ABSTRACT
In this research we focused on Multivariate Generalised Autoregressive Conditional Heteroskedasticity models for volatility series using response vector of variances. The work aimed at developing alternative multivariate GARCH models characterised by either autoregressive or moving average process. Isolated Multivariate Generalised Conditional Heteroskedasticity, ISO-MGARCH (p,0) models and Isolated Multivariate Generalised Conditional Heteroskedasticity, ISO-MGARCH(0,q) models are identified from MGARCH (p,q) model under specific conditions. To ascertain the models applicability, the isolated univariate and multivariate GARCH (2,0) models were fitted to volatility measures of Nigeria average, urban and rural consumer price indices from January 1995 to December 2019 after the series were subjected to stationarity checks using the positive definiteness property of the sub-autocovariance or autocorrelation matrices of individual vector processes as components of the cross-covariance or cross-autocorrelation matrix to ascertain the stationarity of the series,. The volatility series were also subjected to autocorrelation and partial autocorrelation checks, as applicable to stationary autoregressive moving average process, where single autoregressive and moving average models are identified under certain conditions. This justified the isolation of pure autoregressive and pure moving average MGARCH models. Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and Schwarz’s Information criterion (SIC) compare the isolated multivariate GARCH models with the existing univariate GARCH models, and its simulated values, the results revealed the same comparative advantage in capturing volatility series.
TABLE OF CONTENTS
Page
Title Page i
Declaration ii
Certification iii
Dedication iv
Acknowledgments v
Table of Contents vi
List of Tables viii
List of Figures ix
Abstract x
CHAPTER 1: INTRODUCTION
1.1
Background of the Study 1
1.2 Statement
of the Problem 9
1.3
Objective of the Work 9
1.4
Justification of the Study 10
1.5 Scope
of Study 11
1.6 Significance
of the Study 11
CHAPTER 2: REVIEW OF RELATED LITERATURE 12
2.1 Review
of Models 12
2.2
Empirical Review 21
CHAPTER
3: METHODOLOGY 28
3.1 Cross-covariances 28
3.2 Auto-correlations and
Cross-auto-correlations 31
3.3 Positive
Definitness of Auto-correlations and Cross-
auto-correlation
matrices 33
3.4 Univariate
Case 34
3.4.1
Testing for ARCH
Effects 34
3.4.2
Model estimation 36
3.4.3
Post estimation test 36
3.5
Multivariate Case 36
3.6
Volatility Measure 38
3. 7 Conditions
for Model Identification 42
3.7.1 Proof 42
3.8 Model
Selection Criteria 44
3.8.1 Akaike Information Criterion (AIC): 44
3.8.2 Bayesian Information Criterion (BIC): 45
3.8.3 Schwarz’s Information Criterion (SIC): 45
CHAPTER 4: DATA ANALYSIS AND DISCUSSION OF RESULTS 46
4.1 Numerical Verification 46
4.2 Components of the Autocorrelation and
Cross-autocorrelations 47
4.3 Positive Definiteness of 3x3 Component
Autocorrelation Matrices 49
4.4 Positive Definiteness of 9x9 Autocorrelation
Matrix 49
4.5 Graphical
Analysis 53
4.6 Univariate
GARCH (p,0) Model Estimates 55
4.7 Isolated
Multivariate GARCH (p,0) Model Estimates 57
CHAPTER 5: SUMMARY AND CONCLUSION 64
5.1 Summary
and Conclusion 64
5.2 Recommendation 66
References
67
Appendices 71
LIST OF TABLES
4.1 Parameter Estimates of the Univariate
GARCH (2,0) Models 56
4.2 Parameter Estimates of the Multivariate
GARCH(2,0) Models 57
4.3 The
parameter estimates for 3000 Data Points Simulated Values 59
4.4 Information
Criteria 62
4.5 Information
Criteria for Simulated Values 62
LIST OF FIGURES
4.1 Return series of Average CP 53
4.2 Return Series of Urban CPI 53
4.3 Return Series of Rural CPI 54
4.4 Autocorrelation Function of Average
Consumer Price Index 54
4.5 Partial
Autocorrelation Function of Average Consumer Price Index 54
4.6 Autocorrelation function the residual of
MGARCH model 61
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND
OF THE STUDY
In univariate time series analysis, a
process, say
is
described as a function of its lagged variables, that is
F
, k= ,
. These lagged variables are the
predictor time variables, while
is
the response time variable. Box and Jenkins (1970) gave a comprehensive summary
of linear time series, and models which are indespensible in analyzing
stochastic processes. This research is focused on the univariate and multivariate Generalised
Autoregressive conditional Heteroscedasticity MGARCH models with special
interest on identification of some special classes of univariate and
multivariate GARCH models. The Autoregressive Conditional Heteroscedasticity
(ARCH) model proposed by Engle (1982), was first modeled to assume that
variance is not constant, unlike the Autoregressive framework that assumes
constant variance for the series. In the
Autoregressive Conditional Heteroscedasticity (ARCH) model, the Autoregressive
implies that the series depends on its past values and describes a feedback
mechanism that incorporates past observations into the present, while the
Conditional Heteroscedasticity is referred to as the time varying volatility,
where volatility of time series in econometrics refers to Conditional variance.
