ABCs (and Ds) of UnderstandingVARs

By JES1S FERNIANDEZ-VILLAVERDE,JUAN F. RUBIO-RAMiREZ,THOMAS J. SARGENT,AND MARK W. WATSON*

How informative are unrestricted VARs I. Two RecursiveRepresentations of about how particulareconomic models respond Observables to preference, technology, and information shocks01 In the simplest possible setting, this A. Recursive Representationof an paperprovides a check for whethera theoretical Equilibrium model has the property in population that it is possible to infer economic shocks and impulse Let an equilibriumof an economic model or responses to them from the innovations and the an approximationto it have a representationfor associated with a vector au- impulse responses { y,t+ } in the state space form toregression (VAR). We revisit an invertibility issue that is known to cause a potentialproblem for and a (1) = Ax, + Bwt+l, interpreting VARs, present simple xt+1 check for its presence.2We illustrateour check (2) + in the context of a permanentincome model for Yt+1 Cx, Dwt+1, which it can be applied by hand. where xt is an n X 1 vector of possibly unob- served state variables, y, is a k x 1 vector of variables observed by an econometrician, and * Fernindez-Villaverde:Department of Economics, Uni- wt is an m X 1 vector of economic shocks versity of Pennsylvania,3718 Locust Walk, Philadelphia,PA impinging on the states and observables, i.e., 19104,National Bureau of Economic andCentre for Research, shocks to in- EconomicPolicy Research(e-mail: [email protected]); preferences,technologies, agents' Rubio-Ramirez:Department of Economics,Duke University, formation sets, and the economist's measure- P.O. Box 90097, Durham,NC 27008, and FederalReserve ments. The shocks wt are Gaussianvector white Bank of Atlanta(e-mail: [email protected]); Sar- noise satisfying = 0, Eww' = I, and of Ewt gent: Department Economics, New York University,269 = 0 for 0, where the of MercerStreet, New York, NY 10003, and Hoover Institution Ewtwtj j assumption (e-mail: [email protected]);Watson: Departmentof Economics normality is for convenience and allows us to and Woodrow Wilson School, Princeton University, 321 associate linear least squares predictions with Bendheim Hall, Princeton, NJ 08544 (e-mail: mwatson@ conditional expectations. With m shocks in the princeton.edu).We thankV. V. Chari,Patrick Kehoe, James economic model, n states, and k observables,A Nason, RichardRogerson, and the refereesfor very insightful is n X B is n X C is k X and D is k X criticismsof an earlierdraft. Beyond the usual disclaimer,we n, m, n, must note that any views expressed herein are those of the m. In general, k 0 m. The matricesA, B, C, and authorsand not necessarilythose of the FederalReserve Bank D are functions of parametersthat define pref- of Atlantaor the FederalReserve System. This researchwas erences, technology, and economics shocks. supportedby NSF grantsSES-0338997 and SES-0617811. the re- some recent efforts to answer this see They incorporate typical cross-equation 'For question, strictions embedded George Kapetanios, Adrian Pagan, and Alastair Scott in modem macroeconomic (2005), VaradarajanV. Chari, PatrickJ. Kehoe, and Ellen models. R. McGrattan(2005), and Lawrence J. Christiano,Martin Equilibriumrepresentations of the form (1)- Eichenbaum,and Robert Vigfusson (2006). (2) are obtained in one of two used 2 Lars P. Hansen and Marco widely Sargent(1981, 1991), Lippi The first is to a linear or and LucreziaReichlin (1994), ChristopherA. Sims and Tao procedures. compute Zha (2004), and Hansen and Sargent(2007) contain general loglinear approximationof a nonlinearmodel as treatmentsof this problem and furtherreferences. exposited, for example, in HaraldUhlig (1999).

1021 1022 THE AMERICANECONOMIC REVIEW JUNE 2007

It is straightforwardto collect the linear or log (4) [I - (A - BD-'C)L]xt,, = BD-l'yt , linear approximationsto the equilibrium deci- sion rules and to arrange them into the state where L is the lag operator.Consider: space form (1)-(2). A second way is to derive (1)-(2) directly as a representationof a member CONDITION1: The - BD- 1C of a class of dynamic stochastic general equi- eigenvaluesofA librium models with linear transitionlaws and are strictlyless than one in modulus. For see Jaewoo quadraticpreferences. example, When Condition is we that and Sherwin Rosen (2004), Rosen, Kevin (1) satisfied, say Ryoo A - BD-1C is a stable matrix. The inverse of M. and Jose A. Scheinkman (1994), Murphy, the on the left side of this and Robert Topel and Rosen (1988).3 operator equation gives a squaresummable polynomial in L if and if Condition 1 is satisfied.In this only case, xt+ B. The Question satisfies

