Advanced ime-series analysis (Universiy of Lund, Economic Hisory Deparmen) 30 Jan-3 February and 6-30 March 01 Lecure 9 Vecor Auoregression (VAR) echniques: moivaion and applicaions. Esimaion procedure. IRF and moivaions for SVAR. 9.a Defining a VAR, noaion: Vecor auoregressive models can generally be defined as follows: p Y Θ Y βx ε i -i i1 where ( y, y,..., y ) is a vecor k endogenous variables (k-variae VAR). These variables are hough Y 1 k o be deermined wihin he sysem. (Remember wha you learn during he lecure on simulaneous equaion models.) Wha we here define is ha we have k equaions, one for each endogenous variable, δ is a vecor of equaion specific consans (so i is again a vecor wih k elemens). Wha makes he sysem auoregressive is ha each endogenous variable is regressed on is own lagged values plus he lagged values of all oher endogenous variables ( Y ). The lag lengh (p) deermines he order of he VAR. Θi is obviously a kxk marix of coefficiens. We can augmen he equaion by including exogenous variables denoed by x. These variables now included in marix exogenous variables. ε is a vecor of k elemens, denoing he residuals. 1 -i X. β is he coefficien vecor of he This is now a bi oo difficul o see, so le us simplify his sysem o a wo-variae case: Then we have y 1 and y as dependen variables. The sysem, wihou exogenous variables, can be wrien as: y 111 1 1 11 1 11 1 1... 1 1 y y y y p11y1 p p1 y p 1 y y y y y... y y so: 11 1 1 1 1 1 1 p1 1 p p p y1 1 p11 p 1 Y, δ, Θ p, ε y p1 p You can wrie his in a polynomial form: Θ( L) Ik Θ1 Θ... Θ p hen: 1 Θ( L) Y βx ε, his is now an objec ha you can already rea similarly o he univariae ime series models: ( L) ( L) ( L) 1 1 1 Y Θ Θ βx Θ ε his is going o be an infinie order VMA (vecor moving average) model. The expeced value of a VAR(p) is: 1 1 E( Y ) Θ( L) Θ( L) βe( X ) in he presence of exogenous variables and 1 E( Y ) Θ ( L) in he presence of consans only
Obviously, a finie expeced value will require ha Θ( L) is inverible, so here are saionariy condiions here as well. The equaions of he VAR sysem can be esimaed wih an OLS, since hey are ARDLX(p,p) ype models, and he residual will be saionary. You do no have o esimae he coefficiens simulaneously, you can go by equaions. Of course, since he residuals may be correlaed wih each oher, OLS is no guaraneed o be he bes esimaor (you can use a SUR insead). Obviously you have o make some choices when esimaing a VAR. You need o decide which variables are endogenous and which are exogenous. Since he righ hand side of he equaions will conain only predeermined and exogenous variables, you do no need o worry abou simulaneiy bias. You need o choose he order of he VAR sysem (p). This is done wih informaion crieria, jus like you would do in he single equaion case. 9.b. The idea behind a VAR Unil he 1970s macroeconomerics had used simulaneous equaion sysems ha we already covered earlier during his course. This required ha one precisely deermined he relaionship among variables, and also required ha you had a leas he key equaions idenified so ha you can esimae he coefficiens. This approach has been criicized following he Lucas criique ha raised he problem ha even he coefficiens of such equaion (behavioral parameers) may change over ime, so policy recommendaions and forecasing from such sysems may be misleading and invalid. During he 1980s Sims suggesed he VAR as an alernaive soluion. This heoreically reaed he economic sysem you were o esimae as a black box: oupu (endogenous variables) unknown process (modelled as a VAR) inpus (exogenous variables and shocks) The main idea was ha you do no precisely specify a sysem of equaions bu raher you use a very general VAR sysem and you draw conclusions based on he reacion of he endogenous variables on changes in exogenous facors. These are he impulse response funcions. Sims argued ha using a VAR has obvious advanages, like: you need no o be very cerain which of your variables are exogenous and endogenous, i