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
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