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IOSR Journal of Economics and Finance (IOSR-JEF)
e-ISSN: 2321-5933, p-ISSN: 2321-5925.Volume 8, Issue 1 Ver. I (Jan-Feb. 2017), PP 15-26
www.iosrjournals.org
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 15 | Page
Modeling USD/KES Exchange Rate Volatility using
GARCH Models
Cyprian Ondieki Omari1*, Peter Nyamuhanga Mwita2, Antony Gichuhi Waititu2
1Department of Statistics and Actuarial Science, Dedan Kimathi University of Technology, Nyeri, Kenya
2Department of Statistics and Actuarial Sciences Jomo Kenyatta University of Agriculture and Technology,
Nairobi, Kenya
Abstract: In this paper the generalized autoregressive conditional heteroscedastic models are applied in
modeling exchange rate volatility of the USD/KES exchange rate using daily observations over the period
starting 3rd January 2003 to 31st December 2015. The paper applies both symmetric and asymmetric models that
capture most of the stylized facts about exchange rate returns such as volatility clustering and leverage effect.
The performance of the symmetric GARCH (1, 1) and GARCH-M models as well as the asymmetric EGARCH
(1, 1), GJR-GARCH (1, 1) and APARCH (1, 1) models with different residual distributions are applied to data.
The most adequate models for estimating volatility of the exchange rates are the asymmetric APARCH model,
GJR-GARCH model and EGARCH model with Student’s t-distribution.
Keywords: GARCH Models, Volatility clustering, forecasting volatility, Leverage effect, Value-at-Risk
I. Introduction
In the last decade the foreign exchange market has become the most volatile and liquid in all financial
markets in the world. Particularly because of the dynamics of the foreign exchange market, it is essential to
study some of the important historical events relating to currencies and currency exchange. The modeling and
forecasting exchange rates volatility has important implications in a range of areas in macroeconomics and
finance. Value-at-Risk (VaR) is a risk measurement tool based on loss distributions. The Basel III framework
developed by the Basel Committee on Banking Supervision requires that financial institutions such as banks and
investment firms set aside a minimum amount of capital to cover potential losses from their exposure to credit
risk, operational risk and market risk. For measuring market risk they recommend the use of VaR, which is the
worst loss in an asset or a portfolio of assets over a given time horizon at a given confidence level. Inaccurate
portfolio VaR estimates may lead firms to maintain insufficient risk capital reserves so that they have an
inadequate capital cushion to absorb large financial shocks.
A number of models have been developed in empirical finance literature to investigate volatility across
different regions and countries. The most commonly applied models to estimate exchange rate volatility are the
autoregressive conditional heteroscedastic (ARCH) model introduced by Engle (1982) and the generalized
(GARCH) models developed independently by Bollerslev (1986) and Taylor (1986). The purpose of the
autoregressive conditional heteroscedasticity (ARCH) model is to estimate the conditional variance of a time
series. Engle described the conditional variance by a simple quadratic function of its lagged values. Bollerslev
(1986) extended the basic ARCH model and described the conditional variance by its own lagged values and the
square of the lagged values of the innovations or shocks. In many cases, the basic GARCH model provides a
reasonably good model for analyzing financial time series and estimating conditional volatility. However,
GARCH models have been criticized in that they do not provide a theoretical explanation of volatility or what
information flows are in the volatility generating process according to Tsay (2010). The GARCH model also
responds equally to asymmetric shocks, and cannot cope with significantly skewed time series which results in
biased estimates. Another problem encountered when using GARCH models is that they do not always fully
embrace the heavy tails property of high frequency financial time series. To overcome this drawback Bollerslev
et al. (1987) used the Student's t distributions. The GARCH extensions such as Exponential GARCH, Threshold
GARCH, GJR-GARCH model and power GARCH models have been proposed to address some of these
weaknesses. Nelson (1991) formulated the Exponential GARCH (EGARCH) model by extending the GARCH
model to capture news in the form of leverage effects. Afterwards, the GARCH model extension was developed
to test for this asymmetric news impact (Glosten et al., 1993; Zakoian, 1994). Ding et al. (1993) extensions nest
a number of models from the ARCH family. The GARCH family models capture heteroscedasticity and
volatility clustering in financial data.
The main objective of this paper is to model exchange rate return volatility for USD/KES, by applying
different univariate specifications of GARCH type models for daily observations of the exchange rate return
series for the period 3rd January 2003 to 31st December 2015. The volatility models applied in this paper include
the GARCH (1, 1), GARCH-M (1, 1), E-GARCH (1, 1), GJR-GARCH (1, 1), and Power GARCH (1, 1). The
Modeling USD/KES Exchange Rate Volatility using GARCH Models
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 16 | Page
relative performance of the symmetric and asymmetric GARCH family models in estimating and forecasting
Value-at-Risk using the USD/KES exchange rates is also tested.
