VAR¶
Linearization¶
In a model with power utility and a one-period asset with gross return $R_t$, the dynamic equilibrium condition looks like $c_t^{-\sigma} = \beta R_{t+1} c_{t+1}^{-\sigma}$, where $c$ is consumption. Linearize that equation.
Structural VAR¶
The following is a linearized New Keynesian monetary policy model with variables for output, $y$, inflation, $\pi$, and the interest rate $i$. Map it to a structural VAR.
\begin{gather*} \hat{y}_{t-1} = \hat{y}_t - \frac{1}{\sigma}(\hat{i}_{t-1} - \hat{\pi}_t) \\ \hat{\pi}_{t-1} = \beta \hat{\pi}_t + \kappa \hat{y}_{t-1} \\ \hat{i}_t = \phi_\pi \hat{\pi}_t + \phi_y \hat{y}_t + \sigma \varepsilon_t \\ \end{gather*}
Identification¶
Write down a bivariate VAR($1$) where the coefficient matrix on the shocks has been recursively identified (e.g., via a Cholesky decomposition). Why and how does the ordering of the variables matter?
Cholesky Decomposition¶
Suppose you decompose the matrix $\mathbf{A}$ into $\mathbf{A} = \mathbf{L} \mathbf{L}'$, where
\begin{gather*} \mathbf{L} = \begin{bmatrix} 2 & 0 & 0 \\ 1 & 2 & 0 \\ 2 & 3 & 2 \end{bmatrix} \end{gather*}
What was $\mathbf{A}$?
VECM¶
Rank¶
Suppose
\begin{gather*} \mathbf{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 5 & 7 & 9 \end{bmatrix} \end{gather*}
Is $\mathbf{A}$ full rank? Why or why not?
Cointegration¶
Suppose you observe the following data
| $t$ | $y_t$ | $x_t$ |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 2 |
| 2 | 3 | 2 |
| 3 | 4 | 4 |
| 4 | 6 | 7 |
| 5 | 7 | 7 |
| 6 | 9 | 9 |
| 7 | 11 | 10 |
| 8 | 13 | 12 |
| 9 | 15 | 15 |
- Do $y_t$ and $x_t$ appear to be stationary? Why or why not?
- What conditions need to be satisified so that $y_t$ and $x_t$ are cointegrated?
- Do you think $y_t$ and $x_t$ are cointegrated? Why or why not?
Cointegration Test¶
- What are the null and alternative hypotheses of the Johansen cointegration test?
- Suppose you are testing for the number of cointegrating relationships among 3 variables. Describe how to to do that with the Johansen conintegration test.
VECM Model¶
Suppose in an estimated bivariate VECM with variables $y$ and $x$ (in that order), the cointegrating vector is $[1, -3]$ and the loading on $y$ is $\alpha_1 = -0.5$
- What is the long-run relationship between $y$ and $x$?
- Suppose at some point in time $y_t = 6$ and $x_t = 1.5$. What is the error correcion term at $t$? Is the system in long-run equilibrium?
- Interpret the sign and magnitude of $\alpha_1 = -0.5$.