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Multivariate Time Series Models∗
Myles J. Callan
February 21, 2008
Abstract This is the first in a 2 part review of multivariate time series mod- els. It begins by distinguishing between structural and reduced form models, using Samuelson’s “Accelerator” model to motivate the discus- sion. As there is no consensous in macroeconomic theory to suggest the necessary identifying restrictions, VAR models treat endogenous variables symmetrically. The review concludes by discussing how to estimate these models and the restrictions required to identify these models.
1 Structural and Reduced Form Models
We begin by noting the difference between Structural and Reduced Form equations, which was one of the topics that we covered in the first hand-out, as follows: Stochastic dynamic general equilibrium models are examples of systems of difference equations (or differential equations). A simple of ex- ample of this is Samuelson’s model:
yt = ct + it ct = αyt− 1 + ct it = β(ct − ct− 1 ) + it
The final equation is an example of a structural equation—where the endoge- nous variable, it, depends on the current realization of another endogenous variable, ct. We can derive the reduced form of this equation by expressing
∗The source for this material is Cochrane’s Time Series for Macroeconomics and Fi- nance text
it as a function of predetermined variables (its own lags and lags of other endogenous variables), current and past values of exogenous variables (there are no exogenous variables in this model), and stochastic (or disturbance) terms:
it = β(αyt− 1 + ct − ct− 1 ) + it , or it = βαyt− 1 − βct− 1 + βct + it
We can do the same for the GDP variable to get:
yt = α(1 − β)yt− 1 − αβyt− 2 + (1 + β)ct + it − βct− 1
This is an example of a univariate reduced form equation—where yt is expression solely as a function of its own lags and disturbance terms. This provides one motivation for the first part of the course: Univariate Time Series Analysis.
We can use OLS to estimate the reduced form models and, in this case, we showed that the structural parameters in this model are identified. That is, we can derive estimates of the structural parameters by estimating the reduced form model.
2 Vector Autoregression
In general macroeconomics theory provides no consensus on the structural models for general equilibrium models, and so there are insufficient restric- tions suggested by theory to identify the model. This idea can readily be seen in a structural model in which all variables are treated symmetrically—so that no restrictions are imposed. Initially, the role of macroeconomic theory was to suggest the variables that are relevant for the structural model and use these variable in the estimated reduced form model (VAR). However, if the model is not identified, no economic analysis of the results can be performed as, as we shall see, the coefficients and errors of the reduced form are combinations of the coefficients and errors of the structural model. It is the coefficients and errors of the structural model that allows economic analysis - as with Samuelson’s accelerator model above. The only use that the reduced form model can be put to is forecasting.
Where
e 1 t =
ηyt − b 12 ηzt 1 − b 12 b 21 and e 2 t =
ηzt − b 21 ηyt 1 − b 12 b 21
That is, the reduced form errors are linear combinations of the structural errors—and are therefore contemporaneously correlated (but not serially correlated), unless b 12 = b 21 = 0. So the variance/covariance matrix of the reduced form errors, Σ, is not diagonal (as above).
3 Conclusion
Estimation of these models broadly follows that of the univariate models that we have already seen earlier in the course. We test for stationarity^1 (which in this case is determined by A 1 ), select the appropriate lag length (typically by using AIC/SBC), perform diagnostic tests on the residuals. Once the appropriate model is selected, we can use it to (i) forecast, (ii) to derive the impulse response functions, and (iii) to derive the variance decomposition. (ii) and (iii) have economic interpretations, but cannot be performed without identifying restrictions being imposed.
To see this, we can rewrite the structural model using the lag operator notation as: B(L)xt = ηt, where B(0) = B—that is, the structural shocks have a contemporaneous effect on their “own” variables and, indirectly, on all other variables through the contemporaneous relationship between the all endogenous variables in the system. If we were able to estimate B(L), we could derive the impulse response functions (which are equivalent to the MA representation of the structural model): xt = B(L)−^1 ηt = C(L)ηt. However, we can only estimate the reduced form: A(L)xt = et. As et is a linear combination of the structural errors, we can not derive the impulse response functions associated with the structural shocks without imposing identifying restrictions.
(^1) Sims in his original paper (1980) and Sims, Stock and Watson (1990) recommend against differencing, even if the variables contain a unit root. The rationale being that the VAR technique is used to determine relationships between variables and not to estimate parameter values.
