10.4 VAR cointegration

In case of multivariate time-series ($k>2$) there could be none, one or more cointegrating relationships
In bivariate case ($k=2$), there can be either one or no cointegrating relationship
Testing the number of cointegrating relationships within VAR analysis requires Vector Error Correction (VEC) specification of the VAR model
VEC specification of bivariate VAR($1$) model is easy to obtain by simple substitution $z_t=\Delta z_t+z_{t-1}$ on the left hand side

\[\begin{equation} \begin{aligned} z_t&=a_0+A_1 z_{t-1}+u_t \\ \Delta z_t + z_{t-1}&=a_0+A_1 z_{t-1}+u_t \\ \Delta z_t&=a_0+(A_1 - I_k) z_{t-1}+u_t \\ \Delta z_t&=a_0+ \Pi z_{t-1}+u_t~;~~~~~~~~~~~~~~~~\Pi=(A_1 - I_k) \end{aligned} \tag{10.8} \end{equation}\]

Last equation in (10.8) is VEC specification of VAR($1$) model, while matrix $\Pi$ plays key role in finding cointegrating relationship(s)

In case of bivariate VAR($1$) model matrix $\Pi$ is of dimensions $2 \times 2$, and the rank of matrix $\Pi$ coule be $0$, $1$ or $2$. A full rank of the matrix $\Pi$ ($r=2$) indicates that all variables in the VAR($1$) model are stationary, meaning that standard VAR($1$) analysis is appropriate. If the rank of the matrix $\Pi$ is zero ($r=0$), then all variables in the VAR($1$) model are non-stationary, but there is no cointegrating relationship, which indicates that the variables should be differenced beforehand. However, if the matrix $\Pi$ has reduced rank ($r=1$), then all variables in the VAR($1$) model are non-stationary, but there is cointegrating relationship between the variables.

If the matrix $\Pi$ has reduced rank (variable are non-stationary, but cointegrated with one concertinaing relationship due to $r=1$), it also means that the second column of the matrix $\Pi$ can be obtained by multiplying the first column of the same matrix by some constant term

\[\begin{equation} \begin{aligned} \Delta z_t&=a_0+ \Pi z_{t-1}+u_t~~~~~~~\Pi=\begin{bmatrix} \alpha_1 \quad -\beta \alpha_1 \\ \alpha_2 \quad -\beta \alpha_2 \end{bmatrix}=\begin{bmatrix} \alpha_1 \\ \alpha_2 \end{bmatrix} \begin{bmatrix} 1 \quad -\beta \end{bmatrix} \\ \Delta z_t&=a_0+\alpha \beta^T z_{t-1} + u_t \end{aligned} \tag{10.9} \end{equation}\]

The vector $\alpha$ is interpreted as the speed of adjustments to the equilibrium or steady state, while the vector $\beta$ is interpreted as the cointegrating vector
VEC specification of VAR($1$) model in more detail

\[\begin{equation} \begin{aligned} \begin{bmatrix} \Delta y_t \\ \Delta x_t \end{bmatrix}&=\begin{bmatrix} \beta_{1,0} \\ \beta_{2,0} \end{bmatrix} + \begin{bmatrix} \alpha_1 \quad -\beta \alpha_1 \\ \alpha_2 \quad -\beta \alpha_2 \end{bmatrix} \begin{bmatrix} y_{t-1} \\ x_{t-1}\end{bmatrix}+\begin{bmatrix} u_{1,t} \\ u_{2,t}\end{bmatrix} \\ \Delta y_t&=\beta_{1,0}+\alpha_1(y_{t-1}-\beta x_{t-1})+ u_{1,t} \\ \Delta x_t&=\beta_{2,0}+\alpha_2(y_{t-1}-\beta x_{t-1})+ u_{2,t} \end{aligned} \tag{10.10} \end{equation}\]

From the last to equations in (10.10) the common relation is cointegrating relation $y_{t-1}-\beta x_{t-1}$
Testing of the number of cointegrating relationships is based on the Johansen procedure, which can be described in $3$ steps
Step 1) the null hypothesis that the matrix $\Pi$ has a zero rank is tested

\[\begin{equation} \begin{aligned} H_0: rank (\Pi)=0 \\ H_1: rank (\Pi) > 0 \end{aligned} \tag{10.11} \end{equation}\]

The null hypothesis will not be rejected if all eigenvalues of the matrix $\Pi$ are all zero. If the null hypothesis is rejected, the process continues to step 2

Step 2) the null hypothesis that the matrix $\Pi$ has a reduced rank is tested

\[\begin{equation} \begin{aligned} H_0: rank (\Pi)=1 \\ H_1: rank (\Pi) > 1 \end{aligned} \tag{10.12} \end{equation}\]

The null hypothesis will not be rejected if the first eigenvalue of the matrix $\Pi$ is different from zero. If the null hypothesis is rejected, the process continues to step 3
Step 3) the null hypothesis that the matrix $\Pi$ has a rank $r$ is tested

\[\begin{equation} \begin{aligned} H_0: rank (\Pi)=r \\ H_1: rank (\Pi) > r \end{aligned} \tag{10.13} \end{equation}\]

The null hypothesis will not be rejected if the first $r$ eigenvalues of the matrix $\Pi$ are different from zero. If the null hypothesis is rejected, the testing procedure continues iteratively, increasing the value of $r$ by one in each step ($r+1$) until the null hypothesis is not rejected.
When the null hypothesis is not rejected for the first time, the iteration process stops, and it is concluded that there are $r$ cointegrating relationships, i.e. $1 \leq r \leq (k-1)$
Each cointegrating relationship is represented by a column in the matrix $$ of dimension $k \times r$, and all columns of the matrix $\beta$ are linearly independent
This means that the rank of the cointegrating matrix $\beta$ is equal to the number of columns in that matrix
The test the null hypotheses in the Johansen procedure the most commonly trace statistic $\lambda_{trace}$ is used

\[\begin{equation} \lambda_{trace}=-T\sum_{i=r+1}^{k}log(1-\lambda_i)~~~~~~1>\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \dots \geq \lambda_k > 0 \tag{10.14} \end{equation}\]

Additionally, when performing the Johansen test, deterministic components (constant term and/or linear trend) can be included in the VEC model, just like in the ADF test

Exercise 46. Perform Johansen cointegration trace test with constant term but without linear trend using time-series object z and employing ca.jo() command from the urca package. Comment the obtained results.

Solution

Copy the code lines below to the clipboard and paste them into an R Script file opened in RStudio. The test statistics are provided for each possible rank of cointegration, which are compared to critical values at different significance levels ($10\%$, $5\%$, and $1\%$). Based on trace statistic $3.66$ we can not reject the null hypothesis of at most one cointegration relationship and based on trace statistic $16.34$ we can not reject the null hypothesis of no cointegration relationship (whenever test statistic is smaller then critical values null hypothesis is not rejected). The Johansen test finds no evidence of cointegration among growth and production at any significance level.

library(urca) # loading the same package
summary(ca.jo(z,type="trace",ecdet="const",K=2))