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Structural Approaches to Vector Autoregressions Cia It HE VECTOR AUTOREGRESSION (VAR) VAR Model Into a System of Structural Equations

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Structural Approaches to Vector Autoregressions Cia It HE VECTOR AUTOREGRESSION (VAR) VAR Model Into a System of Structural Equations John W. Keating John W. Keating, assistantprofessor, Department of Economics, Washington University in St. Louis, was a visiting scholar at the Federal Reserve Bank of St. Louis while this paper was written. Richard L Jako provided research assistance. Structural Approaches to Vector Autoregressions cia it HE VECTOR AUTOREGRESSION (VAR) VAR model into a system of structural equations. model of Sims (1980) has become a popular tool The parameters are estimated by imposing con- in empirical macroeconomics and finance. The temporaneous structural restrictions. The crucial VAR is a reduced.form time series model of the difference between atheoretical and structural economy that is estimated by ordinary least VARs is that the latter yield impulse responses 1 squares. Initial interest in VARs arose because and variance decompositions that can be given of the inability of economists to agree on the structural interpretations. economy’s true structure. VAR users thought that important dynamic characteristics of the An alternative structural VAR method, developed economy could be revealed by these models by Shapiro and Watson (1988) and Blanchard and without imposing structural restrictions from a Quah (1989), utilizes long-run restrictions to identify the economic structure from the reduced particular economic theory. form. Such models have long-run characteristics Impulse response functions and variance that are consistent with the theoretical restric- decompositions, the hallmark of VAR analysis, tions used to identify parameters. Moreover, illustrate the dynamic characteristics of empirical they often exhibit sensible short-run properties models. These dynamic indicators were initially as well. obtained by a mechanical technique that some For these reasons, many economists believe that believed was unrelated to economic theory.2 structural VARs may unlock economic information Cooley and LeRoy (1985), however, argued that embedded in the reduced-form time series model. this method, which is often described as atheo~ This paper serves as an introduction to this retical, actually implies a particular economic developing literature. The VAR model is shown structure that is difficult to reconcile with eco- to be a reduced-form for a linear simultaneous nomic theory. equations model. The contemporaneous and This criticism led to the development of a long-run approaches to identifying structural “structural” VAR approach by Bernanke (1986), parameters are developed. Finally, estimates Blanchard and Watson (1986) and Sims (1986). of contemporaneous and long-run structural This technique allows the researcher to use eco- VAR models using a common set of macro- nomic theory to transform the reduced-form economic variables are presented. These models ‘A VAR can be derived for a subset of the variables from a 2A Choleski decomposition of the covariance matrix for the linear structural model. Furthermore, it is a linear approxi- VAR residuals. mation to any nonlinear structural model. The accuracy of the VAR approximation will depend on the features of the nonlinear structure. SEPTEMBEP,/OCTOBEFI 1992 are intended to provide a comparison between and attractive assumptions are that shocks have contemporaneous and long-run structural VAR either temporary or permanent effects. If shocks modeling strategies. The implications of the have temporary effects, z1 equals s,, a serially empirical results are also discussed. uncorrelated vector (vector white noise).~That is, (3) z1 = E~. Alternatively, z can be modeled as a unit root process, that is, The standard, linear, simultaneous equations (4) z1 — z1, = model is a useful starting point for understanding Equation 4 imphes that z equals the sum of all the structural VAR approach. A simultaneous past and present realizations of E. Hence, shocks equations system models the dynamic relation- to z are permanent. The assumptions in equations ship between endogenous and exogenous vari- ables. A vector representation of this system is 3 and 4 are not as restrictive as they might ap- pear. If these shock processes were specified as general autoregressions, the VARs would have (1) Ax = C(L)x , + Dz , 1 1 1 additional lags. The procedures to identify struc- tural parameters, however, would be unaffected. where x1 is a vector of endogenous variables and z1 is a vector of exogenous variables. The Under the assumption that exogenous shocks elements of the square matrix, A, are the struc- have only temporary effects, equation 2 can be tural parameters on the contemporaneous en- rewritten as, dogenous variables and C(L) is a kth degree matrix polynomial in the la~operator L, that is, (5) x1 = /3(L); , + e1, C(L)=Co+CIL+CZL2+...+CkL, where all of the C matrices are square. The matrix D measures the where /3(L) = A~C(L)and e1 = A”Dç. The contemporaneous response of endogenous varia- equation system in 5 is a VAR representation of bles to the exogenous variables.