ñòð. 17 |

to 1=(1 Ã€ l). Then for z(t) Â¼ s2 (t) and y(t) Â¼ e2 (t), equation (5.3.11) is

equivalent to equation (5.3.7) with v Â¼ 0.

The GARCH models discussed so far are symmetric in that the

shock sign does not affect the resulting volatility. In practice, how-

ever, negative price shocks influence volatility more than the positive

shocks. A noted example of the asymmetric GARCH model is the

exponential GARCH (EGARCH) (see, e.g., [3]). It has the form

log [s2 (t)] Â¼ v Ã¾ b log [s2 (t Ã€ 1)] Ã¾ lz(t Ã€ 1)Ã¾

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ (5:3:12)

g(jz(t Ã€ 1)j Ã€ 2=p)

54 Time Series Analysis

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ

where z(t) Â¼ e(t)=s(t). Note that E[z(t)] Â¼ 2=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) Â¼ ax(t) Ã¾ 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 & Poorâ€™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) Â¼ (y1 (t), y2 (t), . . . , yn (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

y1 (t) Â¼ a10 Ã¾ a11 y1 (t Ã€ 1) Ã¾ a12 y2 (t Ã€ 1) Ã¾ e1 (t)

y2 (t) Â¼ a20 Ã¾ a21 y1 (t Ã€ 1) Ã¾ a22 y2 (t Ã€ 1) Ã¾ e2 (t) (5:4:2)

55

Time Series Analysis

that can be presented in the matrix form

y(t) Â¼ a0 Ã¾ Ay(t Ã€ 1) Ã¾ Â«(t) (5:4:3)

In (5.4.3), y(t) Â¼ (y1 (t), y2 (t))0 , a0 Â¼ (a10 , a20 )0 , Â«(t) Â¼ (e1 (t), e2 (t))0 ,

a11 a12

and A Â¼ .

a21 a22

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)

y1 (t) Â¼ a11 y1 (t Ã€ 1) Ã¾ a12 y2 (t) Ã¾ e1 (t)

y2 (t) Â¼ a21 y1 (t) Ã¾ a22 y2 (t Ã€ 1) Ã¾ e2 (t) (5:4:4)

The matrix form of this process is

Ã€a12 y1 (t) y1 (t Ã€ 1)

1 a11 0 e1 (t)

Â¼ Ã¾ (5:4:5)

Ã€a21 0 a22 y2 (t Ã€ 1)

1 y2 (t) e2 (t)

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

matrix yields

a12 a22 y1 (t Ã€ 1)

y1 (t) a11

Â¼ (1 Ã€ a12 a21 )Ã€1

y2 (t Ã€ 1)

y2 (t) a11 a21 a22

1 a12 e1 (t)

Ã¾ (1 Ã€ a12 a21 )Ã€1 (5:4:6)

a21 1 e2 (t)

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) Â¼ a0 Ã¾ A1 y(t Ã€ 1) Ã¾ . . . Ã¾ Ap y(t Ã€ p) Ã¾ Â«(t) (5:4:7)

where y(t), a0 , and Â«(t) are n-dimensional vectors and Ai (i Â¼ 1, . . . , p)

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

independent. Let us introduce

Ap (L) Â¼ In Ã€ A1 L Ã€ . . . Ã€ Ap Lp (5:4:8)

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

be presented as

56 Time Series Analysis

Ap (L)y(t) Â¼ a0 Ã¾ Â«(t) (5:4:9)

Two covariance-stationary processes x(t) and y(t) are jointly covar-

iance-stationary if their covariance Cov(x(t), y(t Ã€ s)) depends on lag

s only. The condition for the covariance-stationary VAR(p) is the

generalization of (5.1.11) for AR(p). Namely, all values of z satisfying

the equation

jIn Ã€ A1 z Ã€ . . . Ã€ Ap zp j Â¼ 0 (5:4:10)

must lie outside the unit circle. Equivalently, all solutions of the

equation

jIn lp Ã€ A1 lpÃ€1 Ã€ . . . Ã€ Ap j Â¼ 0 (5:4:11)

must satisfy the condition jlj < 1.

The problem of whether the lagged values of process y can improve

prediction of process x (so-called Granger causality) is often posed in

forecasting. It is said that if y fails to Granger-cause x, then the

following condition holds for all s > 0

MSE(E[x(t Ã¾ s)jx(t), x(t Ã€ 1), . . . ]) Â¼

MSE(E[x(t Ã¾ s)jx(t), x(t Ã€ 1), . . . , y(t), y(t Ã€ 1), . . . ]) (5:4:12)

In this case, y is called exogenous variable with respect to x. For

example, y2 (t) is exogenous with respect to y1 (t) in (5.4.2) if a12 Â¼ 0.

General methods for testing the Granger causality are described in [2].

The last notion that is introduced in this section is cointegration.

Two processes are cointegrated if they both have unit roots (i.e., they

both are I(1) ), but some linear combination of these processes is

stationary (i.e., is I(0) ). This definition can be extended to an arbi-

trary number of processes. As an example, consider a bivariate model

y1 (t) Â¼ ay2 (t) Ã¾ e1 (t)

y2 (t) Â¼ y2 (t Ã€ 1) Ã¾ e2 (t) (5:4:13)

Both processes y1 (t) and y2 (t) are random walks. However the process

z(t) Â¼ y1 (t) Ã€ ay2 (t) (5:4:14)

is stationary. Details of testing the integration hypothesis are de-

scribed in [2]. Implications of cointegration in financial data analysis

are discussed in [3].

57

Time Series Analysis

5.5 REFERENCES FOR FURTHER READING

AND ECONOMETRIC SOFTWARE

A good concise introduction into the time series analysis is given by

Franses [1]. The comprehensive presentation of the subject can be

found in monographs by Hamilton [2] and Green [6]. Important

specifics of time series analysis in finance, particularly for analysis

and forecasting of volatility, are discussed by Alexander in [3]. In this

chapter, only time series on homogenous grids were considered. Spe-

cifics of analysis of tick-by-tick data on non-homogenous grids are

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