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ISSN: 2548-0162 © 2021 Gazi Akademik Yayıncılık
Gazi İktisat ve İşletme Dergisi, 2021; 7(1): 1-16 https://dergipark.org.tr/tr/pub/gjeb
Modelling exchange rate volatility using GARCH models
Basma Almisshala*, Mustafa Emirb
a PhD., Stu- Business Administration,
61220 Trabzon, TURKEY. Email: basmapal@gmail.com
ORCID ID: https://orcid.org/0000-0001-7885-1217
b Pro- Business Administration,
61220 Trabzon, TURKEY. Email: memir@ktu.edu.tr
ORCID ID: https://orcid.org/0000-0002-2891-3085
ARTICLE INFO
ABSTRACT
Received: 21.07.2020
Accepted: 01.02.2021
Available online:15.02.2021
Article Type: Research
article
This paper aims to model the volatility of USD and EUR exchange
rates against TRY for the period from January 2005 to December 2019
using the Generalised Autoregressive Conditional Heteroscedasticity
(GARCH) models. Both symmetric and asymmetric models have been
applied to measure factors that are related to the exchange rate returns
such as leverage effect and volatility clustering. The symmetric
GARCH (1,1) model and the asymmetric EGARCH (1,1), GJR-
GARCH (1,1), and PGARCH (1,1) have been applied to each currency
against TRY. The results of this paper conclude that the most adequate
model for estimating volatility of the USD/TRY exchange rates are the
symmetric GARCH (1,1) and asymmetric GJR-GARCH (1,1) models.
Moreover in USD/TRY returns, GARCH (1,1) and GJR-GARCH (1,1)
models are the most appropriate models along with PGARCH (1,1) in
EUR/TRY as well. Regarding forecasting volatility, Root Mean Square
Error (RMSE), Mean Absolute Error (MAE) and Mean Absolute
Percentage Error (MAPE) tests have been used. Based on the results,
the static forecast of GJR-GARCH (1,1) is the best model in predicting
the future pattern for both USD and EUR.
Keywords:
exchange rate volatility,
leverage effect, ARMA,
GARCH models,
forecasting volatility.
GARCH yöntemleri kullanarak döviz kuru volatilitelerinin modellenmesi
MAKALE BİLGİSİ
ÖZ
21.07.2020
Kabul Tarihi: 01.02.2021
Tarihi: 15.02.2021
makalesi
Bu çalışmada, genelleştirilmiş otoregresif koşullu değişen varyans
(GARCH) modelleri kullanılarak 2005-2019 döneminde ABD Doları
(USD) ve Euro (EUR) döviz kurlarının Türk lirası (TRY) karşısında
volatilitelerinin modellenmesi amaçlanmaktadır. Kaldıraç etkisi ve
oynaklık kümelenmesi gibi döviz kuru getirileri ile ilgili faktörleri
* Corresponding author
Doi: https://doi.org/10.30855/gjeb.2021.7.1.001
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
ISSN: 2548-0162 © 2021 Gazi Akademik Yayıncılık 2
Anahtar Kelimeler:
döviz kuru volatilitesi,
kaldıraç etkisi,
ARMA, GARCH
modelleri, volatilite
tahmini.
ölçmek için hem simetrik hem de asimetrik modeller uygulanmıştır.
Simetrik model olan GARCH (1,1) ve asimetrik modeller olan
EGARCH (1,1), GJR-GARCH (1,1) ve PGARCH (1,1) modelleri her
bir para biriminin TRY karşısında volatilitesini öngörmek için
uygulanmıştır. Çalışma sonuçlarına göre, USD/TRY döviz kurlarının
oynaklığını tahmin etmek için en uygun yöntem simetrik GARCH (1,1)
ve asimetrik GJR-GARCH (1,1) modeller olarak belirlenmiştir.
USD/TRY modelinde olduğu gibi EUR/TRY'de PGARCH (1,1)
modelinin yanı sıra GARCH (1,1) ve GJR-GARCH (1,1) modelleri en
uygun modellerdir. Bununla birlikte EUR/TRY döviz kurlarının
oynaklığını tahmin etmek için PGARCH (1,1) modeli de anlamlı bir
sonuç sunmaktadır. Tahmin oynaklığı ile ilgili olarak, Kök Ortalama
Kare Hata (RMSE), Ortalama Mutlak Hata (MAE) ve Ortalama Mutlak
Yüzde Hata (MAPE) testleri kullanılmıştır. Sonuçlara göre, statik GJR-
GARCH (1,1) modelinin, hem USD hem EUR için daha yüksek bir
volatilite tahmininde bulunabileceği ortaya konulmuştur.
1. Introduction
After the massive stagflation, combined to inflation and recession that the United States was
suffering from during the 1960s; the Bretton Woods system had broken down. Meanwhile, in 1971,
Richard Nixon, the president of the United States, had declared the "temporary" commentary of the
USD ability to convert into gold. Afterwards, the USD replaced the gold standard as a global currency.
Thenceforth, the members of the International Monetary Fund (IMF) were able to pick their preferred
form of currency arrangements, but not pegging their currency’s value to gold’s price. These forms
include; legalizing the currency to float freely, adopting the currency of another country, pegging it to
another currency or a basket of currencies, participating in a currency bloc, or forming part of a
monetary union (IMF.org, 2019). As a result, the exchange rate exposure issues surrounding the
volatility and the risk management techniques against exchange rate loss surfaced periodically
throughout the time since the late 1970s.
