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Estimating Vector Autoregressions Using Methods Not Based on Explicit Economic Theories (P 4 Federal Reserve Bank of Minneapolis Quarterly Review Summer 1979 A Way to Improve Economic Forecasting tP. n Help for the Regional Economic Forecaster: Vector Autoregression <P. 2) Estimating Vector Autoregressions Using Methods Not Based on Explicit Economic Theories (P. 8) District Conditions (p. 16) Federal Reserve Bank of Minneapolis Quarterly Review voi.3,no.3 This publication primarily presents economic research aimed at improving policymaking by the Federal Reserve System and other governmental authorities. Produced in the Research Department. Edited by Arthur J. Rolnick, Kathleen S. Rolfe, and Alan Struthers, Jr. Graphic design by Phil Swenson, Graphic Services Department. Address requests for additional copies to the Research Department, Federal Reserve Bank, Minneapolis, Minnesota 55480. The views expressed herein are those of the authors Articles may be reprinted if the source is credited and the Research and not necessarily those of the Federal Reserve Department is provided with copies of reprints. Bank of Minneapolis or the Federal Reserve System. Estimating Vector Autoregressions Using Methods Not Based on Explicit Economic Theories Thomas J. Sargent Adviser, Research Department Federal Reserve Bank of Minneapolis and Professor of Economics University of Minnesota This paper describes procedures for analyzing in- often must restrict the domain of the questions to terrelated time series1 which are mainly intended which answers are sought if Lucas' criticism is not as an alternative to using structural econometric to be operative. models as forecasting devices. Alternatives to the structural models have been sought because of in- Vector Autoregressions creasingly compelling suspicions that the a priori For the purposes of making forecasts and dis- restrictions used in existing structural models are playing its operating characteristics, a linear not implied by good dynamic economic theory and econometric model is often represented as a par- that the interpretations and policy conclusions ticular set of random difference equations called a 2 based on those faulty a priori restrictions are worth vector autoregression. Thus, let zt be an (Nx\) little. The techniques described in this paper are vector of variables, including both all of the en- not based on economic theory. Instead, the idea is dogenous and all of the exogenous variables in the to estimate vector autoregressions with many free model. Let zt be measured in terms of deviations parameters and to introduce restrictions not di- from means. Further, assume that Zt is a wide- rectly motivated by economic theory but rather sense stationary stochastic process3 which has = th aimed simply at forecasting better, that is, deliver- matrix covariogram Eztz't-k Cz(k). Then the M ing estimators with small mean squared errors. Because these techniques are not based on ec- onomic theory, they do not completely substitute 'These procedures have been developed and applied to mac- for structural models. They cannot appropriately roeconomic data by Christopher Sims of the University of Minnesota and Robert Litterman, now of M. I .T. but until recently an analyst in the be used to analyze the range of policy interven- Research Department at the Federal Reserve Bank of Minneapolis. The tions that structural models were designed to eval- key references are Sims 1975, 1977; Sargent and Sims 1977; and Litter- uate. The techniques are not appropriate for con- man 1979. [Author names and years refer to the works listed at the end of this paper.] This paper is intended only as an introduction to the work ditional forecasting, for predicting the behavior of of Sims and Litterman and makes no claim for originality of any of the the system under what may be a drastic change ideas discussed. from the historical pattern in a feedback rule for a 2Good general references on the time series methods described policy variable, for example. Instead, these tech- here are Anderson 1971, Box and Jenkins 1970, Fuller 1976, Granger and Newbold 1977, and Nerlove, Grether, and Carvalho 1979. For an niques are designed mainly for unconditional fore- introduction to vector autoregressions and some of their uses in mac- casting and for compactly summarizing data. roeconomics, see Sargent 1979, chapter XL :{ Thus, users of the statistical models described in A vector stochastic process zt is said to be wide-sense stationary this paper must acknowledge from the start that if the vector of means Ez, is a constant vector independent of time t and they are vulnerable to Lucas' (1976) criticism of if the matrix covariogram Eztz's depends on only the difference (t~s) and not only t and 5 separately. Wide-sense stationarity is also referred econometric policy evaluation methods, and they to as second-order stationarity and covariance stationarity. 