Vector Autoregression Analysis: Estimation and Interpretation

1 Vector Autoregression Analysis: Estimation andInterpretationJohn E. FloydUniversity of Toronto September 19, 20051 IntroductionThis expositional paper lays out the mechanics of running and interpretingvector autoregressions. It proves no theorems. Rather, it sets out the basicsof how VAR s work and outlines some fundamentals regarding interpreta-tion. For the theoretical details, see Walter Enders,Applied EconometricTime Series, John Wiley & Sons, 1995, pp. 291 353 and earlier material asrequired, Helmut L utkepohl,Introduction to Multiple Time Series Analysis,Springer-Verlag, 1991, pp. 9 27, 43 58, and 97 117, and James D. Hamil-ton,Time Series Analysis, Princeton University Press, 1994, pp. 257 372and earlier material as The Underlying Economic ModelConsider the following economic model with two variablesy1andy2, eachof which depends on itself lagged, on the current and lagged values of theother variable and on aiiderror term:y1(t)=v10+v12y2(t)+a11y1(t 1)+a12y2(t 2)+e1(t)(1)y2(t)=v20+v21y1(t)+a21y1(t 1)+a22y2(t 2)+e2(t)(2) This is written to help students understand how to run VARs.

2 It is not a substitutefor reading the literature cited. I would like to thank John Maheu for system can be written in matrix notation as[y1(t)y2(t)]=[v10v20]+[0v12v210][y1(t) y2(t)]+[a11a12a21a22][y1(t 1)y2(t 1)]+[e1(t)e2(t)](3)or, in general matrix notation withmvariables andplags,yt=v+A0yt+A1yt 1+A2yt 2+A3yt 3+ +Apyt p+et(4)whereyt,vandetarem 1 column vectors andA0,A1,A2, Aparem mmatrices of coefficients. The vectoretis am-element Vector of whitenoise residuals that satisfiesE{etet }=D, whereDis a diagonal matrix. Anappropriate scaling of the elements ofywould makeDan identity VAR estimationEquations (1) and (2), which are called astructuralVAR or aprimitivesystemcan be solved simultaneously to yield thereduced formorstandardformof the VAR:y1(t)=b10+b11y1(t 1)+b12y2(t 2)+u1(t)(5)y2(t)=b20+b21y1(t 1)+b22y2(t 2)+u2(t)(6)or[y1(t)y2(t)]=[b10b20]+[b11b 12b21b22][y1(t 1)y2(t 1)]+[u1(t)u2(t)](7)whereb10=v10+v12v201 v12v21b11=v12a11+a211 v12v21b12=a12+v12a221 v12v21b20=v20+v21v101 v12v21b21=a21+v21a111 v12v21b22=v21a12+a221 v12v212andu1(t)=11 v12v21[e1(t)+v12e2(t)]u2(t)=11 v12v21[v21e1(t)+e2(t)].

3 In the generalmvariable case withplags we have(I A0)yt=v+A1yt 1+A2yt 2+A3yt 3+ +APyt p+et(8)which reduces toyt= (I A0) 1v+ (I A0) 1A1yt 1+ (I A0) 1A2yt 2+ (I A0) 1A3yt 3+ + (I A0) 1 APyt p+ (I A0) 1et.(9)Lettingb= (I A0) 1v,B1= (I A0) 1A1,B2= (I A0) 1A2, etc., andut= (I A0) 1etwe can write the VAR in standard form in thegeneral case asyt=b+B1yt 1+B2yt 2+B3yt 3+ +BPyt p+ut.(10)All this assumes, of course, that the matrix (I A0) has an inverse. GiventhatE{etet }=D, the variance-covariance matrix of the Vector of residualsutequals =E{utut }=E{[(I A0) 1et][(I A0) 1et] }=E{[(I A0) 1]etet [(I A0) 1] }= [(I A0) 1]E{etet }[(I A0) 1] = [(I A0) 1]D[(I A0) 1] The equations in (10) can be estimated by ordinary least squares because the independent variables in all equations are the same, there isno efficiency gain by estimating these equations as a system using the seem-ingly unrelated regression this point it is appropriate to perform a number of tests to deter-mine what variables should be in the VAR, the appropriate number of lags,whether seasonal dummies should be included and, indeed, whether a VARis even appropriate for the research problem at hand.

