Vector autoregression VAR

Transcription

1 Vecor auoregression VAR So far we have focused mosly on models where y deends on as y. More generally we migh wan o consider models for more han on variable. If we only care abou forecasing one series bu wan o use informaion from anoher series we can esimae an ARMA model and include addiional elanaory variables. For eamle if y is he series of ineres, bu we hink migh be useful we can esimae models like y y 0 This model can be fi by leas squares. Our deenden variable is y and he indeenden variables are y and.

2 Once he model is fi, he one se ahead forecas is given by: E y F E y F E F y 0 0 Jus like he simle AR model, he one se 2 ahead forecas variance is. A oin model for and y is required if we are ineresed in mulile se ahead forecass, or if we are ineresed in feedback effecs from one rocess o he oher. E y F E y F E F 2 0 Wha do we use here? Answer: We need a model for as well. 2

3 The VAR() model Suose ha we have 2 variables ha we observe a ime eriod and we consider he oin model: y y y 0 2 y y y y 0 2 Each equaion is like an AR() model wih one oher elanaory variable. Each equaion deends on is own lag and he lag of he oher variable. We also now have wo errors, one for each y equaion: and 3

4 We can wrie he model in mari noaion: 0 2 y y y y y y 0 2 Since deends on y and y deends on, a more horough undersanding of dynamics and forecasing requires us o oinly consider and y in he sysem of equaions. Simlify Noaion By defining he following vecors and maricies we end u wih a very simle form for he VAR() model. Le 0 2 y,, 0, and y y y y y v β β 0 2 y β β y v 0 4

5 Assumions on he errors Assumion : he errors are uncorrelaed hrough y y ime (i.e. is uncorrelaed wih and. is y uncorrelaed wih and for 0). Assumion 2: The s are iid bu may be conemoraneously correlaed. denoes he variance covariance mari of he error erms. Variance of 2 y 2 y y Conemoraneous covariance Variance of y Trades and quoes eamle Le denoe he h rade in a given asse. Le denoe he log ransacion rice associaed wih he h rade and le r denoe he reurn r =. Le denoe wheher he rade was buyer or seller iniiaed (+=buy, =sell). 5

6 Trades and quoes equaions Then we sack r and in a vecor. Now a each oin in ime we have an observaion for he reurn associaed wih he ransacion rice and a variable for wheher he rade was buyer or seller iniiaed. Define: r r r r r 0 2 y,, 0, and v β β 0 2 A Firs order Vecor Auoregression VAR() is hen defined as: y β0 βy v where he elemens of v are iid Normal wih variance covariance mari E vv and E vv 0 for 0 For his model, he deendence of each variable on he as is summarized by he mari, and (condiional on he as) he conemoraneous deendence is deermined by he variance covariance mari of v. In general, y is an n dimensional vecor. 6

7 VAR for rades and quoes Inerreing he dynamics of a VAR() For he VAR model wih one lag we wrie: y β β y v 0 If we subsiue in for y we ge: y β β β β y v v β0 ββ 0 βy2 βv v Doing his subsiuion over and over, k imes, we k k * k * ge: y = β 0 β y k β v where β 0 β β

8 k k y * * = β k 0 β y k β v where β 0 β β Now y is a funcion of y k and a weighed sum of he inervening values of he error vecor v How much does he value of y change when we increase one elemen of y k by one uni, holding all revious y >0 rices and rades fied? The answer is obained by aking he derivaive of y wih resec o y k above. dy i, k d β y i, k, This means (i,) elemen of mari k When we lo β as a funcion of k, we see how i, fuure values of variable i are imaced by a one uni change in variable, k eriods in he as. This is called he imulse resonse funcion of variable i o a change in variable. This is he rimary mehod used o undersand he imlied dynamics of a VAR model. I answers he basic quesion of how a change in one variable affecs he sysem in he fuure. 8

9 So owers of he mari deermine how a change in one variable oday effecs he fuure values. Taking owers accommodaes he feedback effecs from one equaion o he oher in he righ way. A mari algebra rick makes he dynamics and decay of hese effecs a lile more ransaren. The decay is deermined by he eigenvalues. (more in a few slides) Noice ha if we wrie he VAR in is infinie MA reresenaion by coninuing he subsiuion we ge: * = 0 0 y β β v where I β * 0 0 The derivaive of y wih resec o elemens of as values of v k is he same as he derivaive wih resec o y k obained before: dy i, k d β v i, k, 9