The remainder of this paper is organized as follows. Section 2 provides the overview of symmetric and
asymmetric GARCH family models used throughout the paper. Section 3 describes the data and empirical
results and finally, Section 4 concludes the paper.
II. Methodology
The traditional methods of measuring volatility (variance or standard deviation) are unconditional and
cannot capture the characteristics exhibited by financial time series data, such as, time varying volatility,
volatility clustering, excess kurtosis, heavy tailed distribution, leverage effect and long memory properties. The
most commonly used models that capture these properties of financial time series data are the Autoregressive
Conditional Heteroskedasticity (ARCH) model and its generalization, the Generalized Autoregressive
Conditional Heteroskedasticity (GARCH) models. In this paper different univariate GARCH specifications are
applied to model USD/KES exchange rate return volatility and these models are GARCH (1, 1), GARCH-M (1,
1), EGARCH (1, 1), GJRGARCH (1, 1) and APARCH (1, 1). In presenting these different models, there are two
distinct equations or specifications, the first the conditional mean and the conditional variance which are briefly
reviewed in this methodology.
2.1 Conditional Mean Equation
The exchange rate return moving pattern might be an autoregressive (AR) process, moving average
(MA) process or a combination of AR and MA processes i.e. (ARMA) process. For the purposes of this study
the mean equation is modified to include appropriate AR and MA terms to control for autocorrelation in the
data. For example, in ARMA (1, 1) process pattern would be:
jt
q
jjtit
p
iit YY
11
(1)
where
t
Y
is a time series being modeled.
2.2 Volatility Modeling
The existing models of volatility can be divided into two main categories, symmetric and asymmetric
models. In the symmetric models, the conditional variance only depends on the magnitude, and not the sign, of
the underlying asset, while in the asymmetric models the positive or negative shocks of the same magnitude
have different effect on future volatility.
2.3 Symmetric GARCH Models
2.3.1 Generalized Autoregressive Conditional Heteroscedasticity (GARCH) Model
In this model, the conditional variance is represented as a linear function of its own lags. The general
form of the GARCH (p, q) model is given by:
2
1
2
1
2
,
jt
q
jjit
p
iit
ttttt
y
yyr
(2)
where
t
r
are the logarithm returns of the financial time series at time t,
are mean value of the returns,
t
y
are
the error terms (innovations) from the mean equation, and it can split into a stochastic piece
t
and a time
dependant standard deviation
t
characterizing the typical size of the terms.
t
is a zero mean, identical and
independent distribution, which is assumed to have normal distribution, t distribution and skew t distribution and
,0,0,0 ji
with constrains that
1
11
q
jj
p
ii
. In most empirical applications the basic
GARCH (1, 1) model fits the changing conditional variance of the majority of financial time series reasonably
well. The GARCH (1, 1) model is given by the following equation:
211
211
2 ttt y
(3)
To guarantee a positive variance at all instances, the following restricts are imposed
0
and
.0, 11
Modeling USD/KES Exchange Rate Volatility using GARCH Models
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 17 | Page
In many cases, the basic GARCH model provides a reasonably good model for analyzing financial time series
and estimating conditional volatility. However, there are some aspects of the model which can be improved so
that it can better capture the characteristics and dynamics of a particular financial time series. Ever since the
GARCH (p, q) model was introduced by Bolleslev (1986), new GARCH extension models which address the
different weaknesses of the GARCH model and capture different characteristics of the financial time series data
have been invented.
2.3.2 The GARCH-in-Mean (GARCH-M) Model
In finance, high risk is often expected to lead to high returns. To model such a phenomenon one may
consider the GARCH-M Model of Engle, Lilien, and Robins (1987) where “M” stands for GARCH in the mean.
This model is an extension of the basic GARCH framework which allows the conditional mean of a sequence to
depend on its conditional variance or standard deviation. A simple GARCH-M (1, 1) model is given by:
211
211
2
22 ,0~,,
ttt
tttttttt
y
Nyyr
(4)
where
and
are constants. The parameter
is called the risk premium parameter. A positive
indicates that
the return is positively related to its volatility.
