CHAPTER 5: MODEL SPECIFICATION AND METHODOLOGY
5.5 Multivariate Cointegration Analysis
The finding that many macroeconomic time series could possibly contain a unit root has spurred the development of the theory of non-stationary time series analysis. Engle and Granger (1987) postulates that a linear combination of two or more non-stationary time series can be stationary. Under such circumstances, the non-stationary time series are said to be cointegrated. Such a stationary linear combination represented by a cointegrating equation can be inferred as the long run equilibrium relationship between the variables in the model (Stock and Watson 1988). Cointegration analysis is essential in this study to ensure that the basic and threshold models estimated in the following Chapters 6 and 7 are not ‘spurious’ regressions.
According to Engle and Granger (1987), the original definition of cointegration refers to variables that are integrated of the same order. Two variables integrated of different orders cannot be cointegrated. However, while the inclusion of a stationary variable is prohibited, it should not affect the remaining coefficients in the cointegrating relationship (as long as it is not the dependent variable). It follows that the asymptotic critical values of the test statistics should not be affected as well. In addition, the inclusion of dummy variables for regime changes or data corrections are allowed although these variables cannot be taken as I (1) or I (0) (Engle and Granger 1991).
Campbell and Perron (1991) reinforces this idea by giving a more general definition of cointegration. In accordance with the definition given by Campbell and Perron (1991), some or all the series can be trend-stationary in a model. This definition of cointegration does not require that each of the times series be integrated of order one. The authors
27 If the AIC and SIC criteria chooses a different model, it should be noted that the SIC selects a more parsimonious model. Hence, it is important to visually inspect the residuals to determine if they are white noise.
recognise that in practice, empirical research is often faced with a combination of series that can be either be I (1) or I (0).
Therefore, the inclusion of stationary variables like the measures of volatility28 and threshold indicator variables29 are not a problem in this study. Hence, Vector Error Correction Models (VECM) will be estimated in this study for the basic trade and productivity models. However, VECM(s) will not be estimated for the threshold models although cointegration tests will be carried out to ensure that the threshold models are not
“spurious” regressions. It should also be noted that each stationary variable included in the model could possibly cause the number of cointegration relationships to increase accordingly (Harris, R 1995).
5.5.1 Johansen Cointegration Test
This study uses the Johansen (1988) methodology to test for cointegration instead of the
‘two-step’ procedure proposed by Engle and Granger (1987). This is mainly due to the
‘two-step’ procedure having lower power since the procedure is prone to errors introduced in the first step being carried over to the second step of estimation.30 According to Kremers, Ericcson and Doldado (1992), the ‘two-step’ procedure has a tendency to reject the null hypothesis of no cointegration on the borderline. In addition, when the error correction model is estimated using the same set of data, the coefficient on the error adjustment term may be highly statistically significant.
28 Previous empirical studies like Siregar and Rajan (2003) have found that volatility measures generated using ARCH/GARCH approach could be I (0).
29 Threshold indicator variables as defined earlier on in this chapter are essentially dummy variables that capture the threshold effect. As mentioned, dummy variables cannot be taken as I (1) or I (0)
30 This procedure involves saving the residuals from the long-run equilibrium relationship and using it in the second step. In the second step, the following regression is estimated:∆eˆt =a1eˆt−1+εt. The coefficient is obtained b estimating a regression using residuals from another regression. Hence, any errors from first step will be compounded in the second step.
a1
The Johansen cointegration test involves testing for the existence of cointegration as well as to determine the number of cointegrating vectors in the model. This is done within a general n variable VAR framework.
To illustrate its workings, consider the following n dimensional VAR model of order p:
t t
t AX
X = 1 −1+ε (5.28)
So that by differencing (5.27), the error correction formulation takes the form:
t t
t X
X =Π +ε
∆ −1 (5.29)
Where Π= A1−I indicating the number of cointegrating vectors, Xt and εt are vectors, = matrix of parameters,
(
n×1)
A1(
n×n)
I =(
n×n)
identity matrix. If the rank( )
Π =0, it follows that each element in(
A1−I)
must be zero. This implies that there is no cointegration and all the processes are non-stationary (contain unit roots). Hence, indicates that characteristic roots of the system imply convergence to long run equilibrium.Xit
Xt
∆
Similarly, a drift term can be included to account for plausibility of a linear trend in the data generating process (DGP) as seen below:31
t t
t A X
X = +Π +ε
∆ 0 −1 (5.30)
In this case, the rank represents the number of cointegrating relationships in the equation after data has been detrended.
