A common finding
in time series regression however is that the residuals are correlated with
their own lagged or previous values. Since this correlation is sequential in
time, it is then called serial correlation.

ARMA models are
linear in both mean and variance while the GARCH models are either linear in
mean but non linear in the variance or nonlinear in both mean and variance.

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a.      ARMA Models

The serial
correlations for the models which are linear in both mean and variance can take
the form of autoregression (AR), Moving Average (MA) or a combination of the
two (ARMA).

These models are
said to be stationary. Stationarity here implies that a time series has memory
depicted in the lags of the error/residual terms or past observations of the
variable under consideration.

·
In an AR model, the variable in the
current period is also related to the variable’s lag and is given by;

Where

is represent the order of the model and

is the disturbance term
with N(0,?2).

·
In an MA model, the variable

is affected by the
previous prediction errors

and is models as;

Where

is represent the order of the model

previous prediction
error term for

and

is the disturbance term
with N(0,?2).

·
The autoregressive and moving average
specifications can be combined to form an ARMA(p,q) specification:

b.      ARCH and
GARCH models

For most financial data the Classical Linear
Regression Model assumption that the Variance is homoskedastic is often
violated. If the
variance of the errors is not constant, this would be known as heteroscedasticity.

Autoregression Conditional Heteroskedasticity (ARCH) and the Generalized
Autoregression Conditional Heteroskedasticity (GARCH) models are used to model
time-varying volatilities.

Given a stationary mean given by

,

The ARCH Model of the

will be given by;

Where the
Autoregression in the squared residual has an order of

, or

lags.

If

is
a time varying conditional variance with both Autoregression (AR) and Moving
Average (MA), the GARCH can be modeled by;

One
of the advantages of GARCH over ARCH is parsimonious, i.e. less lags are
required to capture the property of time-varying variance in GARCH.

2. Discuss many variations of GARCH and their
relevance to financial modeling.

(10
Marks)

SOLUTION

Many of the variations to the GARCH model have been suggested as
a consequence of perceived problems with standard GARCH(p, q) models
because;

1.      The non-negativity conditions may be
violated by the estimated model. The only way to avoid this for sure would be
to place artificial constraints on the model coefficients in order to force
them to be non-negative.

2.      GARCH models cannot account for leverage
effects, although they can account for volatility clustering and leptokurtosis
in a series.

3.      The model does not allow for any direct
feedback between the conditional variance and the conditional mean.

Variations of the GARCH model remove some of the restrictions or
limitations of the basic model.

The variations to the GARCH Model can either be Univariate or
Multivariate.

1.      UNIVARIATE
MODELS

a.
Model  relating the return on a security to its time-varying
volatility or risk

These models allows for a direct feedback between the
conditional variance and the conditional mean

i.
The
GARCH in Mean (GARCH-M) model

Most models used in finance suppose that
investors should be rewarded for taking additional risk by obtaining a higher
return. One way to operationalise this concept is to let the return of a
security be partly determined by its risk.

When the
conditional variance enters the mean equation for a GARCH process, the GARCH-in-Mean
or simply the ARCH-M model is derived and is given by:

Where

, k
= 1, . . ., m are
exogenous variables which could include lagged

. In the sense of
asset pricing, if

is
the return on an asset of a firm, then

,k = 1, . . ., m would
generally include the return on the market and possibly other explanatory
variables such as the price earnings ratio and the size.

The parameter

captures the
sensitivity of the return to the time-varying volatility, or in other words,

This model can
be used as an extension of the CAPM to model with an allowance for time-varying
risk to identify the trade-off preferences of investor between risk and return.

b.
Asymmetric
GARCH Models

One of the primary restrictions of GARCH models is that they
enforce a symmetric response of volatility to positive and negative shocks.
This arises since the conditional variance in the GARCH is a function of the square
of the lagged residuals and not their signs.

However, it has been argued that a negative shock to financial
time series is likely to cause volatility to rise by more than a positive shock
of the same magnitude. In the case of equity returns, such asymmetries are
typically attributed to leverage effects, whereby a fall in the value of
a firm’s stock causes the firm’s debt to equity ratio to rise.

Two of the most popular asymmetric GARCH models are the GJR and
the Exponential GARCH (EGARCH) models)

i.
GJR
Model (also Called Threshold GARCH -TGARCH)

The GJR model is a simple extension of GARCH with an additional
term added to account for possible asymmetries. The conditional variance is now
given by;

Where

account for the leverage effect in most
financial data.

ii.
Exponential
GARCH (EGARCH)

The model
captures asymmetric responses of the time-varying variance to shocks and, at
the same time, ensures that the variance is always positive. For a GARCH(1,1)
model, the volatility is given by;

Where

is asymmetric response
parameter or leverage parameter. The sign of

is expected to be
positive in most empirical cases so that a negative shock increases future
volatility or uncertainty while a positive shock eases the effect on future
uncertainty. This is in contrast to the standard GARCH model where shocks of
the same magnitude, positive or negative, have the same effect on future volatility.

