Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy
Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he denominaor of he leas squares esimae is y j 3...... j y cross produc erms So E y E he variance of y depends on and explodes.
Wha abou he numeraor? We have: so yy y y yy y y y y Noe: y y y y or y y y y y y y y y y y y y Hence: yy y y Muliplying he las erm by y yy y 3
Since y y ~ N(0, ), ~ p So yy y We ve shown: yy so y yy y ˆ y y y 4
. he numeraor converges o a well defined disribuion. However he denominaor grows wih since we already saw E ˆ y D y Normally for he leas squares esimaor we would muliply by roo and ge ha ˆ converges o a well defined disribuion. Bu ha does no work here: since ˆ 0 y y E y 5
. Ineresingly he rae of convergence of he leas squares esimae is faser ha roo! 3. However, ˆ has a chi squared disribuion in he numeraor, and a denominaor ha doesn explode (bu we don know he acual disribuion ye). Wha does he denominaor converge o? Brownian Moion he sandard Brownian Moion is a coninuous ime sochasic process associaing each dae 0, wih he scalar W() such ha: W(0)=0 For any daes 0... k he changes W W, W3 W,..., Wk Wk are independen Gaussian wih Ws W ~ N0, s 6
he Funcional CL Basic CL says ha if is iid mean zero and variance : N0, D where Consider an esimaor ha only uses he firs half of he daa se he CL sill applies so ha N 0, D More generally, le r be any number beween 0 and. Le X Le s call X (r) he scaled parial sum hen by he CL r X r N 0, or r r X r X r N r D r 0, r r r D and 0, X r N r D or X r N r D 0, 7
Funcional Cenral Limi heorem (FCL) If we consider a sample using observaions r o r for r >r hen X r X r N 0, r r I shouldn be surprising ha D X W he fac ha X he funcional CL. D W resul is known as Coninuous Mapping heorem (CM) Le g be a coninuous funcional and le S S, hen gs gs. D D For example we have X W So X X W D D Or X W D 8
Back o our random walk regression: If y y hen y and le y 0 =0. Le X r his is jus a sep funcion. X r is, as before, he sum of r iid rv s. 0 for 0 r y for r y for r he inegral of he sep funcion is he sum of he recangles, each of widh / so: 0 X y r dr Or 0 y 3 X r dr y 9
Bu we know So by he CM X r W r 0 0 X r dr W r dr So 3 y W r dr D 0 We need o figure ou he disribuion of he sum of he squared y s (he denominaor). Le S r X () r S r 0 for 0 r y for r y for r 0
hen 0 S r dr y And from he CM and he FCL 0 0 S r dr W r dr Which means 0 y W r dr We worked ou previously: ˆ y We now know ha for he denominaor y Wr dr 0 So muliplying he op resul by on boh sides yields
So, ˆ y Wr dr 0 0 0 ˆ Wr dr Wr dr he probabiliy ha a Chi squared() random variable is less han uniy is 68%. So 68% of he ime, he esimaed slope value of a random walk is less han. Also, if he squared Normal in he numeraor is large, hen he square of he denominaor will end o be large as well meaning ha he disribuion will be negaively skewed. his is called he Dickey Fuller disribuion.
Some examples in EViews Wha abou regressing one random walk on anoher random walk? Generally, a non saionary process added o a non saionary process yields a non saionary process. A saionary process added o a saionary process yields a saionary process. 3
Spurious Regression Generally speaking (an excepion will follow) i s a bad idea o regress one uni roo process on anoher uni roo process. ha is, if y and x boh follow independen random walks. y y where y 0 and ~ N 0, y y 0 y x x where x 0 and ~ N 0, x x 0 x x x and independen. And we run he regression: ˆ y x x y 0 y x x x 0 x y x u W r W r dr W r dr ˆ y x y 0 y x x x 0 x W r W r dr W r dr his has expecaion zero since he numeraor is expecaion zero and he denominaor is a posiive number. Bu he esimae does no converge o any number! I is a random draw from a disribuion ha doesn depend on! 4
Simulaions and regressions from random walk models. Generally, we don regress one random walk on anoher random walk, insead, we work wih he differenced series insead of levels. ha is if he wo series x and y follow a random walk, hen ake he firs difference and model he differences in he series. y p 0 j x y β β y v j Regressing saionary series on saionary series no problem. 5
Coinegraion Coinegraion is a special relaionship ha wo non saionary series can exhibi. Someimes a pair of series migh each follow a random walk, bu over he long run heir pahs are ied ogeher. While hey wander around over ime, he wo series can ge far apar. Bid/Ask example. omao example. he $/Euro exchange rae and he price of convering from $/Yen and hen from Yen/$. 6
Price of omaoes per pound Bid and Ask prices 7.8 7.7 7.6 7.5 7.4 7.3 7. 7. 50 500 750 000 50 500 750 ASK BID 7
Formal Definiion of Coinegraion he series y and x are said o be coinegraed if boh y and x have a uni roo bu here exiss a linear combinaion z y x where z is covariance saionary i.e. doesn have a uni roo. describes he coinegraing relaionship, ofen i is (as in he preceding examples). A simple srucural inerpreaion of Coinegraion. Le p be he fair marke value of he asse a ime. a b Le p be he ask price a ime and p he bid price. he ask price should be slighly above he fair marke value and he bid below. he differences beween he prices are compensaion for he risk of leaving shares available (is basically an opion conrac). 8