2.4 Asymmetric GARCH Models
In practice, the price of financial assets often reacts more pronouncedly to “bad” news than “good’
news. Such a phenomenon leads to a so called leverage effect, as first noted by Black (1976). The term
“leverage” stems from the empirical observation that the volatility (conditional variance) of a stock tends to
increase when its returns are negative. The leverage effect causes the asymmetries of variance dynamics and
points out the drawbacks of GARCH model because of its symmetric effect towards the conditional variance. In
order to capture the asymmetry in return volatility (“leverage effect”), a new class of models was developed,
termed the asymmetric GARCH models. This paper uses the following asymmetric GARCH models; EGARCH
GJR-GARCH and Asymmetric Power ARCH (APARCH) model for capturing the asymmetric phenomena.
2.4.1 The Exponential GARCH (E-GARCH) Model
The general form of the Exponential GARCH (p, q) model introduced by Nelson (1991) is given by
2
11
2ln)(ln jt
q
jj
it
it
i
it
it
p
iit yy
(5)
where
is the asymmetric response parameter that can take a positive or negative sign depending on the effect
of the future uncertainty. The simplest form is the EGARCH (1, 1) model, which is given by:
211
1
1
1
1
1
1
2lnln
t
t
t
t
t
tyy
(6)
For a positive shock
,0
1
1
t
t
y
the equation becomes
211
1
1
11
2ln)(ln
t
t
t
ty
(7)
whereas for a negative shock
,0
1
1
t
t
y
the equation becomes
211
1
1
11
2ln)(ln
t
t
t
ty
(8)
2.4.2 The Glosten, Jagannathan and Runkle GARCH (GJR-GARCH) Model
The GJR-GARCH model is another type of asymmetric GARCH models, which was proposed by
Glosten, Jagannathan and Runkle (1993). The variance equation of GJR-GARCH (p, q) is given by
ititijt
q
jjit
p
iit yIy
2
11
2
(9)
Modeling USD/KES Exchange Rate Volatility using GARCH Models
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 18 | Page
where
,
and
are constant parameters, and
I
is a dummy variable (indicator function) that takes the value
zero (respectively one) when
it
y
is positive (negative). If
is positive, negative errors are leveraged (negative
innovations or bad news has a greater impact than the positive ones). The parameters of the model are assumed
to be positive and that
12/
. If all leverage coefficients are zero, then GJR-GARCH model reduces
to GARCH model. This means one can test a GARCH model against a GJR-GARCH model using the likelihood
ratio test.
2.4.3 The Power GARCH (PGARCH) Model
Ding, Engle and Granger (1993) introduced the asymmetric power ARCH model also called APARCH
(p, q) specification to deal with asymmetry. The variance equation of APARCH (p, q) can be written as
jt
q
jjitiit
p
iit yy
11 )(
(10)
where
qjpi jii ,,1,0,,,1,11,0,0,0
.
i
and
i
are the standard
ARCH and GARCH parameters,
i
are the leverage parameters and
is the parameter for the power term. The
symmetric model sets
0
i
for all i. When
,2
Equation (10) becomes a classic GARCH model that allows
for leverage effects and when
,1
the conditional standard deviation will be estimated. In addition, we can
increase the flexibility of the APARCH model by considering
as another coefficient that must also be
estimated.
In this paper, conditional volatility is estimated using the probability distributions that are available in
the rugarch package which include; normal, Student t and skewed Student t-distribution. Engle (1982) assumed
that asset returns follow a normal distribution. However, the asset returns are not normally distributed, so the
normality assumption could cause significant bias in VaR estimation and could underestimate the volatility. A
number of authors evidenced that standard GARCH models with normal empirical distributions have inferior
forecasting performance compared to models that reflect skewness and kurtosis in innovations. To capture the
excess kurtosis in financial asset returns, Bollerslev (1987) introduced the GARCH model with a standardized
Student’s t distribution with
2
degrees of freedom.
The common methodology used for GARCH estimation is maximum likelihood assuming i.i.d. innovations. The
parameters of the GARCH model can be found by maximizing the objective log-likelihood function:
n
ttt zL 1
22 )()(ln)2ln(
2
1
)(ln
(11)
where
is the vector of parameters
),,,( ji
estimated that maximize the objective function
t
zL );(ln
represents the standardized residual calculated as
2
t
t
y
.
Maximum likelihood estimates of the parameters are obtained by numerical maximization of the log-likelihood
function using the Marquardt algorithm (Marquardt (1963)). We use the quasi-maximum likelihood estimator
(QMLE) since, according to Bollerslev and Wooldridge (1992), it is generally consistent, has a normal limiting
distribution and provides asymptotic standard errors that are valid under non-normality.