An important
advantage of the EGARCH specification is that the conditional variance is
guaranteed to be positive at each point in time because the variance is
expressed in terms of log.

2.
MULTIVARIATE
GARCH MODELS (MGARCH)

The multivariate
time-varying conditional variance problem is given by the model;

Where vectors
for a bivariate model (GARCH(1,1) are;

,

, and

a.
Full
Parametization (VECH) Model

The VECH specification is presented by;

Where

,

,

,

and

are matrices in their
conventional form, and

means the procedure of
conversion of a matrix into a vector.

For a bivariate
GARCH;

b.
Positive
Definite Parametization (BEKK) Model

It takes the
form;

The important
feature of this specification is that it builds in sufficient generality,
allowing the conditional variances and covariances of the time series to
influence each other, and at the same time, does not require estimating a large
number of parameters.

In a bivariate
GARCH process, BEKK model has only 11 parameters compared with 21 parameters in
the VECH representation.

Even more
importantly, the BEKK process guarantees that the covariance matrices are
positive definite under very weak conditions.

In the bivariate
model, BEKK takes the form;

3. What is the stochastic volatility model? Discuss
the similarities and differences between a GARCH-type model and a stochastic
volatility model.                          (10
Marks)

SOLUTION

Stochastic volatility models are models that are random in time
and are used to model volatilities which are time-varying. For example a random
walk with a drift model given by;

In a stochastic
model, the best guess for the next value of series is the current value plus
some constant, rather than a deterministic mean value.

Stochastic volatility models contain a second error term, which
enters into the conditional variance equation.

Stochastic volatility models are closely related to the
financial theories used in the options pricing literature.

The primary advantage of stochastic volatility models is that
they can be viewed as discrete time approximations to the continuous time
models employed in options pricing frameworks. However, such models are hard to
estimate.

A GARCH model is deterministic trend rather that stochastic and
is a nonrandom function of time. A GARCH(p,q) model can be modeled as;

Stochastic volatility models differ from GARCH principally in
that the conditional variance equation of a GARCH specification is completely
deterministic given all information available up to that of the previous
period. In other words, there is no error term in the variance equation of a
GARCH model, only in the mean equation.

While stochastic volatility models have been widely employed in
the mathematical options pricing literature, they have not been popular in
empirical discrete-time financial applications, probably owing to the complexity
involved in the process of estimating the model parameters.

Both Stochastic and GARCH models are used in modeling
volatility.

4.
Discuss and comment on the new developments in modeling time-varying
volatilities.

(10
Marks)

SOLUTION

of Multivariate Volatility Models in Finance

Because of the
new variations in the volatility models, most fund managers are now modeling
volatility as a time-varying variable. This is in contrast to for example the
assumptions made in the CAPM and Option pricing models which take volatility as
measured by the standard deviation of returns to be constant.

b.
Time-Varying
Beta Risk

In most
financial models, specifically CAPM model, the beta is always assumed to be
constant over time, thus the use of historical betas. The restriction of
constant beta risk may, however, be unrealistic. For example, beta risks may be
affected by periods of financial crises and economic booms and recessions. GARCH-based
and other stochastic volatility models have been developed to structure the
beta as time-varying which may be given by the function;

Univariate Conditional
covariance models can be used to model the individual variances while
covariances can be modeled using a simple bivariate GARCH model.

c.
Modeling
transmission of volatilities from one global market to another

GARCH Models are
now being extended to study how volatility is transmitted through different
regions of the world (mainly US, UK and Middle East) during the course of a
global financial trading day. The aim is of these GARCH study is to examine
international linkages in volatility between major global financial regions and
investigate in particular two patterns as possible explanation of international
volatility transmission;

Heatwave: These models are based on the premise that volatility
in any one region or market is a function of the previous day’s volatility in
the same region or market.

Meteor Shower: the models are explains that volatility
in one region is driven mainly by volatility in the region immediately
preceding it in terms of calendar times.

d.
Dynamic
Optimal Hence Ratio

Multivariate
GARCH models are now being used to determine the optimal hedge ratio for an
investor who buys or sell futures contracts to hedge against the movement in
the spot prices of an asset. This is achieved by specifying the covariance
between the returns on the futures and the assets, and the variance of the
return on the futures contracts to be time varying