JRSSEM 2021, Vol. 02, No. 4, 485 – 503
E-ISSN: 2807 - 6311, P-ISSN: 2807 - 6494
10.36418/jrssem.v2i04.302 https://jrssem.publikasiindonesia.id/index.php/jrssem/index
FORECASTING INTEREST RATE VOLATILITY IN NIGERIA
IN THE ARCH-GARCH FAMILY MODELS
AYOGU, Ebenezer Chukwuma
Department of Statistics, University of Abuja, Nigeria
*
e-mail: ayoguchuks49@gmail.com
*Correspondence:
ayoguchuks49@gmail.com
Submitted
: 03
th
November 2022
Revised
:14
th
November 2022
Accepted
: 21
th
November 2022
Abstract: Modeling the volatility of interest rates is essential for many areas in finance. However,
it is well known that interest rate series exhibit non-normal characteristics that may not be captured
with the standard GARCH model with a normal error distribution. But which GARCH model and
error distribution to use is still open to question, especially where the model that best fits the in-
sample data may not give the most effective out-of-sample volatility forecasting ability, which we
use as the criterion for the selection of the most effective model from among the alternatives. In
this work, the GARCH family models were employed in modeling interest rate volatility in Nigeria.
A time series of data spanning January 2000 to December 2018 (in-sample data) was used to fit
the models and out-of-sample data running from January to December 2019 to determine the best
conditional volatility forecast. Twenty-four symmetric and eighteen asymmetric models were
estimated and compared using three distribution errors; the normal, student's t, and the
generalized error distributions; while four error loss functions, namely, RMSE, MAE, MAPE, and TIC,
were adopted to determine the best fit and conditional volatility forecast. The result shows that the
symmetric GED-GARCH (1,1) model was considered the overall best fit in both the symmetric and
asymmetric models. The best-fitted GED-GARCH (1,1) model exhibited volatility persistence. The
in-sample and out-of-sample volatility forecast of the GED-GARCH (1,1) model reveals that
unconditional mean and variance will be achieved in the third month of 2019. Some transmission
spillover effects running from the exchange and inflation rates to interest rates were also detected.
Keywords: Interest rate, Volatility, GARCH-Type Models, Best fit, Persistence, forecast
AYOGU, Ebenezer Chukwuma | 486
INTRODUCTION
The fluctuation of a variable over a
period of time is an indication of the
volatility of that variable, and the deviation
from an expected value is often used to
describe volatility. Financial volatility is
defined as the measure of the variation in
the price of a financial instrument over
time (Ezzati, 2013). One of the basic roles
of the Central Bank of Nigeria in controlling
financial institutions is the setting of
interest rates. The economic well-being of
any nation is, to a large extent, determined
by the interest rate fixed by its apex bank.
We know that the interest rate and the
aggregate supply of money in circulation
are the two main instruments of monetary
policy, which can either be achieved by
controlling the growth of money supply or
expanding the supply of money in
circulation, which in turn leads to excess
demand, thereby causing the interest rate
to decline. Although the 1980
economic reforms saw some significant
levels of development, particularly in the
financial system, there are still many
unresolved economic problems,
particularly interest rates, which have had
devastating effects on the cost of
borrowing and investment in Nigeria and
which have been the bane of dissatisfied
foreign investment. The interest rate policy
in Nigeria, for example, has changed within
the time frame of regulated and
deregulated regimes. According to (Okwori
et al., 2014), Nigeria has pursued a two-
interest rate regime starting from the
1960s to the mid-1980s with the
administration of low interest rates, which
was intended to encourage investment.
However, the advent of the Structural
Adjustment Programme (SAP) in the third
quarter of 1986 ushered in an era when
fixed and low-interest rates were gradually
replaced by a dynamic interest rate regime,
where rates were more influenced by
market forces. (Chirwa E.W. and M.
Mlachila, 2004) argued that the major
economic indicator used to boost
investment is interest rates, which have
been found to be higher in Africa, Latin
America, and the Caribbean countries than
in the Organization of European Countries.
The behavior of interest rates, to a large
extent, determines the investment
activities and hence economic growth of a
country. According to (Jhingan, 2003), if
interest rates are high, investment is at a
low level; when interest rates fall, the
investment will rise. The high-interest rate
in Nigeria might be owing to high inflation
that remained at double digits and other
macroeconomic factors like the instability
in the Nigeria currency, even the increased
sub-national government spending and
government high expenditure. On the
basis of the foregoing, it becomes
necessary to investigate the dynamic
nature of interest rates in Nigeria and
whether or not there are external forces
influencing such volatility, so that investors
and government agencies can, on the basis
of the outcome of this research, be
properly informed in making appropriate
investment and policy decisions.
The economic status of any nation is,
to a large extent, determined by the level
487 | Forecasting Interest Rate Volatility In Nigeria In The Arch-Garch Family Models
of investment made by the private sector,
foreign investments, and the national
government. Following the assertion that
interest rates (bank lending rates) are
highly volatile in Nigeria, both local and
foreign investors have become skeptical
about whether to borrow and invest or not
and when. It was also argued that there
may be other independent variables whose
variances may be contributing to the
conditional volatility of interest rates in
Nigeria over the years. It, therefore,
becomes imperative to study this
dynamism or volatility of interest rates in
Nigeria so that these investors can rightly
be advised on what the future holds. To
accomplish this, it is necessary to look for
a statistical package or GARCH-family
model that will provide the best fit for the
best forecast for such advice.
