In this survey, estimation methods for structural vector autoregressive models are presented in a systematic way. Both frequentist and Bayesian methods are considered. Depending on the model setup and type of restrictions, least squares estimation, instrumental variables estimation, method-of-moments estimation and generalized method-of-moments are considered. The methods are presented in a unified framework that enables a practitioner to find the most suitable estimation method for a given model setup and set of restrictions. It is emphasized that specifying the identifying restrictions such that they are linear restrictions on the structural parameters is helpful. Examples are provided to illustrate alternative model setups, types of restrictions and the most suitable corresponding estimation methods.
In a seminal paper Sims (1980) criticized traditional simultaneous equations systems and proposed using vector autoregressive (VAR) models as alternatives. Since then structural VAR models have become a standard tool for macroeconomic analysis. Structural VAR models are estimated with a variety of methods that depend on the model setup and the type of structural (identifying) restrictions. The estimation techniques that have been used in this context are least-squares (LS), method-of-moments (MM), instrumental variables (IV), generalized method-of-moments (GMM), maximum likelihood (ML) and Bayesian methods. All these methods are described in the related literature (Breitung
The study focusses explicitly on models where the identifying restrictions are placed on the impact effects of the shocks or on the structural relations of the variables. It sets up the model as a system of simultaneous equations and directs the reader to the most suitable estimation methods for a given set of restrictions. The presentation of the estimation methods partly follows Kilian and Lütkepohl (2017). The methods are not only presented but also practical advice is given which one to use in a specific situation. The emphasis is on exposing the methods rather than specific estimation algorithms because suitable software for most of the methods is available in the internet. (For example, Matlab code for many of the methods mentioned in this review is available at http://www-personal.umich.edu/~lkilian/book.html. JMulTi is a menu-driven software that includes estimation methods for a range of structural VAR models (Krätzig, 2004). It can be downloaded from http://www.jmulti.de. Moreover, EViews is a commercial software with a structural VAR estimation part (EViews, 2000, http://www.eviews.com/home.html).) Also statistical inference derived from the estimates is not the focus of this survey because inference methods in structural VAR analysis are more important for functions of the parameters such as impulse responses than for the structural parameters. For impulse responses and related quantities frequentist inference is typically based on bootstrap methods that require the computation of a large number of point estimates. Hence, being able to obtain such estimates is important. This survey can help finding suitable estimation techniques that can be used as basis for bootstrap methods.
The study is structured as follows. In Section 2 the model setup is presented and Section 3 discusses the types of restrictions considered. Sections 4–7 describe a range of estimation methods and illustrations are presented in Section 8. Section 9 concludes and mentions some extensions.
The following general notation is used throughout. The natural logarithm is abbreviated as log and Δ is the differencing operator defined such that Δ
The basic model of interest in this survey is a VAR model of order
We consider the
where
The VAR(
has all its roots outside the complex unit circle. If the polynomial (
To represent structural relations between the variables, the equations (
In this structural-form VAR representation, the structural innovations or shocks are
where
Alternatively, the structure can be imposed by defining
In other words, the reduced-form innovations
Occasionally a combination of A- and B-models is of interest to ease the imposition of the identifying restrictions. It is set up as
and is referred to as AB-model (Amisano and Giannini, 1997). In this model, again
For the purposes of estimation it is useful to note that the exposition of the estimation methods for the structural parameters is sometimes simplified by concentrating out the reduced-form slope parameters
if there are no restrictions on
Notice that the standard levels form of the VAR(
where
Denoting the corresponding estimated residuals by
The preferred estimation method depends not only on the model setup but also on the identifying restrictions. In the next section alternative sets of restrictions are presented which are important in this context. The corresponding estimation methods are discussed in Sections 4–7.
It is recommended to use a model setup that facilitates imposing the identifying assumptions in the form of linear restrictions. These are the most popular restrictions in practice. Important examples are exclusion restrictions on the impact effects of the structural shocks or on their long-run effects.