Our question is: underwhat conditions do the o0 economic shocks in the state-spacesystem (1)- (5) = I [A - BD-'CYBD-'yt+ xt+, -j, (2) match up with the shocks associated with a j=o VAR0 That is, under what conditions is so that x,,t is a square summable linear com- (3) w,,, = (y,, - E(y,, ly')), bination of the observationson the history of y - 1 at time t + 1. This means that the complete state where wt+, are the economic shocks in (1)-(2), vector is in effect observed so that var(xlyt) = back one and yt denotes the semi-infinitehistory y, yt-_1,...,I 0. Shifting (5) period substituting into we obtain Yt+1 - E(y,,+ Iy) are the one-step-ahead fore- (2), cast errors associated with an infinite order VAR, and 11 is a matrix of constants that can (6) yt+ potentially be uncovered by "structural"VAR When re- (SVAR) analysis0 (3) holds, impulse C > [A - + from the SVAR match the re- = BD-aCBD-'ytj Dwt+. sponses impulse j=o sponses from the economic model (1)-(2). To begin to characterize conditions under If condition (1) is satisfied, equation (6) defines which (3) holds, consider the prediction errors a VAR for y,t, because the infinite sum in (6) from (2) after conditioningon yt, that is, y,,+ - converges in mean squareand Dw,,+1 is orthog- E(yt+, yt) = C(x, - E(xtlyt))+ Dwt+,,. Evi- onal to Yt-1 for all j > 0. - drives a between If one of the of A - is dently, C(xt wedge eigenvalues BD-1C the VAR errorsE(x/,yt)) - and the struc- than in modulus, this Yt+1 E(y,,t+ y') strictly greater unity argu- tural errors w + 1. What is requiredis a condi- ment fails because the infinite sum in (6) di- tion that eliminates this wedge. In Condition 1, verges. When A - BD-'C is an unstable we offer a simple conditionthat yields (3) in the matrix,the VAR is associated with anothercel- interesting"square case" in which k = m and D ebratedstate space representationfor {y,t+ 1, to has full rank. which we now turn.

C. A Poor Man's InvertibilityCondition D. The InnovationsRepresentation

When D is nonsingular,(2) implies wt,, = Associated with any state space system (A, B, - this into (1) and C, D) for {y,,+ of the form (1)-(2) is D- (yt+ 1 Cx,). Substituting }T= rearranginggives another state space system, called the innova- tions representation:

3 Hansen and (2007) other exam- Sargent provide many (7) = + B,,1 ,, ples of this second approach. t Ai, i+ +1 VOL.97 NO. 3 FERNANDEZ-VILLAVERDEETAL.: ABCs (AND Ds) OF UNDERSTANDINGVARs 1023

