VAR and Cointegration

Question 1

What are the consequences of including unnecessary deterministic terms in the first-step of the Engle-Granger coitnegration testing framework?

Answer 1

You lose power but will still reject if the series are cointegrated The critical value will be more negative that it would be with the correct deterministic terms

Question 2

Explain the Engle-Granger framework

Answer 2

First test if individual variables are I(1) using ADF. If not, they cannot be cointegrated. Then, test whether the residuals from the first-stage regression are covariance stationary, which is an indirect test of the eigenvalue. Johansen directly tests the eigenvalues.

Question 3

Write out a general form VAR(P)

Answer 3

Question 4

What are some recent methods to work with large VARs

Answer 4

LASSO, Machine Learning, Bayesian methods with possibility of imposing structure

Question 5

Define Vector White Noise

Answer 5

1) Zero unconditonal mean 2) Constant covariance (need not be diagonal) 3) Zero autocovariance May be dependent No restrictions on conditional mean

Question 6

When is a VAR(1) stationary?

Answer 6

When the eigenvalues of the transition matrix are less than 1 in absolute value and the errors are white noise

Question 7

What is the effect of scaling variables to unit variance?

Answer 7

It can make the interpretation easier. It has no effect on persistence, inference or model selection

Question 8

What is companion form

Answer 8

Companion form is when a higher order VAR or AR is twritten as a VAR(1). It simplifies computing the autocovariance function

Question 9

Show how an AR(2) can be written as a VAR(1)

Answer 9

Question 10

What is a problem with forecasting with VARs?

Answer 10

Fore more steps ahead than 2, residuals are not white noise and follow and MA(h-1) structure.

Question 11

What are the solutions to autocovariance in forecast errors?

Answer 11

Question 12

Define Granger-Causality

Answer 12

X does not cause Y if conditioning on past values of X does not change the forecast for Y

Question 13

How to test for Granger-Causality

Answer 13

Question 14

Define an Impulse Response Function (IRF)

Answer 14

The IRF of y_i w.r.t. a shock in e_j is the change in y_i,_t+s for s>-0 for a 1 std. shock in ej,t

Question 15

How do we compute orthogonal shocks and their effect?

Answer 15

We need Sigma^(-0.5)

Question 16

What are 4 ways of computing Impulse Response Functions?

Answer 16

1) Assume sigma is diagonal 2) Choleski decomposition 3) Generalized IR 4) Spectral decomposition

Question 17

Explain Choleski decomposition

Answer 17

Assume a causal ordering, so that some variables do not affect others temporaneously. Examples. Nominal variables may react faster than real. No immediate effect of inflation and unemployment on Fed Funds (interest rate)

Question 18

Explain Generalized IR

Answer 18

Repeated application of Choleski decomposition. Sqrt(sigma) is full square matrix. More general than Choleski decomposition, harder to interpret

Question 19

Explain spectral decomposition

Answer 19

Again Sqrt(sigma) is full square matrix, no ordering. Based on eigenvalues and eigenvetors.

Question 20

What are 3 ways of computing IRF confidence intervals?

Answer 20

1) Monte Carlo 1 Estimate parameters, error covariance matrix and the covariance matrix of the parameters Simulate parameter for their assymptotic distribution and compute IRFs Repeat 2) Monte Carlo 2 Estimate parameters and covariance matrix of errors Simulate parameters from their assymptotic distribution and compute IRFs Repeat 3) Bootstrap Estimate parameters and compute residuals Resample residauls with replacement and use this compute new time series Estimate parameters with the new time series and compute IRFs. Repeat

Question 21

When is a varialbe I(1)

Answer 21

A variable is integrated of order 1 is it is non-stationary but its first differences are stationary

Question 22

Define cointegration

Answer 22

A set of k variables y are cointegrated if at least two variables are I(1) and there exists a non-zero, reduced rank k-by-k matrix pi such that: pi times y is stationary

Question 23

Why must pi be reduced rank for variables to be cointegrated?

Answer 23

If pi is full rank and pi times y is stationary, then y must also be stationary. If pi is full rank, it has only non-zero eigenvalues. Hence, phi does not have any unit eigenvalues. So the system contains no unit roots.

Question 24

Explain the Engle-Granger methodology

Answer 24

Tests for unit roots. Can only test if there is 1 cointegrating relationship. Best for only two variables. Variables must be I(1). Estimate the long-run relationship in a cross-sectional regression on levels. Test if errors are stationary by ADF. If so, the variables are cointegrated

Question 25

What is an unbalanced regression?

Answer 25

When order of integreation is not the same on RHS and LHS of regression. It ultimately must be wrong. There can be no tight relationship if 1 is unit root and there other is not

Question 26

What are some specific problems with regressing I(0) on I(1)?

Answer 26

It gives very bad finite sample properties. CLT will not be relevant for realistic sample sizes. Can use approximations. Sometimes incurred in finance when regressing returns (I(0) ) on price/dividend (potentially I(1)).

Question 27

Show how you can rewrite a VAR as VECM

Answer 27

Question 28

When does it matter if you run a VAR on original or transformed variables? Assume same number of variables

Answer 28

We can write the transformed variables in terms of the first

Two second set of variables is of full rank

Question 29

Write pi = [0 0;beta -1] as alpha and beta and interpret

Answer 29

Alpha = [0; 1]; Beta is [Beta -1]