1.1 (a)

Briefly outline Johansen’s methodology for testing for cointegration between a set of variables in the context of a VAR.

A VAR with k lags containing these variables could be set up:

\[\begin{eqnarray} y_{t} &=& \beta_{1}y_{t-1} &+ \beta_{2}y_{t-2} &+...+&\beta_{k}y_{t-k}\\ g\times 1 && g\times g g\times 1&g\times g g\times 1&&g\times g g\times 1 \end{eqnarray}\]

Transform the VAR model into a Vector Error Correction Model (VECM) to incorporate potential long-run relationships.

\[\begin{eqnarray} \Delta y_{t} &=& (\beta_{1}-I)(y_{t-1}-y_{t-2})&+ (\beta_{1}+\beta_{2}+I)y_{t-2}&+...\\ &=& \Gamma_{1} \Delta y_{t-1} &+ \Gamma_{2} \Delta y_{t-2} &+...+ (\beta_{1}+\beta_{2}+..+\beta_{k}-I)y_{t-k} + u_{t} \end{eqnarray}\]

In the long run: \(t \to +\infty\): \(\Pi_{y}=0\)

\(\Rightarrow\) 0 is one eigenvalue of \(\Pi_{y}\)

\(\Rightarrow\ r=rank\ \Pi \le g-1\)

Johansen test \(r = rank \Pi = the\ number\ of\ cointegration\ relationship\)

\[H_{0}: r = 0\ vs\ H_{1}: r> 0\\ H_{0}: r = 1\ vs\ H_{1}: r> 1\]

From \(r = rank\ \Pi\): (\(r=1, g=4\))

\[\begin{eqnarray} \Pi &=& \alpha &\times& \beta'\\ g\times g && g\times r&& r \times g \\ \Rightarrow \Pi_{y}&=& \begin{pmatrix} \alpha_{1} \\ \alpha_{2}\\ \alpha_{3} \\ \alpha_{4} \end{pmatrix}&&\begin{pmatrix} \beta_{1} \beta_{2} \beta_{3} \beta_{4} \end{pmatrix}\begin{pmatrix} y_{1} \\ y_{2}\\ y_{3} \\ y_{4} \end{pmatrix}=0\\ &=& \begin{pmatrix} \alpha_{1} \\ \alpha_{2}\\ \alpha_{3} \\ \alpha_{4} \end{pmatrix} &&\begin{pmatrix} \beta_{1}y_{1} + \beta_{2}y_{2} + \beta_{3}y_{3} + \beta_{4}y_{4} \end{pmatrix} = 0 \\ \Rightarrow&& \beta_{1}y_{1} + \beta_{2}y_{2} + \beta_{3}y_{3} + \beta_{4}y_{4} &&=0 \end{eqnarray}\]

1.2 (b)

A researcher uses the Johansen procedure and obtains the following test statistics (and critical values)

Determine the number of cointegrating vectors.

Since the \(\lambda_{max}\) is less than the 5% critical value, the null hypothesis do not reject.

From the table above,

• For \(r= 0\): \(\Rightarrow\) Reject null hypothesis (0 cointegrating vectors).

• For \(r= 1\): \(\Rightarrow\) Reject null hypothesis (1 cointegrating vectors).

• For \(r= 2\): \(\Rightarrow\) Do not reject null hypothesis \(\Rightarrow\) 2 cointegrating vectors.

Therefore, the number of cointegrating vectors are two.

2.1 (b)

The researcher obtains results for the Johansen test using the variables outlined in part (a) of the question as follows:

Determine the number of cointegrating vectors, explaining your answer.

Since the \(\lambda_{max}\) is less than the 5% critical value, the null hypothesis do not reject.

From the table above,

• For \(r= 0\): \(\Rightarrow\) Reject null hypothesis (0 cointegrating vectors).

• For \(r= 1\): \(\Rightarrow\) Reject null hypothesis (1 cointegrating vectors).

• For \(r= 2\): \(\Rightarrow\) Do not reject null hypothesis \(\Rightarrow\) 2 cointegrating vectors.

Therefore, the number of cointegrating vectors are two.