Recursive models are perhaps the most common structural VAR models identified by short-run restrictions on the impact effects of the structural shocks. They amount to restricting
A recursive setup is also often used in partially identified models where, for example, only one economic relation is identified or only one structural shock is of interest. In that case only one row of
can be set up in this case. The first two equations in this system are clearly not identified but can be identified by setting
For non-recursive models or over-identified structural VAR models it is important whether linear restrictions are formulated for
Generally linear restrictions on
where
and
Alternatively, there may be linear restrictions on
where
where
Structural identification in VAR models is often achieved by imposing restrictions on the long-run effects of the shocks. Since the long-run effects of the shocks in a stable, stationary VAR model are zero, the long-run effects considered in the context of stable models are the accumulated effects of the shocks. Defining
for the AB-model and analogously for the A-model and the B-model. It is more common in this context to use either the A-model or the B-model. In particular, the B-model with long-run multiplier matrix Ξ =
Typical restrictions in this setup are exclusion restrictions. The most common set of restrictions constrains Ξ to be a triangular matrix (e.g., Blanchard and Quah, 1989). More generally, for the B-model, linear restrictions on Ξ can be written as
where
It is easy to combine these restrictions with further linear restrictions on the impact effects.
In a model with integrated and cointegrated variables, a shock can have permanent effects on the variables. Hence, considering such effects for identification purposes makes sense. The long-run effects matrix for the B-model in this case is
where
Again linear restrictions can be imposed easily on
Typically such long-run restrictions are complemented with linear restrictions directly on the impact effects. Such restrictions are in fact necessary for identifying all shocks properly if there is more than one purely transitory shock in which case such shocks can obviously not be distinguished by their long-run effects which are all zero.
All the frequentist estimators considered in the following are consistent and have asymptotic normal distributions under standard conditions. Many of them are even equally efficient asymptotically. However, as mentioned in the introduction, the structural parameters as such are often not of main interest but derived quantities such as the implied impulse responses and forecast error variance decompositions are of interest. Inference for these quantities is typically based on bootstrap methods for which a large number of estimates has to be computed. Thus, it is essential that a computationally efficient and robust estimation method is used for the structural parameters. This survey is meant to direct the reader to suitable methods for a given setup. Asymptotic properties of the estimators and conditions for their validity can be found, for example, in Kilian and Lütkepohl (2017, Chapters 9 and 11).
If the model is just-identified and there are just exclusion restrictions, estimation of the A-model is particularly simple. A popular example is a recursive model where the structural matrix
The latter estimator uses a degrees-of-freedom adjustment and may be preferable in small samples. The estimator Σ̃
The MM estimator of
where the
Generally, if the A-model is just-identified and there are just exclusion restrictions for the off-diagonal elements of
where
Summarizing the model for
where
This estimator is equivalent to the MM estimator based on solving the system of equations
for the unrestricted elements of
LS estimation can also be used if there are over-identifying restrictions of the form
where
where
This estimator can also be interpreted as a GMM estimator based on the moment conditions
In other words, if this GMM objective function is used for estimation,
So far only the estimation of the
by GMM. Note that there may be equations without any parameters to estimate like in the recursive model. Such equations are dropped from the system. The full system has to be estimated jointly, if there are cross-equations restrictions.
It is worth emphasizing again that concentrating out the parameters of the lagged variables and working with the estimated reduced-form residuals instead of the observations is not the same as the standard GMM framework for estimating the structural model including the lagged values. If standard GMM software is used, it is preferable to work with the original observations and include the lagged values in the equations. The same formulas can be used in that case but the symbols have to be adjusted properly. For example,
Estimates of the elements of
Suppose there are just-identifying restrictions of the form
where this set of equations may contain short-run and long-run restrictions. In other words,
For this type of restrictions, MM estimation is the preferred method for just-identified models. Since
∑
is solved by a nonlinear equations solver. It can be shown that the resulting estimates
If there are over-identifying restrictions, Gaussian ML estimates of the structural parameters are obtained by maximizing the concentrated log-likelihood as a function of the structural parameters,
where
Pagan and Pesaran (2008) observe that in a structural VECM the cointegration relations can serve as instruments under certain conditions. They show that a VECM with
where the intercept has been deleted for simplicity,
Since
Kilian and Lütkepohl (2017, Section 11.2) emphasize that this IV method for estimating the structural parameters in the presence of long-run restrictions requires that (1) the number of transitory shocks is equal to the cointegrating rank,
If the B-model is just-identified, a MM approach to estimating
Replacing ∑
If
If
such that, for example,
(see, Lütkepohl, 2005, A.9.3). This MM estimator is also the Gaussian ML estimator if ∑
Estimation by Cholesky decomposition is also possible if there are recursive long-run restrictions, i.e., if Ξ =
is the MM estimator of the structural matrix
It is important to note that this MM estimator involves the inverse of
If the restrictions are just-identifying but not recursive so that
for
If over-identifying restrictions are available for
A set of moment conditions for estimating
for which the empirical counterpart is
Defining
with
the GMM estimator for
subject to all identifying and over-identifying restrictions. Typically numerical algorithms have to be used for minimizing
As mentioned earlier, the MM estimator is equivalent to Gaussian ML estimation if Σ̃
In general the concentrated log-likelihood function can be maximized by a numerical optimization algorithm with respect to
If
where
MM estimation can also be used for just-identified AB-models. The moment conditions are
Given identifying restrictions
the system of equations to be solved for
where
For over-identified models, GMM or Gaussian ML estimation of the AB-model may be used.