These two features combine to give the ARCH model. Thus Autoregressive
Conditional Heteroscadasticity (ARCH) is
a mechanism that includes past variances in the explanation of future variance,
and a time varying technique that allows researchers to model the serial
dependence of volatility. The approach
expects that the series is stationary other than the change in variance,
meaning it does not have a trend or seasonal components. In this research,
stationarity will be ascertained using a more revealing approach of the
positive definiteness of the cross-autocovariance/cross- autocorrelation matrix
of the return series. As asserted by Box and Jenkins, the two fundamental
process for a stationary time series that have been extensively applied in time
series modeling and forecasting are the Autoregressive and moving average
processes. The two processes have wide applications in macro-economics such as
money supply, interest rates, price, inflation, exchange rates, and Gross
domestic products and in financial time series analysis. A change in variance
or volatility over time can cause problem with modeling time series with
classical methods like ARMA. Conditional Heteroscadasticity (ARCH) model
provides a way to model a change in variance of a time series that is time dependent,
such as increasing or decreasing volatility. Autoregressive models can be
developed for univariate time series data that is stationary (AR), has a trend
(ARIMA) and has seasonality (SARIMA). One aspect of a univariate time series
that this Autoregressive do not model is a change in variance over time.
Classically, a time series with modest change in variance, changes consistently
over time. In the context of a time series in the financial domain, this is called volatility. Financial time series such
as stock returns, exchange rates, inflation etc are available at a high
frequency, hence exhibit random walk and time varying volatility. If the change in variance can be correlated
over time, then it can be modeled using an Autoregressive process such as the
Autoregressive Conditional Heteroscadasticity (ARCH) model.
The Autoregressive Conditional
Heteroscadasticity (ARCH) model proposed by Engle (1982) has superior
performance over Autoregressive and moving Average models in the sense that,
some time series that are characterized by high variations are well fitted with
ARCH models. The ARCH is a method that explicitly models the change in variance
over time in a time series, the ARCH model captures variance of the error
residuals referred to as Volatility, which is part of the limitations of the ARMA linear model.
Assuming that the series is volatile from
preliminary diagnostic check, the Autoregressive Conditional Heteroscadsticity
(ARCH) series can be modeled by considering two equations, the mean equation
and the variance equation;
The Mean Equation;
The Variance Equation;
The stationarity condition is given as;
Combining these two leads to;
The ARCH model as proposed by Engle(1982)
Where
Further studies carried out on the ARCH
model revealed some yet basic deficiency in the models, some of which includes,
assuming that positive and negative shocks in the models have the same effect
on volatility. Again the fact that the ARCH model is restrictive and does not
accommodate higher order ARCH model, which limits the ability of the ARCH model
to capture excess kurtosis. The ARCH model is also believed to provide only a
mechanical way to describe the behavior of the conditional variance without any
insight for understanding the source of variation of the financial time series.
Amongst these reasons has led to the development of the Generalized
Autoregressive Conditional Heteroscadasticity (GARCH) model developed by
Bollerslev(1986).
The GARCH model is an extension of the ARCH
model which considers the Autoregressive
and moving Average components of the process, variance and error respectively.
Specifically the model includes lag variance terms together with lag residuals
from the mean process. The introduction of the Moving Average (MA) component
allows the model to both model the conditional change in variance over time as
well as changes in the time dependent variance, that is the conditional
variance equation incorporates lags of the conditional variance as regressors
in the conditional variance equation in addition to the lags of the squared
error. Conventionally, Autoregressive Conditional Heteroscedasticity (ARCH)
model is written in the form ARCH (p). the ‘p’ is the order of the model, which
indicates the number of lag errors as the predictor variables of the ARCH
model. With the Generalized Autoregressive Conditional Heteroscedasticity GARCH
(p,q), ‘p’ assumes the order of the Autoregressive component( the number of
lags of the variance to be included as predictors), while the ‘q’ assumes the
order of the moving Average component(the numbers of lags of the residual error
that predicts the volatility) Gujarati and Porter(1997).
Thus for GARCH models;
Conditional variance = f(ARCH terms,
GARCH terms)
The simplest version of GARCH is the
GARCH(1,1) model;
The (1,1) in parenthensis is a standard
notation for the ARCH and GARCH terms. The first number (1) refers to the AR
term, while the second number (1) refers to the first order GARCH term or first
moving Average term MA(1)
The ARCH and GARCH models are volatility models that are found suitable
in modeling financial and economics time series characterized by time- varying
dispersions from their mean values.