For the GARCH (p, q) model the one-step-ahead conditional variance forecast,
tt |1
ˆ
is:
21
1
21
1
2|1
ˆ
jt
q
jjit
p
iitt y
(12)
For the EGARCH (p, q) model, we get:
21
1
1
1
1
1
1
2lnln
jt
q
jj
it
it
i
it
it
p
iit yy
(13)
Note that the value of
|| t
zE
depends on the density function of
t
z
. For example, for the standard normal
distribution,
/2||
t
zE
, for the Student t-distribution,
Modeling USD/KES Exchange Rate Volatility using GARCH Models
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 19 | Page
,
)2/()1(1
22/)1(2
||
vv
vv
zE t
However, the quantity
,
1
11
it
it
y
both with and without the absolute value operator, for
,1i
can be
computed by the model as the values of both the innovation and its conditional standard deviation are available.
Finally, the corresponding one-step ahead conditional variance forecast in the case of the GJR-GARCH (p, q)
model is:
11
21
1
1
1
2|1
ititijt
q
jjit
p
iitt yIy
(14)
Therefore, to compute the one-step-ahead VaR forecast under all distributions, we compute the corresponding
quantiles, which are then, multiply by the conditional standard deviation forecast, hence;
,
ˆ
)( |1|1 tttt FVaR
(15)
where
)(
F
is the corresponding quantile of the assumed distribution, and
tt |1
ˆ
is the forecast of conditional
standard deviation at time t.
According to Tsay (2010), if one further assumes that
t
z
is Gaussian, then the conditional distribution of
1t
r
given the information available at time t is
tttt
rN |1|1 ˆ
,
ˆ
. Quantiles of this conditional distribution can easily
be obtained for VaR calculation. For example, the 5% quantile is
.
ˆ
645.1
ˆ|1|1 tttt
r
Therefore, if
t
z
of the GARCH model in Equation (12) is a standardized Student-t distribution with v degrees of
freedom and the probability is p, then the quantile used to calculate the 1-period horizon VaR at time index t is
)2/(
ˆ
)(
ˆ|1
|1
vv
pt
rttv
tt
where
)(ptv
is the p-th quantile of a Student-t distribution with v degrees of freedom.
III. Empirical Results
3.1 Data
The data set consists of the daily currency exchange rate of the US Dollar versus Kenyan Shilling
(USD/KSH). These data are obtained from Central Bank of Kenya (CBK) website, (www.cbk.co.ke). The data
set was for the period from January 3, 2003 to December 31, 2015, a total of 2818 observations. A visual
inspection of Figure 1 shows that daily USD/KES exchange rate prices are not stationary. In order to test for
stationarity an Augmented Dickey–Fuller test (ADF) for a unit root in a time series sample is performed. The
computed ADF test-statistic in Table 1 is (-3.0) which greater than the critical values at one per cent significance
level. Therefore, we fail to reject the null hypothesis that there is a unit root and that the series needs to be
differenced in order to make it stationary.
Table 1: Augmented Dickey-Fuller test of the daily returns
The currency exchange rates are then transformed into daily log returns using the following returns formula:
100log
1
t
t
tP
P
r
(16)
Modeling USD/KES Exchange Rate Volatility using GARCH Models
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 20 | Page
where
t
P
is the daily closing value of the USD/KES exchange rate on day t.
Figure 1: Daily USD/KES Currency Exchange Rates
A plot of the log returns series for USD/KES exchange rates given in Figure 2 shows periods of high
volatility, occasional extreme movements and volatility clustering, as upward movements tend to be followed by
other upward movements and downward movements also followed by other downward movements. This
indicates that the logarithm of USD/KES exchange rates is stationary after taking the first-difference, and the
ADF test results in Table 2 confirm the stationarity of the return series data. The computed ADF test-statistic in
Table 2 is (-10.0) which smaller than the critical values at 5% significance level.
Table 2: Augmented Dickey-Fuller test of the daily returns
Figure 2: Daily Logarithmic Returns of the USD/KES Currency Exchange Rates
A summary of the statistics of the return series data is given in Table 3. The mean is positive,
suggesting that exchange returns increase slightly over time. The coefficient of skewness indicates that returns
have asymmetric distribution, i.e., they are skewed to the left. The kurtosis of returns is 73.6776 which is greater
than three, indicating that the distribution of returns follows a fat-tailed distribution, thereby exhibiting one of
the important characteristics of financial time series data, namely that of leptokurtosis. The non-normality
Modeling USD/KES Exchange Rate Volatility using GARCH Models
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 21 | Page
condition is supported by a Jarque-Bera test which shows that the null hypothesis of normality is rejected at the
five per cent level of significance.
