3.3 Estimation of VAR parameters
3.3.3 Checking the model adequacy
Once a VAR(p) model has been estimated, it is of interest to see whether the residuals are compared with model assumption. That is, one should check for stability of the VAR process, the presence of serial correlation, heteroskedasticity and if the error process is Normally distributed.
Stability condition
Using the lag operator and defined the matrix polynomial in the lag operator Φ(L) as Φ(L) = Ik− Φ1L − Φ2L = ... = ΦpLp, the process 3.1 can be equivalently written
as Φ(L)yt= ut.
The VAR process is stable if and only if all included variables are stationary, i.e, all roots of the characteristic equation of the lag polynomial are outside the unit
circle.
det(Φ(z)) = det(Ik− Φ1z − Φ2z2− ... − Φpzp) , 0 for |z| ≤ 1 (3.13)
So the necessary and sufficient condition for VAR stability is that all characteristic roots of Φ(z) = 0 lie outside the unit circle.
Multivariate test for autocorrelation
According to Lütkepohl [77], autocorrelation of the residuals indicates that there is information that has not been accounted for in the model. For testing the lack of serial correlation in the residuals of a VAR(p) model, a Portmanteau test and Lagrange Multiplier (LM) test proposed by Breusch [25] and Godfrey [53] are most commonly applied.
The multivariate Portmanteau test for autocorrelation in a set of residuals is a generalization of the univariate Ljung - Box Portmanteau test for white noise discussed in subsection 2.1.2. The test was proposed by Hosking [61], and it’s applied to the residuals of a multivariate regression, such as VAR model with the following null hypotheses, H0: no residual autocorrelation up to lag h. As a function
of h-lags, the test statistic is.
M LB(h) = T (T + 2) h X i=1 (T − i)−1trCb0iCb−1 00Cb 0 0iCb−1 00 (3.14)
where Cb0i = T−1Pt=i+1T ubtub0t−i is the sample auto-covariance matrix of order i. Under the null hypothesis, the multivariate Ljung-Box test MLB(h) is distributed asymptotically as χ2(K2(h − p)) where K is the number of time series, p the number of lags in the model estimation and h the number of lags chosen after lag p. Rejection the null indicates that at least one series have autocorrelation in the residuals that is, one series is not white noise.
The Breusch-Godfrey LM -statistic is based upon the following auxiliary regres- sion for residuals: ubt = Φ1yt−1+ ... + Φpyt−p+ B1ubt−1+ ... + Bhubt−h+ εt. The null hypothesis is H0 : B1 = ... = Bh= 0, against the alternative H1: ∃Bi , 0 for
i = 1, 2, ..., h. The test statistic is defined as:
LMh= T
K − tr( ˜Σ−1re − ˜Σun)
(3.15)
where ˜Σ−1re and ˜Σun are the residual covariance matrix of the restricted and unre-
stricted models, respectively. The test statistic LMh is distributed as χ2(hK2). In
3 . 3 . E S T I M AT I O N O F VA R PA R A M E T E R S
Multivariate normality test
Testing for normality of log-returns is a common procedure in much applied work and many tests have been proposed. In the literature, there are different procedures for multivariate normality tests, some of them are discussed in Mulenga et al [88]. Here we present the procedure suggested by Mardia [80], and by Doornik and Hansen [36], based on multivariate Skewness and Kurtosis tests.
Let K independent variables and yi= (y1, y2, ..., yk)0 for i = 1, ..., T , where y de-
note the T ×K matrix of observations. In order to test whether the set of K -variables are multivariate normal distributed with mean µ = (µ1, ..., µk)0 and finite covariance
Σ > 0, M V N (µ, Σ). Mardia [80], define the sample measures of multivariate Skew- ness bM,1 and Kurtosis bM,2, as:
bM,1= 1 T2 T X i=1 T X j=1
[(yi−y)0S−1(yj−y)]3; bM,2=
1
T
T
X
i=1
[(yi−y)0S−1(yi−y)]2 (3.16)
where y and S be the sample mean vector and the covariance matrix as follows. y = T1PT
i=1yi and S = T1PTi=1(yj− y)0(yi− y).
The multivariate normality when the population is N (µ, σ), using Mardia mea- sures of multivariate Skewness and Kurtosis is given by:
M J BM = T b M,1 6 + (bM,2− K(K + 2))2 8K(K + 2) (3.17)
The Mardia MJB statistic is asymptotically distributed as χ2α(K(K +1)(k +2)/6) distribution, where K is the number of variables.
An alternative of Jarque-Bera test can be found in Doornik and Hansen [36], Lütkepohl [77] and Srivastava [111]. The Jarque-Bera normality tests for univariate and multivariate series are implemented to the residual of a VAR(p) as well as separate tests for Skewness and Kurtosis. The univariate test is applied to the residual of each equation, while the multivariate version can be computed by using the residuals that are standardized by a Choleski decomposition of the variance- covariance matrix. Note that, the result is dependent upon the ordering of the variables.
Using notation presented in Lütkepohl [77], for orthogonalized VAR residuals, if we call λb1 the multivariate skewness, bλ2 the multivariate kurtosis, the multivariate Jarque - Bera statistics λb3 is defined as:
b
λ3=λb1+λb2 (3.18)
Under the null hypothesis of multivariate Gaussian disturbance, λb1 =
Tbb0 1bb1 6 ∼ χ2(K) and λb2 = T (bb2− 3)0(bb2− 3) 24 ∼ χ 2(K), observe that b 1 and b2 are K × 1
vectors of skewness and kurtosis coefficients, that is third and fourth non-central moment vectors of standardized residuals.
b b1= (bb11, ...,bbk1)0 ; bbk1= 1 T T X i=1 ω3kt b b2= (bb12, ...,bbk2)0 ; bbk2= 1 T T X i=1 ω4kt (3.19)
where ωkt is orthogonalized VAR residual and the λb3 statistic is distributed as
χ2(2K). The corresponding statistics against the null hypothesis that the residuals from the kth equation come from a univariate normal distribution are.
b λ1k =Tbb 2 k1 6 ∼ χ 2(1); b λ2k= T (bb 2 k2− 3) 24 ∼ χ 2(1); b λ3k=λb1k+λb2k∼ χ2(2) (3.20) Note that, this procedure the second is most used in statistical packages such as STATA and EViews.
Since the VAR model is verified that it is adequate to describe the dynamic in the system formed by the variables, follows other purposes of multivariate analysis of time series such as Forecast in-sample or out-of-sample, the Granger causality test, impulse response functions to shocks and forecast error variance decomposition.