Various forms of statistical models have been evolved to capture the volatility effect. These
models are often applied for estimating the degree of the exchange rate instability. Autoregressive
Conditional Heteroskedasticity-ARCH by Engle( 1982), had been modelled as the serial correlation of
returns throughout the inclusion of conditional variance as a function of the past errors and changing
time. This had been done during Engle’s attempts to explain the dynamic of inflation occurred in the
United Kingdom. Generalised GARCH models had been developed independently by Bollerslev
(1986) and Taylor (1986) respectively. Bollerslev(1986) added to Engle’s model by inserting a long
memory and created a more flexible lag structure by adding lagged conditional variance to the original
model. The use of ARCH model in the exchange rate was first applied by Hsieh (1988) to compute
daily data of five-currency’s exchange rates. Friedman and Stoddard (1982) had used non-standard
techniques to uncover the underlying patterns. Increasingly, the volatility in exchange rates became an
interest for many researchers through using Heteroskedasticity models to study a higher volatility time
series of currency exchange rates because they are considered much favorable against other stable
variance models. Below is a brief on these studies in different regions.
Dritsaki (2019) examined the EUR/USD monthly exchange rate return from August 1953 until
January 2017, a data set of 763 observations. Applying ARCH, GARCH, and EGARCH models the
results showed that ARIMA (0,0,1)-EGARCH(1,1) model was the recommended choice in terms of
describing exchange rate returns and leverage effect. Also, ARIMA(0,0,1)-EGARCH(1,1) model static
procedure provides better results on the forecasting rather than the dynamic. Nguyen (2018) in the
paper, conducted an empirical study of the exchange rate volatility in Vietnam. A volume of 330
observations covering the period from January 1990 to Jun 2017 and monthly data on exchange rates
of Vietnamese Dong comparing to USD, British Pound, Japanese Yen, and Canadian Dollar were
used. The outcomes showed that ARMA(1,0)-GARCH(1,2) models were fit well in terms of capturing
the mean and volatility trend of USD/VND and GBP/VND exchange rate returns. At the same time,
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
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the study indicated that ARMA(1,0)-GARCH(1,1) model better fitted in a quite reasonable manner to
capture the mean and volatility trend of JPY/VND and CAD/VND exchange rate returns. The findings
stated that the return on exchange rates failed in the Gaussian distribution test at a significant level of
1%, and all four currencies return time-series to keep a high level of volatility clustering. In Epaphra’s
(2017) paper, they examined the behaviour of the exchange rate in Tanzania. A data set of daily
Tanzanian Shilling against USD exchange rate over the period starting from 4 January 2009 till 27
July 2015 was used. The exchange rate movement had been modelled using ARCH, GARCH (1,1),
and EGARCH models. The paper concluded that TZS/USD exchange rates’ volatility would be
modelled successfully with GARCH (1,1) model. Also, as for the forecasted volatility the results, it
showed that GARCH (1,1) had a noticeable predictive power. Ganbold et. al. (2017) in their study
discussed a forecasting method in the use of ARCH models accompanying ARIMA, SARIMA
(seasonal ARIMA) and semi-structural-SVAR in the Turkish context. By using a data set covering the
period from 2005-2017 ARCH and GARCH family models (EGARCH, IGARCH, and PARCH) had
been applied to forecast exchange rate volatility. The results showed EGARCH (1,1) model after
including dummy, was the best model in terms of forecasting exchange rate volatility. The model also
succeeded to control the leverage effect. The three models of forecasting, ARIMA, SARIMA, and
SVAR had been evaluated. The comparison of prediction techniques through RMSE and MAE
formulas showed that SARIMA model was much accurate against the rest. And; Omari et. al. (2017)
in their paper applied GARCH family models in modelling exchange rate volatility of the USD against
Kenyan Shilling exchange rate of a data set of daily prices of USD/KES over a timeframe from
January 2003 to December 2015. The performance of GARCH (1,1) and GARCH-M symmetric
models in addition to EGARCH (1,1), APARCH (1,1), and GJR-GARCH (1, 1) asymmetric models
were captured. The results of the study indicated that the asymmetric APARCH, GJR-GARCH, and
EGARCH models were better fit models to estimate volatility. Moreover Bosnjak et. al. (2016) in their
paper examined the attitude derived by the number of ARCH models for the EUR and USD against
the Croatian Kuna on daily observations of a timeframe between 1997 to 2015. The models had been
evaluated using standard information criteria. The findings showed that GARCH (2,1) was the most
appropriate model for the EUR/HRK, and GARCH (1,1) for the USD/HRK. For the estimated models,
there was no significant evidence that positive and negative shocks affected the volatility of
EUR/HRK and USD/HRK exchange rate returns. And finally; Karuthedath et. al. (2012) in their study
aimed to understand the behaviour of foreign exchange rate in India and the appearance of volatility
through using a day-to-day price of Indian Rupee against USD during a period up to forty years, from
the second quarter of 1973 until 2012. The results of exchange rate volatility of Indian rupee against
USD conducted through hiring ARCH family models like ARCH (1,1) GARCH (1,1) EGARCH(1,1)
TGARCH(1,1) revealed that symmetric GARCH (1,1) model which had the volatility of Indian
foreign exchange rate was highly persistent during the forty-year timeframe.