Table 3: Summary Statistics for the returns of USD/KES exchange rates
Number of Observations
2817
Max.
9.5776
Min.
-9.4458
Mean
0.0069
Median
0.0053
Variance
0.3054
Std.Dev
0.5527
Skewness
-0.0736
Kurtosis
73.6776
Jarque-Bera
638102.8932
JB p-value
0.0000
The Ljung-Box test is applied to the daily log returns of the USD/KES exchange rates and the test
results are shown in Table 4. The null hypothesis of the Ljung-Box is rejected for the returns, squared returns
and absolute returns, at lags 1, 6, 10, 15 and 20. The test statistics are statistically significant with p-values not
greater than 0.01, indicating that the returns are not white noise. Indeed, the daily exchange rate returns exhibits
correlation.
Table 4:
p
-values based on the Ljung-Box test for of the USD/KES exchange rates
m
1
6
10
15
20
Returns
m
Q
p
-value
10
(0.0004)
40
(0.0000)
50
(0.0000)
60
(0.0000)
90
(0.0000)
Squared returns
m
Q
p
-value
6
(0.01)
20
(0.0004)
30
(0.002)
40
0.003
600
(0.0000)
Absolute returns
m
Q
p
-value
200
(0.0000)
800
(0.0000)
1000
(0.0000)
1000
(0.0000)
2000
(0.0000)
From the results of Ljung-Box test in Table 4 and the autocorrelation (ACF) and partial autocorrelation
(PACF) plots in Figure 3, for the exchange rate return series, absolute and squared return series shows that the
return series exhibit autocorrelation at some lags at 5% level of significance. The presence of autocorrelation
detected in the log return can be removed by fitting the simplest plausible ARMA (p, q) model to the data. On
the other hand, the autocorrelation detected in the squared log returns, indicate that there exists conditional
heteroskedasticity of the exchange rate returns series which could be removed by fitting the simplest plausible
GARCH model to the ARMA filtered data.
3.2 Estimated Mean Equation
An ARMA (p, q) model is used to fit the mean returns, as it provides a flexible and parsimonious
approximation to conditional mean dynamics. The Autocorrelation Function (ACF) and Partial Autocorrelation
Function (PACF) are used to determine the order of ARMA (p, q) models. The ACF and PACF plots given in
Figure 3 suggest that the returns may be modeled by an ARMA (2, 2) process. Tsay and Tiao (1984) proposed
the extended autocorrelation function (EACF) technique to identify the orders of a stationary or non-stationary
ARMA process based on iterated least square estimates of the autoregressive parameters. The output of EACF is
a two-way table, where the rows correspond to AR order p and the columns to MA order q. Therefore, the
EACF suggests that the daily log returns of USD/KES exchange rate follow an ARMA (2, 0) model. This is in
agreement with the result in Table 5 suggested by the best fitting model selected based on Bayesian Information
Criterion (BIC) values. The criterion is to choose a model with minimum AIC and BIC and largest log-
likelihood function. BIC always gives penalty for the additional parameters more than AIC does. So the ARMA
(2, 0) is selected as the mean equation that mainly takes account of the BIC.
Modeling USD/KES Exchange Rate Volatility using GARCH Models
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 22 | Page
Table 5: Criterion for ARMA (p, q) Order Selection
Figure 3: ACF/PACF Plots for USD/KES returns
3.3 Estimated Volatility Model
The results of the fitted AR (2)-GARCH (1, 1) and AR (2)-GARCH –M (1, 1) models to the USD/KES
log return series with normal distribution, Student’s t distribution and skewed t distribution for the standardized
residuals are presented in Table 6. The estimates of the model parameters are all significant for normal,
Student’s t and skewed t distribution except for the
parameter which is not significant for all the distributions.
The estimates of
1
and
2
are significant, supporting the use of the AR (2) model for the returns. Volatility
shocks are persistent since the sum of the ARCH and GARCH coefficients are very close to one. The Box-
Pierce Q statistics is insignificant up to lag 20, indicating that there is no excessive autocorrelation left in the
residuals. Comparing the log-likelihood and information criterion in Table 6 within the three conditional
distributions, the model with conditional distribution of skewed t has larger log-likelihood and smaller
information criterion statistics than estimated by normal and t distribution which means this model is better
fitted.