Statement of the Problem
The economic status of any nation is,
to a large extent, determined by the level
of investment made by the private sector,
foreign investments, and investment by the
national or federal government. Following
the assertion that interest rates (bank
lending rates) are highly volatile in Nigeria,
both local and foreign investors have
become skeptical about whether to borrow
and invest or not and when. To advise these
investors, a statistical package or GARCH-
family model that provides the best fit for
such advice is required. If interest rates are
indeed volatile, are we certain that the
variances of other related economic
variables are not equally affecting the
conditional variance of interest rates in
Nigeria, i.e. the spillover effect? It is
therefore critical to investigate the
dynamism or volatility of interest rates in
Nigeria so that these investors can be
properly advised on what the future holds.
However, while modeling the volatility of
interest rates is essential for many areas in
finance, it is well known that interest rate
series (like many other variables) exhibit
non-normal characteristics that may not be
captured with the standard GARCH model
with a normal error distribution. But which
GARCH model and error distribution to use
is still open to question, especially where
the model that best fits the in-sample data
may not give the most effective out-of-
sample volatility forecasting ability, which
we use as the criterion for the selection of
the most effective model from among the
alternatives.
1. Theoretical Review of Related
Literature
The first breakthrough in volatility
modeling was (Engle, 1982), where it was
shown that conditional heteroscedasticity
can be modeled using the Autoregressive
Conditional Heteroscedasticity (ARCH)
model. The ARCH model relates the
conditional variance of the disturbance
term to the linear combination of the
squared disturbance in the recent past
Determining optimal lag length is
cumbersome, often times engendering
parameterization. However, (Bollerslev,
1986) and Taylor (1986) independently
proposed the extension of the ARCH
model with an Autoregressive Moving
Average (ARMA) formulation, with a view
to achieving parsimony. The model is
called the Generalized Autoregressive
Conditional Heteroscedasticity (GARCH)
model, which models conditional variance
as a function of its lagged values as well as
AYOGU, Ebenezer Chukwuma | 488
squared lagged values of the disturbance
term. Although the GARCH model has
proven useful in capturing symmetric
effects of volatility, it is bedeviled with
some limitations, such as the violation of
non-negativity constraints imposed on the
parameters to be estimated.
To overcome these constraints, some
extensions of the original GARCH model
were proposed. This includes asymmetric
GARCH family models such as Threshold
GARCH (TGARCH) proposed by (Zakoian,
1994).
Exponential GARCH (EGARCH) was
proposed by Nelson (1991) and Power
GARCH (PGARCH) was proposed by Ding
et al (1993). The idea of the proponents of
these models is based on the
understanding that good news (positive
shocks) and bad news (negative shocks) of
the same magnitude have different effects
on conditional variance. The EGARCH
model which captures asymmetric
properties between interest rate and
conditional volatility was proposed to
address major deficiencies of GARCH
model. They are (i) Parameter restrictions
that ensure conditional variance positivity;
(ii) non-sensitivity to asymmetric response
to shocks and (iii) difficulty in measuring
persistence in a strongly stationary series.
The log of the conditional variance in the
EGARCH model signifies that the leverage
effect is exponential and not quadratic.
2. Empirical Review of Related Literature
Several empirical works have been
done since the seminar paper of (Engle,
1982) on volatility modeling, especially in
finance, even though a number of
theoretical issues still remain unresolved
(Franses & McAleer, 2002).
(Olweng, 2011) studied the best fit of
short-term interest rate volatility in Kenya.
The ARCH(q) and GARCH (p,q) models
were considered and the result revealed
that the GARCH (p,q) model is better for
explaining the conditional volatility of
short-term interest rates in Kenya. The
result further indicated the presence of
volatility clustering that exists as a link
between the level of short-term interest
rate and volatility of interest rate in that
country. He recommended that the study
be extended to asymmetric GARCH
models.
in their work to
determine the best fit applied the discrete-
time
GARCH (1,1) and continuous-time
COGARCH (1,1) models to analyze the
interest rate dynamics in Turkish market.
The result shows that the COGARCH (1,1)
model provided the best and excellent
result in modeling interest rate series, as
they capture the characteristics of the
volatility process and yielded a better
conditional volatility estimate than the
discrete-time counterpart GARCH (1,1)
model.
(Okoro et al., 2017) applied two
asymmetric models, EGARCH (1,1) and
GJR-GARCH(1,1) models in the forecasting
of USDNGN Exchange rate in Nigeria,
under error distributions such as the
-
-
distribution, Generalized error distribution
and skewed Generalized error distribution.
The result obtained indicates that all the
models performed fairly well in capturing
the volatility fluctuation of Nigeria
Exchange rate returns with slight
advantage of GED-EGARCH (1,1) and GJR-
489 | Forecasting Interest Rate Volatility In Nigeria In The Arch-Garch Family Models
GARCH (1,1) for the in-sample fit. The two
models have the lowest AIC and the
highest log likelihood values. For out-of-
sample forecasting, the EGARCH (1,1)
analyzed with Generalize Error Distribution
have the minimum MSE and MAE
respectively. The empirical results of the
study however revealed evidence of
leverage effects in USDNGN Exchange rate
return within the period under study.