Conclusively, this study aims to determine the US Dollar and EURO against Turkish Lira TRY
exchange rate behaviour pattern using GARCH models and to make a comparison between them. USD
and EUR have been chosen as the two currencies, which are widely used, and trusted currencies in the
business world and both are among the world’s currencies that are accepted for most international
transactions. For that purpose, the paper applies part of GARCH family models using daily
observations quoted from Central Bank of the Republic of Turkey TCMB from January 2005 until
December 2019. So, the research hypothesis is that both USD and EUR versus TRY exchange rate
volatilities can be determined using GARCH models. The volatility models applied are ARMA,
ARCH, GARCH, GJR-GARCH, EGARCH, and PGARCH. In the end, the paper testes the best model
for future forecasting of time series volatility. The paper is organised as follows; section 2 discusses
the methodology; section 3 illustrates the data and empirical results; and finally, section 3 states the
conclusion of the paper.
2. Research methodology
The main characteristic of financial time-series which are high-frequency values, volatility
clustering, excess kurtosis, heavy-tailed distribution, leverage effect, and long memory properties
(Omari et al, 2017) have been examined using the Autoregressive Conditional Heteroscedasticity
(ARCH) and its Generalised form GARCH models. In this paper different models under the GARCH
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
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family models have been used. In presenting these models, it is important to mention in this
methodology the equations of conditional mean and the conditional variance which shall be conducted
before applying the GARCH family models.
2.1 Autoregressive moving average (ARMA)
ARMA model is a forecasting variable technique used as one of the known methods used for
employing the information generated from the available variables to predict its movements. ARMA is
a combination of two separate models. These models normally explain the behaviour of a time series
from two different perspectives: the autoregressive (AR) models and the moving average (MA)
models. The majority of economic data are non-stationary. Thus, they must be imposed on a
transformation process called “differencing” before they turn to be stationary. The transforming
process in some literature is known as “integration”. In brief, ARMA model presents that the series in
question are being imposed to an “integration process “before being used for any analysis (Adeleye,
2019).
(1)
(2)
(3)
Where is is the time series is being modelled.
2.2 Volatility modelling
The main purpose of modelling volatility is being able to forecast future trends. Typically, a
volatility model is used to forecast the returns’ absolute largeness (Engle and Patton, 2001). The
symmetric and asymmetric effect of GARCH family models have been used in this paper in modelling
the volatility of the exchange rate return time series of USD/TRY. Symmetric effect models such as
GARCH (p,q) and asymmetric effect have been captured through hiring GJR-GARCH (p,q),
EGARCH (p,q), and PGARCH(p.q).
2.3 Conditional variance equation for ARCH (q)
The conditional variance equation is calculated as a constant + the previous value of the squared
error:
+… (4)
It should be noted that α1 has to be positive since itself and has to be positive as they are
squared terms. Increasing the value of q in ARCH(q) model where q is the number of lags in
conditional variance equation, would eventually remove ARCH effect from residuals, and this
probably is not the most parsimonious model. The parsimonious model however simply means
accurately modelling a variable’s DGP with the fewest possible parameters. Instead of estimating
ARCH (7) model for USD and ARCH (3) for EUR, it would be better to estimate GARCH (1,1) model
since this is more parsimonious (uses fewer parameters in the conditional variance equation).
2.4 Symmetric GARCH models
The Generalised Autoregressive Conditional Heteroscedasticity-GARCH (p,q) Model, is another
form of ARCH model that combines moving average element (MA) together with the autoregressive
element (AR). Fundamentally, the model put in a new parameter (p) which is the number of lag
variances. However, parameter (q) is the number of lag residual errors in GARCH model. In short, the
model has the lag variance terms along with lag residual errors from a mean process (Bollerslev,
1986).
This model represents the conditional variance as a linear function of its lags. The general equation
of the GARCH (p, q) model is given by the following formula;
(5)
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
ISSN: 2548-0162 © 2021 Gazi Akademik Yayıncılık 5
(6)
Where is the logarithm value of returns of the time series at time mean value of the returns,
the error term. In most empirical applications the basic GARCH (1, 1) model fits the dynamic
conditional variance of the bulk of the time-series statistics in a reasonable way. GARCH (1, 1)
model’s equation shown below;
(7)
To guarantee a positive variance in all instances, the following restricts are imposed >0 and
In several instances, GARCH model describes a rational model for modelling time-series and
estimating volatility. Nevertheless, weaknesses of the ARCH model could be developed to capture the
characteristics and dynamics of the time series much better. Thereafter, the GARCH (p, q) model was
first presented by Bolleslev (1986). New GARCH related models have been invented to include the
incompetence of the original GARCH and capture the different characteristics of the financial time
series. (Omari et. al., 2017).
2.5 Asymmetric GARCH models
Practically, financial assets’ returns are expressed as “bad” news rather than “good’ news in
general. This phenomenon known as leverage effect was first mentioned by Black (1976). The term
“leverage” stems from the empirical observation that conditional variance (volatility) of an asset tends
to increase when its returns marked negative. In the purpose to capture the asymmetry in return
volatility or “leverage effect”, another extension of GARCH models has been developed. It is widely
known as the asymmetric GARCH models (Stokes et. al., 2004). This paper uses the following
asymmetric GARCH models; GJR-GARCH, Exponential Generalised ARCH (EGARCH), and Power
Generalised ARCH (PGARCH) model for capturing the asymmetric phenomena.