0.00 0.04 0.08 0.12
-0.10 0.00
ACF for log Returns
Lag
ACF
0.00 0.04 0.08 0.12
-0.10 0.00
Lag
Partial ACF
PACF for log Returns
0.00 0.04 0.08 0.12
0.0 0.2 0.4
ACF for Squared log Returns
Lag
ACF
0.00 0.04 0.08 0.12
-0.2 0.1 0.4
Lag
Partial ACF
PACF for Squared log Returns
0.00 0.04 0.08 0.12
-0.05 0.15
ACF for Absolute log Returns
Lag
ACF
0.00 0.04 0.08 0.12
-0.05 0.15
Lag
Partial ACF
PACF for Absolute log Returns
Modeling USD/KES Exchange Rate Volatility using GARCH Models
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 23 | Page
Table 6: Estimation of AR (2)-GARCH (1, 1) and AR (2)-GARCH-M (1, 1) with Different Distributions
AR (2)-GARCH (1, 1)
AR (2)-GARCH-M (1, 1)
Normal
t
Skew t
Normal
t
Skew t
0.000137
(0.00230)
0.000079
(0.01642)
0.000093
(0.017355)
0.000141
(0.005763)
0.000082
(0.018374)
0.000094
(0.017024)
AR(1)
0.132993
(0.00000)
0.139786
(0.00000)
0.139081
(0.00000)
0.131115
(0.00000)
0.139499
(0.00000)
0.138878
(0.00000)
AR(2)
-0.097567
(0.00000)
-0.044787
(0.012145)
-0.045087
(0.011554)
-0.098027
(0.000010)
-0.044807
(0.012114)
-0.045080
(0.011577)
Omega
0.00000
(0.658234)
0.00000
(0.469487)
0.00000
(0.472896)
0.00000
(0.675454)
0.00000
(0.470185)
0.00000
(0.473492)
1
0.120459
(0.00000)
0.318323
(0.00000)
0.317466
(0.00000)
0.118585
(0.00000)
0.318264
(0.00000)
0.317520
(0.00000)
1
0.878420
(0.00000)
0.680677
(0.00000)
0.681534
(0.00000)
0.880377
(0.00000)
0.680736
(0.00000)
0.681480
(0.00000)
Skew
1.011980
(0.00000)
Shape
3.146979
(0.00000)
1.013131
(0.00000)
3.147627
(0.00000)
3.150416
(0.00000)
LLF
12002
12511
12512
12002
12511
12512
AIC
-8.5166
-8.8778
-8.8772
-8.5167
-8.8778
-8.8772
BIC
-8.5039
-8.8630
-8.8603
-8.5167
-8.8630
-8.8603
*P-values are shown in parentheses.
To capture the asymmetry dynamics and the presence of the “leverage effect” in the USD/KES
exchange rate returns, the nonlinear asymmetric models; AR (2)-EGARCH (1, 1), AR (2)-GJR-GARCH (1, 1)
and AR (2)-APARCH (1, 1) with conditional distributions; normal distribution, Student’s t distribution and
skewed t distribution are fitted to the exchange returns. Table 7 gives the results of the parameter estimates for
the AR (2)-EGARCH (1, 1), AR (2)-GJR-GARCH (1, 1) and AR (2)-APARCH (1, 1) models. The parameters
estimates for these three models are all significant except for the mean under the AR (2)-EGARCH (1, 1) for the
normal and skew t distribution, also the coefficient of the second term of autoregressive process under the skew
t distribution and the coefficients of
1
under the Student’s t and skew t distribution are not significant. For
both the AR (2)-GJR-GARCH (1, 1) and AR (2)-APARCH (1, 1)
is not significant for all the distribution.
The parameter
is not significant for the AR (2)-APARCH (1, 1) under the t distribution. The coefficient
in
the case of AR (2)-APARCH (1, 1) is statistically significant at level of significance of 5% implying that there is
an asymmetry under the normal distribution. On the other hand, its negative value indicates the presence of the
“leverage effect”. The coefficient
in the AR (2)-E-GARCH (1, 1) and AR (2)-GJR-GARCH (1, 1) is
significantly different from zero, which indicates the presence of asymmetry. The value of
which is less than
zero implies presence of the “leverage effect”. According to the log-likelihood value and information criterion
of the estimated models, the APARCH model has the larger log-likelihood value and smaller information
criterion compared with E-GARCH model and GJR-GARCH model. Secondly, comparing within the APARCH
models under normal distribution, and Student’s t distribution, the model with conditional Student’s t
distribution outperforms the normal distribution which means this model is superior in modeling the USD/KES
exchange rate returns with asymmetry and fat tail.