GMM estimation of the AB-model is a straightforward extension of GMM estimation of the B-model. If there are identifying restrictions on both
subject to all restrictions on both
If there are no restrictions on the reduced-form parameters, Gaussian ML estimation is again executed by concentrating on the structural parameters. The relevant concentrated log-likelihood function is
(see Lütkepohl, 2005, Chapter 9).
In general the concentrated log-likelihood function can be maximized by a numerical optimization algorithm with respect to
In other words, ML estimation and MM estimation provide identical results if the same estimator for ∑
In practice, structural VAR analysis is often based on Bayesian estimation methods. Bayesian estimators are obtained by evaluating the posterior distribution of the parameters of interest or functions of the structural parameters. The first step in constructing the posterior distribution of the structural parameters is the specification of a prior. In the framework of structural VAR analysis the prior is either imposed on the reduced-form parameters or on the structural parameters. The latter approach is more plausible if the structural parameters are of main interest and prior believes are available for the structural parameters. Priors imposed on the reduced-form parameters often reflect the limited structural knowledge of the investigator. They are formulated with the objective to obtain posterior distributions which are easy to sample from and to induce little distortion for the structural analysis.
Gaussian-inverse Wishart priors are conventional priors for reduced-form VAR models. The mean of the prior for the VAR coefficients is typically specified as for a so-called Minnesota prior which shrinks the VAR slope parameters to zero or a random walk process depending on the persistence properties of the data (Kilian and Lütkepohl, 2017, Chapter 5).
Suppose that the VAR(
and
where ℐ (
where
and
Here
and
Since the posterior is from the same distributional family as the prior, the prior (
Using a prior that gives rise to a known form of the posterior distribution of the reduced-form parameters is convenient because it makes it easy to draw samples from the reduced-form posterior. For just-identified models, reduced-form posterior draws can then be transformed to draws of structural parameters (e.g., Canova, 1991; Gordon and Leeper, 1994). For example, for a recursive identification scheme, a Cholesky decomposition of the
Given that the prior in this approach is not specified for the structural parameters which are supposedly the parameters of interest, this approach is unsatisfactory. It is even more problematic if there are over-identifying restrictions for
Canova (2007, Section 10.3) compares the structural priors discussed in Sims and Zha (1998) with more traditional Minnesota priors on the reduced-form parameters and points out that there are important differences. Thus, imposing the prior on the structural parameters can make a difference for the posterior of the structural parameters.
In this section some illustrative examples for estimating structural VAR models are provided. The ISLM example data from Breitung
A VAR(4) model is fitted to the data, as in Breitung
As discussed in the previous sections, the choice of estimation method depends on the type of model and restrictions. The frequentist estimation methods most suitable for specific model types and restrictions are summarized in Table 1. To illustrate the methods, the following three alternative A-models with
are considered. They all have diagonal structural covariance matrix
The asterisks denote unrestricted elements while 0 and 1 stand for restricted parameters. The first model is recursive. The second identification scheme is not recursive but just-identified and the third scheme is over-identified with just two structural parameters to be estimated in the
Based on Table 1, MM via Cholesky decomposition is the recommended estimation method for the recursive model. The resulting estimates are presented in Table 2. Equivalently, the unknown elements of
and
where
LS is also the recommended method for estimating the nonrecursive just-identified model. In other words, LS is applied to the two equations
and
The resulting estimates are also presented in Table 2.