is heory free (now, i is dispuable wha his means), and you need no o worry much abou idenificaion. (My personal view is ha he las wo saemens are simply false.) Bu wha is a VAR? So far VAR has been defined as a compleely new ype of model bu his is misleading: VAR is acually a se of reduced form equaions form a sysem of simulaneous equaions. (This is he reason why i is beer if you learn simulaneous equaions mehods firs ) Le us assume ha we have wo macroeconomic variables, say, real aggregae income (y ) and rae of inflaion (π ). We believe ha hese are in simulaneous relaionship bu have a degree of hyseresis (so hey can parly be explained by heir pas values). y 10 11 1 y 1 u, u IID(0, ) 0 1y 1 v, v IID(0, ) u v
Of course, if you have some pre-knowledge, you see ha boh equaions are exacly idenified (if you do no see his, please read he noes for lecure 8 again). Le us derive he reduced form equaions: y y 1 11 1 1 1 1 y 1 1 1 Wha you have obained is a VAR(1) sysem wih: u 11v 10 110 1 11 v 1u 1, 1, 11, 1,, 1 1 1 1 1 11 1 0 110, 1 11 1 11 1 1 1 1, 1 111 11 1 1 11 1 So all VAR sysems are already based on an assumed sysem of equaions, which necessarily involves some heoreical consideraions and a se of assumpions. A mos you are no aware of hem, bu ha will no make he mehod free of heory. This is one principal reason why VAR should no be labeled heory free. You should be aware ha 1 and are correlaed, since hey boh depend on u and v. The pracical applicaion of VAR is like follows: firs you choose a proper VAR represenaion and you draw conclusions on he relaionship among he endogenous variables based on IRFs. The IRFs for y y y are simply:, and for π:, for i=0,,t. In words, he IRF will show you how 1 i i 1 i i an endogenous variable reacs on innovaions in is own value (own error erm) or on an innovaion in anoher endogenous variables (error erm of anoher equaion). The problem of simulaneiy and IRFs becomes apparen when you look a he values of he η s. IRF is only useful if he wo eas can be subjec o a shock independenly. Bu how is i possible ha only η 1 experiences a shock if i is correlaed wih η? Well, he answer is ha i is no possible. So, unless your endogenous variables are uncorrelaed (so hey are no simulaneously correlaed ) you need o ell how hey are relaed. When you ake accoun wih he conemporaneous correlaion srucure of your shocks, wha you are doing is a Srucural Vecor Auoregression or SVAR. For a SVAR you need o have some ideas (a heory) in order o explain he conemporaneous correlaion srucure of your residuals. This is he second reason why you canno call a VAR heory free. 9.c Basics of a SVAR 11 1 1 11 For hinking in erms of an SVAR we use he above example of real GDP and inflaion. you have he following primiive (or srucural) form of he VAR sysem: y 10 11 1 y 1 u y v 0 1 1 This can be rewrien as: y 10 0 11 y 0 11 y 1 u 0 0 v 0 1 1 1 We can rearrange his as follows: 3
1 11 y 10 0 11 y 1 u 1 1 0 1 0 1 v The square marix a he lef-hand side conains he coefficiens of he conemporary relaionships (his is ofen referred o as marix A. You can arrive a he VAR (or reduced) form by aking he inverse of his marix A and premuliply boh sides by i. 1 1 1 y 1 11 10 1 11 0 11 y 1 1 11 u 1 1 0 1 v 1 0 1 1 1 1 1 11 1 1 11 1 1 11 1 111 1 11 1 We know ha 1 1 1 111 1 1 1 1 1 111 1 11 1 Which will give you acually he same coefficiens ha we already had earlier (check i!). So in general, all VAR can be wrien in a primiive and in a reduced form: Primiive form: p AY α B Y u i i i1 reduced form (which is he VAR): p p 1 1 1 i i i1 i1 i -i Y A α A B Y A u δ Θ Y e Obviously, unless marix A is an ideniy marix (having ones in he main diagonal and nulls elsewhere), your residuals in he reduced form (your VAR) will be correlaed. In he simples case you do no care abou his, his is called he basic case of VAR. This is very useful if you wish o make forecass, bu once you wish o know somehing he relaion of your variables (and his is wha you in mos cases wan) he IRF will be misleading. So acually you need o know he elemens of marix A, and when you do his, you are doing a Srucural VAR. And SVAR involves a lo of heory. Using VAR will no save you from having some ideas abou he underlying srucure of your daa. 9.d A basic VAR Le us ge down o business! We have daa for inflaion (infl) and he log of real GDP (lny). We will now esimae a wo-variae VAR sysem. Firs we need o choose he order of he sysem. For his purpose we esimae a simple VAR(), which is he defaul seing in Eviews: 4