2.5.1 Test of asymmetry, Engle and Ng test:
For the purpose of testing the GARCH specification against asymmetry, Engle and Ng (1993)
initiated the use of the sign-bias and size-bias tests. These tests are common in testing the effect of
good and bad news on the stock returns volatility.
The best test involves the following regression:
(8)
Where denotes the squared residuals of a GARCH model fitted to the return. is constant,
is a dummy variable that takes the value 1 if < 0 otherwise, is an error term and is
defined as 1- . If is significant, then sign bias is present. If either or is significant, then
size bias is also present.
2.5.2 The Glosten, Jagannathan and Runkle GARCH (GJR-GARCH model)
The GJR-GARCH model proposed by Glosten, Jagannathan, and Runkle in 1989 as a different
form of asymmetric GARCH model. GJR-GARCH model allows for variances to react differently
depending on the sign of the shock size it might receive. The conditional mean equation is the same as
previous ARCH and GARCH models. However, the conditional variance equation is different to
capture asymmetric volatility. The variance equation of GJR-GARCH (p, q) is as below;
(9)
Where parameters are constant, I is a dummy variable or an indicator function that
takes the value of zero if is positive, and one otherwise. When the value of I is one it indicates
bad news and negative errors are leveraged. Negative or bad news has a greater effect than the positive
ones (Omari et. al., 2017). The parameters of the model have to be positive and that /2 <1.
If all leverage coefficients are zero, then GJR-GARCH model reduces to GARCH model. In other
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
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words, GARCH model can be compared with a GJR-GARCH model using the likelihood ratio
method.
2.5.3 The exponential GARCH (EGARCH) model
Although GARCH models are successfully a useful tool in investigating fat-tail return distributions
and the volatility clustering they are poor models in capturing the leverage effect as the conditional
variance is a function of the past values not their sign only (Abdalla, 2012). Normally, stock returns
are more responsive to negative shocks than positive shocks (Black, 1976). This asymmetric
behaviour is defined as” leverage effect” term. It describes how the negative shock affected the rise of
volatility and what if the positive shock with the same volume occurred. Meanwhile, EGARCH
model, developed by Nelson (1991), captures asymmetric responses of the time-varying variance to
shocks and ensures that the existence of positive sign of variance. The main form of the conditional
variance is as follows;
- (10)
The EGARCH model is asymmetric because the level included with coefficient . As the
coefficient has a negative sign, it leads to the fact that positive shocks cause less volatility than
negative shocks. In macroeconomic analysis, negative shocks usually imply bad news in financial
markets, that leading to a higher uncertainty in the future (Wang, 2008). To capture asymmetric
responses of the fluctuated variance to shocks, the paper considers EGARCH (1,1) model, which has
the following mean and variance equations;
Mean Equation (11)
Variance Equation - (12)
2.5.4 Measuring forecast accuracy
The Root mean square error (RMSE), Mean Absolute Error (MAE) and Mean Absolute Percentage
Error (MAPE) are well-known models in measuring forecast accuracy of a timeseries. The formulas of
the three models are as bellow;
RMSE = (13)
Where , with being the actual value of returns, and being the fitted value from
one of the estimated models with the same date. Finally, is the number of forecasted observations
(10) days in this case.
MAE = (14)
Where is prediction, is the value and is the number of forecasted observations.
MAPE = (15)
Where is actual value, is forecast value and is the number of forecasted observations.
There are two different forms of each model; dynamic and static. Dynamic Models allow you to
make a multiple-step-ahead forecast, while Static Models consist of a series of rolling single-step-
ahead forecasts.
3. Bilgi ekonomisinin göstergeleri
The return on the exchange rate of USD/TRY and EUR/TRY has been chosen as a dataset of this
paper: a total of 3,709 observations for both currencies represents the working days of the period from
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
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1 January 2005 until 31 December 2019 in Turkey. During the timeframe of fourteen years fluctuation
in the Turkish Lira’s has been observed. The primary data sourced from the database of the Central
Bank of the Republic of Turkey TCMB. A time series in daily format has been generated and tested
for the presence of Unit Root as a stationary test before proceeding in applying GARCH family
models. The Augmented Dickey-Fuller (ADF) unit root test is used with the following formula;
(16)
Table 1
ADF Unit Root Test on USD/TRY Daily
Exchange Rates
Table 2
ADF Unit Root Test on EUR/TRY Daily
Exchange Rates
t-Statistic
Prob.*
Augmented Dickey-
Fuller
1.712721
0.9997
Test critical
values:
1% level
-3.431900
5% level
-2.862110
10% level
-2.567117
*MacKinnon (1996) one-sided p-values.