PermanentIncome (8) Yt-1 = Cti + Et + 1, II. Example l A state space representationfor the surplus where = xo - (ko, o),,k E(x,l{yi}'=1), -- Yt+, - c,t,1 for the permanent income con- 1) = is anotherYt+ sumption model (e.g., see Sargent 1987, chap. E(yt+ {yi= D,+1Et+1, Et+1 i.i.d. Gaussianprocess with mean zero and iden- XII) is: tity covariance matrix, and the matrices B,,, and D can be the = + - +1 recursively computed by (13) c,1, c, R-l)wt+,, Kalmanfilter. Under a general set of conditions, ow(1 for any positive semi-definite1o, as t --> +oo, (14) yt+l - c,+l = -c, + wR-lw,+1l, the matrices Bt+1 and D,,1 converge to limits B and D that satisfy the equations:4 where Yt+I1 = owt +1 is an i.i.d. labor income (9) 1 = AlA' + BB' process and R > 1 is a constant gross interest rate on financial assets. Equations(13) and (14) - (AIC' + BD')(CIC' + DD')-'(AIC' correspondto (1) and (2), where ct is the unob- served state and yt - ct is the variableobserved + BD')', by the econometrician. The impulse responses for the model are shown in Figure 1 for the case K = + + that R = 1.2 and = 1. show the (10) (AIC' BD')(CIC' DD')-', o-w They familiarpatterns: consumption increases perma- the value of the (11) = DD' + CXC', nently by annuity transitory II' increase in income; this leads to a large positive (12) B = KD, impact effect of wt on yt - ct and small negative values for all other periods. For this example, it is easy to compute that whereI = var(x,ly).5When A - BD-1C is A - BD-1C = R > 1, so that Condition 1 does unstable, I > 0, meaning that at least some not hold. This failure of Condition 1 is part and parts of the state xt are hidden. This means the parcel of the permanentincome model because one-step-aheadforecast errorscomputed by the it is needed to verify that the present value of - contain the shocks the coefficients the of the VAR, yt+l E(yt+ lyt), describing response Dw,+1 and the error from estimating the state surplus Yt+1 - c,,+ to an endowment innova- C(x, - xi). Thus, (3) does not hold. These two tion must be zero, an outcome that embodies the of - are uncorre- value balance that is built into components yt+1 E(yt+ 1yt) present budget lated, so that the variance of the VAR innova- the permanentincome model. tions Det+1 is larger than the variance of the The innovations representationof the model economic model disturbancesDw,, (Equiva- is lently, from equation (11) DD' > DD'.)61. (15) Ct+1= C, + ouw(R1-- 1) , = 4 - - + Alternative conditions for the existence of this time (16) Yt+ Ct+ o-we,,.+1 invariantinnovations representation and for convergence of iterationson the Riccati equation are stated in Brian D. O. Anderson and John B. Moore (1979, chap. 4), Sargent Equations(15) and (16) correspondto the steady- 5 and Evan Anderson et al. and (1980, chap. 6), (1996), state versionof (7) and (8), where = - Hansen and Sargent (2007). ct E(ctlyt is the estimate of constructed 5 With m shocks in the economic model, n states, and k ct) consumption observables,K, the steady-stateKalman gain, is n X k, D)is fromthe historyof Yt- ct. BecauseCondition 1 is k x k, and B is n X k. not satisfied,c, cannotbe estimatedperfectly from 6 Hansen and discuss the inno- - Sargent (2007, chap. 9) yt so that , c,. Indeed,a simplecalculation vations representation,prove that A - BD-1C is a stable Ct, +/: shows that , - = - matrix, and derive a general formula that describes the = var(clyt ct) o2(1 R-2). mapping from the economic shocks w,,, to the VAR BecauseA - BD- C = R', which is stable,the shocks Et+1. errors a VAR for - are computedby Yt ct owt + 1. 1024 THE AMERICANECONOMIC REVIEW JUNE 2007

1.2r.

0.8

Yt+h

0.6

Response

Impulse0.4

Ct+h

0.2

Yt+h-Ct+h i

0.2 0 1 2 3 4 5 6 7 Period(h)

FIGURE1. IMPULSERESPONSES OF Yt+ h, Ct+ h, AND Yt+ h - Ct+ h TO A SHOCKIN Wt

(Of course, since y, - c, is a scalar,the VAR is falls short of the present value of the impulse just a univariateautoregression.) response of income.7 Figure 2 shows the impulse responses of 9,, This example can be modified in instructive what is observed.For c,, and Yt - c, with respect to the VAR shocks, ways by altering example, Et. These are markedly different from the im- if c,, y, 1, or if the value of the consumer's pulse responses shown in Figure 1. Notably, accumulatedassets were observed, then Condi- while ct respondspositively and permanentlyto tion 1 would be satisfied.8 a shock in wt in Figure 1, respondsnegatively and permanentlyto a VAR,I shock et. The reason III. ConcludingRemarks is simple: forecast errors in y, - ct arise from shocks to income, oaJ or from estimation We hesitate to draw sweeping conclusions 1wt, errors in past consumption, ct - t,, and the about VARs. Some applications of VARs Kalmanfilter optimally allocates e, to these two possible sources. In Figure 1, the impulse re- 7 sponse of the surplusyt - ct has a presentvalue For more discussion of this example, see Sargent of value balance; (1987), Hansen, William Roberds, and Sargent (1991), and zero, implying present budget Roberds in the of the (1991). Figure 2, impulse response surplus 8 See Watson (1994) for a more extensive discussion of has a positive present value so that the present how what is observed affects whetherCondition 1 is likely value of the impulse response of consumption to be satisfied. VOL.97 NO. 3 FERNANDEZ-VILLAVERDE ET AL.: ABCs (AND Ds) OF UNDERSTANDINGVARs 1025

1.2

1

0.8

0.6

Response

Impulse0.4

Yt+h-Ct+h

0.2 Yt+h

Ct+h

0.2 0 1 2 3 4 5 6 7 Period(h)

FIGURE 2. IMPULSE RESPONSES OF AND - A Yt + h, t+h, Yt+h Ct + h TO SHOCK IN

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