The estimated power parameter
in the APARCH model is 2.44 which is slightly different from the
estimated result of Ding, Granger and Engle (1993)’s under the normal distribution which is 1.43. This may be
caused by the time period of the data is different and then mean equation is also different to model the data. But
in this paper is still significantly different from 1 (GJR-GARCH) or 2 (GARCH). When the conditional
distribution changes to t distribution
is getting smaller to 0.73, however, using the same test as in Ding,
Granger and Engle (1993)’s paper, let
0
l
be the log-likelihood of value under the GARCH model which is set as
the null hypothesis, while the alternative hypothesis is APARCH model with log-likelihood is l, then
)(2 0
ll
have a
2
distribution with 2 degrees of freedom when
0
H
is true. Then, under the Student’s t distribution
,72)1251112547(2)(2 0ll
which means we can reject the null hypothesis that the data is generated
from GARCH model. And also in the same way we can reject that the data is generated from E-GARCH model
and GJR-GARCH model.
Modeling USD/KES Exchange Rate Volatility using GARCH Models
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 24 | Page
Table 7: Estimation of AR (2)-EGARCH (1, 1) and AR (2)-GJR-GARCH (1, 1) with Different Distributions
AR (2)-EGARCH (1, 1)
AR (2)-GJR-GARCH (1, 1)
AR (2)-APARCH (1, 1)
Normal
t
Skew t
Normal
t
Skew t
Normal
t
-0.000138
(0.103591)
0.000077
(0.010444)
0.000085
(0.052036)
0.00154
(0.000768)
0.000084
(0.011170)
0.000100
(0.010824)
0.000100
(0.031989)
0.00059
(0.00000)
AR(1)
0.121891
(0.000036)
0.128796
(0.00000)
0.128516
(0.00000)
0.134449
(0.00000)
0.139695
(0.00000)
0.139224
(0.00000)
0.134804
(0.00000)
0.091779
(0.00000)
AR(2)
-0.046525
(0.000491)
-0.029849
(0.038131)
-0.029962
(0.076620)
-0.097030
(0.000011)
-0.043918
(0.013838)
-0.043998
(0.013590)
-0.100342
(0.000004)
-0.011551
(0.00000)
Omega
-0.535835
(0.00000)
-0.599479
(0.000097)
-0.599915
(0.000098)
0.000000
(0.665489)
0.000000
(0.482739)
0.000000
(0.489444)
0.000000
(0.981524)
0.000000
(0.962086)
1
0.111569
(0.00000)
0.022818
(0.453770)
0.022900
(0.452603)
0.13854
(0.00000)
0.354876
(0.00000)
0.354889
(0.00000)
0.087753
(0.00000)
0.394055
(0.00000)
1
0.942070
(0.00000)
0.944689
(0.00000)
0.944655
(0.00000)
0.879810
(0.00000)
0.686124
(0.00000)
0.687931
(0.00000)
0.879502
(0.00000)
0.855587
(0.00000)
Gamma
0.484044
(0.00000)
0.554240
(0.00000)
0.554074
(0.00000)
-0.025328
(0.026778)
-0.084006
(0.049376)
-0.086881
(0.041340)
-0.051172
(0.035613)
-0.085833
(0.066427)
Delta
2.437594
(0.00000)
0.727418
(0.00000)
Skew
1.006334
(0.00000)
1.014788
(0.00000)
Shape
2.386515
(0.00000)
2.387376
(0.00000)
3.145621
(0.00000)
3.148859
(0.00000)
2.100008
(0.00000)
LLF
11557
12505
12505
12004
12513
12514
11997
12547
AIC
-8.2004
-8.8728
-8.8721
-8.5177
-8.8785
-8.8780
-8.5119
-8.9014
BIC
-8.1857
-8.8559
-8.8531
-8.5029
-8.8616
-8.8590
-8.4950
-8.8825
*P-values are shown in parentheses.
The GARCH models with the innovations of Student’s t and skewed Student’s t distributions have a
better fit in general than the models with normal distribution innovations since they have the highest log-
likelihood function (LLF) and smallest AIC and BIC. Secondly, the values of the AIC, BIC and LLF for all the
models with Student’s t and skewed Student’s t distributions innovations are not significantly different. This
implies that the models with Student’s t and skewed Student’s t distributions innovations would result in the
same conclusions. The volatility (conditional variance) process and standardized residuals for the AR (2) -
APARCH (1, 1) model with Student t distribution is plotted in Figure 4.
Figure 4: Volatility (conditional variance) process and standardized residuals of exchange rate returns derived
from the AR (2)-APARCH (1, 1) residuals.