*MacKinnon (1996) one-sided p-values.
t-Statistic
Prob.*
Augmented Dickey-
Fuller
1.503260
0.9994
Test critical
values:
1% level
-3.431900
5% level
-2.862110
10% level
-2.567117
Source: Authors’ calculation
According to the result illustrated in Table 1 and Table 2, the ADF test of USD/TRY daily rate
series is 1.713 >1 and 1.503 >1 respectively. Also, both statistics are greater than the critical p-values
at all significant levels of (1%, 5%, and 10%). Therefore, the null hypothesis of the ADF test H0 that
the data have a unit root is accepted for both USD and EUR. The time series of daily exchange rate
observations is non-stationary. To overcome this issue; the return on USD/TRY and EUR/TRY
exchange rates are generated.
Thus, the series is converted into the exchange rate return by following a logarithmic
transformation;
(17)
Where is the percentage of the return on the exchange rate, and is the exchange rate
of the current and the previous period that the exchange rate is being observed.
1
2
3
4
5
6
7
8
05 06 07 08 09 10 11 12 13 14 15 16 17 18 19
EUR middle rate USD middle rate
Fig. 1. Time series of the USD/TRY and EUR/TRY daily
middle exchange rates for the period 01.01.2005 till
31.12.2019. Source: Central Bank of the Republic of Turkey
USD/TRY & EUR/TRY
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
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-.08
-.04
.00
.04
.08
.12
.16
05 06 07 08 09 10 11 12 13 14 15 16 17 18 19
-.08
-.04
.00
.04
.08
.12
.16
05 06 07 08 09 10 11 12 13 14 15 16 17 18 19
Fig. 2. Return on Exchange Rate of USD/TRY for the
period 01.01.2005 till 31.12.2019.
Fig. 3. Return on Exchange Rate of EUR/TRY for the
period 01.01.2005 till 31.12.2019.
Figure 1 illustrates the time series of the daily exchange rates of USD/TRY and EUR/TRY. It is
obvious from the graph that the plots are non-stationary for the two currencies against TRY and follow
a random walk. However, a modified plot of the log-returns series for the exchange rates of USD/TRY
in Figure 2 and EUR/TRY in Figure 3 show that return rates are not constant over the timeframe of
this study. One is at the end of October 2008 and the other is on the mid of August 2018. The two
figures also show overall massive increases of volatility. As a result of that, decreasing slope of
exchange rate is always followed by high volatility. This percentage is known as the leverage effect
(Abdalla, 2012). The ADF test has been recalculated to test the stationarity in the return series. The
results in Table 3 and Table 4 show ADF of (-57.02) and (-37.36) which are much smaller than all the
significance p-values and the H0 is (return) rejected.
Table 3
ADF Unit Root Test on the Modified Time
Series of USD/TRY Exchange Rate Return
Table 4
ADF Unit Root Test on the Modified Time
Series of EUR/TRY Exchange Rate Return
*MacKinnon (1996) one-sided p-values.
t-Statistic
Prob.*
Augmented Dickey-Fuller
-57.02171
0.0001
Test critical
values:
1% level
-3.431898
5% level
-2.862109
10% level
-2.567116
*MacKinnon (1996) one-sided p-values.
t-Statistic
Prob.*
Augmented Dickey-Fuller
-37.35721
0.0000
Test critical
values:
1% level
-3.431899
5% level
-2.862110
10% level
-2.567117
Source: Authors’ calculation
Table 5 shows the descriptive statistics of the return series. A positive mean value is shown in both
USD and EUR. The series exhibits volatility clustering considering that USD/TRY and EUR/TRY
exchange rate returns witnessed periods when large changes are following with further large changes
and periods when small changes are followed by further small changes. The coefficient of skewness of
1.1857 and 1.4124 respectively indicate that returns have asymmetric distribution in both cases. The
kurtosis of returns in both statistics is bigger than 3. Distributions with kurtosis greater than 3 are said
to be leptokurtic. The non-normality condition is supported by a Jarque-Bera (JB) test. The results are
high, which lead to the rejection of the null hypothesis of normality at the significance level of 5%. In
other words, the errors are not normally distributed.
Return on Exchange Rate USD
Return on Exchange Rate EUR
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Table 5
Descriptive Statistic of the USD/TRY and EUR/TRY Exchange Rate Return
N
Min
Max
Mean
Std. D.
Skewness
Kurtosis
JB test
3,773
-0.1194
0.1471
0.000395
0.0091
1.1857
35.855
170,579.0
3,773
0.0719-
0.1402
0.000343
0.0087
1.4142
28.485
103,364.1
Source: Authors’ calculation
3.1 Ljung-Box test
The Ljung-Box Q test was named after Greta Marianne Ljung and her university’s advisor George
E. P. Box. Ljung-Box Q test is a portmanteau test that is based on the autocorrelation plot. It is a
widely used test in economics to test for serial correlation in time series and to determine whether
there is a structure in a time series worth modelling or not. It also applies to residuals after a forecast
model has been fit to data. The Ljung-Box test formula is;
(18)
Where is the Ljung-Box statistics, is the estimated autocorrelation between observations
separated by time and is the degree of freedom. According to the results of Ljung-Box and Q-test
where the null hypothesis H0 is that all correlation up to lags is equal to 0, the series of exchange
rate return show significant autocorrelation as shown in p-value which at any is less than 0.05.
Therefore the null hypothesis is rejected and the series are correlated.