8 Federal Reserve Bank of Minneapolis Quarterly Review/Summer 1979 order vector autoregression for the zt process is the number of parameters that have to be esti- mated from (N2 x M) to a much smaller number of M theoretically more fundamental parameters of j=i which the Df's are functions. The argument is that the vector autoregression is a "profligately par- where the Df's are (NxN) matrices and the ameterized" representation7 and that estimation (Nx 1) stochastic error process rjf satisfies the or- proceeds much more efficiently by focusing on the 4 thogonality conditions structural parameters about which something is known in advance of estimation. (2) Evfzl-k = 0 k= 1,... ,M. To make this argument more precise, we use the representation of a linear econometric model Post-multiplying (1) by zl-k and using (2) gives the described by Lucas and Sargent (1979). The struc- least squares normal equations (or Yule-Walker tural equations are equations) (4) A{)yt + A xyt-x + ... + A myt —m M (3) Cz(k) - 2 Df Cz (k-j) = 0 *=1,...,A#. j = l = B(}Xt + BxXt-x + ... + BnXt-n + €f. The normal equations (3), in general, uniquely de- The random error generating equations are termine the matrices Df in terms of the population values of the second-moment matrices Cz (/:), k = (5) R{)et + + ... + Rret-r = ut R{) = /. 0,1 Under the assumptions given here, least Here yt is an (Lxl) vector of endogenous vari- squares estimates of the Df's are known to be ables, xt is a (Kx 1) vector of exogenous variables, 5 statistically consistent. But the vector autore- and e, and ut are each (Lxl) vectors of random 2 gressive system (1) has (N x M) free parameters disturbances. The matrices A} are each (LxL), the in the Df matrices, so that for even moderate sizes Bf s are (LxK), and the Rf s are each (LxL). We of M and N, least squares estimation either is assume that L + K = N. The (Lx 1) disturbance simply not feasible due to exhaustion of degrees of process ut is assumed to be serially uncorrelated freedom or else is unwise due to the large sampling with Eut = 0 and with contemporaneous covariance errors present when the number of parameters to matrix Eutui = 2 and Eutu's = 0 for t * s. be estimated nearly exhausts all degrees of free- The defining characteristic of the exogenous dom. For this reason, systems of vector autore- variables xt is that they are uncorrelated with the gressions with the sizes N and M usually encoun- es at all lags so that Eutx's is an (LxK) matrix of tered in economics have typically been estimated zeros for all t and s. The xt process is itself as- by methods other than least squares. sumed to be generated by the vector autoregres- Until recently, the most popular method of es- sion timating vector autoregressions was to apply classical simultaneous equation estimators to the (6) xt = Cx xt-x + ... + Cpxt-P + at structural model that presumably underlay the vector autoregression.6 Simultaneous equation es- timators have the virtue of permitting the model 4 builder to bring to bear a priori information of cer- These orthogonality conditions uniquely identify 2/L, Df zt~j as tain kinds to produce parameter estimates with the least squares projection of z, onto the linear vector space spanned by _i, ..., Z,-m}- smaller sampling 5For proofs, see Ljung 1976 and Anderson and Taylor 1976. errors of the Df's than can be HThe classical simultaneous equation model and estimators are produced by least squares. In statistical jargon, described in any good book on econometrics, for example, Goldberger use of this prior information produces more "effi- 1964 or Maddala 1977. cient estimators." In the present context, these 7This is Sims' terminology. techniques can be viewed as a device for reducing 9 where Eat = 0 and Eatxt~j = 0 for j ^ 1. about which economic theory has something di- As in Lucas and Sargent 1979, the reduced rectly to say. Generally, the restrictions used take form of this system is the form of sets of simple linear restrictions on the A/s, B/s, and R/s. Most often, these assume the (7) yt = -PXyt-X ~ ... ~ Pr+m Jf-r-m form simply of setting many, indeed most, of the coefficients in Aj9 Bj, and Rj to zero a priori. Q()Xt + ••• + Qr+n-Xt-n-r Another set of exclusion restrictions is evident in (6), in which lagged yf s are assumed not to appear. where The asymmetrical treatment of lagged jc's and y's 00 in (6) and (9) is what distinguishes between en- 1 Ps = A() £ Rj As-j dogenous and exogenous variables. For now, we J=-00 simply note that the exclusion of lagged y's from (8) (6) in most applications is done on an entirely a J=-oo priori basis. From the somewhat narrow viewpoint of es- In these expressions for and it is to be un- timating vector autoregressions, the virtue of using derstood that matrices not previously defined (for this body of a priori exclusion restrictions on lag- example, any with negative subscripts) are zero.
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