Review 2A

Review 2A#

by Professor Throckmorton
for Time Series Econometrics
W&M ECON 408/PUBP 616
Slides

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 \(\alpha_1 = -0.5\).

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