3.2 Estimating mean equation
Autoregressive Moving Average model (ARMA) is used to describe the mean returns, as it
provides flexible and parsimonious approximation to conditional mean dynamics. To choose the
appropriate model, it can be chosen among several factors like having the most significant coefficient,
the least volatility, the highest adjusted R-squared, and finally the lowest of both Akaike Information
Criterion AIC and Schwarz criterion SBIC. In this paper as shown in Table 6 and Table 7, the essential
statistics for ARMA (p,q) are listed in which among wider analysis best four ARMA (p,q) models
have been chosen per each currency against TRY to choose the best ARMA equation.
Table 6
Comparison for ARMA (p, q) for Best Selection – USD/TRY
Return on
Exchange rate
ARMA (3,0)
ARMA (0,1)
ARMA (3,1)
ARMA (5,2)
Significant
Coefficient
2
1
4
5
Sigma2 Volatility
8.305197
8.322311
8.292686
8.238736
Adj. R2
0.005625
0.004059
0.007518
0.012331
AIC
-6.557868
-6.555578
-6.559509
-6.565166
SBIC
-6.552907
-6.553926
-6.552894
-6.553585
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
ISSN: 2548-0162 © 2021 Gazi Akademik Yayıncılık 10
Table 7
Comparison for ARMA (p, q) for Best Selection – EUR/TRY
Return on Exchange rate
ARMA (0,3)
ARMA (1,3)
ARMA (2,2)
ARMA (3,1)
Significant Coefficient
3
3
4
3
Sigma2 Volatility
7.484738
7.471631
7.486616
7.472274
Adj. R2
0.013981
0.015452
0.014890
0.015519
AIC
-6.660598
- 6.661753
-6.661111
-6.662241
SBIC
-6.655640
-6.655140
-6.654498
-6.655626
Source: Authors’ calculation
As for USD/TRY shown in Table 6, the statistics indicate that ARMA (5,2) has the most appropriate
ARMA model in this case with the highest number of significant coefficients at 5% level, and highest
adjusted R-squared, in contrast, ARMA (5,2) has the lowest volatility and AIC and the second-lowest
SBIC after ARMA (0,1). Likewise, EUR/TRY statistics in Table 7 also assigned that ARMA (3,1) is
the most appropriate ARMA model that minimizes the short information criteria.
3.3 Testing for heteroscedasticity
The highest priority before testing GARCH model is to look for evidence of heteroscedasticity
throughout examining the residuals of the return series. To test for it, the Lagrange Multiplier (LM)
test proposed by Engle (1982) has been applied. Based on the result of ARCH effect test the null
hypothesis H0 is that there is no ARCH effect presented in the model. It has been rejected and ARCH
effect is considered at a significant level of 5%.
Now, the following Table 8 and Table 9 summarize the GARCH models results of both USD and
EUR volatility against TRY respectively;
Table 8
Symmetric and Asymmetric Models’ Results of USD/TRY Exchange Rates
Model
Parameter
Coefficient
Std. Error
z-statistic
p-value
ARCH LM
Schwartz
Criterion
GARCH (1,1)
2.300544
2.359925
9.748377
0.0000
1.491967
(0.2219)
-6.972931
0.168218
0.009543
17.62781
0.0000
0.805366
0.007397
108.8831
0.0000
GJR-GARCH
(1,1)
1.296788
1.500463
8.642583
0.0000
3.46283
(0.0628)
-6.987574
0.164868
0.008817
18.69895
0.0000
ϒ
-0.099352
0.010516
-9.447524
0.0000
0.867781
0.006427
135.0209
0.0000
EGARCH (1,1)
-0.425897
0.024950
-17.06974
0.0000
9.650463
(0.0019)
-6.985476
0.214947
0.011047
19.45789
0.0000
ϒ
0.078262
0.006984
11.20570
0.0000
0.973421
0.002365
411.5768
0.0000
PGARCH
8.488113
5.052334
1.680038
0.0929
5.275598
(0.0216)
-6.986743
0.115732
0.008500
13.61593
0.0000
ϒ
-0.280704
0.037323
-7.521040
0.0000
0.873271
0.006160
141.7679
0.0000
1.633621
0.113898
14.34289
0.0000
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
ISSN: 2548-0162 © 2021 Gazi Akademik Yayıncılık 11
Table 9
Symmetric and Asymmetric Models’ Results of EUR/TRY Exchange Rates
Model
Parameter
Coefficient
Std. Error
z-statistic
p-value
ARCH
LM
Schwartz
Criterion
GARCH (1,1)
2.809435
2.636117
10.65748
0.0000
3.307463
(0.0690)
-7.040759
0.187783
0.009452
19.86739
0.0000
0.782566
0.010113
77.37913
0.0000
GJR-GARCH
(1,1)
2.617629
2.587766
10.11540
0.0000
1.680007
(0.1949)
-7.056927
0.234311
0.012600
18.59580
0.0000
ϒ
-0.156353
0.012396
-12.61348
0.0000
0.805673
0.010726
75.11279
0.0000
EGARCH
-0.636669
0.041681
-15.27473
0.0000
8.303869
(0.0040)
-7.054059
0.264540
0.013286
19.91051
0.0000
ϒ
0.093485
0.006826
13.69581
0.0000
0.956075
0.003753
254.7617
0.0000
PGARCH
2.002498
1.135208
1.763993
0.0777
3.725402
(0.0536)
-7.056097
0.149117
0.010128
14.72283
0.0000
ϒ
-0.319507
0.032489
-9.834242
0.0000
0.823922
0.009752
84.48431
0.0000
1.580491
0.111050
14.23228
0.0000
Note: ω is constant, α is ARCH term, ϒ is Leverage term, β is GARCH term and δ is Power term.
