Methodology: Structural Vector Auto-Regression (SVAR)

The method that is commonly used to analyse the transmission mechanism of monetary policy in contemporary times is the SVAR method. SVAR has significantly impacted and enhanced macroeconomic research, and thus research analyses of the effects of monetary policy in the last decade have been enormous.80 Bernanke & Mihove (1995), Sims (1980) and Watson and Stock (2001) are some of the works which showed that S(VAR) methods can be used to summarize macroeconomic time series to make forecasts, and provide valuable policy advice for economic decisions. Vector auto regression (VAR) from which SVAR is a component is well known for its remarkable contribution to the field of macro-econometrics in contemporary times. The main contributions lie in its ability to answer important questions about the dynamic interactions conditional upon the future path of variables of interest such as monetary policy rate or tax rate. Through VAR, macro-econometricians have been able to investigate the effects of monetary policy on real GDP and inflation, and the relative strength of individual channels of monetary policy. First, structural VAR provides avenues through which econometricians are able to pin down the effects of monetary policy shocks and trace their expected impact on various macroeconomic variables (Bjornland & Jacobensen, 2010). Through this method we can recover the true structure of the economy and macro-economy behaviours of time series from the national account data. SVAR has become one of the main tools for testing and evaluating the effectiveness of economic policies over time. It is, therefore, well accepted by many as one of the means to empirically test theoretical models with real data. Finally, it is argued that SVAR avoids incredible restrictions in single equations and strict restrictions in DSGE models. In all, this method helps to test formal theories and helps to learn more about the dynamics of the macroeconomics over time.

2.3.1 S(VARs) Descriptions

SVAR is defined as a system of k-equations and k-variables of stationary linear relation, where current variables are explained by contemporaneous terms, their own lags and the lags of ¢ − 1 remaining variables (Stock and Watson, 2001). A general formal SVAR appears in the following format:

£= ¤R+ Ψ@+ £.∗/.+ £¦∗/¦+ ⋯ + £∗¨/¨+ >€ (2.1)81

80 _{Christopher Sims won the Nobel Prize in Economics (2011) for his works on VAR and for its usefulness in}

diagnosis of dynamic economic behaviours through impulse response functions and variance decomposition. Most questions about dynamic behaviour, interactions and the effects of monetary policy shock on variables such as GDP and inflation are answered through SVAR analysis.

The matrix A with ¢ × ¢ dimensions is called an invertible matrix of contemporaneous coefficient relations on Y_t ; and Y_t is column vector ¢ × 1 of endogenous variables. Generally, contains non-policy macroeconomic variables and policy variables assumed under of policy makers. In addition, £∗ (for i=1, 2…p) are matrices of structural coefficients on the lagged variables in the model. The entries in these matrices represent the dynamic properties of the system while the interaction of variables is represented by cross-variables coefficients. Ris ¢ × 1 vector that contains all deterministic terms e.g. a linear trend, seasonal and other user specified dummies to capture the structural breaks, and intercept. While @ is a vector of exogenous variables. € is a ¢ × 1 vector of structural shocks normally distributed with mean zero and its variance-covariance matrix

I

=

Ω . The matrix > is ¢ × ¢ –dimensional matrix that specifies which variables are to what extend directly affected by structural shocks. This matrix > is usually set as a diagonal matrix.

One immediate problem with the SVAR method is that it cannot be estimated as it is in (2.1) using the Ordinary Least Squares (OLS) method. This is because the main standard assumptions about the system in are that the variables in are stationary, and the variance-covariance p¡V(_c, ›_c) ≠ 0 are violated in the basic VAR and SVAR models. SVAR in its primitive system violates the OLS assumption of no relation between structural shocks € and independent variables in matrix A. Thus, using OLS to estimate the matrix A will produce inconsistent parameter estimates and incorrect impulse response functions. Circumventing this problem requires that we exclude some contemporaneous effects by restricting them to zeros; in this way the system will fulfil the assumption of no correlation and become identified. Explicitly, in order to overcome this problem, econometricians have devised procedures to recover the true structural parameters for the underlying structural VAR model from the standard reduced form VAR model – (see Enders, 2010, pp. 325-338).

A short run SVAR without R and Ψ then can be written as follows:

£ F1 − £.() − £¦(¦) − ⋯ − £¨(¨)G = £E = >€ (2.2)

And its standard form reduced form is given as follows:

= £.∗/.+ £¦∗/¦+ ⋯ + £¨∗/¨+ £/.>€ (2.3)

Whereby £_/c∗ = £/._& £_/c

Enders (2010) points out that the departure point to analyse SVAR is to estimate the compact reduced form model, which mimics the predictable movements of variables within the system82 (see Robinson & Robinson, (1997). From the reduced form VAR we obtain the residuals E. Using equation (2.3), we do linear mapping of residuals E into the £/.>€, hence this can be used to identify the structural shocks by imposing identification restrictions on matrices £ and/or B. Procedurally, we want to express the non-orthogonal Efrom the VAR(p) reduced form model as a linear combination of orthogonal structural shocks (€) in order to obtain the innovation model:

