Review 2 - Time Series

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

Unit Root Test¶

Write down the null and alternative hypotheses of the augmented Dickey-Fuller test (ADF test). You run an ADF test on a time series and obtain a test statistic of -1.8. If the critical value at the 5% level is -2.86, what is your conclusion?

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*}

(1)

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*}

(2)

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*}

(3)

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 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 $\alpha_1 = -0.5$ .