for regression Y ˆ ˆ corr Z, X 0 and

Transcription

1 Lecure Noes 9 Econ B2000, MA Economercs Kevn R Foser, CCNY Fall 202 Insrumenal Varables Regresson vald nsrumen, some Z for regresson Y ˆ ˆ 0 X u corr Z, X 0 and o relevance: o exogeney: corr Z, u 0 o nsrumen explans X bu NOT Y can be excluded from ls of varables explanng Y Two-Sage Leas Squares (TSLS or 2SLS) o X 0Z v, Y 0 X u o ge X 0 Z and regress Y on o ˆ szy s ZX X General Case: Y ˆ ˆ ˆ ˆ 0 X 2X 2 k X k o ˆ ˆ ˆ kw k2w2 krwr u o X are endogenous regressors o W are exogenous regressors Z, Z, Z are nsrumens o 2 m o f m>k hen "overdenfed"; f m=k hen jus denfed; f m<k hen undenfed o sll need: E u W, W, W 0 2 r X, W, Z are all..d. wh fourh momens W no perfecly collnear Insrumen Relevance and Exogeney Two-Sage Leas Squares: o regress X on Z o ge ˆX o hen regress Y on W and ˆX Evaluang Insrumens n he Real World o Weak nsrumens: check frs-sage regresson F-sa bgger han 0? o Examples: cgaree ax o fnd effec of prce prson capacy n place of jal erms

2 random varaon n brhs for class sze geography for hear aack reamen number of mmgrans 0 years ago for mmgran ncrease Marel boalf, oher polcy shfs deploymen of polce afer 9/ o esmae effecs of polce on crme o Bad examples of poor nsrumens: weak nsrumen: monh of brh on wage earnngs o Many bad examples where nsrumens needed: wage explaned by schoolng healh nsurance explaned by wage wage explaned by wegh (dscrmnaon agans fa people?) vs wage explaned by race/ehncy (dscrmnaon agans mnores) Heckman 2-sep for 2-par quesons: frs, "yes or no?"; nex "how much?" Lke 2SLS bu frs sage s a prob! Agan need an excluson resrcon, some varable ha explans he frs sep bu no he second. Two-Sage Leas Squares n SPSS: o run frs-sage regresson, save he predced values o use predced values n he second-sage predcon Expermens and Quas-Expermens deal: double-blnd random sor no reamen and base ses dfferences esmaor Problems can be nernal: o ncomplee randomzaon o falure o follow reamen proocol o aron o expermen (Hawhorne) effecs or exernal o non-represenave sample o non-rep program o reamen/elgbly o general equlbrum effecs Tme Seres Basc defnons: frs dfference Y = Y Y-

3 Y % Y percen change s Y and s approxmaely equal o ln(y) ln(y-) hs log approxmaon s commonly used lags: he frs lag of Y s Y-; second lag s Y-2, ec. Auocorrelaon: how srong s las perod daa relaed o hs perod? The cov Y, Y j j var auocorrelaon coeffcen s Y for each lag lengh, j. Somemes plo a graph of he auocorrelaon coeffcens for varous j. Saonary: a model ha explans Y doesn' change over me he fuure s lke he pas, so here's some pon o examnng he pas a crucal assumpon n forecasng! Bu hs s why we usually use sock reurns no sock prce he prce s no lkely saonary even f reurns are. If auocorrelaons are no zero, hen OLS s no approprae esmaor f X and Y are boh me seres! The sandard errors are a funcon of he auocorrelaon erms so canno properly evaluae he regresson. Seasonaly s bascally a regresson wh seasons (monhs, days, whaever) as dummy varables. So could have Y January February March November u - remember o leave one dummy varable ou! Or Y Monday Tuesday Saurday u. 0 2 Types of Models AR() auoregresson wh lag Y 0 Y u Forecas error s one-sep-ahead error Noe ha can re-wre he AR() equaon, by subsung Y 0 Y 2 u, as Y Y u u Y u u, hen subsue n for Y Y u, and so on. So he curren value s a funcon of all pas error T 2 T T Y 0 u u u 2 u T Y T. Noe ha as long as, he las erm drops and he sums converge as T. erms, 2 T Remnder of convergen seres: look a, noe ha 2 T 2 T T. Add and subrac parenheses o wre 2 T Noae ha ugly erm Z T Z and fddle he. 2 T 2 T T T, hen he equaon says ha T Z. Solve, Z Z Z, and hs no he prevous equaon for Y Z T. Subsue