We can wrie his as: p p a p p b So, and are paymens received by hose who pos shares a he ask and bid. I is compensaion for risk, i s he cos of being able o ransac immediaely (Zhang Russell say 007). I would be naural o hink of and as mean revering processes. If he paymens ge large enough, new orders will be placed closer o he fair marke value. b p a p p Noice hen ha: p p p p a b So while he ask and he bid are nonsaionary he difference beween hem is a saionary processes (acually he difference is he spread!) 9
esing for Coinegraion. CASE, KNOWN COINEGRAING RELAIONSHIP. Firs, es o see ha boh y and x have a uni roo. Creae sequence of z y x known value es he z series using a sandard Augmened Dickey Fuller es. If z is saionary hen x and y are coinegraed. CASE : UNKNOWN COINEGRAING RELAIONSHIP. es o see ha boh y and x have a uni roo.. Regress y on x and esimae. Creae he residual series y ˆ x z z y ˆ x hese esimaes of urn ou o no suffer from he spurious regression problem when x and y are coinegraed. 0
his is he one ime is OK o run his regression and in fac we ge super consisen (rae ) convergence. 3. es o see if z has a uni roo. We can use he sandard Dickey Fuller because he esimaion of used o consruc he z series messes hings up. We mus use anoher special disribuion ha is obained from simulaions. Eviews will perform his es for us. 3.765 Bid ask prices 3.760 3.755 3.750 3.745 3.740 3.735 3.730 05 79 495 376 4933 6099 770 8433 964 0693 85 95 435 577 647 7573 879 9887 06 9 3395 466 5806 6905 7990 9077 300 350 365 3367 LASK LBID
Case : Creae z y x series.006 Z.005.004.003.00.00.000 05 79 495 376 4933 6099 770 8433 964 0693 85 95 435 577 647 7573 879 9887 06 9 3395 466 5806 6905 7990 9077 300 350 365 3367 Perform he Augmened Dickey Fuller es P-value Null is ha z has a uni roo which means ha here is no coinegraion. We rejec he null ha z has a uni roo and conclude ha he series x and y are coinegraed.
Wha abou unkown coinegraing relaionship? Esimae from daa he value of by leas squares regression. Perform special augmened Dickey Fuller. Eviews Engle Granger Single Equaion es here are wo ways ha he es can be done depending on which variable is pu on he lef hand side. Each one has a p-value I use he firs p-values here. 3
Error Correcion Model (ECM) If y and x are coinegraed, hen i mus be he case ha he model for y and x follows wha is called he Error Correcion Model ECM (his was proven in he original Engle Granger paper (Economerica 987)). he model is specified as a VAR in changes in y and x, bu i includes a special erm on he righ hand side. Error Correcion Model (ECM) If he x and y are coinegraed hen you should fi an error correcion model. Since x and y follow a random walk, ake he firs difference of x and y. p y 0 z x y β β y α v j j j his is he usual VAR(p) par. z y x Where is he error correcing erm ha is new in he VAR specificaion and α. (recall ha bold y is a vecor, unbold y is no) his is he new par 4
So his is jus a regular VAR model for he changes in x and y bu i has he error correcion erm αz y x Ofen is, so his erm ells us how o updae changes in y and x as a funcion of he difference beween y and x a ime. Sep ) esimae coinegraing relaionship via regression of y on x Sep ) creae series Sep 3) esimae error correcion model, equaion by equaion. z would be an addiional righ hand side variable (exogenous). 5
If x and y are coinegraed hen he correc model is he error correcion model. If we simply fi a VAR on he changes, he model will be misspecified since we need boh changes and levels on he righ hand side of he equaion. he fac ha he VAR depends on z allows he series o correc for misalignmens hence he erm error correcion. A simple error correcion model for logarihmic bid and ask prices is given by: ln( ask) 0 ln( ask ) 3 Spd ln( bid) 0 ln( bid ) 3 where Spd z ln ask ln bid So 3 and 3 deermine how he bid and ask prices change as we vary he spread. hese dicae wheher he ask will rise or he bid will fall when he spreads ge wide. 6
Marke forces should force wide spreads o narrow. We should expec ha a wide spread should lead o an increase in he bid and a decrease in he ask. his is rue empirically wih 3 around.5 and 3 around.5. (we ge a lile larger values in our sample). ECM VAR 7
An addiional variable x can be included in he Error Correcion Model. ln( ask) 0 ln( ask ) 3 3 Spd x ln( bid) 0 ln( bid ) 3 3 A by sell indicaor (+ for buy, for sell) he (signed for buy or sell) size of he previous rade. Muliple variables can be included. Buys end o raise boh he bid and he ask. Sells end o decrease boh he bid and he ask. However, he ask ends o raise by more han he bid following a buy and he bid ends o fall by more han he ask following a sell. rade size maers. Larger rades have a larger price impac. he effec increases a a decreasing rae. 8