(Omari-Sasu et al., 2015), studied the
volatility of stock market in Ghana and
employed the GARCH family model to
determine the best model that will best
explain the stock market in that country.
The result shows that the GARCH (1,1)
model was the best fit among others in the
analysis of three equities examined. The
work further revealed that though there is
a presence of volatility, but not persistence
in the three stock markets examined.
(Kosapattarapim et al., 2012), in evaluating
the volatility forecasting performance of
best-fitting GARCH models in emerging
Asian stock markets concluded that out of
six different types of error distributions
employed in the analysis, the GARCH (p,q)
model with non-normal error distribution
tend to provide out-of-sample forecast
performance than a GARCH (p,q) model
analyzes with normal error distribution.
(Tobia, 2011) inferred that there is a
relationship between interest rate and
interest rate volatility in Kenya. The work
further noted that GARCH (1,1) model is
ideal for modeling interest rate volatility in
Kenya compared to other GARCH family
models studied.
(Ahmed & Suliman, 2011), examined
modelling stock market volatility using
GARCH models evidence from Sudan. The
symmetric and asymmetric behavior of the
stock was analyzed and the result revealed
that the conditional variance process was
highly persistent and as such provided
evidence of risk premium for the KSE index
stock series which showed that the
asymmetric model provided a better fit
than the symmetric model, which
confirmed the presence of leverage effect.
(Maqsood et al., 2017) in their work
employed the GARCH model to analyze
the stock market volatility of Nairobi
Securities Exchange (NSE). According to
the report, the GARCH process captured
the symmetric and asymmetric properties
of the models and in agreement with the
work done by inferred that the volatility
process is highly persistent, showing
evidence of risk premium for the NSE index
return series. The report further revealed
that the symmetric model provided a
better fit than the asymmetric model.
(Ahmed & Suliman, 2011), in their
work applied the GARCH models to
forecast the stock market volatility.
Contrary to the works of (Maqsood et al.,
2017) and, inferred that on the basis of out-
of-sample forecasts and a majority of
evaluation measures, the asymmetric
GARCH model performed better in
forecasting conditional variance of the
BSE-SENSEX returns than the symmetric
GARCH model, confirming the presence of
leverage effect. Dedi and Yavas, (2016)
used the Augmented GARCH model to
detect the spillover effect between
markets. Similarly, (Edwards, 1998) and
Zouch, Abbes, and Boujelbene (2011) used
the Augmented GARCH model and
detected the presence of capital
transmission/spillover effect Mexico to
AYOGU, Ebenezer Chukwuma | 490
Chile for the Mexico bonds during 1994
crisis.
MATERIALS AND METHODS
In financial time series, modelling real
data needs proper attention, and suitable
model selection is also required to better
understand the structure of the statistical
data which ultimately helps in better
forecasting. This is because these selected
models are later used for policy-making
whether in finance or economics. The
reason for this care is the non-linear
dynamics present in such data. For
financial data, it is sometimes obvious to
find volatility clusters in a given set of data.
According to (Mandelbrot, 1963) volatility
clustering refers to the observation where
large changes tend to be followed by large
changes of either sign, and small changes
tend to be followed by small changes. In
other words, volatility depends more on
recent past values than distant past values.
1. Model Specification.
Financial markets react nervously to
political disorder, economic crises, wars or
natural disasters. Similarly, following the
rate of inflation, volume of money supply,
exchange rate and other regulatory
instruments employed by Central Bank of
Nigeria (CBN) to ensure sustained
economic growth, the interest rate no
doubt fluctuates at low and high rates at
interval of times. To model time series
volatility, the symmetric ARCH-GARCH
family models shall be employed in the
analysis of this work.
2. Autoregressive Conditional
Heteroscedasticity (ARCH) Model
The ARCH method provides a way to
model a change in variance in a time series
that is time dependent, such as increasing
or decreasing volatility. Autoregressive
therefore describes a feedback mechanism
that incorporates past observations into
the present, that is, the series depends on
its past values. In other words, it implies
that (unequal variance) observed in the
series over different time periods may be
auto-correlated. Conditional implies that
variance is based or depends on past errors
(shocks). ARCH therefore simply conveys
that series in question which has a time-
varying variance that depends on the
lagged effects (autocorrelation). The ARCH
model originally proposed by (Engle, 1982)
is given by;
2
=
(3.1) where
~ (0, 1),
and equation (3.1) is called the mean
equation while the ARCH variance
equation is given by;
(3.2)
~ (0, ) and
is the
dependent variable,
0
is the constant
term,
is the disturbance term,
is the
information set available at time t, q is the
lag length of ARCH model and
s
vectors
of unknown parameters in the variance
equation.
3. Test for ARCH-GARCH Effect.
Though it is assumed that financial and
economic variables change rapidly from
time to time in an apparently unpredictable
manner, it therefore becomes necessary to
determine periods when large changes are
followed by further large changes and
periods when small changes are followed
491 | Forecasting Interest Rate Volatility In Nigeria In The Arch-Garch Family Models
by further small changes, popularly known
as volatility clustering.