The ARCH model as proposed by Engle(1982)
is
Where
Bollerslev (1986) proposed the GARCH
model as
And
Where 
is
the conditional variance of the GARCH model, 
is
the squared error term. Similar to the ARMA model,
and
are parameters of the lagged variance and squared terms respectively. This
implies that the GARCH model subsumes the ARCH. As with ARCH, the GARCH model
predicts the future variance and expects that the series is stationary other
than the change in variance, meaning it does not have trend or seasonal
components. The ARCH(q) model proposed by Engle(1982) could also be expressed
as GARCH(0,q) as a component of GARCH (p,q) model Emenogu, Adenomon and Nweze
(2009). It follows that GARCH(p,0) is a component of GARCH (p,q). The earlier
represents the moving Average component, while the later indicates the
Autoregresssive part of GARCH (p,q).
Box and Jenkins (1978) provided a
condition for the isolation of AR(p) and MA(q) models from the mixed ARMA(p,q)
model, t his is also applicable to the Bilinear Autoregressive moving Average,
BARMA and Bilinear Autoregressive moving Average Vector BARMAV model, where
BAR, BMA and BARV, and BMAV models were isolated respectively, Usoro, Omekara
(2008), Usoro (2017) and Anthony and Clement (2018). In this work, this
isolation principle will be applied to the univariate and multivariate cases of
the Autoregressive Conditional Heteroscadasticity (ARCH) and the Generalized
Autoregressive Conditional Heteroscadasticity (GARCH) models to identify some
special classes of these models and its application to real life data.
It didn’t take long for the GARCH model
to progress from univariate to multivariate settings. Multivariate extentions
of ARCH and GARCH models may be defined in principle similar to VAR and VARMA
models. In financial economics, its rare to have only one asset of interest, if
you are analizing bonds, there are different maturities, of exchange rates,
multiple currencies and of course, there are thousands of equities, not only is
the volatility for each, likely to be described fairly well by a GARCH process,
but within a class of assets, the movements are likely to be highly correlated.
As a result, there will be expected to be substancial gain in statistical efficiency
in modeling them jointly, hence the need for the Multivariate Autoregressive
Conditional Heteroscedasticity(MGARCH) model, Kraft and Engle (1982) was the
first attempt. Engle et al. (1984) put forward Bivariate ARCH model, the move
to financial application was done by Bollerslev et al (1988), who also extended
multivariate ARCH to GARCH. The multivariate GARCH (MGARCH) also known as the
VEC model, has too many parameters to be useful for modeling more than two
assets returns jointly, until a milestone breakthrough with the BEKK model of
Engle and Kraft in 1995, the factor model of Engle et al.(1990) and so on . a
VARMA representation of the MGARCH model may be obtained in the same way as in
the univariate case as;
Where 
is
a triangular (K x K) matix
and the coefficient matrices 
are also (K x K) given the similarity of the MGARCH and the VARMA models it is obvious
that restrictions have to be imposed on the coefficients matrices to ensure
uniqueness of the parametization.
Engle and Kroner (1995) showed that for
the MGARCH to be stationary, all the eigenvalues of the matrix 
must have modulus less than one.
The multivariate form of the GARCH model
can also be given as ;
In this research work, interest is on the
identification of some special classes of multivariate GARCH models. Like the ARMA,
BARMA and BARMAV models, the conditions for identification of the Isolated
univariate and multivariate GARCH(p,q) model with their proofs and ascertaining
the stationarity of the return series, through the positive definiteness
condition of the cross-autocovariance/cross-autocorrelation matrix of the
return series, before the application of the established model to real life
data.
There are many statistical procedures which
verify stationarity state of a time series. These are carried out through some
statistical tests/investigations of some stationarity properties. Assessing the
structure of the autocovariances and autocorrelations is very prominent in
investigating the stationarity of time series process, Box and Jenkins (1976),
Kendall and Ord (1990), Gujarity and Porter (2009). In univariate time series,
positive definiteness or semi-positive definiteness property is investigated to
ascertain the stationarity of the autocovariance structure with the
autocorrelation matrix at different time lags. This procedure has an extension
to multivariate time series, Engle and Kroner (1995). In multivariate time
series, the n-dimensional cross-autcovariance or cross-autocorrelation matrix
composes of sub-matrices of individual vector processes with distributed lags.
