If time-series are I(0), as shown in Figure 4.1, a vector autoregression model (VAR) can be estimated. As already noted, the main purpose of VAR models is that they are useful for forecasting systems of interrelated stationary time-series data and for analyzing the dynamic impact of random disturbances on the system of variables. This applies even if the VAR models do not represent the truth in economics, but are nevertheless useful for gaining understanding in the interactions between variables, by providing relevant descriptions of the data.
The basic form of a p-th order bivariate VAR model is shown in Equation 4.10 and Equation 4.11 which may be also estimated using standard OLS regression techniques (Wooldbridge, 2008).
Following Sims (1980) let us assume:
(4.10)
(4.11)
Where as well as must be stationary, and are uncorrelated white-noise error terms. Equations 4.10 and 4.11 therefore represent the first order VAR model since the longest lag length is equal to unity in its non-reduced form where has a contemporaneous impact on and the reverse relationship is valid too.
In its matrix form Equation 4.10 and Equation 4.11 can be written as: (4.12) B zt 0 ut
90 Thus:
(4.13)
The reduced (standard) form of VAR model is achieved by multiplying both sides of the Equation 4.13 by :
(4.14)
Therefore the VAR model can be written in the form:
(4.15)
(4.16)
The system of Equations 4.15 and 4.16 represent the VAR model where and are composites of the two shocks.
When estimated in this form, the VAR model has a good potential to provide empirical evidence on the response of economic variables to monetary policy shocks (Christiano et al., 1996).
4.5.1 VAR critique
It is convenient to note that every model has some limitations, but an important question is whether the advantages of the model overcome its limitations. VAR models are usually preferable due to the flexibility in formulating data without restrictions on the dynamic relations between variables approached by economic theory, as well as in testing economically meaningful hypothesis. In comparison with univariate time-series models there is no need to specify exogenous and endogenous variables since all variables in VAR models are endogenous. Also the value of a variable can be dependent on more than its own lags (Brooks, 2008). Nevertheless the most appreciated advantage of VAR models compared to traditional models is their simplicity and better forecast results. As showed by Sims (1980) and McNees (1986), VAR models produced more accurate forecasts for variables than large- scale structural models. On the other hand, the main critique of VAR models is based on the difficulty of interpreting the coefficients. The interpretation problem arises due to the lack of restrictions on parameters causing the causality to float in both directions (Figure 4.2).
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Figure 4.2: The identification problem of VAR models
Source: Author
Therefore, the shock in the system can affect both the variables. The difficulty of interpreting coefficients leads to using VAR models in their reduced form (Equation 4.14) which allows for testing the formation of expectations (Strongin, 1995).
The most common approach for overcoming the problem of interpreting coefficients is to use an impulse response function (discussed later in Section 4.7.4) which examines the response of variables to the shock. The complication in this case is how to define the shock (Asteriou and Hall, 2007). However, impulse responses which are driven from VAR in their reduced form lack the structure to be easily interpreted. To overcome this problem, Sims (1980) proposed a transformation by triangulating the system which enables the interpretation of the system. A different and popular approach for dealing with the problem of interpreting VARs is to develop a structural vector autoregression (SVARs) which introduces ‘theoretical’ restrictions to identify underlying shocks.
4.5.2 Structural vector autoregression model (SVAR)
As noted before, the VAR models are usually used for forecasting and perform well in investigating the effects of shocks on a system of variables. The interpretation of estimated results from an unrestricted VAR model (Equation 4.15 and Equation 4.16) is not an easy task. The output of an unrestricted VAR cannot provide information about how the economy reacts to different shocks. Thus, to investigate the effect of shocks on a system of variables it is necessary to impose restrictions on the model. The need for restrictions led to the development of structural vector autoregression models SVARs. However, an important implication in SVAR models is that only restrictions common to a variety of theoretical models can be applied (Bank of England, 1999). The advantage of SVAR models over VAR models or traditional models is that the focus is on obtaining information about the shocks’ driving movements in the endogenous variable and the identification of the effect of these shocks on movements between these endogenous variables.
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SVAR models become especially beneficial when the different effects of structural shocks need to be examined. Bernanke (1986) introduced an identification scheme for SVAR models which is widely used nowadays. The approach is based on imposing n(n-1)/2 zero restrictions on the matrix where n is a number of variables in the SVAR model. Blanchard and Quah (1989) suggest imposing zero restrictions on levels only on the long-run effects of structural shocks in endogenous variables.
Assume the three variable SVAR model from Equation 4.14 and Equation 4.15. The zero restrictions on matrix will, according to Blanchard and Quah (1989), have a form:
(4.17)
However, this approach is based on the assumption of a non-stationarity series I(1) with no cointegrated relationships. Nevertheless, not all series are stationary and cointegration cannot always be rejected, therefore King et al. (1991) suggest taking into account any cointegrating relationships in the system method by estimating the VAR model in its error correction form. The identification of structural parameters requires restrictions on the elements of A and B matrixes. The SVAR model can be identified either by imposing restrictions that economic variables do not react to monetary variables simultaneously but this is not rejected to be the other way round. Or alternatively, by imposing restrictions on variables that represent monetary policy in order to reflect operational procedure. In case of monthly data, these restrictions can either be driven from the theory or based on institutional analysis. Once the monetary policy is identified, a VAR model helps to identify the deviations from the policy rule, thus it is possible to detect the response of macroeconomic variables to the shock in monetary policy (Bagliano and Favero, 1998).
4.5.3 SVAR critique
However, there are several potential problems with SVARs. First, as Blanchard and Quah (1989) recognised, the economy may be hit by higher number of shocks than identified in the SVAR system. SVARs will therefore produce reliable results only if the most important types of shock are identified in the system. This could be overcome, but not completely avoided, by imposing restrictions which are widely accepted in the theory. Another limitation of SVAR models is that due to the exact identification of the system, not all tests usually applied to VAR models can be implemented. The Granger causality test can be an example since it does
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not provide any information about contemporaneous causality. SVAR models as well as VAR models are particularly useful when the determinants of endogenous variable movements are unobservable. However, there are situations when the advantages of traditional modelling overcome the advantages of SVAR models, especially in the case of identifying restrictions which can be inappropriate, unrealistic or insufficient. Moreover, though VAR and SVAR models have been used widely to examine the reaction of monetary policy to defined shocks, they are difficult to use for particular policy analysis such as estimating Taylor rules. The choice of using autoregressive models must therefore be determined case by case (Bank of England, 1999).