You could add exogenous variables as well, like seasonal dummies, addiional dummies for policy changes, rend or any oher variables as your model requires. We do no do so for his exercise, bu feel free o experimen! We have significan coefficiens, high R-squared (do no be surprised, he log of real GDP is likely o be non-saionary). Obviously, a good sraegy would be o add lags as long as he informaion crieria are falling. Forunaely, Eviews has a buil-in process for his: 5
You can see ha he sofware pus a sar o he lag-lengh ha has been found o yield he lowes informaion crierion (or predicion error) value. Mos crieria sugges a VAR(4) model be he bes, excep for he Schwarz, bu his is no surprising, his crierion ofen sugges a more parsimonious model han he res. Since we have quarerly daa, common sense also sugges ha we should move for a VAR(4). I is worhwhile o es if he residuals fulfill he assumpion of he classical linear model: 6
I seems ha he residuals have no serial correlaion a 1% level of significance, bu are heeroscedasic and in case of he second equaion (inflaion) hey are also no normally disribued. We can live wih hese, especially since we canno do much abou hese (you could idenify possible breakpoins). If you find he residual exhibiing serial correlaion you can experimen wih adding furher lags. Now, if you are ineresed in he relaionship beween he rae of inflaion and he log level of real GDP, you normally esimae an impulse-response funcion (IRF). 7
You can deermine he ime horizon of he IRF, you can ask for CIRF. In he Impulse Definiion able you can ell Eviews wha ype of shocks you have in mind. If you have good reason o believe ha he residuals are uncorrelaed (no conemporary relaion beween he wo endogenous variables or he marix A is an ideniy marix), you can go for a simple one sandard deviaion innovaion in he residuals. Of course, you can have a one uni shock as well, bu do no forge ha increasing lny by one uni means a growh by a facor of.71, while in he case of he rae of inflaion i means one percenage poin. So i seems a beer idea o se he magniude of he shock o one sandard deviaion of he variable, which is abou 0.49 for he log GDP and 3.4 percenage poins for inflaion. If we ask for a horizon of 30 quarers (7.5 years) we obain he following IRFs: 8
The resuls ells us ha in his case of a one-ime shock of one sd. deviaion in log GDP he effec on he log of GDP will no rever o zero even afer 30 periods (i is non-saionary ), while i seems o have a ransiory posiive effec on he rae of inflaion. In case of a one-ime shock in inflaion, we find ha i has a negaive, seemingly permanen effec on log GDP, and a ransiory effec on iself. 9.e. Srucural facorizaion Bu are hese resuls saisfacory? I depends on our beliefs abou he underlying economic process. Do you believe ha real GDP does no affec inflaion and vice versa in he same quarer? If he answer is no, you need o have a srucural facorizaion (ha is, you need o ell how he marix A looks). From his poin on, you are doing SVAR. Wha are he possible soluions? Firs, you can assume, ha boh inflaion and GDP affec each oher. Then your marix A is: 1 a 1 A his will cause a problem, however, since his means ha you canno separae he 1 a 1 wo residuals. We already found ou ha he residuals (e ) from he VAR sysem are going o be: 1 A u e from his you arrive a he following: u Ae. You already know ha in he primiive form 0 u he residuals u were independen of each oher, so E( uu ).