10 The reason is ha if we hold all as values of y fied and change an elemen of y by one uni, ha is equivalen o making he ime error esilon one uni larger. y -k y -k y β β y v k 0 k k hold consan Recall our eigenvalue decomosiion β T T β T T Where is he diagonal mari wih and 2 he eigenvalues of he mari T is a mari wih corresonding eigenvecors and 2. * = y0 0 T T 0 y β β v 0

11 The resonse of elemen i, o a shock in elemen, k eriods ago is given by: dy dv β T T a b i, k k k k i, i, i i 2 k, Where a i and b i are deermined by he elemens of he eigenvecors in T Using he lag oeraor noaion, we can wrie I βl y β0 v If he eigenvalues of lie inside he uni circle, hen I βl has an inverse given by: I β L I βlβl βl since I β Lβ L β L β L I β L I β L k k k k k k lim I β L I k

12 dr k Problem, T T assumes ha when d,2 k you change one innovaion (error) you hold he oher conemoraneous values of he innovaions fied. Changes in one error will be correlaed wih changes in he oher errors in he same ime eriod if he variance covariance mari is no diagonal he errors are correlaed so ha movemens in one innovaion end o be associaed wih movemens in he ohers.? For eamle if I ell you ha he rice wen u more han eeced (a osiive error) ha migh also make us more likely o hink ha he rade was a buy (a osiive error). We have o make a decision abou how o handle his conemoraneous correlaion. Ignore? Assume ha we change one error wihou changing ohers. Tha may be he quesion you wan o ask in a olicy siuaion where you believe you can conrol one variable. Make some oher assumion? 2

13 Common soluion: Take a sand on he way ha he shocks roagae. Tha is, allocae he correlaion o one direcion only. Do rades conemoraneously cause rices o change, or he oher way around? The answer o his quesion canno be assessed urely wih saisics. We mus bring economic ideas o bear on he quesion. Choosing an order ha shocks roagae is equivalen o a choice of orhogonalizing. Marke srucure of rades and quoes: There is a naural ordering in he marke. The bid and ask are osed. The rader rades a he revailing bid or ask rice. Prices influence rades, bu because rices are se rior o he rade, rades don conemoraneously affec rices. 3

14 How do we imose he one direcional causaliy in he innovaions? We creae a vecor of orhonormal innovaions and hen re mulily by a lower riangular mari. We wan v =Pu where Euu I and P is lower riangular. Since P is lower riangular, a movemen in he firs elemen of u changes all elemens of v while movemens in he second elemen of u only affec he second elemen of v. Any variance covariance mari can be decomosed ino =PP where P is a unique lower riangular mari (Choleski Decomosiion). Noice ha P P I n Ne, we can ge he vecor u by remulily he error vecors by P. u =P v. Le s verify: uu vv E E P P P P P P I n So he u are now a vecor of shocks wih uncorrelaed elemens wih uni variance. 4

15 Finally, u =P v so v =Pu So 0 u u v y 2 22 u 2 2u 22u 2 u and u 2 are uncorrelaed. If we move u by one uni, changes by and y changes by 2. If we move u 2 by one uni, doesn change a all and y changes by 22. Imlicily, we have imosed ha he reason ha and y are correlaed is because changes in cause changes in y, bu changes in y do no cause changes in. Hence he lower riangular form of P means ha changes in he variables ha aear in he o of he vecor conemoraneously affec variables below. We can choose he ordering and herefore he order in which he shocks roagae. 5

16 In Eviews, he ordering of he variables in he VAR equaion definiion deermines he ordering of he shocks. The firs variables affec he las variables. * * = y0 0 = y0 0 P 0 0 y β β β v y β β β u So P deermines how moving one variable in eriod affecs oher variables conemoraneously. Powers of, deermine how fuure values of y will change. 6

17 Back o our rade Eamle r r r e e ˆ P Eeced resonse of y given a one s.d. increase in r k Recall * = y0 0 P 0 y β β β u * y= βy0 β u k dy E F r du k k How ha shock affecs y k-eriods ahead. Iniial sd shock o v When we make he reurn one sandard deviaion larger, we also make he rade indicaor larger. 7

18 k k Eigenvalues of ˆ Eeced resonse of y given a one s.d. increase in k * y= β u k dy E F k du k k dy E F k du k where When we make he rade indicaor larger we don change he value of he reurn. k k 8

