Lecture 05 1 
Lecture 5: Multivariate Models
Financial Time Series, Spring 2015
MQF at Rutgers University
Heng Sun
February 24, 2015
1/46
Today’s Topics
Vector time series basics
VARMA(p,q)
Cointegration
References
Ruey Tsay, Analysis of Financial Time Series, Chp 8
Ruey Tsay, Multivariate Time Series Analysis
2/46
Vector Time Series
Each observation at time t
r1t
r2t
rt = .
..
is a column vector in Rk
T
= [r1t , r2t , · · · , rkt ]
rkt
Example of vectors of different time series.
Multiple stocks and ETFs
Interest rates at different tenors
Foreign exchange rates among different currencies
Futures of different expiration times and their underlying spot
Economic variables and prices of financial instruments
Key questions:
Is there a co-movement pattern?
Is there a correlation between one variable at next time and another at previous time?
3/46
0
5
10
15
US 1Y and 30Y CMT Yield
1953−04−30
1969−12−31
1986−08−31
2003−04−30
4/46
CCM (Cross Correlation Matrix)
Assume rt is weak stationary, i.e., the first two moments are time invariant. Define: µ = E (rt )
Γl
= E (rt − µ) (rt−l − µ)T
ρl
= D −1 Γl D −1
where D = diag{ Γ11 (0), · · · , Γkk (0)}. We use convention
Γl = Γ(l), ρl = ρ(l). Note each LHS is independent of t though
RHS does. Write down cross correlation explicitly: ρij (l) =
cov (rit , rj,t−l ) std (rit ) std (rjt )
ρ(l) is lag-l CCM. It is not necessarily symmetric. But we have ρ(−l) = ρ(l)T . Hence it is enough to consider l ≥ 0.
5/46
Linear Dependence
Understand the information provided by ρ:
Diagonal {ρii (l) : l = 0, 1, 2, · · · } are the autocorrelation function of rit .
Off diagonal ρij (0) measures the concurrent relationship between rit and rjt .
Off diagonal ρij (l) for l > 0 (cross-lag correlation) measures dependence of rit on past rj,t−l .
Cross lag correlations can be useful for prediction.
If ρij (l) = 0 for all l > 0 but ρji (v ) = 0 for some v > 0, then rit leads rjt . (Unidirectional relationship)
If ρij
Financial Time Series, Spring 2015
MQF at Rutgers University
Heng Sun
February 24, 2015
1/46
Today’s Topics
Vector time series basics
VARMA(p,q)
Cointegration
References
Ruey Tsay, Analysis of Financial Time Series, Chp 8
Ruey Tsay, Multivariate Time Series Analysis
2/46
Vector Time Series
Each observation at time t
r1t
r2t
rt = .
..
is a column vector in Rk
T
= [r1t , r2t , · · · , rkt ]
rkt
Example of vectors of different time series.
Multiple stocks and ETFs
Interest rates at different tenors
Foreign exchange rates among different currencies
Futures of different expiration times and their underlying spot
Economic variables and prices of financial instruments
Key questions:
Is there a co-movement pattern?
Is there a correlation between one variable at next time and another at previous time?
3/46
0
5
10
15
US 1Y and 30Y CMT Yield
1953−04−30
1969−12−31
1986−08−31
2003−04−30
4/46
CCM (Cross Correlation Matrix)
Assume rt is weak stationary, i.e., the first two moments are time invariant. Define: µ = E (rt )
Γl
= E (rt − µ) (rt−l − µ)T
ρl
= D −1 Γl D −1
where D = diag{ Γ11 (0), · · · , Γkk (0)}. We use convention
Γl = Γ(l), ρl = ρ(l). Note each LHS is independent of t though
RHS does. Write down cross correlation explicitly: ρij (l) =
cov (rit , rj,t−l ) std (rit ) std (rjt )
ρ(l) is lag-l CCM. It is not necessarily symmetric. But we have ρ(−l) = ρ(l)T . Hence it is enough to consider l ≥ 0.
5/46
Linear Dependence
Understand the information provided by ρ:
Diagonal {ρii (l) : l = 0, 1, 2, · · · } are the autocorrelation function of rit .
Off diagonal ρij (0) measures the concurrent relationship between rit and rjt .
Off diagonal ρij (l) for l > 0 (cross-lag correlation) measures dependence of rit on past rj,t−l .
Cross lag correlations can be useful for prediction.
If ρij (l) = 0 for all l > 0 but ρji (v ) = 0 for some v > 0, then rit leads rjt . (Unidirectional relationship)
If ρij