A number of equations are provided for data simulation.
Artificial Data 1
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).
Artificial Data 2
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.
Tasks
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, \(Y_{t}\).
- 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.