1
CHAPTER ONE INTRODUCTION 1.1 Background to the Study
Time varying parameter models have a long history in statistics. Modelling unequal variances in non-linear time series is a challenging task. The importance of non-linear models in time series cannot be over emphasized. These models are useful in Statistics and in fields that apply Statistics, such as Economics, Biology and Medicine. The ideas and techniques of modelling time series in these diverse areas of science and other related disciplines are proved to be useful and innovative by Box and Jenkins (1976), Engle (1982), Gilks and Berzuini (2001) and Posedel (2005). Bilinear Generalized Autoregressive Conditional Heteroskedasticity (BL-GARCH) model is one of the major tools that Economists use to model financial markets‟ behaviour in the presence of political disorders, economic crises, wars or natural disasters. In such stress periods, prices of financial assets tend to fluctuate very profusely, Posedel (2005). In Medicine, medical sectors react promptly to epidemic such as sudden general outbreak of tuberculosis, diarrhea and poliomyelitis. In such periods, comprehensive records of patients are kept together with vaccines to combat the epidemic.
For instance, the tuberculosis patients‟ data collected from Lagos General Hospital (1998-2007) can be analyzed to prove non-constant variance as seen in Chapter 4 of the study. The results obtained are similar to Zhu (2011) who studied comprehensive polio patients‟ case files recorded by the Centre for Disease Control in the United States from 1970 to 1983. Though, this research work is aware of some of these applications, its main focus is on the empirical point of view. That is, to establish the generalization of BL-GARCH model using various parametric distributions: Normal, Student-t, Generalized Student-t and Generalized Error distributions.
2
Let (1,2,,n), be a time series, collected over equally spaced time period, t0, t1,,tn1, tn such that tn t0it. It is assumed that the time series is a realization of a stationary stochastic process{t}, where t, is the indexing set. Following Box and Jenkins (1976), this can be represented as:
()yt ()(1)d f(yt)()t (1.1) where
p
p
( ) 1 1 2 2 , is an autoregressive operator of order p, such that the roots of the polynomial () = 0 lie outside the unit circle for stationarity.
p
1, 2,, are the autoregressive parameters.
) ( ) 1 ( )
(
d is the generalized autoregressive operator; it is a non-stationary operator with d of the roots of ()0 equal to unity.
) (yt
f is the observed time series data.
q
q
( ) 1 1 2 2 , is a moving average operator of order q, such that the roots of the polynomial ()0 lie outside the unit circle for invertibility.
q
1, 2,, are the moving average parameters.
t is normally independently distributed white noise with mean zero and variance
2. B is a backward shift operator such thatdyt ytd d is a nonnegative integer.
When d = 0, the model (1.1) represents a stationary process, that is, the autoregressive moving average (ARMA) model. A criticism of Box-Jenkins‟ model in economic applications is that it is
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free from any economic theory. Thus, when forecasts are in error, one is at a loss as to what economic explanations to give for such errors. However, Bollerslev (1986) showed that the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models used in financial time series belong to the class of autoregressive moving average models. This is because the generalized autoregressive conditional heteroskedasticity models use autocorrelation functions and partial autocorrelation functions in model identification.
Consider a stochastic process
{ y
t}
t1 defined by a certain statistical model disturbances{
t}t1 and conditional variances t2 E(t2),that are generated by a GARCH (1, 1) model. The relationship between this GARCH (1, 1) model and an ARMA (1, 1) model is given as
t2
t21
t21
t2
(
)
t21
vt1 vt (1.2) where
2
t = the conditional variance = the conditional mean
t21 = news about volatility from the previous period, measured as the lag of the squared residual from the mean equation (autoregressive conditional heteroskedasticity (ARCH) term).
t21 = last period‟s forecast variance (GARCH term).
,
0
0, 0, vt
t2
t2
t2(zt2 1) andz
is a standardized random variable with mean zero and scaling factor, one; for example,z
~ N (0, 1). The difference between GARCH model and a standard ARMA model refers to the parameters:
,
and the disturbancesv
t. Theparameters
and
are restricted to be non-negative in GARCH model, which have some consequences for the stationarity condition, and although the disturbancesv
t have mean 0, they are4
clearly not white noise because of their time-varying asymmetric probability density functions.