4 Y u u u u Y T 0 2 T T 2 T T. As T, he frs erm goes o 0, he las erm goes o zero, and he mddle erm s u. If hen none of he erms converge he model becomes a random walk or negraed wh order, I() or has a un roo. (Can es for hs, mos common s Augmened Dckey-Fuller ADF.) Random walk means ha AR coeffcens are based oward zero, he -sascs (and herefore p-values) are unrelable, and we can have a "spurous regresson" wo me seres ha seem relaed only because boh ncrease over me AR(p) auoregresson wh lag p Y 0 Y 2Y 2... py p u ADL(p,q) auoregressve dsrbued lag model wh p lags of dependen varable and q lags of an addonal predcor, X. Need usual assumpons for hs model Lag lengh? Some ar; some scence! Varous crera (AIC, BIC, gven n ex) o selec lag lengh. Granger Causaly jargon meanng ha X helps predc Y; more precsely X does no Granger-cause Y f X does no help predc Y. If X does no help predc Y hen canno cause Y. Trends provde non-saonary models Random walk non-saonary model: Breaks can also gve non-saonary models es for breaks, sup-wald es Can model me seres as regresson of Y on X, of ln(y) on ln(x), of Y on X, or of %Y on %X (where, recall, %Y = lny snce he dervave of he log s he recprocal) hs s where he ar comes n! Dsrbued lag models can be complcaed (Chaper 5) and so we wan a a mnmum Heeroskedascy and Auocorrelaon Conssen (HAC) errors lke he heeroskedascy-conssen errors before (Newey-Wes) VAR Vecor AuoRegresson, ncorporae k regressors and p lags so esmae as many as k*p coeffcens works bes wh los of daa! GARCH models Generalzed AuoRegressve Condonal Heeroskedascy models allow he varance of he error o change over me, dependng on pas errors allows "sorms" of volaly followed by que (low-varance) 0 Facor Analyss Anoher common procedure, parcularly n fnance, s a facor analyss. Ths asks wheher a varey of dfferen varables can be well explaned by common facors. Somemes when 's no clear abou he drecon of causaly, or where he modeler does no wan o mpose an assumpon of causaly, hs can be a way o express how much varaon s common. As an

5 example. one prce ha people ofen see, whch changes very ofen, s he prce of gasolne. If you have daa on he prces a dfferen gas saons over a long perod of me, you would bascally see ha whle he prces are no dencal, hey move ogeher over me. Ths s no surprsng snce he prce of ol flucuaes. There mgh be neresng varaon ha a some mes ceran saons mgh be more or less responsve o prce changes bu overall he sory would be ha here s a common nfluence. Facor Analyss (and he relaed echnque of Prncpal Componens Analyss, PCA) are no model-based and can be useful mehods of exploraon. An example mgh be he eases way o see how works. I have daa from he US Energy Informaon Admnsraon (EIA) on he spo and fuures prces of gasolne from (Spo prces are he prce pad for delvery oday; fuures prces are prces agreed now for delvery n a few monhs.) The prces also dffer dependng on where hey were delvered snce he prce of gasolne vares over dfferen pars of he counry alhough we usually only hear abou when somehng goes wrong wh he sysem (e.g. a refnery mus be closed or a sorm damages a por or ppelne) and he varaon becomes large. We would have every reason o expec ha hese prces ough o be hghly correlaed. Wh SPSS we can use "Analyze \ Dmenson Reducon \ Facor". Ths gves us oupu lke hs: Toal Varance Explaned Componen Inal Egenvalues Exracon Sums of Squared Loadngs Toal % of Varance Cumulave % Toal % of Varance Cumulave % Exracon Mehod: Prncpal Componen Analyss. If you've aken lnear algebra you'll recognze he egenvalue as deermnng he common varaon. In hs case, lookng a he hrd column, "% of Varance," we see ha he frs componen explans % of he varaon n he 6 varables. The addonal facors (up o 6) make lle addonal conrbuon. So n hs case s reasonable o represen hese 6 prce seres as beng mosly (more han 98%) explaned by a sngle common facor. So from he oupu, Componen Marx a Componen

