Problem 1: Consider the following stationary data generation process for a random variable y t. e t ~ N(0,1) i.i.d.

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1 A/CN C m Sr Anal Profor Òcar Jordà Wnr conomc.c. Dav POBLM S SOLIONS Par I Analcal Quon Problm : Condr h followng aonar daa gnraon proc for a random varabl - N..d. wh < and N -. a Oban h populaon man varanc auocovaranc and auocorrlaon. nc { } aonar. Furhr noc ha. Nong onl h nonzro rm: 6 6 Fnall k k COV b Drv [ ] [ ] [ ]

2 A/CN C m Sr Anal Profor Òcar Jordà Wnr conomc.c. Dav v No: 6. Drv [ ] [ ] [ ] [ ] O V c Drv h lmng drbuon of h ampl man. Hn: { } no..d. bu rgodc for h man. Snc aonar h onl dffcul n h drvaon of h drbuon of h ampl man ha { } no an..d. proc alhough rgodc. A gnral rong law of larg numbr and a cnral lm horm can b appld o uch proc o ha h ampl man convrg a.. o h populaon man whch zro and normall drbud around zro.. ung h cnral lm horm for marngal dffrnc qunc w dcud n cla. Howvr h problm can alo b olvd from fr prncpl a follow: ; j j j V A lm o p V. Howvr V

3 A/CN C m Sr Anal Wnr Profor Òcar Jordà conomc.c. Dav Nx a lnar funcon of ndpndn normall drbud componn { } wh a fn non-zro varanc n larg ampl o ha: d N d Oban h lmng drbuon of h la quar maor of. Hn: calcula and hn drv drbuon. ng h abov rul and Sluk horm: Nx from Mann-Wald p lm p lm p lm ng Cramr horm w hu hav d N d N

4 A/CN C m Sr Anal Profor Òcar Jordà Wnr conomc.c. Dav Problm : Condr h A modl of h prvou xrc bu uppo nad ha. N h condonal dn for obrvaon hrfor log log ; log π f L b h unrrcd ML ma of θ and l b h rrcd ML ma ubjc o h conran c whr and c ar known conan. Alo l Σ Σ ; a Vrf ha mnmz h um of quard rdual o ha f h am a h OLS maor. h problm a lo mplr f w procd wh h condonal lklhood whch log log π θ L and can b vwd a a QML problm. h fr ordr condon ar L whch h OLS maor b Vrf ha mnmz h um of quard rdual ubjc o c o ha h rrcd LS maor. h rrcd lklhood vr mpl and ak h form log log c L λ π θ

5 A/CN C m Sr Anal Wnr Profor Òcar Jordà conomc.c. Dav whr h lagrang mulplr rcald b for convnnc and whou lo of gnral. h fr ordr condon ar from whch L λ λ Of cour n h xampl h conran can b rvall mpod n h problm and would no rqur an maon. h rrcd la quar maor would b calculad b mnmzng mn λ λ c who fr ordr condon can b hown o concd wh ho calculad va rrcd maxmum lklhood. c Wha aumpon dd ou mak on f aumng unobrvd? Dcu h mporanc of an mplfng aumpon and h mporanc ha h paramr ha f valu wr unrrcd. W hav bn ung h condonal lklhood for mplc. Howvr f w had ud h xac lklhood h nbl aumpon rgardng f ha drbud N / -. No howvr ha h xac lklhood onl vald f < ohrw h log-lklhood bcom unboundd for. d L Show ha Q θ log f ; 5

6 A/CN C m Sr Anal Wnr Profor Òcar Jordà conomc.c. Dav Q θ logπ log Q θ logπ log Whr and. Hn: Show ha and. o how and all ha ou bd o compu h fr ordr condon of h lklhood and h rrcd lklhood wh rpc o. Subuon of and no Q θ rvall dlvr h wo xpron for Q θ and Q θ. Vrf ha h Σ gvn abov alhough no h am a H ; θ conn for [H ; θ ]. Vrf ha Σ alhough no h am a H ; θ conn for [H ; θ ]. Hn: ou ma aum ha θ conn for θ and θ conn for θ undr h null. h hould mak provng connc of and ar. h a approach o calcula h han for h condonal and h rrcd lklhood. ng h hn hn bcom rval o how ha boh H ; θ [ ; θ ] and H H ; θ [ ; θ ] and H Σ [ H ; θ ] and Σ [ H ; θ ] 6

7 A/CN C m Sr Anal Profor Òcar Jordà Wnr conomc.c. Dav 7 f Show ha h Wald LM and L ac ung Σ and Σ can b wrn a LM L c W ' ' ' ' log log No cr hr ju drc applcaon of h formula. g Show ha h hr ac can alo b wrn a LM L W log h nvolv mpl algbrac manpulaon.

8 A/CN C m Sr Anal Wnr Profor Òcar Jordà conomc.c. Dav Problm : Condr h ochac proc {x } whch dcrb h numbr of rad pr nrval of m of a parcular ock. hu x ngr-valud and non-ngav. h Poon drbuon commonl ud o dcrb h p of proc. I dn ha h form: wh condonal man f j Ω ; θ x λ λ j j! x x... λ xp α x Anwr h followng quon: a ndr wha paramrc rrcon of α and wll {x } b aonar? h a rck quon nc w hav a pcfcaon for h condonal man drcl rahr han for {x }. Howvr can b hown ha h modl xplov for >. b Wr down h log-lklhood funcon for h problm condonal on x. { x α x xp α x ln x! } L θ c Compu h fr ordr condon and oban h maor for α and. L α L { x xp α x } { x [ α x xp α x ]} 8

9 A/CN C m Sr Anal Wnr Profor Òcar Jordà conomc.c. Dav d Suppo x no Poon drbud. Ar α and connl mad b h condonal log-lklhood n b? h Poon drbuon an xampl of lnar xponnal funcon who QML propr hav bn ablhd b Gourroux Momfor and rognon 98. Chck Camron 998 book for h rfrnc. Compu h on-p ahad forca x. Nx dcrb h Mon Carlo xrc ha would allow ou o compu h wo-p ahad forca. Fnall dcrb wha approach would ou ak f h daa wr no Poon drbud bu ou wand o produc mul-p ahad forca wh h condonal man mad n b. h on p ahad forca x λ xpα x. o compu h wo-p ahad forca no ha w nd o compu x x. o do h w wll draw from a Poon drbuon wh man paramr λ xpα j x. Suppo ou draw n m from h Poon ou wll hav n valu for x whch ou can hn plug no h xpron of h condonal man o compu h forca a: n x xp α n x Compung h h-p ahad forca would con on drawng n m from a Poon drbuon who condonal man gvn b h h -p ahad forca and calculang h avrag a don abov. No ha a long a h condonal man corrcl pcfd mul-p ahad forca compud a dcrbd abov would b conn alb no ffcn f h drbuon no Poon. 9

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