Thus, this stationary process is not capable of capturing well known phenomena present in financial markets such as marginal distributions having heavy tails and thin centres (Leptokurtosis); return series appearing to be almost uncorrelated over time but to be dependent through higher moments.
The possibility of having dependence between higher conditional moments, most notably variances, involves examining non-linear stochastic processes from a more realistic perspective in time series data. This motivates the consideration of nonlinear models. Since the last two decades, a new area of “Nonlinear time series modelling” is fast coming up. For their analysis, there are basically two possibilities, namely Parametric or Non-parametric approach. Evidently, if in a particular situation, we are quite sure about the distributional form, we should use the former; otherwise the latter may be employed. However, in this study, we shall confine our attention only to “Parametric approach”.
Statistically speaking, if the conditional variance of t given t1, t2, that is,
t t1,t2,
Var of a time series is not constant over time then the process t is conditionally heteroskedastic. Heteroskedasticity refers to the random errors having unequal variances. In particular, a heteroskedastic model has Cov(e)Diag(t2), Christensen (1996). Let Y = (
n
1, 2, , ) be a Gaussian vector with mean vector and variance matrix
.
2 2
1
21 2
2 21
1 12
2 1
n n
n
n
5
Otherwise, if the expected value of all error terms when squared is the same at any given point, then the vector is homogeneous (homoskedastic) that is,
1
2
n. When this assumption does not hold, the vector is heteroskedastic, see Box, Jenkins and Reinsel (1994), Bollerslev, Engle and Nelson (1994). There are several approaches to dealing with heteroskedasticity. If the error variance at different times is known, weighted regression is a good method. If as is the case with financial time series, the error variance is unknown, and must be estimated from the data, we can model the changing error variance with the bilinear generalized autoregressive conditional heteroskedasticity model.1.2 Statement of Problem
Some of the major problems faced in high frequency financial time series data are:
(i) the parameter estimation of conditional variance.
(ii) the problem of unconditional disturbances having heavy tails and thin centres.
It is noted that while many authors have worked on the existence of nonlinear time series models for modelling and estimation of high frequency of stock returns in Gaussian framework, see Engle (1982), Bollerslev (1986) and Nelson (1990), works are on-going on the simultaneous estimation of volatility clustering and leverage effect in non-Gaussian framework.
Several researchers, Bollerslev (1986), Engle and Bollerslev (1986) as well as Nelson (1990) have worked extensively on the estimation of GARCH, integrated GARCH and exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models respectively in the Gaussian framework. Under some modifications, the bilinear generalized autoregressive conditional heteroskedasticity (BL-GARCH) model generalizes various GARCH-type models including the standard GARCH and IGARCH models among others.
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One of the major problems faced by statisticians and econometricians in time series analysis and applications is how to model and forecast volatility. This applies in other fields like Economics, Biology and Medicine. Therefore, this study sets out to analyze the risk of holding an asset or the value of an option by modelling the variance of the innovation (disturbances). The risk of holding an asset approximates the size of the disturbances of the model. Thus, this research work is undertaken to provide some of the solutions to these challenges as already identified.
1.3 Purpose of the Study
The overall aim of this work is to generalize the estimation of bilinear models, BL-GARCH model particularly, in connection with high frequency financial time series data. The specific objectives are:
i to extend the ability of BL-GARCH model to capture the empirical characteristics of high frequency financial time series data beyond those previously considered by various authors.
ii to establish the positivity conditions for the conditional variance in the BL- GARCH model.
iii to develop the practicability and sample performance analysis of the BL-GARCH (1, 1) model under elliptical distributions.
iv to evaluate parameters of BL-GARCH (1, 1) model from Gaussian and non-Gaussian frameworks.
1.4 Significance of the Study
This study follows such previous works as Bollerslev (1986), Storti and Vitale (2003) and Diongue, Guegan and Wolff (2010) in the estimation of GARCH and BL-GARCH models using the method of Maximum likelihood.