4 To focus strictly on themechanics at this point, however, these model-selection issues are postponedto a later The Moving Average RepresentationThe standard form system given by (10) can be manipulated to express thecurrent value of each variable as a function solely of the Vector of residualsut. This is called itsmoving average representation ytis a moving averageof the current and past values +C1ut 1+C2ut 2+ +Csut s+y0.(11)wherey0is some initial value see how we can do this, suppose for the moment that we have onlyone lag of each variable in the VAR ( , a VAR(1) process). Under thisassumption, (10) reduces toyt=b+Byt 1+ut.(12)Lagging (12)ntimes, we obtainyt 1=b+Byt 2+ut 1yt 2=b+Byt 3+ut 2yt 3=b+Byt 4+ut 3 yt s=b+Byt s+ut s 1 Successive substitution into (12) yieldsyt= [1 +B+B2+B3+B4+ +Bs]b+ut+But 1+B2ut 2+B3ut 3+ +Bsut s= (1 B) 1b+ut+But 1+B2ut 2+B3ut 3+ +Bsut s.

5 (13)In terms of (11) this yieldsy0= (1 B) 1bandCk=Bk,k= 0 there arep >1 lags, we first convert the VAR(p) system into aVAR(1) system of the form4 ytyt 1yt 2yt p+1 = + B1B2B3 BP 1 BPIm0 0 000Im0 000 0Im 0 0 Im0 yt 1yt 2yt 3yt p + which can be expressed more simply asYt= +BYt 1+ t.(14)Here,Yt, , and taremp 1 column vectors andBis anmp system is formed by taking the expression (10) as the first equation(more correctly, set of equations) and adding thep 1 equations (sets ofequations)yt 1=yt 1yt 2=yt 2yt 3=yt 3 yt p+2=yt p+2yt p+1=yt p+1sequentially analogy with equation (13), the moving average representation of(14) is seen to beYt= [Imp B] 1 + t+B t 1+B2 t 2+B3 t 3+ +Bn t s(15)whereImpis anmp mpidentity turns out that the moving average representation of our original VAR(p)system is represented by selected parts of the topmequations of the system5(15).

6 We can strip off these terms by operating on (15) with them mpmatrixJ= [Im0 0 0 0].We thereby obtain (11), reproduced below,yt=C0ut+C1ut 1+C2ut 2+ +Csut s+y0.(11)wherey0=J C0=JB0J =JImpJ C1=JB1J =JBJ C2=JB2J C3=JB3J Cs=JBsJ To fully understand how the above procedures work it is necessary toapply them. Consider the age-old problem of modelling the behaviour ofmonetary and other aggregates over the business cycle. We focus the analysison conditions in the United States because that is one of the few economiesthat one might treat, at considerable risk, as if it were a closed is probably defensible because the authorities pay little attentionto the effects of their policies on the rest of the world and the country islarge enough that changes in the domestic money supply and output canhave a significant effect on world levels of these variables. Moreover, to theextent that other countries are concerned about the effects of monetaryshocks on their exchange rates with respect to the dollar, and adjusttheir monetary policies to offset these effects, their monetary conditions willmimic those in the , whose authorities will then effectively control worldmonetary reasons that will be outlined later when issues regarding the inter-pretation of VARs are discussed, we select four variables the detrendedlogarithms of base money and real GDP, the year over year rate of inflation6of the implicit GDP deflator, and the level of the market interest rate on 90-day commercial data are used with no seasonal dummyvariables.