~In theory, the structural model because the last term in some exogenous variables are observable while this expression is serially uncorrelated and each others are not. Observable exogenous variables variable is a function of lagged values of all the typically do not appear in VARs because Sims variables.6 The VAR coefficient matrix, /3(L), is a (1980) argued forcefully against exogeneity. nonlinear function of the contemporaneous and Hence, the vector zisassumed to consist of un- the dynamic structural parameters. observable variables, which are interpreted as If the shocks have permanent effects, the VAR disturbances to the structural equations, and x1 model is obtained by applying the first difference and z are vectors with length equal to the num- 1 operator (A = 1— L) to equation 2 and inserting ber of structural equations in the model.~ equation 4 into the resulting expression, to obtain A reduced-form for this system is (6) Ax, = /3(L)Ax1, + e1, (2) x = A”C(L) x , + A~Dz . 1 1 1 with /3(L) and e1 previously defined. A panicular structural specification for the “er- This is a common VAR specification because many ror term” zisrequired to obtain a VAR macroeconomic time series appear to have a representation. Two alternative, commonly used unit root.7 Because of the low power of tests ~Thismodel can accommodate lags of z; this feature is omitted, ‘This model can also be written in levels form: however, to simplify the discussion. 1 1 x =[AC(L) + I]x,_ —A C(L) x _ + A DE . if observable exogenous variables exist, they are included 1 1 1 2 1 as explanatory variables in the VAR. 5 Sims, Stock and Watson (1990) show that this reduced form The individual elements in a vector white noise process, in is consistently estimated by OLS, but hypothesis tests may theory, may be contemporaneously correlated. In structural have non-standard distributions because the series have VAR practice, they are typically assumed to be independent. unit roots. 6 The last term represents linear combinations of serially uncorrelated shocks, and these are serially uncorrelated as well. See any textbook covering the basics of time series analysis for a proof of this result. 39 for unit roots, their existence is controversial. a contemporaneous structural VAR, however, VARs can accommodate either side of the de- generally does not impose restrictions on the bate, however.8 reduced form. The VAR is a general dynamic specification An alternative approach of Doan, Litterman because each variable is a function of lagged and Sims (1984) estimates the VAR in levels with values of all the variables. This generality, how- a Bayesian prior placed on the hypothesis that ever, comes at a cost. Because each equation each time series has a unit root. The Bayesian has many lags of each variable, the set of varia- VAR model permits more lags by imposing bles must not be too large. Otherwise, the mod- restrictions on the VAR coefficients, reducing el would exhaust the available data? If all the number of estimated parameters (called shocks have unit roots, equation B is estimated. hyper-parameters). The reduction in parameters If all shocks are stationary, equation 5 is used.b0 contributes to the Bayesian model’s propensity If some shocks have temporary effects while to yield superior out-of-sample forecasts compared others have permanent effects, the empirical with unrestricted VARs with symmetric lag model must account for this. structure, Recently, Blanchard and Quah (1989) have es- timated a VAR model where some variables rp2~/’fJ21%Jp~flR4,29j.2i)jt12/S’171iUC~ were assumed to be stationary while others had ‘L1JR.A1..~VAR 9/flfl9L 9~ unit roots. Alternatively, ICing, Plosser, Stock 1~ and Watson (1991) use a cointegrated model, It is clear from equations 5 and 6 that, if the where all the variables are difference stationary contemporaneous parameters in A and D were but some linear combinations of the variables known, the dynamic structure represented by are stationary. The stationary linear combinations the parameters in C(L) could be calculated from are constructed by cointegration regressions prior the estimated VAR coefficients, that is, C(L) = to VAR estimation. They impose the cointegration A/3(L). Furthermore, the structural shocks, E,, constraints using the vector error-correction could be derived from the estimated residuals, model of Engle and Granger (1987). Sometimes that is, ç = D - ‘Ae,. Because the coefficients in unit-root tests combined with theory suggest the A and D are unknown, identification of structural coefficients for stationary linear combinations. parameters is achieved by imposing theoretical Shapiro and Watson (1988), for example, present restrictions to reduce the number of unknown evidence that the nominal interest rate and in- structural parameters to be less than or equal flation each have a unit root while the differ- to the number of estimated parameters of the ence between these two variables is stationary. variance-covariance matrix of the VAR residuals. They impose the cointegration constraint by Specifically, the covariance matrix for the residuals, selecting this noisy proxy for the real rate of interest as a variable for the model.
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