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Hence, the study will establish that the parameter estimates of BL-GARCH model are efficient and consistent under Gaussian and non-Gaussian distributions.
The study will show that the generalized student-t distribution helps to fit the data in the tail regions which are critical to the risk management and other financial economic applications.
The study will demonstrate how GARCH (1, 1) and BL-GARCH (1, 1) models adequately capture the volatility clustering and leverage effect of stock returns of selected banks in Nigeria within the sampled period (2007-2011). So, financial institutions can always rely on the empirical findings of the study to control volatility of stocks.
It will provide functional values for economic policy makers and researchers alike by depicting the existence of special relationships between economic and financial data which is highly essential for designing policy instruments in order to curb inflationary pressures and stimulate economic growth with employment generation as an economic dividend. In addition, researchers can use these special relationships as a vehicle for estimating models required for appraising the performance of the economy.
1.5 Research Questions
This research is carried out to answer the following questions:
i How can the BL-GARCH model be extended to capture empirical characteristics in the high frequency financial time series data?
ii How can the results of Posedel (2005) which was established for symmetric model be extended to asymmetric model and still retain the conditions for positivity of the conditional variance?
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iii In what ways can the practicability and sample performance of the BL-GARCH (1, 1) model be developed using elliptical distributions?
iv In what ways can the parameters of BL-GARCH (1, 1) model be evaluated from Gaussian and non-Gaussian frameworks?
1.6 Operational Definition of Terms
Model: This is a mathematical statement showing a linear (or non-linear) relationship between the dependent (response) variable and the independent variable(s).
Sequence: An ordered list of objects or events.
Stationarity: This occurs when a value is stable or when there is no significant difference between the actual and approximated values. That is, E()0 and conditional variance t2 are approximately constant.
Covariance Stationary: A stochastic process is called weak stationary or covariance stationary when the mean, the variance and the covariance structure of the process is stable over time.
White Noise: A stochastic process
t with t belonging to positive integers or natural numbers (tZor tN) is called white noise if the
t‟s are i.i.d with mean zero,E(t) = 0, positive definite covariance matrix
E(tt) and finite fourth order moments.Autoregressive Model: A model where the current value of the process is expressed as a finite, linear aggregate of previous values of the process and a shock,
e
t.Moving Average Model: is a linear combination of present and past values of random disturbances.
Heteroskedasticity: It refers to the random errors having unequal variances.
Autoregressive Conditional Heteroskedasticity (ARCH) Models: These are autoregressive models in which error variances are not homogeneous.
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1.7 Terminologies and Notations
The trending properties of the variables make special terminologies useful. Hence, the following general terminologies and notations are used in this work:
ARCH: Autoregressive Conditional Heteroskedasticity
GARCH: Generalized Autoregressive Conditional Heteroskedasticity
IGARCH: Integrated Generalized Autoregressive Conditional Heteroskedasticity EGARCH: Exponential Generalized Autoregressive Conditional Heteroskedasticity NARCH: Non-linear Autoregressive Conditional Heteroskedasticity
SWARCH: Switching Autoregressive Conditional Heteroskedasticity
BL-GARCH: Bilinear Generalized Autoregressive Conditional Heteroskedasticity ARMA: Autoregressive Moving Average
QMLE: Quasi-Maximum Likelihood Estimation ML: Maximum Likelihood
LR: Likelihood Ratio MSE: Mean Squared Error
The natural logarithm is abbreviated as log.
The lag operator B is defined such that for a time series variable yt,
y
t y
t1, that is it shifts the time index backward by one period.The differencing operator = 1- B is defined such that
y
t y
t y
t1.
The symbol „~ (,
)‟ abbreviates „has a distribution with mean (vector) and (co)variance (matrix)
and N(,
)denotes a (multivariate) normal distribution with mean (vector) and (co)variance (matrix)
.a.s. also abbreviates almost surely.
Independent, identically distributed is abbreviated as i.i.d.