Source: Authors’ calculation
3.3.1 GARCH (1,1) model
As for GARCH (1,1) model; both the ARCH and GARCH parameters in both USD and EUR
statistics are highly significant with a p-value of 0.000 in both of them. The sum of the coefficient of
the ARCH and GARCH parameters (0.168218+0.805366), (0.187783+0.782566) respectively are very
close to 1 which means that the shocks to the conditional variances will be highly persistent. Since the
GARCH parameter is significant, a large return value (either negative or positive) will lead future
forecasts of the variance to be high for a prolonged period. This means the GARCH model will be a
better forecasting model than ARCH model in periods of high volatility.
The Ljung-box test shows that GARCH (1,1) has no ARCH effect in both currencies. After
estimating the GARCH (1,1) model asymmetric GARCH models have been estimated for both
USD/TRY and TYR/EUR return on exchange rates series. For that purpose The Engle and NG test has
been conducted. The Engle and NG test is a good way of determining whether there is sign bias or size
bias present in the volatility of the returns of the variables. If any significant value is found from the
test, this is a good justification for estimating some asymmetric GARCH models afterwards.
Table 10
Results of Engle and NG Test – TRY Versus USD and EUR
Engle and NG Test
Variable
Coefficient
Std. Error
t-Statistic
P-value
USD/TRY
C
-6.082912
1.354203
-4.491874
0.0000
4.395050
1.940208
2.265247
0.0236
-0.014654
0.001707
-8.587342
0.0000
0.025697
0.001357
18.94095
0.0000
EUR/TRY
C
-5.386889
1.031686
-5.221442
0.0000
5.186443
1.502688
3.451443
0.0006
-0.011442
0.001379
-8.298124
0.0000
0.024156
0.001105
21.85540
0.0000
Source: Authors’ calculation
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
ISSN: 2548-0162 © 2021 Gazi Akademik Yayıncılık 12
The coefficient for has p-values of 0.0236 and for 0.0006 which demonstrates that it
is significant at 5% level. This is a strong indication of sign bias. Moreover, the coefficient for
and are both in USD and EUR statistics significant with p- values of 0.000.
This is a strong indicator of size bias. The test result in Table 10 serves as a good justification for
estimating GARCH models which allow for asymmetric volatility.
3.3.2 GJR GARCH (1,1) model
In asymmetric GJR GARCH model (1989), importantly, the coefficient is noticed negative of -
0.099352 and -0.156353 respectively; as shown in Table 8 and Table 9. This suggests that the leverage
effect is not present, and it is statistically significant with a p-value of 0.000. The other terms in the
conditional variance equation are statistically significant. In contrast, the ARCH LM test and the Q-
test both are not significant at level of 5% with heteroscedasticity and autocorrelation affecting the
series. Therefore, a weak argument can be made that the leverage effect is presented.
3.3.3 EGARCH (1,1) model
If is statistically significant and has a negative sign, this implies that a fall in returns results in
greater volatility than an increase in returns of the same magnitude (leverage effect).
The significance of the terms which is the ARCH term, has a p-value of 0.000. Therefore, the
size of the shock has a significant impact on the volatility of returns. (The leverage effect term) is
significant at a level of 5%, which encourages an argument that the sign of the shock has an impact of
volatility of returns. has a p-value of 0.000. Therefore, past volatility helps to predict future
volatility.
The signs of the terms which is positive, shows there is a positive relationship between the past
variance and the current variances in absolute value. This means the bigger the magnitude of the shock
to the variance, the higher the volatility. is also positive which indicates that both good and bad news
will increase the volatility of the small size evidence of leverage effect.
3.3.4 PGARCH (1,1) model
The Power GARCH (PGARCH) model of Ding, Granger, and Engle (1993) represents a flexible
alternative that allows a bigger range of power transformations rather than adopting the absolute value
or squaring the data as in other classical heteroskedastic models (Anѐ, 2006). In Table 8 and Table 9, α
is ARCH term, ϒ is Leverage term and is negative and approves leverage effect. β is GARCH term
and δ is Power term and is significant at 5% of the p-value of 0.000, like in GJR-GARCH model
which shows autocorrelated series in Q-test result.