An estimation test to determine whether a
particular variable or series is volatile (has
ARCH effect) or not becomes necessary. To
achieve this, a methodology to test for lag
length of ARCH errors using the Lagrange
Multiplier (LM) test was proposed by
(Engle, 1982).
The Lagrange Multiplier (LM) test statistic
is defined as;
LM = T
|
R
2
~
2
(q)
(3.3)
Where T
|
is the number of equations in the
model which fits the residual versus the
lags, that is, T
|
= T-q, where q is the lag
length of the ARCH model. The null
hypothesis is rejected in favour of the
alternative if the p-value is less than one or
five percent level of significance.
4. Generalized Autoregressive
Conditional Heteroscedasticity
(GARCH) Model
The generalized autoregressive conditional
heteroscedasticity (GARCH) model, is an
extension of the ARCH model that
incorporates a moving average
component together with the
autoregressive component. The
introduction of the moving average
component is to allow the model to both
model the conditional change in the
variance over time as well as changes in the
time-dependent variance.
(Bollerslev, 1986)
work by developing a technique that
allows conditional variance to be an ARIMA
process. If we allow the error process to be
such that;
2
=
(3.4)
where
~ (0,1), and
(3.5)
is defined as the generalized ARCH (p, q)
model.
2
is the conditional variance,
0
is
the constant term,
are the coefficients of
the squared error of the ARCH component
while
are the coefficients of the
conditional variance of the GARCH term.
2
1
measures the shock on volatility. The
conditional term compared with the
ARCH(q) model is the forecasted variance
from the previous period given by .In
other words, the GARCH model is a model
that attempts to explain the conditional
volatility using the past lagged squared
errors and the past conditional
variance . However, a typical GARCH
(1,1) model Equation is given by;
(3.6)
5. Non-negativity Constraints and
Stationarity in GARCH (1,1) Model
Brooks (2008) inferred that the values of a
conditional variance must always be strictly
positive; a negative variance at any point in
time would be meaningless. The variable
on the RHS of the conditional variance
equation are all squares of lagged errors,
and so by definition will not be negative. In
order to ensure that these always result in
positive conditional variance, all of the
coefficients in the conditional variance are
required to be non-negative. This non-
negativity therefore implies that
0
> 0,
1
1
. Going by the above, the
stationarity condition of a standard GARCH
model states that
0
> 0,
1
,
1
, and,
1
+
1
< 1. The achievement of these
AYOGU, Ebenezer Chukwuma | 492
conditions implies the model is well-
defined. The unconditional variance under
GARCH model specification indicated that
the conditional variance of
is constant
and given by;
(3.7)
6. Asymmetry Volatility Models
However, neither the ARCH (q) nor the
GARCH (p, q) is able to incorporate the
asymmetry volatility. To adjust for this
condition, several models have been
developed using the GARCH model as their
foundation. Other GARCH family models
employed to determine the Asymmetric
properties of the model and best fit are as
follows:
(A) Exponential Generalized
Autoregressive Conditional
Heteroscedasticity (EGARCH)
The EGARCH model was the first model to
incorporate asymmetry volatility. Empirical
studies have shown that the EGARCH
provides a more accurate result compared
to the conventional symmetric ARCH and
GARCH models, (Alberg. shalit, Yosef,
2008).
(Adeleye, 2019), inferred that news,
incidents, merger of companies,
acquisition of companies, wars, terrorist
attacks, launch of new discoveries,
secession or independence etc, have
strong and powerful influence on the
decision making of financial investors,
hence, have asymmetric impact on
financial investors across the globe.
Hence, the impact of good and bad news
on financial market is asymmetric (not the
same). In asymmetric models, positive
shocks do not have exactly the same
magnitude with negative shock and vice
versa.
The EGARCH variance equation with
normal distribution is hereby stated
(Brooks, 2014).
(3.8)
Where is the intercept for the vaeiance,
is the coefficient for the logged GARCH
term, is logged GARCH term, is
the scale of the asymmetric volatility, is the
last period shock which is standardized,
and parameter that takes into account the
shock. It replaces the regular ARCH term.
The model captures the asymmetry
volatility through the variable gamma ().
The sign of the gamma determines the size
of the asymmetric volatility, and if the
asymmetric volatility is positive or negative
(Brooks, 2014).
If = 0, implies is symmetric or no
asymmetric volatility. If < 0, implies
negative shock will increase the volatility
more than positive shocks. If > 0, implies
that positive shock increases the volatility
more than negative shocks. According to
previous studies in the subject, the
coefficient is often negative, this implies
that negative shocks have more impact on
volatility than positive shocks (good news).
Given that the model uses the log of the
variance (
2
), this means that even if the
parameters are negative, the variance will
still be positive. Therefore, the model is not
subject to the non-negative constraints. (B)
Threshold GARCH Model (TGARCH)
The GJR-GARCH also called Treshold
493 | Forecasting Interest Rate Volatility In Nigeria In The Arch-Garch Family Models
GARCH (TGARCH) model was developed
by Glosten, Jagannathan & Runkle (1993).