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The rest of this research work is organized as follows: In Chapter 2, the study dwells on literature review and theoretical framework of the model under study. Chapter 3 deals on the methodology used. In Chapter 4, the results and discussion on the previous chapters are stated, proved and applied to evaluate the parameters of BL-GARCH model in the simulation data, while empirical analyses were carried out using the data on stock prices of selected banks in Nigeria and tuberculosis patients from Lagos General Hospital for the sampled period. Chapter 5 is the summary of findings, conclusion, contributions to knowledge and suggestions for further research.
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CHAPTER TWO LITERATURE REVIEW
In the past decades, many researchers have introduced informal and ad-hoc procedures to take account of the changes in the variance. One of the first authors to address variance changing over time was Mandelbrot (1963), who used recursive estimates of the variance for modelling volatility.
Klein (1977) used rolling estimates of quadratic residuals. Black (1976) observed that “… there is a lot of commonality in volatility changes across stocks: a 1% market volatility change typically implies a 1% volatility change for each stock. Well, perhaps the high volatility stocks are somewhat more sensitive to market volatility changes than the low volatility stocks. In general, it seems fair to say that when stock volatility change, they all tend to change in the same direction.”
Engle (1982) proposes Autoregressive Conditional Heteroskedasticity (ARCH (q)) models that seem to capture the empirical characteristics in the financial time series. The models have non- constant variances conditioned on the past, which is a linear aggregate of recent past disturbances.
This means that the more recent news will be the fundamental information that is relevant for modelling the present volatility. Some leading scholars who studied ARCH models include:
Geweke (1986, 1989), Diebold and Nerlove (1989), Engle (1990), Jones, Kaul and Lipson (1994) and Gourieroux (2007). Harvey, Ruiz and Sentana (1992) established the existence of a few common factors explaining exchange rate volatility movements. Engle, Ng and Rothschild (1990) show that U. S. bond volatility changes are closely linked across maturities. This commonality of volatility changes holds not only across assets within a market, but also across different markets.
For example, Schwert (1989) established out that U. S. stock and bond volatilities move together, while Engle and Susmel (1993), Hamao, Masulia and Ng (1990) discovered close links between volatility changes across international stock markets. Lamoureux and Lastrapes (1990) deduced that conditional heteroskedasticity may be caused by time dependence in the rate of information arrival
12
to the market. They use the daily trading volume of stock markets as a proxy for such information arrival and confirm its significance. Mizrach (1990) associates ARCH models with the errors of the economic agent‟s learning processes. Cai (1994) proposed the switching ARCH or SWARCH model in which there are several different ARCH models and that the economy switches from one to another following a Markov chain. In this model there can be extremely high volatility process which is responsible for events such as the stock market crash in 2009. That volatilities move together should be encouraging to model builders, since it indicates that a few common factors may explain much of the temporal variation in the conditional variances and covariances of asset returns.
2.0.1 The ARCH Process
Let
y
t be a sample realization of a stochastic process that is observed only for a finite number of equally spaced periods, indexed by t1 , ,T. A stochastic process {y }t t can be partially characterized by the first and second moments, that is, the set of means,t E(yt), and the set of variances, t2 Var(yt yt1,) and covariance cov(yt,yh)E(yt t)(yh h), t,h. In order to get consistent forecast methods, the study requires that the underlying probabilistic structure would be stable over time. So a stochastic process is called weak stationary or covariance stationary when the mean, the variance and the covariance structure of the process is stable over time, that is:E(yt)
E(yt
)2
2 For covariance stationary process, t, t0, 1, 2, ..., suppose that, almost surely for example, Given t
et1
2et2 etE(t \t1)E(et12et2 et \t1)0
13 so that
t has a constant mean of zero. Also,
) 1
(
) ( )
( )
(
) (
) (
4 2 2
2 4
1 2
2 2 1
t t
t
t t
t t
e Var e
Var e
Var
e e
e Var Var
By summing the geometric series, we have 2
2
1
For convergence, it is required that 1. and
) 1
(
) , (
) , (
) ,
( )
, (
4 2 2
2 5 2 3 2
2 2 2 1
1
1 3
2 2 2
2 1 1
t t t
t
t t
t t t
t t
t
e e Cov e
e Cov
e e
e e e
e Cov Cov
By summing the geometric series, we have 2
2
1
This series will also converge if 1.
