Some aspects of structural VAR analysis

5.2.2.1 About shocks

As is well known, VAR models concentrate on shocks. Structural shocks occupy then a central place in the SVAR model.

These structural shocks are the input of a linear dynamic system, which represent the driving force behind the stochastic dynamics of the n-dimensional time series yt. They are unpredictable with regard to the past of the process. These structural shocks are generally attached with a certain economic meaning such as an oil price shock, a productivity shock, an exchange rate shock or a monetary shock. Note that such shocks are not thought of as disastrous singular events. On the contrary, it is assumed that the economy is hit regularly by such shocks. The size of these shocks is, however, usually small.

Since such structural shocks can not be directly observed, assumptions are needed to identify them. A consensus has come into being which states that structural shocks should be mutually uncorrelated (orthogonal). To explain this orthogonality restriction in SVAR models, Bernanke (1986) thinks of these structural shocks ‘as ‘primitive’ exogenous forces … which buffet the system and cause oscillations. Because these shocks are primitive, i.e., they do not have common causes, it is natural to treat them as approximately uncorrelated.’ This assumption is necessary to consider the dynamic impact of an isolated shock. If the shocks were correlated, the relationship between the shocks must be taken into account and, as a result, it is impossible to distinguish the effects of different shocks.¹³

5.2.2.2 SVARs in comparison with simultaneous equation systems

Regarding the issue of identification, there are important differences between a SVAR model and a simultaneous equation model.

At first, systems of simultaneous equations are usually identified by linear (exclusion) restrictions. SVAR models, instead, assume orthogonal shocks and,

13 In fact, the decomposition into orthogonal components has a long tradition in statistical analysis.

hence, the structure is identified also using restrictions on the covariance matrix of the errors. This makes the estimation of such systems considerably complicated.

Secondly, much more restrictions than necessary are generally employed to identify the traditional system of simultaneous equations models. That means that these models are often highly over-identified. It is just these over restricted models that are qualified by Sims (1980) as ‘incredible’ in his famous critique.

SVAR proponents instead try to avoid such an over-simplification of the structure and thus impose just as many restrictions as necessary to identify the structure.

Consequently, most SVAR models are just identified.

However, note that just identified models are no more than convenient reformulations of the reduced form. As a result, so long as the reduced form and the just identified structure are correctly specified, it is impossible to decide between alternative identified structures on empirical grounds. SVAR models are thus used to quantify prior views of the economy and to assess the plausibility of the outcomes.

5.2.2.3 SVAR as an extension of the traditional VAR analysis

Advocated by the influential Cowles Commission, the simultaneous equation approach surely dominated the empirical research in econometrics until the late 1970s. The initial optimism about the potential of the simultaneous equation system was, however, not fulfilled. The inability of large macroeconomic models to compete with ‘a-theoretic’ Box-Jenkins ARIMA models on predictive grounds has led to an increased interest in time series analysis. Particularly, the seminal paper by Sims (1980) paved the way for the ultimate success of the VAR approach in empirical macroeconometrics.

The development of this approach was thus promoted by both the inability of economists throughout the 1970s to agree on the true underlying structure of the economy and the Lucas critique, which states that changes in policy systematically alter the structure of econometric models, with a resultant shift away from the use of large scale macroeconomic models as forecasting tools.

Attempting to overcome the problem of ‘incredible identification restrictions’

in traditional macroeconomic modeling, particularly in the determination of exogenous variables, the VAR approach takes all variables as endogenous.

Since economists could not get an agreement on the true structure of the economy, VAR models were developed with the belief that such models could reveal important dynamic characteristics of the economy, without imposing structural restrictions from a particularly economic theory. Impulse response functions and variance decompositions, which illustrate the dynamic characteristic of empirical models, were initially obtained by a mechanical technique which was often believed to be unrelated to economic theory.

However, as argued forcefully by, e.g., Cooley and LeRoy (1985), traditional VARs are of a ‘reduced form’ status and hence only summarize the dynamic properties of the data. Without referring to a specific economic structure, such a reduced form is difficult to interpret. For example, it is generally impossible to draw any conclusion from the bulk of coefficient estimates in a VAR system. So long as such parameters are not related to ‘deep’ structural parameters characterizing preferences, technologies, and optimization behavior, these parameters do not have an economic meaning.

The criticism of this traditional a-theoretical VAR method has led to the development of a new kind of econometric model, which is now known as the

‘Structural’ VAR approach. SVAR analysis is an extension of traditional unstructured VAR analysis. It is the imposition of a certain structure that makes SVAR different from the traditional VAR analysis.

SVAR approach allows researchers to use economic theory to transform the reduced form VAR model into a system of structural equations. The difference between the two is that within a SVAR framework, it is attempted to identify a set of independent shocks by means of restrictions under the guide of economic theory; whereas a-theoretical restrictions are used in traditional VAR analysis. As a result, SVAR approach yields impulse responses and variance decompositions that can be given structural interpretations, which is in sharp contrast with traditional VARs.

The first strand of SVAR analysis made use of economic theory to impose restrictions on the observed values of estimated residuals et to recover underlying

structural shocks εt.¹⁴ Such kind of SVAR approach estimates structural parameters by imposing contemporaneous structural restrictions based on economic theory, which is different from the arbitrary method of imposing restrictions used in traditional VAR analysis. These structural restrictions can be considered as short-run restrictions in that shocks are considered to have temporary effects. A representation of the standard VAR model in levels, as shown in equation (5.5), would apply in this context.

An alternative SVAR approach allows shocks to have permanent effects.¹⁵ In this framework, long-run restrictions are utilized to identify the economic structure from the reduced form. This would imply that the variables are non-stationary since shocks continue to accumulate through time resulting from their permanent effects. The presence of a unit root in the variables can lead to spurious regression if the VAR is estimated in levels. It is thus necessary to use first differences to ensure stationarity when shocks have permanent effects. Here a standard VAR in first differences like that in equation (5.6) is appropriate. Such models have long-run characteristics that are more readily acquired from economic theory. They usually exhibit sensible short-run properties as well.

In short, a structural VAR is a standard VAR where restrictions needed to identify the underlying structural model are provided by economic theory. These restrictions can be either contemporaneous or long-run, depending on whether economic theory suggests the effects of the shocks are temporary or permanent.