The advantage of the model is that the
variance is directly modelled and does not
use the natural logarithm like the EGARCH
model. The main target of the TGARCH
model is to capture asymmetries in terms
of negative and positive shocks. To do that
is simply to add into the variance equation
of the GARCH model a multiplicative
dummy variable
to check whether
there is statistically significant difference
when the shocks are negative. Hence the
conditional variance for a TGARCH or GJR-
GARCH is given by:
(3.9)
However, the form of GJR-GARCH (1, 1) is
given by:
(3.10)
In equation (3.9) above, is the
conditional forecasted variance, is
the intercept for the variance, while
is the variance that depends on previous
lag error term. is the coefficient of
previous period forecasted variance and
is the previous period forecasted
variance. Moreso, is the scale (coefficient)
of the asymmetric volatility. is a
dummy variable that is only activated if the
previous shock is negative ( < 0),
allowing the GJR-GARCH to take the
leverage effect into consideration. Glosten
et al (1993).
From equation (3.10), good news, (positive
shock) and bad news, that is, negative
shock
( < 0) have different impacts on the
conditional variance. A positive shock is
captured by the coefficient , ie have an
-effect on the conditional variance ,
while negative shock (Bad news) has an
effect on the conditional variance
, (volatility). If = 0, the GJR-
GARCH model becomes a linear symmetric
GARCH model but if
If < 0, positive shock will increase
volatility more than
negative shocks.
(C) The Power GARCH (PGARCH) Model
Ding et al (1993), expressed conditional
variance using PGARCH (p,d,q) as;
(3.11)
Here, d > 0 and establishes the
existence of leverage effects. If d is set at 2,
the PGARCH (p , q) replicates a GARCH(p ,
q) with a leverage effect. If d is set at 1, the
standard deviation is modelled. The first
order of the above PGARCH equation is
PGARCH (1, d, 1) expressed as:
(3.12)
The failure to accept the null hypothesis
that shows the presence of leverage
effect. The impact of news on volatility in
PGARCH is similar to that of TGARCH when
d is 1
6. Volatility Transmission (Spillover Effect)
The transmission of shocks from one
AYOGU, Ebenezer Chukwuma | 494
market or variable to another was well
documented by (Ewing, 2002). Co-
movement across volatilities (co-volatility)
due to common information that
simultaneously affect expectations and
information spillover caused by cross-
market hedging are some of the reasons
for volatility transmissions. In addition to
endogenous events or variables,
exogenous variables, deterministic events
(macroeconomic announcements) may all
have influence on the volatility process. To
determine volatility transmission
(Spillover) between variables (markets), we
use the Augmented GARCH model as
developed by (Duan, 1997). The model is
defined as follows:
(3.13)
Where
is the residual squared error of
ARMA model and the term that measures
the magnitude of volatility spillover
(transmission) across the variables
(markets). Two variables, namely, exchange
and inflation rates were introduced to
determine whether they have any spillover
effect on the conditional volatility of
interest rate in Nigeria within the period
under review.
7. Model Selection Criteria
Finding optimal of a model that will fit a
particular data set has always been
favourite for researchers. Reinhard and
Lunde (2001), inferred that there is not a
unique criterion for selecting the best
model, rather it depends on preferences,
example, expressed in terms of a utility
function or loss function. The standard
model selection criteria of (Akaike et al.,
1973) and Schwartz are often applied.
Bieren, H, J (2006) recommended the
following modification, if the model
includes ARCH type errors.
log(
2
) + 2
log(2) (3.14)
Shibata (1976) showed through empirical
evidence that AIC has the tendency to
choose models which are over
parameterized. Various modifications have
been produced to overcome this lack of
consistency. (Schwarz, 1978) developed a
consistent criterion for models defined in
terms of their posterior probability
(Bayesian approach) which is given by;
(
2
) +
() (3.15)
Where is the estimated model error
variance, k is the number of free
parameters in the model, n is the number
of observations. In ARCH context, this form
will look like;
(3.16)
8. Loss Functions-Measure of Forecast
Performance
Although in literature, several methods of
measuring the performance conditional
variance,
(Liu, 2009) inferred that the Root Mean
Square Error (RMSE), Mean Absolute Error
(MAE), Mean Absolute Percent Error
(MAPE) and Theil Inequality Coefficient
(TIC) are most appropriate in determining
the best forecast performance. (Clements,
2005), on the predictive ability of volatility
models proposes that out-of-sample
forecasting ability remains the criterion for
selecting the best predictive model, hence
495 | Forecasting Interest Rate Volatility In Nigeria In The Arch-Garch Family Models
shall be adopted in this study. If
2
and
represents the actual and forecasted
volatility of interest rate at time t, then;
(3.17)
(3.18)
(3.19)
(3.20)
The volatility model with the least RMSE,
MAE, MAPE and TIC statistic is the best
forecasting model.
9. Distribution of Errors
t
As far as error distribution is concerned,
GARCH model theory suggests three
assumptions about the distribution of
residuals. These three assumptions may
follow normal law, a student law or a
generalized Error distribution (GED). In this
work three different distributions namely;
-
Distribution and the Generalize Error
distribution were employed to determine
the best fit and forecast of the conditional
variance. The Normal distribution:
2
(3.21)
Student t-distribution:
+1)
< (3.22)
Generalized Error Distribution:
(3.23)
V > 0 is the degree of freedom or the tail-
tickness.
