currency volatility

7/21/2019 Currency Volatility

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The IUP Journal of Applied Finance, Vol. 21, No. 1, 201522 2015 IUP. All Rights Reserved.

Exchange Rate Volatility Estimation

Using GARCH Models,with Special Reference to Indian Rupee

Against World Currencies

This study is an attempt to estimate the dynamics (volatility) of Indian rupee instability against four major world

currencies, i.e., US dollar, pound sterling, euro and Japanese yen, using 3,340 daily observations over a period of 13

years from January 3, 2000 to September 30, 2013. This paper uses the Generalized Autoregressive Condit ionalHeteroskedastic (GARCH) models to estimate volatility (conditional variance) in the daily log rupee value. The

models include both symmetric and asymmetric that capture the most common stylized facts about rupee exchange

returns such as volatility clustering and leverage effect. It is evident from the findings that asymmetric models are

superior to symmetric models in providing a better fit for the exchange rate volatility because of leverage effect.

Krishna Murari*

* Assistant Professor, Department of Management, School of Professional Studies, Sikkim University, 6thMile,

Samdur, PO-Tadong, Gangtok, Sikkim 737102, India. E-mail: [email protected]

Introduction

The currency exchange rates volatility is among the most examined and analyzed economic

measures by the government. Recently, India had a big concern about rupee value with respect

to US dollar due to its all-time lowest (depreciated) value. On August 28, 2013, the Indianrupee touched up to 68.825 against the dollar. It is not only the rupee depreciation but also

rupee appreciation that is causing concern to the economic imbalance of the country. Ahmed

and Suliman (2011) pointed out the importance of currency exchange rate volatility because

of its economic and financial applications like portfolio optimization, risk management, etc.

It is a well-known fact that the exchange rate volatility is not observed directly. A number of

models have been developed to get the accurate estimate of the volatility. Out of these,

conditional heteroskedastic1models are frequently used. The foundation for building these

models is to make a good forecast of future volatility which would be helpful in obtaining a

more efficient portfolio distribution, better foreign exchange exposure management and

more accurate currency derivative prices.

Surrounded by these models, the Autoregressive Conditional Heteroskedasticity (ARCH)

model proposed by Engle (1982) and its extension, Generalized Autoregressive Conditional

Heteroskedasticity (GARCH) model by Bollerslev (1986) and Taylor (1986) are the first

1 A financial time series is said to be heteroskedastic if its variance changes over time, otherwise it is called

homoskedastic.


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23Exchange Rate Volatility Estimation Using GARCH Models,with Special Reference to Indian Rupee Against World Currencies

models that have become popular in enabling the analysts to estimate the variance of a series

at a particular point in time (Enders, 2004). Since then, there have been a great number ofempirical applications of modeling the conditional variance of a financial time series (Diebold

and Nerlolve, 1989; Nelson, 1991; Bollerslev et al., 1992; West and Cho, 1995; Engle and

Patton, 2001; Evans and Lyons, 2002; Shin, 2005; Charles et al., 2008; Jakaria and Abdalla,

2012; and Rossi, 2013). The focus of these studies was to design explicit models to forecast

the time-varying volatility of the series using past observations. The findings have been

applied successfully in the financial market research.

Many empirical studies have been done on modeling the exchange rate volatility by

applying GARCH specifications and their large extensions, but most of these studies have

focused on developed currencies, and to the best of our knowledge, there are no such practical

studies for estimating the volatility of Indian rupee against US dollar, pound sterling, euroand Japanese yen (world major currencies); therefore, the current paper attempts to fill this

gap. The main objective of this paper is to model exchange rate return volatility for Indian

Rupee (INR) by applying different univariate specifications of GARCH type models for daily

observations of the rupee log differenced exchange rate return series. The volatility models

applied in this paper include the GARCH(p, q), Exponential GARCH(p, q), Threshold

GARCH(p, q), and Power GARCH(p, q).

Data and Methodology

The time series data for rupee exchange rate against most monitored world currencies is used

for modeling volatility. The daily rupee exchange rate against US dollar, pound sterling, euro

and Japanese yen for the period January 3, 2000 to September 30, 2013 is used to estimate the

volatility, resulting in total observations of 3,339, excluding public holidays. These data

series have been obtained from one of the most reliable sources in India, i.e., RBI online

database. In this study, daily returns are the first difference in logarithm of closing prices of

rupee exchange rate of successive days.

Volatility

It is helpful to give a brief explanation of the term volatility before starting the description of

volatility models. Statistically, volatility is frequently measured by the standard deviation

which reflects the degree of fluctuations of the observed values from the mean. Generally,

volatility means the stretch of all possible outcomes of a variable. Sometimes, variance is alsoused as a volatility measure. In foreign exchange markets, we use the term volatility to reflect

the spread of currency returns over a time period. In this paper, we use the variance as a

measure of volatility.