7 The base money and real GDP variables are detrended to keepour analysis of the interactions between the variables uncontaminated by therelationships between their trends. The purpose at this point is to demon-strate the calculations model selection criteria will be considered the best statistical package to use for VAR analysis is , RATS is expensive and one can, with some inconvenience,program VARs using a free platform for statistical computing called Xlisp-Stat, which was written by Luke Tierney at the University of VAR presented immediately below is programmed in detail in the Xlisp-stat code , available from my web-site location noted infootnote 3. While VARs can also be programmed in raw fashion in RATS, itis quicker to use the canned RATS procedures for making these are contained in the adjusted real and nominal gross domestic product figures were ob-tained from the International Monetary Fund publicationInternational Financial Statis-tics(111 is the mnemonic for nominal GDP and 111 is the mnemonic forreal GDP), and the implicit GDP deflator was obtained by dividing the nominal series bythe real series.

8 A seasonally unadjusted base money series was obtained fromDRI-Citibase(mnemonic FZMFB) and seasonally adjusted using the SAS X11 procedure. The 90-daycommercial paper rate was obtained from theCANSIM data base (mnemonic B54412).2 For details about RATS and how to obtain it, for all major operating systems can be obtained from Tierney s web-site luke/xls/ The MS-Windows ver-sion also be downloaded from my web-site by anonymousftp with some official documenta-tion ( ), a short manual that I have put together ( )and some additional functions that will be helpful for time-series anal-ysis ( ). A complete discussion of how to use the program is contained inLuke Tierney,Lisp-Stat: An Object Oriented Environment for Statistical Computing andDynamic Graphics, Wiley Series in Probability and Mathematical Statistics, John Wi-ley & Sons, 1990. LINUX versions of XlispStat are bundled with most of the RATS program , along with the two data files it requires( , ) and other RATS program files subsequently referred to in this pa-per, is available in the zip the ftp site notedin the previous footnote.

9 The code the necessary plots to thescreen and writes them to file. When an output-filename is added to to the command linethe entire output is redirected to that file. The output in the , which is also contained in the IdentificationThe moving average representation (11) does not give a proper indication ofhow the system responds to shocks to the individual structural problem is that the shocks to the equations contained in the vectorutare correlated with each other. We therefore cannot determine what theeffects on themvariables of a shock to an individual structural equationalone would be an observedutwill represent the combined shocks to anumber of equations. This can be seen from the fact that from (9)ut= (I A0) order to determine the effects of a shock to an individual structural equa-tion of the system we have to be able to solve the system forA0and therebyobtain (I A0) 1. This will enable us to operate on (11) to transform theut j s in intoet j s.

10 In the process, of course, the matricesCjwill also betransformed into a useful representation of the way to obtain the matrixA0is to statistically estimate the struc-tural model (4). Were we to do this, we would not be running a , the reason for VAR analysis is to avoid multi-equation approach used to identifyA0in VAR analysis is to find the matrixthat will orthogonalize the errors , transform theut j s in into theet j s, which are uncorrelated with each any matrixGthat has an inverse, equation (11) can be rewrittenyt=C0GG 1ut+C1GG 1ut 1+C2GG 1ut 2+ +CsGG 1ut s+y0.(16)Our task is to find theGfor whichG= (I A0) +Z1et 1+Z2et 2+ +Zset s+y0(17)whereZj=CjGandet j=G 1ut j== ut j=Get Choleski DecompositionsSuppose that the matrixA0takes the following form: 000 00a02100 00a031a0320 00a041a042a043 a0m(m 1)0 This will mean that the structural equations will take the form:y1t=a111y1(t 1)y2t=a021y1t+a121y1(t 1)+a122y2(t 1)y3t=a031y1t+a032y2t+a131y1(t 1)+a132y2(t 1)y4t=a041y1t+a042y2t+a043y3t+a141y1(t 1)+a142y2(t 1) = = = None of the current year values ofy2,y3,y4, ,ymenter into thedetermination of the current year level ofy1.