10. Mean Reversion
(John et al., 2019), stated that mean
reversion means that current information
has no influence on the long run forecast
of the volatility. Persistence dynamics in
volatility is generally captured in the
GARCH coefficients of a stationary GARCH-
type model. In stationary GARCH-type
models, the volatility mean reverts to its
long-run level, at a rate given by sum of
ARCH and GARCH coefficients, which is
usually close to one (1) for financial time
series. The average number of time periods
for the volatility to revert to its long run
level is measured by the half-life of the
volatility shock. The mean reverting form
of the GARCH (1,1) model is given by;
2
2
= ( + )(
2
1
2
) +
1
+
1
+
(3.24)
Where , is the
unconditional long-run level of volatility
and . The magnitude of the
mean reverting rate + (speed of
adjustment) controls the speed of the
mean reversion.
𝑅𝑀𝑆𝐸
=
√
1
𝑇
∑
(
𝜎
𝑡
2
𝑇
𝑖
=
1
−
𝜎
𝑡
2
)
AYOGU, Ebenezer Chukwuma | 496
RESULTS AND DISCUSSION
In financial time series, modelling real
data needs proper attention, and suitable
model selection is also required to better
understand the structure of the statistical
data which ultimately helps in better
forecasting. This is because these selected
models are later used for policy making
whether in finance or economics. The
reason for this care is the non-linear
dynamics present in such data. For
financial data, it is sometimes obvious to
find volatility clusters in a given set of data.
According to (Mandelbrot, 1963) volatility
clustering refers to the observation where
large changes tend to be followed by large
changes of either sign, and small changes
tend to be followed by small changes. In
other words volatility depends more on
recent past values than distant past values.
1. Model Specification.
Financial markets react nervously to
political disorder, economic crises, wars, or
natural disasters. Similarly, following the
rate of inflation, volume of money supply,
exchange rate and other regulatory
instruments employed by the Central Bank
of Nigeria (CBN) to ensure sustained
economic growth, the interest rate no
doubt fluctuates at low and high rates at
intervals of times. To model time series
volatility, the symmetric ARCH-GARCH
family models shall be employed in the
analysis of this work.
2. Autoregressive Conditional
Heteroscedasticity (ARCH) Model
The ARCH method provides a way to
model a change in variance in a time series
that is time-dependent, such as increasing
or decreasing volatility. Autoregressive,
therefore, describes a feedback mechanism
that incorporates past observations into
the present, that is, the series depends on
its past values. In other words, it implies
that (unequal variance) observed in the
series over different time periods may be
auto-correlated. Conditional implies that
variance is based or depends on the past
errors (shocks). ARCH therefore simply
conveys that series in question which has
time-varying variance that depends on the
lagged effects (autocorrelation). The ARCH
model originally proposed by (Engle, 1982)
is given by;
2
=
(3.1) where
~ (0, 1),
and equation (3.1) is called the mean
equation while the ARCH variance
equation is given by;
(3.2)
~ (0, ) and
is the
dependent variable,
0
is the constant
term,
is the disturbance term,
is the
information set available at time t, q is the
lag length of ARCH model and
s
vectors
of unknown parameters in the variance
equation.
3. Test for ARCH-GARCH Effect.
Though it is assumed that financial
and economic variables change rapidly
from time to time in an apparently
unpredictable manner, it therefore
becomes necessary to determine periods
when large changes are followed by
further large changes and periods when
small changes are followed by further small
changes, popularly known as volatility
clustering.
497 | Forecasting Interest Rate Volatility In Nigeria In The Arch-Garch Family Models
An estimation test to determine whether a
particular variable or series is volatile (has
ARCH effect) or not becomes necessary. To
achieve this, a methodology to test for lag
length of ARCH errors using the Lagrange
Multiplier (LM) test was proposed by
(Engle, 1982).
The Lagrange Multiplier (LM) test statistic
is defined as;
LM = T
|
R
2
~
2
(q)
(3.3)
Where T
|
is the number of equations in the
model which fits the residual versus the
lags, that is, T
|
= T-q, where q is the lag
length of the ARCH model. The null
hypothesis is rejected in favour of the
alternative if the p-value is less than one or
five percent level of significance.
4. Generalized Autoregressive
Conditional Heteroscedasticity
(GARCH) Model
Generalized autoregressive
conditional heteroscedasticity (GARCH)
model, is an extension of the ARCH model
that incorporates a moving average
component together with the
autoregressive component. The
introduction of the moving average
component is to allow the model to both
model the conditional change in the
variance over time as well as changes in the
time dependent variance.
(Bollerslev, 1986) extended
work by developing a technique that
allows the conditional variance to be an
ARIMA process. If we allow the error
process to be such that;
2
=
(3.4)
where
~ (0,1), and
(3.5)
is defined as the generalized ARCH (p, q)
model.
2
is the conditional variance,
0
is
the constant term,
are the coefficients of
the squared error of the ARCH component
while
are the coefficients of the
conditional variance of the GARCH term.
