Structural VEC model specification and testing

Before estimating the parameters of the VEC or SVEC it is necessary to verify if the required conditions for the adequacy of the selected model are satisfied. Once the parameters of the model has been estimated, is necessary to check if the model is an adequate description of the data. For this purpose, we perform the test for stability

3 . 6 . A P P L I C AT I O N O F V E C T O R AU T O R E G R E S S I V E M O D E L S

condition of the system, the residual autocorrelation test, the residual normality test and finally the cointegration test.

Regarding multivariate tests we use the multivariate Portmanteau test for au- tocorrelation proposed by Hosking [61], as described in Lütkepohl [77], which is a generalization of the univariate Ljung-Box test. For multivariate normality test, we use the statistic proposed by Doornik and Hansen [36], which is a generalization of Jarque and Bera normality test for univariate case. The test is based on sample measures of multivariate Skewness and Kurtosis. In this case, before estimation the VEC and SVEC models, it is necessary to know the number of cointegrating vectors of the series. In this application we use the Johansen [63] procedure which uses likelihood ratio (LR) tests to determine the number r0 of cointegrating vectors, as

the trace test. We reject the null hypothesis of r0 cointegrating vectors if the LR

statistics is greater than the critical value.

Next we unfold an important task in the multivariate analysis of time series or in the study of the VAR models, that of verifying if there is any relationship between the pairs of variables using the Granger - causality test (Granger [54]). In our study we test Granger-causality between a pair of variables using the Dolado and Lütkepohl [35] procedure, since our series are integrated and eventually cointegrated. That is, for a VAR(p) model we fit VAR(p+1) system to the data and perform a Wald test on the coefficients of the first p lags only.

Because in our system of variables, the Augmented Dickey and Fuller test in- dicate that the series are nonstationary individually in level, the VAR analyse is not adequate. Therefore, situations where there are r cointegrating vectors, which describe the long-run relationships between variables, the VEC model, equation 3.28, is an appropriate.

According to Breitung et al. [24], Zivot and Wang [123], the general SVEC model include deterministic terms and exogenous variables. However, for our present purpose the deterministic terms are of no importance because they are do not shape the impulses in the system. Similarly, exogenous variables under the control of some policy markets may not react to stochastic shocks, therefore they are ignored.

Hence, instead of considering a complete model we will consider the simplified one.

A∆yt=Π∗yt−1+Γ∗1∆yt−1+ ... +Γ∗p−1∆yt−p+1+ Bεt (3.45)

where Π∗and Γ∗_i are structural form parameters matrices and εt is a structural error

term, B is a K × K of impact of shocks and matrix A models the instantaneous relations among the variables in the system.

External activity, we assume that all variables are integrated of order one, with rank r = 2, which suggests two transitory shocks and one permanent shock. The permanent shock we identify by difference m = K − r = 3 − 2 = 1, which corresponds to one column with unrestricted elements on the long-run matrix and remaining two columns with hard zeros. The two restrictions mean that the demand and external shocks have no long term effects on variables, while the supply shock is allowed to have on all variables. Because the total number of restrictions is

K(K − 1)/2 = 3(3 − 1)/2 = 3, we replace the third restriction in B and we need r(r − 1)/2 = 1 additional contemporaneous restriction to identify the transitory

shocks. So the two restrictions on the short-run matrix are that the first and third shocks have no instantaneous effect on prices, that is, b21=b23=0. In general we

impose two long-run independent restrictions and additional two contemporaneous restrictions, to identify the supply shocks, demand shocks, and external shocks to be traced in an impulse response analysis, as illustrated ahead in the application.

Short − run =      b11 b12 b13 0 b22 0 b31 b32 b33     

and Long − run =      c11 0 0 c21 0 0 c31 0 0     

As described in subsection 3.5.3, given yt∼ I(0), the set of residuals from the

regression equations can be represented as MA model where the coefficients are known as response to the impulse of structural shocks in the system.

For our study, we are interested in having accumulated effects, so we add the long-run effects of the matrices for all considered periods and because we apply the SVEC model, the impulse response functions are computed based on the VAR model with cointegrated variables, or a level version of VEC model, which implies that the impulse responses Ψs may not converges to zero as s → ∞, consequently, some

shocks will have permanent effects.

In practice, obtain forecast error impulse responses based on matrices related to long-run and short-run is appropriate if the covariance matrix of residuals Σu

is diagonal, otherwise, the forecast error impulse responses are not valid, due to underlying shocks that are not likely to occur in isolation if the components of ut

have instantaneously correlations. To avoid this problem, orthogonal shocks are preferred for impulse response analysis.

Following Breitung et al. [24], one way to obtain orthogonal shocks is to use Choleski decomposition of the covariance matrix Σu. If B is a lower triangular

matrix such that Σu= BB0 the orthogonal shocks are given by ut= Bεt , where the

forecast errors (ut) are linear functions of the structural innovations (εt), assuming

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Hence, the MA representation is yt =Θ0ε1+Θ1εt−1+ ..., where Θi = ΨiB

for i = 1, 2, ..., here Θ0= B is lower triangular matrix with 1’s along the diagonal

and the triangular structural model, equation 3.39, imposes the recursive causal ordering y1t→ y2t→ ... → ykt. This ordering means that, an ε shock instantaneous

in variable y1t produce effects for all variables, while a shock in y2t cannot have

impact in y1t but only on other variables, and so on. If there is no order imposed by

the relations between the variables, Sims [110] suggests trying different orthogonal orders in the identification until finding the one that is most appropriate. Regarding impulse response statistical inference, we based on Bootstrap Efron Percentile and construct 95% confidence intervals.

The forecast error variance decomposition, are another tool for investigating the impacts of shocks in SVEC system. The FEVD gives information for a portion of the variance of the forecast error in predicting yit+h which is due to variability in

the structural shocks between times t and t+h. Using the orthogonal shocks the

h-step ahead forecast error vector, with known SVEC coefficients, may be expressed

as y_t+h− y_t+h|t= Σh−1_j=0Θjεt+h−j. Since the structural errors are orthogonal, the

variance of the h-step ahead forecast error is: var(y_it+h− y_it+h|t) = σ2_ε1Σh−1_j=0(θ_i1j )2+

... + σ_εk2 Ph−1

j=0(θ

j

ik)2, where σεj2 = var(εj) and the proportion of var(yit+h− yit+h|t)

due to shock εj is then F EV Dij(h) = var(εj)/var(yit+h− yit+h|t).

According to Breitung et al. [24], Zivot and Wang [123], when we want to make an identification using Choleski decomposition, it is important to take into account the order in which the variables are arranged in the system yt, because this order

influences significantly the results. In particular, impulse response functions as well as forecast error variance decomposition are affected by the identification choice.