19 Resonse o Cholesky One S.D. Innovaions ± 2 S.E. Resonse of DMIDPRICE o DMIDPRICE Resonse of DMIDPRICE o XX Resonse of XX o DMIDPRICE Resonse of XX o XX When is he VAR() model weakly saionary? Le I β 2 3 L I βlβlβ L... which eiss as long as he eigenvalues of lie inside he uni circle. β k T k T The variance of each series will eis if all he eigen values lie inside he uni circle. So he VAR() model is saionary as long as he eigenvalues of lie inside he unie circle. 9

20 VAR() model y β β y v where he elemens of v 0 Normal wih variance covariance mari E are In general, y is an n-dimensional vecor and is nn. vv Dynamics of he VAR() Rewrie he VAR() model as: I β Lβ L β L... β L y β v Facor he lag olynomial: I LI 2L... I L I BL B L B L B L where are a nn maricies. n is he dimension of y (2 in he rades and quoes eamle) 20

21 As long as every mari has eigenvalues inside he uni circle, he VAR() model is weakly saionary. We can re wrie he VAR() model as an infinie order vecor MA model as in: 0 nn y = μ v where I μ β β β β 0 Imulse Resonse Funcions for VAR() models We sill mus ake a sand on he order he shocks roagae = P 0 y μ u 2

22 How do we esimae VAR s? Esimaion is sraigh forward. Firs, ick. Can be he same for all variables in y or we can have a differen for each variable. Usually sa ackages use he same value of for all equaions. Esimae he model equaion by equaion using OLS us as we did for he univariae models. Sandard errors For a saionary VAR() model, under he usual regulariy condiions (see Proosiion. Hamilon). Le and vec Then Where 0 T ˆ ~ N0, Q nn ( ) nn ( ) where is he Kronecker oeraor Q E and nn ( ) nn y y 22

23 If i are he arameers of he model for y i hen we ge he usual leas squares resul: T T ˆ 0, ˆ i i N i, i Where ˆ ii, T e T n i Imulse resonse sandard errors The imulse resonses are read off he elemens of he MA reresenaion. = 0 y μ v The elemens of he are us (nonlinear) funcions of he arameers in 23

24 The dela mehod (roosiion 7.4 Hamilon) ells us ˆ ˆ ˆ T vec ˆ 0, ˆ N G Q G 2 2 nn dvec where ˆ G 2 d ˆ n n VAR() eamle for rades and rices Se one esimae model for ransacion rices using lagged values of ransacion rices and lagged rades. Se wo esimae model for rades using lagged values of ransacion rices and lagged rades. 24

25 25 Covariance mari of errors is esimaed by: We have he esimaed model: r r r e c r c c e b r b b r 0 0 ' ˆ 2 T ee T Where b and c corresond o esimaed arameers. Does he model fi he daa? If he model is well secified, he residuals should be uncorrelaed. We can eamine he residuals of each equaion and check if hey are uncorrelaed wih heir own as. We can also check o see ha hey are uncorrelaed wih he residuals of he oher equaions.

26 Eamine los of samle correlaions of he residual vecor: T ˆ l ee T n ' l Noe ha his conains cross correlaions. Criical value is given by T 2 n Forecasing VAR s Le E denoe he k se ahead forecas yk of y +k i.e.. Le denoe he esimae of he h mari and 0 denoe he esimae of he inerce vecor. The one se ahead forecas is given by: E E y β0 β E y y β β y 0 26

27 The 2 se ahead forecas is given by: E E y 2β0 β E y 2 E y β β y β y The k se ahead forecas is given by: E E y β0 β E y k k k 0 E y β β y β y k k k k Granger Causaliy Causaliy is difficul o define. Granger Causaliy is one way of rigorously defining causaliy in a saisical seing. Buil on wo assumions: The fuure canno cause he as. A causes B if A has unique informaion ha elains B. 27

28 From a general oin of view hen, le F denoe he universe ALL informaion available a ime. Le F Z denoe all informaion a ime ece Z. Then Z fails o cause Y if for all s>0: Pr Y F Pr Y F Z. s s To make his feasible o imlemen, we mus define he universe in a way we can handle. To his end we usually secify a vecor of informaion. We evaluae differences in forecass using a mean squared error crieria. 28

29 Consider he bivariae sysem of and y. We say ha fails o Granger Cause y if: MSE Ey y, y2,... MSE Ey y, y2,...,, 2,... A naural feasible es is herefore o esimae wo models for y, one ha includes lagged and y and one ha only includes lagged y. Then comare he MSE of he wo models. This is he usual GC es. Resriced model: GC rocedure y y 0 Unresriced model y y 0 Le RSS 0 T ˆ 2 and T 2 RSS ˆ Then he es sa is: T RSS RSS S RSS 0 2 ~ ASM 29