(his is he covariance 0 v marix of u ). Le us no forge ha you do no know u so here you have wo unknowns. We also know ha he VAR residuals ( e ) can be correlaed if A is no an ideniy marix: c c u c u c v e c u c v, e c u c v A 1 u 11 1 11 1 c1 c v c1 u c v. Which yields k equaions: 1 11 1 1 obviously his will no be idenifiable, since e 1 and e are boh unknown. Now you need o ake he covariance marixes o idenify he SVAR: A uua ee A E uu A Σ where: 1 1 1 1 e 1 e1e Σ E( ee) and e e 1 e his yields: E( uu ) 0 c11 u c1 v c11c 1 u c1c v c11c 1 u c1c v c1 u c v u 0 v e 1 e1e c 11 c1 0 u c11 c1 c11 u c1 v c11 c1 c e 1 c 0 c1 c c1 c e 1 e v c1 u c v Which leads o four equaions: 9
c c e1 11 u 1 v c c c c e1e 11 1 u 1 v c c c c ee1 11 1 u 1 v c c e 1 u v Due o symmery he nd and 3 rd equaion are he same. You have now six unknowns (c 11,c 1,c 1,c,, ) and hree equaions so you canno solve his. Of course, some resricions are u v sraighforward: he main diagonal has only ones (his inroduces wo resricions: c 11 = c =1), so now you have hree equaions for four unknowns. This is sill no enough you need a leas one resricion ye. Possible soluion sraegies: 1. Arguing ha he wo variables (log of GDP and rae of inflaion) are nor conemporaneously correlaed. This is anamoun wih seing A=I (c 1 = c 1 =0, c 11 = c =1) hen you have:, 0, 0,. Now you have wo addiional resricions, yo you e1 u e1e ee1 e v overidenify your SVAR. This can be done, so his would lead o valid (bu no necessarily correc) resuls.. Arguing ha changes in one variable can affec he oher variable immediaely, bu no vice versa. This is he lower riangular marix approach, a.k.a., Cholesky facorizaion. If we assume ha he firs variable affecs he second bu no vice versa. Then A is: A 1 1 0 so a1 1 1 0, so he equaions are: e 1 u, e 1e ee a 1 1 u, e a 1 u v a1 1 This also means ha you argue ha he a shock in he value of he firs variable is compleely responsible for is measured variance, while observed variaion in he second variable is a linear combinaion of he shocks o boh variables. Tha is, e depends on e 1, while e 1 does no depend on e. Now you have idenified your SVAR exacly, as you inroduced only a single addiional resricion. 3. Finally, you can have any heoreically based resricion as long as he SVAR is idenified. In his case you should add one resricion. If you had 3 endogenous variables, you needed o have 3 resricions on A besides seing he elemens in he main diagonal o zero. The general rule is o have k(k+1)/ resricions. The Eviews has a bi differen noaion: Ae Buwhere e denoes you residuals from he VAR and u are he shocks. We assume ha 1 0 0 u where E( uu ) I 0 1. So marix B is se in ha was ha: E( BB ). Acually 0 v wha we called u earlier is now denoed as Bu. For he res i is he same as before. Le us apply his knowledge o our daa! Srucural facorizaion is done as follows. You can define resricions on A in wo ways: by equaions or by marices. 10
Now I show you he equaion version (for he marix version you need o define a marix A and anoher Marix B, wih NAs for he elemens ha you wan o esimae). @e denoes elemens of he VAR residual vecor, @u are he shocks (basically, he error erms from he primiive form equaions). When you assume no conemporary relaionship you would wrie: @ e1 c(1)*@ u1 @ e c()*@ u Tha is, he observed VAR residuals for each equaion depend only on he shocks of heir respecive errors. The oupu ha you receive shows you ha he SVAR is now overidenified (you needed a leas 1 resricion bu we have wo). Plus, you obain he resricions in equivalen marix form (marix A and marix B) as well. Now you obain he IRFs as follows: 11
No surprisingly his is exacly he same as before, wihou srucural facorizaion, since his is he baseline case. You can assume ha innovaions in he log of GDP affec he innovaion sin inflaion bu no vice versa. This is wrien as: @ e1 c(1)*@ u1 @ e c()*@ u c(3)*@ e1 Now we ge: 1
