Fall 2005 Economics and Econonic Methods Prelim. (Shevchenko, Chair; Biddle, Choi, Iglesias, Martin) Econometrics: Part 4

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1 Fall 2005 Economics and Econonic Mthods Prlim (Shvchnko, Chair; Biddl, Choi, Iglsias, Martin) Economtrics: Part 4 Dirctions: Answr all qustions. Point totals for ach qustion ar givn in parnthsis; thr ar 100 points possibl. Within a qustion, ach part rcivs qual wight. You may us a calculator, but only for computations, not for storag or rtrival of information. You must show all your working to gt crdit for your solutions. Th tabls of th critical valus of th t-distribution, F-distribution and chi-squar distribution ar attachd. 1. (15 points) W stimat through OLS thr modls with sampl siz T = 4 whr th Gauss-Markov assumptions hold in th thr modls, E (4) = a2 # 0, t is a tim trnd, and Xt has man zro. For ach of th thr claims blow, prov if thy ar tru or fals. (a) Th rsidual sum of squars from modl (1) is strictly largr than 0. (b) Aftr w stimat modl (2) with OLS, thn th varianc of th OLS stimator of 7 is $. (c) If 3 = (3,, 8,) 'is th OLS stimator in modl (3), thn 3, and 3, ar uncorrlatd. 2. (25 points) Lt XI,..., XT b iid random variabls with dnsity function f (4 = o-lx(i-l) if x~(0,l) othrwis whr 0 E (0,l) is an unknown paramtr, 5,' o-~x(~-') ln (x) dx = -19 and 2 5,' B~X(+-~) (In (x)12 dx = 20.

2 (a) Driv th maximum liklihood stimator of 8. Call this stimator 2. (b) Show that 2 is unbiasd. (c) Find out if 2 is th most fficint stimator among thos unbiasd stimators of (20 points) In ordr to stimat th paramtr a2 in a modl Y = XP + u that satisfis all th Gauss-Markov assumptions with normal rrors, with X having rank k, w know that th xprssion of th OLS stimator is 3bLs = whr RSS is th rsidual sum of squars coming from stimating th modl by OLS, var (a;,,) = 2 ~ ~5 - k and also 3iLs is unbiasd. Howvr, w can considr an altrnativ stimator 3: = y, whr s is som natural numbr. Not that this stimator can also b xprssd as 3: = (a) Find var (3:). (b) Among all th stimators of th form 3: (with diffrnt natural numbrs s), which on has th smallst man squar rror? 4. (25 points) An conomist wants to rlat th litrs of ptrol (Lt) that a truck nds in rlation to th mils (Mt) that runs. H/sh constructs a simpl linar rgrssion modl of th form Mi=P 1 + PL 2 t+%. W assum that this modl holds all th Gauss-Markov assumptions with normal rrors. Th sampl siz w hav availabl corrsponds to 7 obsrvations, and w know that (a) Find th 95% confidnc intrval for th y paramtr, whr y ="mils that w xpct that th truck will run whn 140 litrs ar introducd in its tank". (Not: you may us th fact that ( ) ( (b) Th conomist xpcts that th mils that th truck will run whn 100 litrs ar introducd in its tank ar xactly th sam as 100 minus th mils that th truck will run whn 200 ar introducd into its tank. Tst that hypothsis at th 5% significanc lvl. 7

3 5. (15 points) Th modl Yt= PlXlt + P2X2t + Ut satisfis all th classical linar modl assumptions. W dfin th (T x 2) matrix X = ( ~ ( ~ 1, whr T is th sampl siz. Bsids, X(1) = (X11,... X~T) know that - - x(~)), ', X(2) = (X21,... X2T) ' and,b2 # 0. W also W dfin 3, to b th OLS stimator of,b1 whr w only rgrss Y on X(l) (a) Is 3, consistnt for PI? (b) In cas in th prvious qustion (part (a)) you hav found an asymptotic bias, can you provid an xprssion for that asymptotic bias only as a function of paramtrs?

4 Appndix G Statistical Tabls TABLE 6.2 Critical valus of th t Distribution - Sigoificanc Lvl 1-Taild:.10 2-Taild:.20.OS.10, O 1.02.a05.O1 D g r S 0 f r. d CI m rn ,? Exampls- Th 1% critical valu for a on-taild tst with 25 df is Th 5% critical for a two-taild tst w~th larg (> 120) df 1s Sourc. This tabl was gnratd using th Statam function invt.

5 8 18 Appndix G Statistical Tabls TABLE G.3a % Criticat Valus of th F Distribution r- Numrator Dgrs of Frdom Erampl: Th 10% critical valu ior numrator df = 2 and dnominator df = 30 is 2.44 Sourc: This tabl was snrard usins th Stalaa function invfprob.

6 Appndix G Statistical TabLs TABLE G:3b 5% CriticaL Valus of th F Distribution I Numrator Dgrs of Frdom 1 Eranlpl: Th 5% critical valu for numrator df = 4 and larg dnominator dj(m) is 2.37 Sourc: This tabl was gnratd using th Stata" function invfprob.

7 820 Appndix C Statistical Tabls TABLE G.3c l0lo Critical Valus of th F Distribution Numrator Dgrs of Frdom :; D n 0 rn i a 0 r D g I d o rn Exampl: Th 1% critical valu for numrator df = 3 and dnominator df = 60 is Sourc: This tabl was gnratd using th Stata0 function invfprob.

8 Appndix G Statistical Tabls TABLE 6.4 Critical Valus of th Chi-Squar Distribution Significanc Lvl Erantp/: Th 5 4 critical valu wirh df = 8 is Sor~rc: This tabl was grrratd using th Statas function invchi.. -