For a cross-autocorrelation matrix to be positive definite, it is a justifiable
assumption that the individual sub-autocorrelation matrices meet the positive
definiteness condition. That means, their determinants and principal minors
have positive values. Borlerslev et al (1998) considered k=2 ARCH (1) process
and symmetric positive definite matrix of cross-covariances. Kiyang and Cyrus
(2005) investigated stationarity of multivariate time series for
correlation-based data analysis. Carsten and Subba Rao (2015) adopted Discrete
Fourier Transform as a tool to test for second order stationarity of
multivariate time series. In addition to the assumption of positive
definiteness of the n-dimensional cross-autocovariance matrix, it becomes more
revealing to look at the positive definiteness of the sub-autocovariance
matrices of individual vectors as the components of the cross-covariance and
cross-correlation matrix. This is considered the first step of verifying
stationarity property before the larger cross-covariance matrix.. This implies,
each univariate autocovariance structure making up the cross-autocovariance
matrix is verified, and the final verification of stationarity concluded in the
cross-autocovariance or cross-autocorrelation matrix. This research uses both
the sub-covariance/correlation matrices and
cross-autocovraiance/autocorrelation matrix to ascertain stationarity of the multivariate
time process.
1.2 STATEMENT OF THE PROBLEM
Considering the multivariate time series
from the Heteroscadasticity framework, an intrinstic assessment on the
multivariate GARCH, reveals the possibility of a more parsimonious model.
Hence, the need to identify some special classes of MGARCH models, through some
special conditions. Since there exist conditions for the isolation of the
Autoregressive(AR) and moving Average (MA) from Autoregressive moving average (ARMA), and the isolation of
Billinear Autoregessive (BAR) or Billinear moving average (BMA) from the
Billinear Autoregressive moving average (BARMA) models. Therefore it is of
interest in this research, to carry out further study on Multivariate GARCH
models, consider the proof and identify the conditions for the isolation of
each of the processes as independent Autoregressive and Moving Average
components, which make up the Generalized MGAR CH (p,q) models.
Furthermore,
it becomes more revealing to ascertain the stationarity of the return
series as a precondition to the model
application by using a more thorough approach ,in this work the positive definiteness of the
sub-autocovariance matrices of individual vectors was used as the components of
the cross-covariance and cross-correlation matrix, were the
covariances and autocorrelation matrices of individual vector return series
constituted the components of cross-covariance and cross-autocorrelation
matrices, and the positive definiteness of the sub-autocorrelation matrices
were investigated, to ascertain stationarity..
1.3
OBJECTIVE OF THE WORK
Our major concern
in this work, is amongst others
1. To
identify the conditions necessary to establish special classes of multivariate Generalized Autoregressive
Conditional Heteroscadasticity MGARCH models
2. Show
the proofs and the special classes of
MGARCH models established
3. To
use the positive definiteness property of the
cross-autocovariance/cross-autocorrelation functions of the return series to
ascertain stationarity, before the
applicability of the established models
4. To
apply these special classes of MGARCH models established to the Nigeria
macroeconomic data.
5. To
compare these established models with simulated results.
1.4 JUSTIFICATION
OF THE STUDY
The existing gap
between this research and previous researches is the identification of isolated
univariate and multivariate GARCH models from the generalised form whose
response vector of variances (volatility measures) could be expressed as linear
combinations of its distributed lagged variance from the original series
precluding its dependence on the squared errors and simulated values of the
standard normal random variable (
).
The isolation of
the established pure Autoregressive and pure Moving Average MGARCH model from
the multivariate MGARCH(p,q) model, is justified by subjecting the series to
stationarity checks, using the positive definiteness property of the sub-
autocovariance or autocorrelation matrix of individual vector processes as
components of the cross-covariance or cross- autocorellation matrix to
ascertain stationarity of the series. The volatility series were also subjected
to the autocorrelation and partial autocorrelation checks, as applicable to
stationary autoregressive and moving average process.
The revealing of
the same comparative advantage by comparing the established model with the
existing univariate GARCH model and simulated values, in capturing volatility
further justifies this work
1.5 SCOPE OF STUDY
The scope of this research is within
the heteroscadasticty framework, that assumes non constant variance for the
time series. Unlike the homoscedasticity framework that assumes constant
variance for the series. In ascertaining stationarity of the series before
application, we will be limiting ourselves to the positive definiteness
condition of the cross- autocovariance/cross-autocorrelation matrices of the
return series. This research will also examine the multivariate GARCH with
special interest in identifying some special classes of this model under, under
some specified conditions. The result of the real data approach adopted in this
work will also be compared with that of the simulated values for further
justification.
1.6 SIGNIFICANCE OF THE STUDY
The significance of this research
work, ranges from filling in the research gap in multivariate GARCH analysis,
through the established model, to building a more parsimonious model with
interactive advantage, hence modifying Bollerslev (1988), devoid of interactive
effects, and further show that a multivariate process can as well be defined by
a single independent autoregressive process or an independent moving average
process.
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