6 FuuresMonh.996 Fuures2Monhs.997 Fuures3Monhs.995 Fuures4Monhs.989 NYGasSpo.993 GulfGasSpo.985 Exracon Mehod: Prncpal Componen Analyss. a. componens exraced. Ths gves he "loadng" of he facor on each of he varables, whch s he correlaon of he facor wh he varable. In hs case s dffcul o perceve much dfference. For anoher example, consder daly daa on US neres raes a varous maures (from he Federal Reserve webse). The maures are he Fed Funds (overngh), 4 weeks, 3 and 6 monhs, year Treasures, and swap raes a, 2, 3, 4, 5, 7, 0, and 30 years. The oupu shows, Inal Egenvalues Toal Varance Explaned Exracon Sums of Squared Loadngs % of Cumulave % of Cumulave Componen Toal Varance % Toal Varance % E E Exracon Mehod: Prncpal Componen Analyss. We see ha wo prncpal componens explan over 95% of he varaon. The nal componen correlaon s Componen Marx a Componen

7 2 Federal funds effecve rae monh Treasury bll secondary marke rae dscoun bass 6-monh Treasury bll secondary marke rae dscoun bass 4-week Treasury bll secondary marke rae dscoun bass -year Treasury bll secondary marke rae^ dscoun bass swap wh maury of one swap wh maury of wo swap wh maury of hree swap wh maury of four swap wh maury of fve swap wh maury of seven

8 swap wh maury of en swap wh maury of hry Exracon Mehod: Prncpal Componen Analyss. a. 2 componens exraced. Whch s a b dffcul o nerpre. We can ask SPSS o roae he facors (clck he buon for "Roaon" and check "Varmax" whch s he mos common). For hose rememberng some lnear algebra, hs s an orhogonal roaon. The pon of roaon s o help nerpre he facors. A roaed facor loadng s: Roaed Componen Marx a Componen 2 Federal funds effecve rae monh Treasury bll secondary marke rae dscoun bass 6-monh Treasury bll secondary marke rae dscoun bass 4-week Treasury bll secondary marke rae dscoun bass -year Treasury bll secondary marke rae^ dscoun bass swap wh maury of one swap wh maury of wo

9 swap wh maury of hree swap wh maury of four swap wh maury of fve swap wh maury of seven swap wh maury of en swap wh maury of hry Exracon Mehod: Prncpal Componen Analyss. Roaon Mehod: Varmax wh Kaser Normalzaon. a. Roaon converged n 3 eraons. Where we can clearly see ha he frs componen s a shor-erm nnovaon wh effecs ha de off over longer maures whle he second componen s a long-erm nnovaon wh small effecs on shor raes bu larger effecs on long-erm raes. Ths nerpreaon s convenen and helps us undersand how neres raes n he US move. If one were hedgng neres rae rsk, here are a wde varey of nsrumens bu wo man componens so a frm could hedge 95% of s exposure wh wo secures. Economercs goes on and on here are housands of echnques for new suaons and new condons, especally now ha compung power quckly ncreases he amoun of calculaons ha can be done. There s so much o learn!