Lecture 9: Multivariate Time Series Analysis

1
Lecture 9 Multivariate Time Series Analysis

• The following topics will be covered
• Modeling Mean
• Cross-correlation Matrixes of returns
• VAR
• VMA
• VARMA
• Cointegration
• Modeling Volatility
• VGARCH models

2
Lag-0 Cross-correlation Matrix
3
Lag-l Cross-correlation Matrix
4
Linear Dependence
5
Sample Cross-Correlation Matrixes (CCM)
6
Multivariate Portmanteau Test

• For a multivariate series, the null hypothesis is
  H0 ?1?m0 and the alternative hypothesis H0
  ?i ne 0 for some i. The statistic is used to test
  that there are no auto- and cross-correlations in
  the vector series rt. Portmanteau test is listed
  on page 308, where T is the sample size, k is the
  dimension of rt.

7
VAR (1)
8
VAR (1) Reduced Form System
9
Stationarity Condition of VAR(1)
10
VAR(p) Models
11
Building VAR(p) Model
12
Building VAR(p) Model
13
VMA and VARMA
14
Unit Root Nonstationarity and Co-integration
15
Error-Correction Form
16
Procedure in Cointegration tests
17
Conditional Covariance Matrix
18
Use of Correlations
19
Cholesky Decomposition
20
Bivariate GARCH

• For a k-dimensional return series rt, a
  multivariate GARCH model uses exact equations
  to describe the evolution of the
  k(k1)/2-dimentional vector over time. By exact
  equation, we mean that the equation does not
  contain any stochastic shock. However, the exact
  equation may become complicated. To keep the
  model simple, some restrictions are often imposed
  on the equations.
• Constant-correlation models cross-correlation is
  a constant.
• see (9.16) and (9.17) on page 364
• proc varmax dataall model ibm sp / p1
  garch(q1) nloptions techqn
• output outfor lead5 back3 run (all
  contains two sets of returns)
• (2) Time-Varying Correlation models

21
Exercises

• Ch8, problem 2
• Replicate Goeij and Marqliering (2004, J. Fin.
  Econometrics), Modeling the conditional
  covariance between Stock and Bond Returns A
  multivariate GARCH Approach, 2(4), 531-564.