where z(t) ¼ e(t)=s(t). Note that E[z(t)] ¼

ﬃﬃﬃﬃﬃﬃﬃﬃ

2=p

p

. Hence, the last term

in (5.3.12) is the mean deviation of z(t). If g > 0 and l < 0, negative

shocks lead to higher volatility than positive shocks.

5.4 MULTIVARIATE TIME SERIES

Often the current value of a variable depends not only on its past

values, but also on past and/or current values of other variables.

Modeling of dynamic interdependent variables is conducted with

multivariate time series. The multivariate models yield not only new

implementation problems but also some methodological difficulties.

In particular, one should be cautious with simple regression models

y(t) ¼ ax(t) þ e(t) (5:4:1)

that may lead to spurious results. It is said that (5.4.1) is a simultan-

eous equation as both explanatory (x) and dependent (y) variables are

present at the same moment of time. A notorious example for spuri-

ous inference is the finding that the best predictor in the United

Nations database for the Standard & Poor’s 500 stock index is

production of butter in Bangladesh [5].

A statistically sound yet spurious relationship is named data

snooping. It may appear when the data being the subject of research

are used to construct the test statistics [4]. Another problem with

simultaneous equations is that noise can be correlated with the ex-

planatory variable, which leads to inaccurate OLS estimates of the

regression coefficients. Several techniques for handling this problem

are discussed in [2].

A multivariate time series y(t) ¼ (y

1

(t), y

2

(t), ...,y

n

(t))

0

is a vector

of n processes that have data available for the same moments of time.

It is supposed also that all these processes are either stationary or

have the same order of integration. In practice, the multivariate

moving average models are rarely used due to some restrictions [1].

Therefore, we shall focus on the vector autoregressive model (VAR)

that is a simple extension of the univariate AR model to multivariate

time series. Consider a bivariate VAR(1) process

y

1

(t) ¼ a

10

þ a

11

y

1

(t 1) þ a

12

y

2

(t 1) þ e

1

(t)

y

2

(t) ¼ a

20

þ a

21

y

1

(t 1) þ a

22

y

2

(t 1) þ e

2

(t) (5:4:2)

54 Time Series Analysis

that can be presented in the matrix form

y(t) ¼ a

0

þ Ay(t 1) þ «(t) (5:4:3)

In (5.4.3), y(t) ¼ (y

1

(t), y

2

(t))

0

, a

0

¼ (a

10

,a

20

)

0

, «(t) ¼ (e

1

(t), e

2

(t))

0

,

and A ¼

a

11

a

12

a

21

a

22

.

The right-hand sides in example (5.4.2) depend on past values only.

However, dependencies on current values can also be included (so-

called simultaneous dynamic model [1]). Consider the modification of

the bivariate process (5.4.2)

y

1

(t) ¼ a

11

y

1

(t 1) þ a

12

y

2

(t) þ e

1

(t)

y

2

(t) ¼ a

21

y

1

(t) þ a

22

y

2

(t 1) þ e

2

(t) (5:4:4)

The matrix form of this process is

1 a

12

a

21

1

y

1

(t)

y

2

(t)

¼

a

11

0

0a

22

y

1

(t 1)

y

2

(t 1)

þ

e

1

(t)

e

2

(t)

(5:4:5)

Multiplying both sides of (5.4.5) with the inverse of the left-hand

matrix yields

y

1

(t)

y

2

(t)

¼ (1 a

12

a

21

)

1

a

11

a

12

a

22

a

11

a

21

a

22

y

1

(t 1)

y

2

(t 1)

þ (1 a

12

a

21

)

1

1a

12

a

21

1

e

1

(t)

e

2

(t)

(5:4:6)

Equation (5.4.6) shows that the simultaneous dynamic models can

also be represented in the VAR form.

In the general case of n-variate time series, VAR(p) has the form [2]

y(t) ¼ a

0

þ A

1

y(t 1) þ ...þ A

p

y(t p) þ «(t) (5:4:7)

where y(t), a

0

, and «(t) are n-dimensional vectors and A

i

(i ¼ 1, ...,p)

are n x n matrices. Generally, the white noises «(t) are mutually

independent. Let us introduce

AA

p

(L) ¼ I

n

A

1

L ... A

p

L

p

(5:4:8)

where I

n

is the n-dimensional unit vector. Then equation (5.4.7) can

be presented as

Time Series Analysis 55

Get *Quantitative Finance for Physicists* now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.