Table 11
Results of Modelling Metrics USD/TRY
Modelling Metrics
GARCH(1,1)
GJRGARCH(1,1)
EGARCH(1,1)
P-GARCH (1,1)
Adjusted R-squared
0.003733
0.005454
0.005961
0.005685
Log likelihood
13178.17
13209.88
13205.93
13212.43
Akaike info criterion
-6.989476
-7.005774
-7.003676
-7.006598
Schwarz criterion
-6.972931
-6.987574
-6.985476
-6.986743
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
ISSN: 2548-0162 © 2021 Gazi Akademik Yayıncılık 13
Table 12
Results of Modelling Metrics EUR/TRY
Modelling Metrics
GARCH(1,1)
GJRGARCH(1,1)
EGARCH(1,1)
P-GARCH (1,1)
Adjusted R-squared
0.010911
0.009531
0.009472
0.009325
Log likelihood
13300.65
13335.25
13329.84
13337.8
Akaike info criterion
-7.052335
-7.070157
-7.06729
-7.070982
Schwarz criterion
-7.040759
-7.056927
-7.054059
-7.056097
Source: Authors’ calculation
Based on the results in Table 8 the ARCH LM test results for both EGARCH and PGARCH
models indicated rejection of the null hypothesis in a significant level of 5%. Thus, looking at the
results in Table 11 symmetric GARCH (1,1) and asymmetric GJR-GARCH (1,1) achieve the
requirements and has been marked as best volatility models for USD/TRY exchange rate return in this
paper based on highest Adjusted R2 and lower AIC and SC, with a predilection in GJRGARCH (1,1)
model over the GARCH (1,1). However, the literature stated that when it comes to model the
exchange rate volatility, symmetric GARCH models perform better than asymmetric GARCH models
(Arachchi, 2018).
Furthermore, in Table 9 as for EUR/TRY, EGARCH (1,1) has been eliminated as the p-value of the
ARCH LM test is below the significant level of 5% in which the null hypothesis is rejected for the
presence of heteroscedasticity. Among the remaining models Table 12 illustrated symmetric GARCH
(1,1) and both asymmetric models GJR-GARCH (1,1) and PGARCH (1,1) achieve the requirements
with highest Adjusted R2 and lower AIC and SC metrics and have been marked as best volatility
models for EUR/TRY exchange rate return in this paper.
3.4 Forecasting volatility
Ultimately the whole point of estimating all of these ARCH/GARCH models is to see whether
future returns of any kind of asset can be predicted or not. A good format for evaluating which model
is best three method has implemented; Root Mean Square Error test (RMSE), Mean Absolute Error
(MAE) and Mean Absolute Percentage Error (MAPE). The results of each test in both dynamic and
static forecasts are as follows;
Table 13
Forecasting the Best GARCH of USD/TRY Model of Volatility
Forecast Type
RMSE
MAE
MAPE
GARCH (1,1)
Dynamic
0.003210
0.002456
99.757690
Static
0.003160
0.002456
108.353400
GJR-GARCH (1,1)
Dynamic
0.003189
0.002418
90.522390
Static
0.003122
0.002376
92.186660
EGARCH (1,1)
Dynamic
0.003251
0.002437
89.772840
Static
0.003199
0.002409
96.034140
PGARCH (1,1)
Dynamic
0.003192
0.002419
89.775040
Static
0.003127
0.002381
92.434300
Almisshal, B. & Emir, M. G 2021; 7(1):1-16
ISSN: 2548-0162 © 2021 Gazi Akademik Yayıncılık 14
Table 14
Forecasting the Best GARCH of EUR/TRY Model of Volatility
Forecast Type
RMSE
MAE
MAPE
GARCH (1,1)
Dynamic
0.003898
0.003102
98.894920
Static
0.003738
0.003051
149.449900
GJR-GARCH (1,1)
Dynamic
0.003864
0.003075
98.414420
Static
0.003697
0.003007
142.631500
EGARCH (1,1)
Dynamic
0.003908
0.003123
116.042800
Static
0.003761
0.003076
164.079700
PGARCH (1,1)
Dynamic
0.003862
0.003074
98.381290
Static
0.003698
0.003007
140.545800
Source: Authors’ calculation
Looking at Table 13 and Table 14 of the tests’ results, RMSE, MAE and MAPE in dynamic and
static forecast types have been evaluated for the studied GARCH models. Although the results are
critically near to each other in both cases, the best model to obtain the best forecasting model is the
model with lowest RMSE, MAE and MAPE. In this case, GJR-GARCH (1,1) static with the lowest
RMSE ratio of 00.003122 and 00.003697 respectively for both USD and EUR and lowest MAE ratio
of 0.002376 and 0.003007 respectively for both USD and EUR, is the best GARCH model. In regards
to MAPE, it records the lowest ratio in EGARCH (1,1) model by 89.772840 and PGARCH (1,1)
model by 98.381290 for USD and EUR respectively. Actually, MAPE is quite well-known as a poor
accuracy indicator, as it tends to divide each error over demand one by one. As a result, high errors
during low-demand periods will significantly impact MAPE thus it is skewed. However, MAPE
causes strange forecast (Vandeput, 2019). That is the reason the paper has ignored MAPE results.
In Figure 4 and 5 the graphs show how accurate is the selected model (GARCH dynamic) in this
study for predicting future volatility in USD/TRY and EUR/TRY exchange rate returns.
-.004
-.002
.000
.002
.004
.006
.008
18 19 20 23 24 25 26 27 30 31
2019m12
GJRGARCHUSD11STATIC
Return on USDexrate
-.006
-.004
-.002
.000
.002
.004
.006
18 19 20 23 24 25 26 27 30 31
2019m12
Return on EURexrate
GJRGARCH11EURSTATIC
Fig. 4. Shows The Best Forecast Model Of 10 Days
Period for USD/TRY –GJR-GARCH(1,1)
Fig. 5. Shows The Best Forecast Model Of 10 Days
Period for EUR/TRY –PGARCH(1,1)
Source: Authors’ calculation
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