2
1
measures the shock on volatility. The
conditional term compared with the
ARCH(q) model is the forecasted variance
from the previous period given by .In
other words, the GARCH model is a model
that attempts to explain the conditional
volatility using the past lagged squared
errors and the past conditional
variance . However, a typical GARCH
(1,1) model Equation is given by;
(3.6)
5. Non-negativity Constraints and
Stationarity in GARCH (1,1) Model
Brooks (2008) inferred that the values
of a conditional variance must always be
strictly positive; a negative variance at any
point in time would be meaningless. The
variable on the RHS of the conditional
variance equation are all squares of lagged
errors, and so by definition will not be
negative. In order to ensure that these
always result in positive conditional
variance, all of the coefficients in the
conditional variance are required to be
non-negative. This non-negativity
therefore implies that
0
> 0,
1
1
0. Going by the above, the stationarity
condition of a standard GARCH model
states that
0
> 0,
1
,
1
and,
1
+
1
< 1. The achievement of these
conditions implies the model is well
defined. The unconditional variance under
AYOGU, Ebenezer Chukwuma | 498
GARCH model specification indicated that
the conditional variance of
is constant
and given by;
(3.7)
6. Asymmetry Volatility Models
However, neither the ARCH (q) nor the
GARCH (p, q) is able to incorporate the
asymmetry volatility. To adjust for this
condition, several models have been
developed using the GARCH model as their
foundation. Other GARCH family models
employed to determine the Asymmetric
properties of the model and best fit are as
follows:
(A) Exponential Generalized
Autoregressive Conditional
Heteroscedasticity (EGARCH)
The EGARCH model was the first
model to incorporate asymmetry volatility.
Empirical studies have shown that the
EGARCH provides a more accurate result
compared to the conventional symmetric
ARCH and GARCH models, (Alberg shalit,
Yosef, 2008).
(Adeleye, 2019), inferred that news,
incidents, merger of companies,
acquisitions of companies, wars, terrorist
attacks, launch of new discoveries,
secession or independence etc, have
strong and powerful influence on the
decision making of financial investors,
hence, have asymmetric impact on
financial investors across the globe.
Hence, the impact of good and bad
news on financial market is asymmetric
(not the same). In asymmetric models,
positive shocks do not have exactly the
same magnitude with negative shock and
vice versa.
The EGARCH variance equation with
normal distribution is hereby stated
(Brooks, 2014).
(3.8)
Where is the intercept for the vaeiance,
is the coefficient for the logged GARCH
term, is logged GARCH term, is
the scale of the asymmetric volatility, is the
last period shock which is standardized,
and parameter that takes into account the
shock. It replaces the regular ARCH term.
The model captures the asymmetry
volatility through the variable gamma ().
The sign of the gamma determines the size
of the asymmetric volatility, and if the
asymmetric volatility is positive or negative
(Brooks, 2014).
If = 0, implies is symmetric or no
asymmetric volatility. If < 0, implies
negative shock will increase the volatility
more than positive shocks. If > 0, implies
that positive shock increases the volatility
more than negative shocks. According to
previous studies in the subject, the
coefficient is often negative, this implies
that negative shocks have more impact on
volatility than positive shocks (good news).
Given that the model uses the log of the
variance (
2
), this means that even if the
parameters are negative, the variance will
still be positive. Therefore, the model is not
subject to non-negative constraints. (B)
Threshold GARCH Model (TGARCH).
The GJR-GARCH also called Treshold
GARCH (TGARCH) model was developed
499 | Forecasting Interest Rate Volatility In Nigeria In The Arch-Garch Family Models
by Glosten,
Jagannathan & Runkle (1993). The
advantage of the model is that the variance
is directly modelled and does not use the
natural logarithm like the EGARCH model.
The main target of the TGARCH model is to
capture asymmetries in terms of negative
and positive shocks. To do that is simply to
add into the variance equation of the
GARCH model a multiplicative dummy
variable
to check whether there is
statistically significant difference when the
shocks are negative. Hence the conditional
variance for a TGARCH or GJR-GARCH is
given by:
(3.9)
However, the form of GJR-GARCH (1, 1) is
given by:
(3.10)
In equation (3.9) above, is the
conditional forecasted variance, is
the intercept for the variance, while
is the variance that depends on previous
lag error term. is the coefficient of
previous period forecasted variance and
is the previous period forecasted
variance. Moreso, is the scale (coefficient)
of the asymmetric volatility. is a
dummy variable that is only activated if the
previous shock is negative ( < 0),
allowing the GJR-GARCH to take the
leverage effect into consideration. Glosten
et al (1993).
From equation (3.10), good news, (positive
shock) and bad news, that is, negative
shock.
( < 0) have different impacts on the
conditional variance. A positive shock is
captured by the coefficient , ie have an
-effect on the conditional variance ,
while negative shock (Bad news) has an
effect on the conditional
variance , (volatility). If = 0, the GJR-
GARCH model becomes a linear symmetric
GARCH model but if
If < 0, positive shock will increase
volatility more than
negative shocks.
(C) The Power GARCH (PGARCH) Model
Ding et al (1993), expressed conditional
variance using PGARCH (p,d,q) as;
(3.11)
Here, d > 0 and establishes the
existence of leverage effects. If d is set at 2,
the PGARCH (p, q) replicates a GARCH(p ,
q) with a leverage effect. If d is set at 1, the
standard deviation is modelled. The first
order of the above PGARCH equation is
PGARCH (1, d, 1) expressed as:
(3.12)
The failure to accept the null hypothesis
that shows the presence of leverage
effect. The impact of news on volatility in
PGARCH is similar to that of TGARCH when
d is 1.
