Lecture 1: Empirical economic relations

Transcription

1 Ecoomcs 53 Lctur : Emprcal coomc rlatos What s coomtrcs? Ecoomtrcs s masurmt of coomc rlatos. W d to kow What s a coomc rlato? How do w masur such a rlato? Dfto: A coomc rlato s a rlato btw coomc varabls.

2 Ecoomcs 53 Eampls: Producto fucto: rlato btw output of frm ad puts of labor, captal, matrals. Egl curv: rlato btw pdtur o a commodt ad houshold com. Phllps curv: rlato btw flato ad umplomt rats. Eargs fucto: rlato btw args ad ducato, work prc. All ths rlatos ca b prssd as mathmatcal fuctos f (,, K ) wth th dpdt varabl th rlato ad varabls.,, K th dpdt 2

3 Ecoomcs 53 Eampl: Eargs fucto args 2 2 ducato 3

4 Ecoomcs 53 : straght l, lar rlato + 2: o-lar rlato Ecoomc thor usuall dos ot spcf fuctoal form, but t ma sg drvatvs. From th Natoal Logtudal Surv of Youth (NLSY) w obta (usual) wkl args ad ars of ducato for a sampl of 935 dvduals logarthm of usual wkl args of ars of ducato of W plot th 935 pars (, ) a dagram that s calld a scattrplot (s fgur) 4

5 Ecoomcs 53 5

6 Ecoomcs 53 Not Rlato s ot a smpl mathmatcal fucto. Rlato s ot v a mathmatcal fucto, bcaus for sam lvl of ducato w ma hav 2 or mor lvls of pdtur. 6

7 Ecoomcs 53 Frst attmpt to masur a coomc rlato sms a total falur! Th problm s that coomc rlatos hold ctrs parbus,.. holdg all othr rlvat varabls costat. I ampl args also dpds o work prc, ablt tc. Th act rlato s f (, 2,, M ) but th scattrplot,, 2 M ar omttd or assumd to b costat. Scattrplot s th two-dmsoal projcto of a rlato volvg ma varabls. 7

8 Ecoomcs 53 Eampl: Eargs fucto M M ducato othr rlvat vars Ev f rlato btw args ad ducato s lar, th obsrvd valus of ths varabls satsf + + wth M M th cotrbuto of th uobsrvd, but rlvat varabls. 8

9 Ecoomcs 53 Fttg a straght l Assum that th rlato btw ad s dd lar ctrs parbus. How do w masur t,.. how do w masur,? 9

10 Ecoomcs 53 ŷ

11 Ecoomcs 53 Ida: choos, such that th l fts th obsrvatos as wll as possbl. Qualt of ft s masurd b dvatos,,,,, Dvatos ca b postv or gatv. As ovrall masurs of ft w ma cosdr Sum of absolut dvatos Sum of squard dvatos 2

12 Ecoomcs 53 For rasos that wll bcom clar latr o, w prfr th sum of squard dvatos. W obta, as th soluto to m, 2 2

13 Ecoomcs 53 Df (I omt subscrpt o ) S 2 2 ) ( ), ( W obta, b solvg th frst-ordr codtos ) ( 2 ), ( S ) ( 2 ), ( S Ths frst-ordr codtos ar calld th ormal quatos. 3

14 Ecoomcs 53 Soluto: ˆ ( )( ( ) 2 ) ˆ ˆ wth, th sampl avrag of,. Ths mthod to obta, s calld (Ordar) Last Squars (OLS) 4

15 Ecoomcs 53 Som proprts of th OLS soluto: Df OLS rsduals b ˆ ˆ ad th fttd or prdctd valu of b ˆ ˆ ˆ + 5

16 Ecoomcs 53 From th frst ormal quato ad from th scod ) )( ( Cocluso: sampl avrag of ad covarac of ad ar. From soluto for ˆ ˆ ˆ ˆ + Hc th sampl avrag of th fttd valus s qual to that of. 6

17 Ecoomcs 53 Eampl: Eargs fucto OLS soluto for th bst fttg lar rlato btw ˆ 5.45 ˆ.6673 Itrprtato of slop coffct: rtur to o ar of addtoal ducato s 6.7%. 7

18 Ecoomcs 53 Fttg a lar fucto Cosdr a lar rlato wth K varabls K K Trmolog s dpdt varabl varabl to b plad rgrssad s ( k k th) dpdt varabl plaator varabl rgrssor covarat 8

19 Ecoomcs 53 Wth obsrvatos w hav K K +,,, 9

20 Ecoomcs 53 Matr otato -vctor -vctor K K X ( +) K matr K + K -vctor 2

21 Ecoomcs 53 Hc w ca wrt X + ad sum of squard rsduals ) )'( ( ) ( ) ( 2 X X S K K 2

22 Ecoomcs 53 OLS soluto S ( ) ' 2' X + ' X ' X Frst-ordr codto for mmum S ( ) 2X ' + 2X ' X Hc, th OLS stmator ˆ satsfs th ormal quatos X ' X ˆ X ' ad hc f X ' X has full-rak, th ˆ ( X ' X ) X ' 22