A number of equations are provided for data simulation.
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:
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:
Consider the case with:
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:
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:
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:
Using the estimated model, forecast the series 20 steps ahead.
Examine the impulse-response function of the model and comment the results.