Recursively, we obtain for lag k as 2
2
) 1 ,
(
t tk k
Cov . Note that the process defined in this
way is stationary and the autocovariance structure depends only on time lag and not on absolute time.
Let
yt()
refer to the univariate discrete time-valued stochastic process to be predicted (for example, the rate of return of a particular stock or market portfolio from time t - 1 to t ) where is a vector of unknown parameters andE
yt()t1
E
yt()
t() denotes the conditional mean given the information set t1 available in time t - 1. The innovation process for the14
conditional mean,
t()
, is then given by t() yt()t() with corresponding unconditional variance V(t())E(t2())2(), unconditional mean E(
)0 and the covariance0 )) ( ) (
(t h
E , t h as observed in linear models. The conditional variance of the process t given the past Y values, t1, t2,,measures the uncertainty in the deviation of t from its conditional mean E(tt1, t2,). That is, V
yt()t1
Vt1
yt()
Et1(t2())t2(). Since investors would know the information set, t1 when they make their investment decisions at time t - 1, the relevant expected return to the investors and volatility are t() and t2(), respectively. Hence, an ARCH process,
t()
, can be presented as:t() ztt() (2.1)
d i i
zt . .
~ N(0, 1)
t2() g(t1(),t2(), ...; t1(),t2(), ... ; vt1,vt2,...). (2.2) where
, 0 ) (zt
E Var(zt)1. )
(
t is a time-varying positive and measurable function of the information set at time t – 1.
vt is a vector of predetermined variables included in information set, t. g(.) is a linear or nonlinear functional form.
) (
t is serially uncorrelated with mean zero, but a time varying conditional variance equal to
t2(
).The conditional variance is a linear or nonlinear function of lagged values of t and
t, andpredetermined variables
( v
t1, v
t2,...)
included in
t1. In the sequel, for notational convenience, no explicit indication of the dependence on the vector of parameters, , is given when it is obvious from the context.15
2.0.2 ARCH Models
The ARCH (q) model for the series
{
t}
is defined by specifying the conditional distribution of
tgiven the information available up to time t − 1. Engle (1982) introduced ARCH models to model stock return series as a linear function of past q squared innovations to capture serial correlation in volatility. The models assume that the conditional variance at time t depends on past errors and variance. They are designed to model time varying volatility in particular volatility clustering – a feature often displayed by high frequency financial market series (Engle, 1990). The variance at time t is expected to be high when past errors and variances were high in the past and vice versa.
Let {zn} be a sequence of i.i.d random variables. zt ~ N(0, 1) and
t is a non-negative process such that
t2 =Vart1(
t) Et1(
t2)=
1
t21
q
t2q (2.3)
q
i
i t i 1
2
where
q
1, ... , and are scalar parameters to be estimated.
is the conditional mean.2 ti
are news about volatility from the previous period, measured as the lag of the squared residual from the mean equation.For t2 to be valid conditional variance, it is necessary that > 0,
1 0, …,
q 0, in which case t2 > 0 for all t. The necessary and sufficient condition for the existence of stationary varianceis
1
1
q
i
i . When this condition is satisfied, the process t is weakly (covariance) stationary and has a finite unconditional variance16
q
i i t
t E
E
1 2
2 2
1 ) ( ) (
. (2.4)This implies that the process fluctuates about the long run value
2 and conditional variance converges to this value as the forecast horizon lengthens.2.0.3 The GARCH (p, q) Model
The GARCH model is characterized by a symmetric response of current volatility to positive and negative lagged errors
t1 since
t is uncorrelated with its history, it could be interpreted conveniently as a measure of news entering a financial market at time t. In empirical applications of Engle‟s autoregressive conditional heteroskedasticity model, ARCH (q) models have a long lag length and a huge number of parameters are called for. To minimize this problem, Bollerslev (1986) proposed generalized autoregressive conditional heteroskedastic (GARCH) process to allow for past conditional variances in the current equation. This model differs from the ARCH model in that it incorporates squared conditional variance terms as additional explanatory variables.
