A number of equations are provided for data simulation.

Consider an artificial two-dimensional \(\rm VAR(2)\) process of the form: \[ \vec{Y}_t = \begin{pmatrix} Y_{1t}\\ Y_{2t} \end{pmatrix} = \begin{pmatrix} 5\\ 10 \end{pmatrix} + \begin{pmatrix} 0.5 & 0.2\\ -0.2 & -0.5 \end{pmatrix} \begin{pmatrix} Y_{1,t-1}\\ Y_{2,t-1} \end{pmatrix} + \begin{pmatrix} -0.3 & -0.7\\ -0.1 & 0.3 \end{pmatrix} \begin{pmatrix} Y_{1,t-2}\\ Y_{2,t-2} \end{pmatrix} + \begin{pmatrix} \epsilon_{1,t}\\ \epsilon_{2,t} \end{pmatrix} \]

consider two separate cases where:

- \(\begin{pmatrix}\epsilon_{1,t}\\\epsilon_{2,t}\end{pmatrix} \sim \mathcal{N}\left( \begin{bmatrix}0\\0\end{bmatrix}, \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} \right)\)
- \(\begin{pmatrix}\epsilon_{1,t}\\\epsilon_{2,t}\end{pmatrix} \sim \mathcal{N}\left( \vec{\boldsymbol 0}, \Sigma \right)\) and \(\Sigma\) is not diagonal (think of your own value for the
**correlation**between \(\epsilon_{1,t}\) and \(\epsilon_{2,t}\) and some variance values).

Consider a trivariate cointegrated system with two cointegrating vectors and one common stochastic trend. A Phillips’ triangular representation for this system with cointegrating vectors \(\beta_1 = \begin{pmatrix} 1, 0, -\beta_{13}\end{pmatrix}^T\) and \(\beta_2 = \begin{pmatrix} 0, 1, -\beta_{23}\end{pmatrix}^T\) is: \[ \begin{cases} Y_{1t} &= \beta_{13} Y_{3t} + u_t\\ Y_{2t} &= \beta_{23} Y_{3t} + v_t\\ Y_{3t} &= Y_{3,t-1} + \epsilon_{3t} \end{cases} \] where:

- the first two equations describe two
**long-run equilibrium relations**; - the third equation gives the common stochastic trend.

Consider the case with:

- \(\beta_1 = \begin{pmatrix} 1, 0, -1\end{pmatrix}^T\),
- \(\beta_2 = \begin{pmatrix} 0, 1, -1\end{pmatrix}^T\),
- \(u_t = 0.7 u_{t-1} + \epsilon_{1t}\), \(v_t = 0.6 v_{t-1} + \epsilon_{2t}\), where \(\epsilon_{it} \sim iid N(0, 0.4^2)\).

An example in finance of such a system is the term structure of interest rates where \(Y_3\) represents the short rate and \(Y_1\) and \(Y_2\) represent two different long rates. The cointegrating relationships would indicate that the spreads between the long and short rates are stationary.

Depending on the Task:

- If the equation is in matrix form - rewrite it as a system of equations for the
**levels**, \(Y_{t}\). - If the equation is in a system of equations - rewrite it in VAR form for the
**levels**, \(Y_{t}\).

- If the equation is in matrix form - rewrite it as a system of equations for the
Write the VECM form of the equation.

Simulate 150 observations of the series.

Plot the series, their \(\rm ACF\), \(\rm PACF\) and \(\rm CCF\) plots - what do they say about the stationarity and cross-correlation of the series?

Check whether a unit root is present in the series.

Select the appropriate VAR order for the series. Then:

- Perform a Granger Causality test (see the end of “Model Comparison” section ) on the variables. Interpret the results.
- If a unit root is present in the series - carry out any appropriate cointegration tests (Johansen, or, if appropriate, Phillips-Ouliaris, or Engle-Granger cointegration tests). Similarly to unit root testing, do not forget to select one or more of the different cases - no constant, constant, trend, both, etc. - which are
**appropriate for this data**. See slide 53

Estimate the relevant (either VAR, or VEC) model, depending on your results from the previous tasks.

Write down the fitted model. Then, write the model for the levels, \(\vec{Y}_t\):

- if the estimated model is a VECM - write the VAR representation of the estimated model;
- if the estimated model is a VAR model for differences, \(\Delta \vec{Y}_t\) - re-write it in levels.
- Are the estimated coefficients close to the true values?

Using the estimated model, forecast the series 20 steps ahead.

Examine the impulse-response function of the model and comment the results.