte up the mgf, e, E e x of the codtoal dstrbuto of Y gve X x ad fd the (margal) mgf of Y usg the law of total expectato Approach : Show that ( XY, ) s jotly bvarate ormally dstrbuted Oe way to do ths s frst to wrte up the geeral formula for the bvarate ormal pdf, the substtute the fve parameter values you foud secto A, ad show that the resultg expresso reduces to the actual jot pdf for ( XY, ) that you fd from the formato secto A ] Problem A Let X be Pareto dstrbuted wth parameters ( b, ) ad cumulatve dstrbuto fucto (cdf) ( b x) for x b F( x) 0 for x b where b s a kow postve umber, ad 0 a parameter Show that V l X s expoetally dstrbuted wth parameter ( short: b V ~ exp() ) Let Y l X V Expla why Y ~ exp( ) b B Suppose Y s gamma dstrbuted wth shape parameter 0 ad scale parameter 0 ( short: Y ~ (, ) ) Use the momet geeratg fucto (mgf) of Y, MY ( t) t for t, to show that the expected value s EY ( ) Y Use the mgf of Y to show that ~ (, c), where c s a arbtrary postve c costat C Itroducto Let X be the come of a radomly chose woma Norway wth come over kr O several occasos the course we have assumed that X s 3

4 Pareto dstrbuted ( b, ), where b , ad 0 a parameter We wat to test ths Pareto assumpto here ths secto The data s a radom sample of 36 wome, all wth come over kr, draw from SSB s database 998 Oe way to test the Pareto assumpto s to embed the Pareto model to a larger model ad test the Pareto assumpto as a sub-model: Accordg to secto A, assumg X X ~ Pareto( b, ) s equvalet wth assumg Y l ~ exp( ) Sce the exp( ) b dstrbuto s the same as the (, ) dstrbuto, a wder model would be to assume X Y l ~ (, ), where 0 ad 0 are parameters To test the Pareto b assumpto would the be equvalet to test H : 0 the wder model Questo () Model: Let Y, Y,, Y be d wth Y ~ (, ) for,,,, X where Y l, b , ad 36 b I the appedx you wll fd (part of) the Stata output from a maxmum lkelhood estmato (mle) of model () Use the output to test H0: versus H: based o the mle ˆ ad ts stadard error Calculate the p-value (approxmately) ad commet o the result [Ht You may cosder the stadard errors for the coeffcets estmated the output as cosstet estmates of the theoretcal stadard errors ] D Alteratvely, we may test the Pareto model (e, the expoetal model for Y) versus the gamma model for Y, as secto C, usg a lkelhood rato test (LR): The Stata output the appedx cotas the maxmum log lkelhood for the gamma model () uder the headg Log lkelhood, based o the avalable data Y l X b Usg the same data ad the maxmum lkelhood method to for estmate the expoetal model for Y (that correspods to the Pareto model for X), gves, accordg to Stata, a maxmum log lkelhood equal to Use ths formato to perform the LR test for testg the ull hypothess of a expoetal dstrbuto for Y versus the gamma model () Use the level of sgfcace 5% ad formulate a cocluso Suppose that V s ch-square dstrbuted wth degree of freedom SSB s Statstcs Norway 4

5 ( short, V ~ ) The tables for the ch-square dstrbuto Rce are ot suffcet to determe probabltes lke P( V c) for arbtrary postve umbers c However, show that P( V c) G( c), where Gz () s the cdf of the stadard ormal, N(0, ), dstrbuto [Ht You may use the well-kow fact that ] Calculate the p-value approxmately for the LR-test () Z ~ N(0, ) Z ~ Appedx for problem From the Stata output of maxmum lkelhood estmato of the model () Maxmum lkelhood estmato Log lkelhood = Number of obs =, Coef Std Err z P> z [95% Cof Iterval] /alpha *** *** *** *** /lambda *** *** *** ***