E = £/.>€ or simply £E = >€. (2.5)

Equation (2.5) gives the general class of innovation model defined as above in (2.1). The structural shocks are identified by placing identifying restrictions on the contemporaneous matrix A and the matrix B. u_tdenotes VAR residual vector of dimension ¢ × 1, normally independently distributed with full variance-covariance matrix Σ_u . It is commonly acknowledged that the reduced form in (2.4) does not tell us anything about the structure of the economy. Thus, it is necessary to show the mapping of the structural representation in (2.1) into the residuals from the reduced form equation. Equations (2.4) and (2.5) show how the non-orthogonal observable residuals are related to the unobservable structural innovations – that is, expressed as a linear combination of structural shocks. Further, the relation between the variance-covariance matrices of E and € is derived as follows:

%(EE) = £/.>%(€€)>£/.= Σ¬ (2.6). A crucial factor in working with SVAR is that without imposing some identifying restrictions, the system of equations remains unidentifiable – there are no unique solutions for the coefficients in the system.83 Pfaff (2008) shows that there are three common short run identifications of SVAR models, which are all distinguished by the types of restrictions placed on them. SVAR A-model: sets matrix B to ­_“ד. The minimum restriction that must be imposed for exact identification is “(“/.)

¦ . SVAR B-Model: sets matrix A to ­“ד and

the minimum restriction that must be imposed for exact identification is the same as in SVAR model A type. SVAR AB-model: places the restrictions on both A and B matrices.

82 _{See also Robinson & Robinson (1997).} 83 _{This means that given the values of R}

, £¨, and Σ¬ in the reduced form (2.5) it is not possible to uniquely solve structural parameters of the SVAR in (2.3) without placing some identifying restrictions on matrix A0.

The number of restrictions for exact identification on this model is given by ¢¦+ ¢(¢ −

1)/2. In this study we applied the SVAR A-model and AB-model procedures for structural VAR to extract these structural parameters. The meaning of these sets of identifications in a form of zero restrictions is discussed explicitly in Section 2.4 where we set out the structural economic representation of the model.

2.3.1 SVAR: Impulse Response Functions and Forecast Error Variance Decomposition

We mentioned in Section 2.3 that the aim of SVAR is to test formally the theories that form a general structure of the vector auto-regression, and to learn about the historical dynamic behaviours of the economy. However, Enders (2010) pointed out that individual coefficients from VARs or SVARs are of little use in themselves. Hence, we considered two main important outputs of SVAR: the structural impulse response functions (SIRF) and the structural forecast error variance decompositions (SFEVD). Many macro- econometricians agree that these two outputs give a better picture in a palatable manner. The former helps us to show the dynamic response of current and future values of each variable to a one unit change in the current value of one structural shock while assuming that other shocks are equal to zero. The second is the forecast error variance decomposition that provides the relative importance of each structural shock in influencing endogenous variables in the SVAR. Using the VAR in (2.3) the impulse response functions are derived as follows:

Let us take L as the lag operator, and£() = ∑ £¨_{c;. c}c; then (2.3) can be transformed into a structural vector moving average (SMA) as follows:

£&− £() = >€ (2.7)

⇒ = £&− £()/.>€ ; Let¤ = £_&− £()/.

= ¤&€+ ¤.€/.+ ⋯ + ¤¯€/¯ (2.8)

= ∑: ¤_¯€_/¯

¯;& (2.9)

The SMA is based on an infinite moving average of the structural innovation € in (2.9). The (i,j)-th element in matrices Ds stands for the dynamic multipliers - the expected partial impacts of a random change in j-th variable in the system at time t, on the i-th variable within the system at time t+s. In simple terms the matrices Ds constitute marginal effects of the innovations in the system on yt+s. This is expressed as follows:

t s t s _e Y D =∂ + _∂ . (2.10)

It is very important to emphasize here that as s increases we will observe the dynamic path of variable i-th in response to innovation in variable j. Hence, the structural impulse responses are the plots of k_cv(¯)°¤_¯ vs. for i,j=1,2. Generally, these expected partial impacts are only meaningful when all other shocks at time s are set equal to zero (Favero, 2001). This is naturally true in terms of interactions between foreign variables and domestic variables of a small country; however, it is false for the interaction between domestic variables. To overcome this problem, we place restrictions on some of the variables in the system so that the interactions we allow for are justified by economic theory.

Another output that is of interest from the SVAR for our analysis is called the variance of decomposition. This analysis explains the variation in all variables within the system. Under this analysis, we want to find out what portion of the total variance of yt is

attributed to the random shock in the j-th shock. This analysis helps us to assess the relative importance (strength) of each variable in the system. Thus, this result will give us the quantitative picture about the relative strengths of interest-rate and credit channels.