The SVAR is now exacly idenified, bu we find he esimaed elemen of he A marix saisically insignifican which suggess ha our assumpion regarding he conemporaneous correlion srucure may be wrong. The IRFs are jus a lile bi differen. You can have Cholesky-ype facorizaion direcly from your response opion, wihou he need o esimae a facorizaion like his. Do no forge, however, ha wih his ype of facorizaion he order of your variables couns. So you should have he mos exogenous variable firs and so on. 9.f. Anoher example Le us generalize wha we learn here o a more complex model. We will esimae a sysem wih log real gdp, log of M1, log of CPI and bilrae. (You can guess ha his is now a sandard money demand equaion.) For lag lengh we obain: 13
Now we prefer a VAR sysem of order 6. Do no forge o look a he residual saisics! For he IRF we obain: This is now under he assumpion ha he endogenous variables of our sysem are conemporarily uncorrelaed, so i is possible o have a shock in one of hem wihou necessarily having a shock in any oher. Le us hink abou he feasibiliy of his! We can indeed assume ha he reasury bill rae is indeed conemporary exogenous, since is value is se for 3 monhs. You can assume ha lncpi depends on lnm1, he log of real GDP and lncpi depends on ineres rae. 14
Of course a lo of oher sraegies are possible, bu le us ake his now. Wha we had jus said abou he variables yields he following marix A: 1 0 0 a14 a1 1 0 a4 A or wrien as equaions: 0 a3 1 0 0 0 0 1 We obain: 15
Since we used 8 resricions (and we need a leas 4*3/=6), we overidenifed he sysem. We find ha only one of he expeced correlaions (beween lncpi and lnm1) is no significan, we could even se i o zero. Feel free o experience furher wih reasonable resricions! The IRFs are: You find jus small differences compared o he baseline case, bu now, for example, he reacion of real GDP on a shock in reasury bill rae is differen han in he firs case. 9.g. Long-run resricions Resricing marix A was abou defining immediae of conemporary relaionships. For his reason his is called a shor-run idenificaion approach of SVARs. You can alernaively go for defining longrun relaionships as well. Using he marix A we could wrie a VAR(1) sysem in he following form: Y A δ A Θ Y A u 1 1 1 1-1 In order o be able o say somehing abou he IRFs, you need o conver his o an VMA( ) form: ( I A Θ L) Y A δ A u Y ( I A Θ L) A δ ( I A Θ L) A u So: 1 1 1 1 1 1 1 1 1 1 1 1 e ( I A Θ L) A u A u A u A u... 1 1 1 1 1 1 1 1 1 e e e... I A Θ u u u 1 Le 1 Ψ -1-1 16 which is he accumulaed IRF or CIRF. This says us o how much he effec of a shock in he residual is going o sum up in he long-run. Or alernaively,
his is he long-run effec of a permanen increase in one of he endogenous variables. We call his marix ha has he long-run effec C. -1 C ΨA, marix C will be a kxk marix wih each elemen having he long-run impac of shock in variable j on variable i. Le us reurns o our log GDP inflaion example, where k=. If we believe ha inflaion should have no effec on he log of real GDP in he long-run (classical dichoomy) han you should use a long-run resricion marix like his: NA NA C remember, by seing an elemen of a marix o NA in Eviews, you ask he sofware 0 NA o esimae he value of ha elemen. So here you make a single resricion. Firs, define a new marix called lr (you canno use C as a name of a new objec since i is preserved for consan): Now you can use i for idenifying he VAR: We obain he following soluion: 17
Comparing hese resuls wih he baseline case you will find a major difference! Higher rae of inflaion will in he shor-run conribue o a higher real GDP bu his effec wears ou (his is wha we forced on our daa by he long run resricions). Shock in real GDP also leads o a posiive surge in inflaion rae and his is also ransiory. 18
Feel free o play wih differen resricions. Using long-run resricions can be very useful, because someimes we can be unsure abou he conemporaneous correlaion srucure of he daa, while we can be quie sure abou long-run responses (classical dichoomy was a quie cerain poin of deparure in his example). 19