31 It should be noted that the five different options are available in the Johansen Cointegration Test in order to cater to different types of data and cointegrating relations. Johansen (1992) provides more information on the specifications of the five different options.
5.5.2 Specification of number of lags
To specify the number of lags to include in the VECM, a VAR model has to be tested for the variables in the model. The selection of the number of lags for the VECM is based on different information criteria.32 The chosen lag length should be minimised in more of the information criteria relative to other lag lengths. In this study, the various information criteria are utilised up to a lag length of 6.33 It should also be noted that according to Lutkepohl (1993), the AIC and FPE are argued to have better properties compared to HQ and SIC. This means that the AIC and FPE are more likely to choose the correct order more often as compared to HQ and SC. This study will utilise the autocorrelation LM test to ensure the lag length chosen for the VAR model hence VECM is optimal. In addition, the plausibility of the VECM(s) estimates will be also be checked to further ensure that the appropriate lag length has been chosen for the VECM(s).
5.5.3 Conducting the Johansen Cointegration Test
This next step involves determining which deterministic components to include in the VECM. Table 5.1 shows the different alternatives available under the Johansen test. The Johansen test involves estimating π using a ‘reduced rank regression’ (Harris, R 1995).
Johansen (1988) suggested two tests for this purpose:
(i). The trace test34
( ) ∑ (
+
=
−
−
= m
i r i
i
trace r T λ
λ ln1 ˆ
)
(5.31)
32 The information criteria used in this paper includes: sequential modified LR test statistic (LR), Final prediction error (FPE), Akaike (AIC), Schwarz (SIC) and Hannan-Quinn(HQ).
33 Most researchers would start with a lag length of approximately T1/3(Enders, 2004). In this study, a lag length of 6 appears to be appropriate.
34 The hypotheses for this test are: H0 :r =0vs.H1:r = k−1, where the value of r is increased until the null is no longer rejected.
(ii). The Maximum Eigenvalue test35
( ) ( 1)
max r,r+1 =−Tln1−λˆr+
λ (5.32)
Where λˆ ,....λˆmare the estimated characteristic roots of the matrix
1 π .
Table 5.1: Deterministic components considered in the VECM
Model Characteristics Cointegrating Relations Level of the data
1 No intercept No deterministic trend
2 Intercept No deterministic trend
3 Intercept Linear trend
4 Intercept and Linear trend Linear trend
5 Intercept and Linear trend Quadratic trend
Source: Based on Harris (1995, p. 92)
Following Harris (1995), Model 1 is too restrictive and hence not considered as the intercept is needed to account for the units of measurement of the variables. This study also follows the Pantula (1989) principle as suggested by Johansen (1992), which involves testing the joint hypothesis of both the rank order and the deterministic components.
Models consisting of all combinations of the deterministic components as seen in Table 5.1 are estimated from the most restrictive Model 2 to the least restrictive Model 5. The model that is selected based on this technique will be the one that the null hypothesis is not rejected under both theλtrace and λmax tests.
35 The hypotheses for this test are:H0 :r =r* vs.H1:r = r*+1, where the null is rejected if the maximum Eigen value test statistic is greater than the critical values. MacKinnon-Haug-Michelis (1999) p-values that are provided by EViews are used instead of the critical p-values provided in Johansen-Juselius tables.
5.5.4 VECM Estimation
The concept of cointegration indicates that a regression is not spurious but meaningful. As mentioned earlier, the stationary linear combination of variables imply that there is a long run equilibrium relationship. However, in the short run, disequilibrium can occur in the model. Hence, according to Sargan (1984), this ‘equilibrium error’ can be treated as the error term. In accordance, the error correction term (ECT) should be incorporated into the basic VAR, resulting in a VECM. The VECM includes variables of the VAR in first differences, a long run relationship estimated in levels (from the Johansen test), and error correction terms.
A simple two variables
(
Y ,t Xt)
error correction model can be specified as follows:t t t
t X u
Y =α +α ∆ +α +ε
∆ 0 1 2 −1 (5.33)
where ∆ is the first difference operator, εt is the random error term and ut−1 =Yt −β0 −β1∆Xt−1 , which represents the previous period error from the cointegrating relationship. If α2 is significant, then it can be interpreted as the speed adjustment coefficient, which indicates what proportion of the disequilibrium in is corrected for in the next period.
Yt 36 The absolute value of α2 indicates the speed of adjustment back to equilibrium. Therefore, it follows that a VECM is necessary a restricted general VAR model where restrictions are placed on the coefficients (Enders 2004).