t is called the generalized autoregressive conditional heteroskedastic (GARCH (p, q) process if2 2
2 2 2
1 1 2
2 2 2 2
1 1 2
p t p t
t q
t q t
t
t
(2.5)
q
i
p
j
j t j i
t i
1 1
2
2
where
p j
q
i j
i 0, 1, 2, , 0, 1, 2, ,
,
0
. (2.6)These conditions on parameters ensure strong positivity of the conditional variance (2.5). If the study writes the equation (2.5) in terms of lag-operator B, it gets
17
t2
t2
t2 (2.7) where
1
2
2
q
qand
1
2
2
p
p (2.8) The model is covariance stationary if all the roots of
()
()1 lie outside the unit circle, or equivalently if
p
j j q
i i
1 1
1
. That is, if
2 1 1
1 2
1 2
p t
j j q
i t i
t
211 1
2 1 2
t p
j j q
i t i t
p
j j q
i i t
1 1
2 1
Then, its long-run average variance (unconditional variance) is equal to
p
j j q
i i
1 1
2 2
1
. (2.9)
The process {t} which follows a GARCH (p, q) model is a martingale difference sequence Bollerslev and Wooldridge (1992). In order to study second-order stationarity, it is sufficient to consider that:
Var[t]Var[Et1(t)]E[Vart1(t)]E[t2]
and show that it is asymptotically constant in time (that is, it does not depend upon time). Bollerslev (1986) defined as follows:
18
The process {t} which satisfies a GARCH (p, q) model with positive coefficient p
j q
i j
i 0 1, , , 0 1, ,
,
0
is covariance stationary if and only if
(1)(1)1 (2.10) This is a sufficient but not necessary condition for strict stationarity. This is because ARCH processes are thick tailed, the conditions for covariance stationarity are often more stringent than the conditions for strict stationarity.
It is known that
p j
q i
j i
, , 1 0
, , 1 0
, 0
are independent conditions, Rossi (2004). This gives the covariance stationarity as
E
ln
11zt2
0 (2.11)Proposition 2.1
Covariance stationarity of (2.10) (2.11) but the converse is not true.
Proof
Using the multiplicative specification of the GARCH (1, 1) model, which is convenient for the construction of the weak stationarity conditions we have
1
1 1
2 1 1
2 1 ( )
t
k k
i
i t
t z
and considering auto regressively, we have When k = 0 (present value),
19
1 1
1 2 1
2 1 ( )
k k
i
i t
t z
t2 t21(1zt211) When k = - 1 (lag 1),
t21t22(1zt221)
When k = -2 (lag 2),
t22 t23(1zt23 1) That is,
) )(
)(
(
) )(
)(
( ) (
) )(
( )
(
) )](
( [
1 2
1 1 1 2
2 1 1 2
3 1 2
3
1 2
1 1 1 2
2 1 1 2
1 1 1
2 1 1
1 2
1 1 1 2
2 1 2
2 1
2 1 1
1 2
1 1 1 2
2 1 2
2 2
t t
t t
t t
t t
t t
t t
t t
t t
z z
z
z z
z z
z z
z
z z
This shows that when 0, t2 a. s. and {t,t2} is strictly stationary if and only if
ln 1 1zt2
0E .
1 1
21 1 2
1
1 ln ln
ln zt E zt
E .
When 11 1, the model is strictly stationary. E
ln
11zt2
0 is a weaker requirement than1 1
1
.
This ends the proof.
