A number of equations are provided for data simulation.
Artificial Data 1
Consider an artificial two-dimensional VAR(2) process of the form: →Yt=(Y1tY2t)=(510)+(0.50.2−0.2−0.5)(Y1,t−1Y2,t−1)+(−0.3−0.7−0.10.3)(Y1,t−2Y2,t−2)+(ϵ1,tϵ2,t)
consider two separate cases where:
- (ϵ1,tϵ2,t)∼N([00],[1001])
- \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.