7. Volatility Transmission (Spillover Effect)
The transmission of shocks from one
market or variable to another was well
documented by (Ewing, 2002). Co-
movement across volatilities (co-volatility)
AYOGU, Ebenezer Chukwuma | 500
due to common information that
simultaneously affect expectations and
information spillover caused by cross-
market hedging are some of the reasons
for volatility transmissions. In addition to
endogenous events or variables,
exogenous variables, deterministic events
(macroeconomic announcements) may all
have influence on the volatility process. To
determine volatility transmission
(Spillover) between variables (markets), we
use the Augmented GARCH model as
developed by (Duan, 1997). The model is
defined as follows:
(3.13)
Where
is the residual squared error of
ARMA model and the term that measures
the magnitude of volatility spillover
(transmission) across the variables
(markets). Two variables, namely, exchange
and inflation rates were introduced to
determine whether they have any spillover
effect on the conditional volatility of
interest rate in Nigeria within the period
under review.
8. Model Selection Criteria
Finding optimal of a model that will fit a
particular data set has always been
favourite for researchers. Reinhard and
Lunde (2001), inferred that there is not a
unique criterion for selecting the best
model, rather it depends on preferences,
example, expressed in terms of a utility
function or loss function. The standard
model selection criteria of (Akaike, 1974)
are often applied. Bieren, H, J (2006)
recommended the following modification,
if the model includes ARCH type errors.
log(
2
) + 2
log(2) (3.14)
Shibata (1976) showed through empirical
evidence that AIC has the tendency to
choose models which are over
parameterized. Various modifications have
been produced to overcome this lack of
consistency. (Schwarz, 1978) developed a
consistent criterion for models defined in
terms of their posterior probability
(Bayesian approach) which is given by;
(
2
) +
() (3.15)
Where is the estimated model error
variance, k is the number of free
parameters in the model, n is the number
of observations. In ARCH context, this form
will look like;
(3.16)
9. Loss Functions-Measure of Forecast
Performance
Although in literature, several methods of
measuring the performance conditional
variance,
(Liu, 2009) inferred that the Root Mean
Square Error (RMSE), Mean Absolute Error
(MAE), Mean Absolute Percent Error
(MAPE) and Theil Inequality Coefficient
(TIC) are most appropriate in determining
the best forecast performance. (Clements,
2005), on the predictive ability of volatility
models proposes that out-of-sample
forecasting ability remains the criterion for
selecting the best predictive model, hence
shall be adopted in this study. If
2
and
represents the actual and forecasted
volatility of interest rate at time t, then;
(3.17)
𝑅𝑀𝑆𝐸
=
√
1
𝑇
∑
(
𝜎
𝑡
2
𝑇
𝑖
=
1
−
𝜎
𝑡
2
)
501 | Forecasting Interest Rate Volatility In Nigeria In The Arch-Garch Family Models
(3.18)
(3.19)
(3.20)
The volatility model with the least RMSE,
MAE, MAPE and TIC statistic is the best
forecasting model.
10. Distribution of Errors
t
As far as error distribution is concerned,
GARCH model theory suggests three
assumptions about the distribution of
residuals. These three assumptions may
follow normal law, a student law or a
generalized Error distribution (GED). In this
work three different distributions namely;
-
Distribution and the Generalize Error
distribution were employed to determine
the best fit and forecast of the conditional
variance. The Normal distribution:
2
(3.21)
Student t-distribution:
+1)
< (3.22)
Generalized Error Distribution:
(3.23)
V > 0 is the degree of freedom or the tail-
tickness.
11. Mean Reversion
(John et al., 2019), stated that mean
reversion means that current information
has no influence on the long run forecast
of the volatility. Persistence dynamics in
volatility is generally captured in the
GARCH coefficients of a stationary GARCH-
type model. In stationary GARCH-type
models, the volatility mean reverts to its
long-run level, at a rate given by sum of
ARCH and GARCH coefficients, which is
usually close to one (1) for financial time
series. The average number of time periods
for the volatility to revert to its long run
level is measured by the half-life of the
volatility shock. The mean reverting form
of the GARCH(1,1) model is given by;
2
2
= ( + )(
2
1
2
) +
1
+
1
+
(3.24)
Where , is the unconditional
long-run level of volatility and
. The magnitude of the
mean reverting rate + (speed of
adjustment) controls the speed of the
mean reversion.
CONCLUSIONS
The target variable (interest rate) is the
bank's lending rate. A total of forty-two
models were estimated, with twenty-four
symmetric and eighteen asymmetric
models. The GED-GARCH (1,1) model
emerged as the best fit, predicting that
AYOGU, Ebenezer Chukwuma | 502
unconditional variance (homoscedasticity)
would be achieved in the third month of
the following year (March 2019). However,
two independent variables, exchange and
inflation rates, that were incorporated as
external factors, were discovered to have
an influence on the conditional variance of
interest rates in Nigeria within the period
under review. The interest rate in Nigeria is
indeed volatile and the rate of decay of the
shocks is very slow. The volatility is
persistent and, as such, the best GARCH
family model to adopt in analyzing the
volatility of interest rates in Nigeria
remains the symmetric GARCH (1,1) while
the best error distribution is the
generalized error distribution (GED).
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