2.0.4 The IGARCH (p, q) Model
In many applications with high frequency financial data, the estimate for (1)(1) turns out to be very close to unity. This provides an empirical motivation for the so-called integrated GARCH (p, q) or IGARCH (p, q) model introduced by Engle and Bollerslev (1986):
20
t2 A(L)t21B(L)t21, for A(L)B(L)1. (2.12) Recall that the GARCH (p, q) process is characterized by the first two conditional moments:
E
t1[
t] 0
p
j
j t j i
t q
i i t
t
t E
1 2 2
1 2
1
2 [
]
where
0 ,
i 0
and
j 0 for all i, j and the polynomial 1
(x)
(x)0 has d 0 unit root(s) and max{p,q}d roots outside the unit circle is said to be(i) Integrated in variance of order d if 0
(ii) Integrated in variance of order d with trend if
0.The Integrated GARCH (p, q) models, both with or without trend, are therefore part of a wider class of models with a property called “persistence variance” in which the current information remains important for the forecasts of the conditional variances for all horizons. So we have the Integrated GARCH (p, q) model when (necessary condition)
(1)
(1)1To illustrate, consider the IGARCH (1, 1) which is characterized by
1
1 1
) (
) 1
(
2 1 2
1 1 2
1 2
2 1 1 2
1 1 2
t t
t t
t t
t
01 1
For this particular model the conditional variance k steps in the future are
E [
t2k]
t2k
t2 k
(2.13) which looks very much like a linear random walk with drift,
. A linear random walk is strictly non-stationary and has no unconditional first or second moments. In the case of IGARCH (1, 1), the conditional variance is strictly stationary even though its stationary distribution generally lacks unconditional moments. In the case where
= 0, equation (2.13) reduces to
t2k
t2, a21
bounded martingale as it cannot take negative values. According to the martingale convergence theorem, Ding and Engle (2001), a bounded martingale must converge and, in this case, the only value to which it can converge to is zero. Thus, the stationary distributions for t2 and
t are to have moments, but they are all trivially zero.The above study reveals that in many studies of the time series nature of asset volatility, the question has been how long shocks to conditional variance persist? If volatility shocks persist indefinitely, they may move the whole term structure to risk premia, Engle and Lee (1993), Gordon, Salmond and Smith (1993), Fan and Yao (2003), Liu, Longstaff and Pan (2003). There are many notions of convergence in the probability theory (almost surely, in probability, in distribution), so whether a shock is transitory or permanent may depend on the definition of convergence. In linear models it typically makes no difference which of the standard definitions we use, since the definitions usually agree. In GARCH models the situation is more complicated. In the IGARCH (1, 1):
t2
1
t21
1
t21where
1
1 1
. Given that
t2 z
t2
t2,
we can rewrite the IGARCH (1, 1) process as
t2
t21[(1
1)
1zt21] 01 1.Moreover, Nelson (1991) showed that the GARCH (1, 1) model is strictly stationary even if
1 1
1
, as long as E
log(
1
1zt2)
0. Based on the nature of persistence in linear models, 0 and 0 is analogous to random walks with or without drift, respectively, and are therefore natural models of “persistent” shocks. This turns out to be misleading, however, the conditional variance,
t2 in IGARCH (1, 1) with 0, collapses to zero almost surely while
t222
with 0 is strictly stationary and ergodic and therefore does not behave like a random walk, since random walks diverge almost surely. Rossi (2004) gives two notions of persistence as:
(i) Suppose t2 is strictly stationary and ergodic. Let F(t2) be the unconditional cumulative density function (cdf) for t2, and Fh(t2) the conditional cdf for t2, given information at time h t. For any hF(t2)Fh(t2)0 at all continuity points as t . There is no persistence when {t2} is stationary and ergodic.
(ii) Persistence is defined in terms of forecast moments. For some 0, the shocks to t2 fail to persist if and only if for every h, Eh(
t2) converges, as t to a finite limit independent of time h information set.The persistence of shocks to the conditional variance process {t2} depends very much on which definition is adopted. The conditional moment may diverge to infinity for some , but converge to a well-behaved limit independent of initial conditions for other , even when the {t2} is stationary and ergodic.
The GARCH (p, q) model successfully captures several characteristics of high frequency financial time series, such as thick tailed returns and volatility clustering, Hall and Yao (2003). On the other hand, its structure imposes important limitations. This is because it operates best under relatively stable market conditions. The model is explicitly designed to model time-varying conditional variances but it often fails to capture highly irregular phenomena (crashes and later rebounds) and other unanticipated events that can lead to significant structural change. This means that the variance only depends on the magnitude and not the sign of
t, which is somewhat at odds with the empirical behaviour of stock market prices where the “leverage effect” may be present. This model is symmetric in that the magnitude and not the positivity or negativity of innovations determines2