The plot in Figure 4 shows that the model is well specified. The ACF of the square standardized
residuals compares well with the ACF of the square returns in Figure 3. This shows that AR (2)-APARCH (1, 1)
Student-t model sufficiently explains the heteroscedasticity effect in the returns, thus we can conclude that the
model fit the USD/KES returns well. The Ljung-Box test of the standardized residuals at different lags confirms
that standardized residuals have no correlation.
Modeling USD/KES Exchange Rate Volatility using GARCH Models
DOI: 10.9790/5933-0801011526 www.iosrjournals.org 25 | Page
The future return rate and volatility for one-day-ahead based on the estimated parameters of the models are
obtained. These forecasted values are necessary for the estimation of Value at Risk (VaR). The estimated values
of the VaR parameters for one-day-ahead as well as the probabilities of 95% and 99% are exhibited in Table 8.
Table 8: Econometric Estimation of the parameters of VaR for One-day-ahead period
Model
GARCH
t distribution
GARCH-M
t distribution
GJR-GARCH
t distribution
EGARCH
t distribution
APARCH
t Distribution
Forecasted return
0.00005066
0.0000522
0.000055
0.00004574
0.00002786
Forecasted
conditional variance
0.00127
0.00127
0.001269
0.00159
0.002049
VaR 0.95
-0.16%
-0.16%
-0.16%
-0.16%
-0.16%
VaR 0.99
-0.33%
-0.33%
-0.33%
-0.35%
-0.36%
The estimated VaR values obtained with the GARCH approach are negative. The negative sign is usually
ignored since it’s an indicator of loss. With probability of 0.95 the expected maximum loss due to having to
change 1 US Dollars to KES is around in one day period.
IV. Conclusions
Modeling and forecasting the volatility of exchange rate returns has become an important field of
empirical research in finance. This is because volatility is considered as an important concept in many economic
and financial applications like asset pricing, risk management and portfolio allocation. This paper attempts to
explore the comparative performance of different econometric volatility forecasting models in the terms of their
ability to estimate VaR in the USD/KES exchange rates. A total of five different models were considered in this
study. The volatility of the USD/KES returns have been modeled by using a univariate Generalized
Autoregressive Conditional Heteroscedastic (GARCH) models including both symmetric and asymmetric
models that captures most common stylized facts about exchange returns such as volatility clustering and
leverage effect, these models are GARCH (1, 1), GARCH-M (1, 1), exponential GARCH (1, 1), GJR GARCH
(1, 1) and APARCH (1, 1) following three residual distributions namely; normal, Student’s t-distribution and
Skewed Student’s t-distribution. The first two models are used for capturing the symmetry effect whereas the
second group of models is for capturing the asymmetric effect. The study used the USD/KES exchange rates
data from the Central Bank of Kenya (CBK) for the period 3rd January 2003 to 31st December, 2015. Based on
the empirical results presented, the following can be concluded:
The paper finds strong evidence that daily returns could be characterized by the above mentioned
models. The USD/KES data showed a significant departure from normality and existence of conditional
heteroscedasticity in the residuals series. Descriptive statistics for the USD/KES exchange rates show presence
of negative skewness and excess kurtosis. The results of the conducted ARCH-LM test point out significant
presence of ARCH effect in the residuals as well as volatility clustering effect. Standardized residuals and
standardized residuals squared were white noise. The econometric estimation of VaR can be related to the
chosen GARCH model. Therefore a first step in estimation of the parameters of VaR is a detailed specification
analysis of the potential models. Based on the estimated model, a 1-step-ahead forecasting is taken to forecast
the future value of the exchange rate returns and the conditional volatility. The values are used to estimate VaR.
The empirical results have indicated that the most adequate GARCH models for estimating and forecasting VaR
in the USD/KES exchange rates are the asymmetric APARCH, GJR-GARCH and EGARCH model with
Student’s t-distribution. These models have a better fit of the exchange returns, since they have the largest log-
likelihood function and smallest AIC and BIC. We also compared the one step-ahead VaR estimate from the
asymmetric models with Student’s t-distribution and from the results the conclusion was that the AR (2)-
APARCH (1, 1) model is also superior in the estimating the one-step-ahead VaR.
The findings in this paper have important implications regarding VaR estimation in volatile times,
market timing, portfolio selection etc. that have to be addressed by investors and other risk managers operating
in emerging markets. However, the limitation of the study is that the empirical research focused only on the
USD/KES exchange rate and therefore the findings cannot be generalized to other exchange rates in the market.
In the future research a wider sample of exchange rates should be used to compare the performance of the most
commonly used foreign currencies in the market and the inclusion of other asymmetric GARCH-type models,
testing and comparing their predictive performance.
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