Finally, 2SLS may be considered for estimating the second equation of the over-identified A-model,
The estimates are also given in Table 2, where the third equation of the over-identified model is estimated by LS.
To illustrate the estimation of B-models, three such models with
and ∑
To illustrate an AB-model, I note that an A-model can be cast in AB form by specifying the
MM estimation is a suitable method for this type of model (Table 1). It is equivalent to Gaussian ML. The estimates for the example model are given in Table 4.
Finally, to illustrate the estimation of a structural VAR model in the presence of long-run restrictions, consider a B-model with lower-triangular long-run multiplier matrix Ξ,
Note that, in practice, for the example model such restrictions are not recommended because the variables may be integrated in which case
This estimate was also computed with the software JMulTi.
This study reviews methods for estimating structural VAR models set up as A-models, B-models or AB-models. A range of alternative estimation methods are considered. It is stressed that the identifying restrictions should be placed such that easy estimation methods can be used. More precisely, it is useful if the restrictions can be imposed as linear restrictions on
Many estimators allow to concentrate out the reduced-form parameters and estimate the structural parameters from reduced-form residuals or the residual covariance matrix estimator directly. Gaussian ML and GMM are the most general estimation methods which can be used even if there are over-identifying restrictions on
It is also possible to estimate structural VAR models by Bayesian methods. The conventional approach of formulating a prior for the reduced-form parameters and generating posterior draws for the structural parameters
There are a number of extensions of the basic model setup considered in this survey. First of all, linear identifying restrictions are emphasized because they facilitate estimation of the structural parameters. Some of the methods can in principle be adopted to nonlinear restrictions as well. For example, Gaussian ML and GMM estimation can be used in principle in conjunction with nonlinear restrictions. The computational challenges may become more burdensome for that case, however.
The basic model in this survey is a finite order VAR(
Another extension of the basic model setup allows for heteroskedasticity or conditional heteroskedasticity. Clearly, time-varying residual volatility is a common feature of financial data and therefore it also plays an important role in structural VAR analysis. Such features are in principle easy to deal with by using, for example, generalized LS methods rather than LS or by adjusting the likelihood function appropriately. Since the structural parameters are related to the residual covariance matrix, it is in fact possible to use a time-varying covariance structure for the identification of structural shocks. A broad literature addressing the topic of identification through heteroskedasticity has evolved following proposals by Rigobon (2003), Lanne and Lütkepohl (2008) and others. Recent surveys of the related literature are provided by Lütkepohl (2013) and Kilian and Lütkepohl (2017, Chapter 14).
This review focusses on point-identified structural VAR models. In practice set-identified models based on sign restrictions for the structural parameters or the derived impulse responses are often considered (see Uhlig (2005) for an important contribution to this literature and Kilian and Lütkepohl (2017, Chapter 13) for a recent survey). Although extensions of the setup discussed in this review to such models are not straightforward, some of the estimation algorithms presented in the current study are important building blocks of the related algorithms used in the sign restriction literature. Therefore being familiar with the methods discussed in the current study is useful.
Structural restrictions and frequentist estimation methods
Restrictions | Estimation method |
---|---|
Recursive model (triangular | Cholesky decomposition, MM |
Just-identifying linear restrictions on | LS |
Over-identifying linear restrictions on | IV, 2SLS, GMM |
Just-identifying linear restrictions on | MM |
Over-identifying linear restrictions on | GMM, ML |
Just-identifying linear restrictions on | MM |
Over-identifying linear restrictions on | GMM, ML |
MM = method-of-moments; LS = least-squares; IV = instrumental variables; 2SLS = two-stage LS; GMM = generalized method-of-moments; ML = maximum likelihood.
Estimated A-models
Model | Â | Σ̃ |
---|---|---|
Recursive | ||
Nonrecursive | ||
Over-identified |
Computations were done with suitably adjusted Matlab code provided at http://www-personal.umich.edu/~lkilian/book.html.
Estimated B-models
Recursive model | |
Nonrecursive model | |
Over-identified model |
Computations were done with JMulTi (see http://www.jmulti.de).
Estimated AB-model
Â | B̂ |
---|---|
Computations were done with suitably adjusted Matlab programs provided at http://www-personal.umich.edu/~lkilian/book.html.