Stylized Facts About Volatility of Exchange Rates

Mostly, financial time series, including exchange rate returns, are well known to show signs

of certain stylized patterns which are essential for proper model specification, estimation and


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The IUP Journal of Applied Finance, Vol. 21, No. 1, 201524

forecasting. Mandelbrot (1963) and Fama (1965) were pioneers in documenting the empirical

regularities regarding these series. Since then many researchers have found similar regularities

about the financial time series (Baxter, 1991; Guillaume et al., 1997; and Cont, 2001). Due to

a large body of empirical evidence, these regularities can be considered as stylized facts. The

most common stylized facts are the following:

Heavy Tails

When the distribution of foreign exchange return time series is compared with normal

distribution, heavy tails are observed in terms of excess kurtosis. The standardized fourth

moment for a normal distribution is 3, whereas for many financial time series, a value well

above 3 is observed (Cont, 2001). A similar observation is witnessed in the present study also

(see Table 1).

Volatility Clustering

Similar values/changes in the long run tend to accumulate and this is termed as volatility

clustering. In most of the time series, large and small values in the log-returns have a tendency

to occur in clusters. When volatility is high, it is likely to remain high, and when it is low, it

is likely to remain low. According to Engel and West (2005), volatility clustering is nothing

but accumulation or clustering of information. Volatility clustering is well evident in Figures

1 to 4.

Leverage Effects

The leverage effect refers to the negative correlation between an asset return and its volatility,

i.e., rising asset prices are accompanied by declining volatility and vice versa (Nelson, 1991;

Gallant et al., 1992; Campbell and Kyle, 1993; and Longmore and Robinson, 2004). In foreign

Table 1: Descriptive Statistics

Mean 77.66 45.96 57.38 46.84 0.01 0.01 0.02 0.01

Median 77.84 41.43 57.42 46.17 0.02 0 0.02 0

Maximum 106.03 72.12 91.47 68.36 3.68 5.76 4.15 4.02

Minimum 63.96 32.69 38.79 39.27 5.7 5.12 3.89 3.01

SD 6.89 9.70 9.76 4.19 0.64 0.83 0.69 0.44Skewness 0.28 1.01 0.03 1.31 0.43 0.20 0.04 0.28

Kurtosis 2.92 2.90 2.65 6.02 7.85 6.89 5.16 10.94

JB 45.62 564.75 17.8 2,221.66 3,376.36 2,125.5 652.03 8,822.54

Prob. 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Obs. 3,340 3,340 3,340 3,340 3,339 3,339 3,339 3,339

INR

per

Poun

d

Sterl

ing

(RPS)

INR

per

100

Japanese

Yen

(RJY)

INR

per

Euro

(RE)

INR

per

US

Do

llar

(RD)

Log

Difference

ofRPS(LRPS)

Log

Difference

ofRJY

(LRJ)

Log

Difference

ofRE(LRE)

Log

Difference

ofRD

(LRD)


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Figure 1: INR per US Dollar (RD) and Log Difference of RD (LRD)

70

60

50

40

300 02 04 06 08 10 12

6

4

2

0

2

4

RD

LRD

Figure 2: INR Against Euro (RE) and Log Difference of RE (LRE)

100

80

60

40

200 02 04 06 08 10 12

6

4

2

0

2

4

RE

LRE


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Figure 3: INR per 100 Japanese Yen (RJ) and Log Difference of RJ (LRJ)

80

70

60

50

40

300 02 04 06 08 10 12

6

4

2

0

2

4

6RJ

LRJ

Figure 4: INR per Pound Sterling (RPS) and Log Difference of RPS (LRPS)

110

100

90

80

70

600 02 04 06 08 10 12

4

2

0

2

4

6

RPS

LRPS


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exchange markets, it is repeatedly witnessed that currency depreciation is followed by higher

volatility.

Co-Movements in Volatility

Financial time series across different markets exhibit parallel fluctuations in terms of direction

of movements; for example, an upward movement in stock returns in Bombay Stock Exchange

(BSE) being matched by an upward movement in National Stock Exchange (NSE). These co-

movements in rupee exchange rate volatility are also observed in Figures 1 to 4.

Calendar Effects

Generally, the volatility of asset returns or exchange rate returns are lower during weekends

and holidays compared to normal trading days. The most common calendar anomalies that

affect the exchange rate volatility are the January effect and the day-of-the-week effect.

Many studies attribute this phenomenon to the accumulative effects of information during

weekends and holidays (Miller, 1984; Theobald and Price, 1984; Abraham and Ikenberry,

1994; Kaur, 2004; and Cai et al., 2006).

Volatility Estimation Models

Based on the literature reviewed, the volatility models can be divided into two main groups:

symmetric and asymmetric. The symmetric models reflect that the conditional variance

depends on the magnitude only, while the shocks of the same magnitude, positive or negative,

have different effect on future volatility in the class of asymmetric models.

Symmetric GARCH Model

ARCH models are specifically designed to model and forecast conditional variances. UnderGARCH modeling, the variance of the dependent variable is modeled as a function of past

values of the dependent variable.

The GARCH(p, q) Model: Higher order GARCH models, denoted by GARCH(p, q), can

be estimated by choosing eitherp or qgreater than 1, wherep is the order of the autoregressive

GARCH terms and qis the order of the moving average ARCH terms. The representation of

the GARCH(p, q) variance is:

q

j jtj

p

i itit 1

2

1

22 ...(1)

We begin with the simplest GARCH (1, 1) specification:

ttt XY ...(2)

21

21

2 ttt ...(3)

where

= Constant term;


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21t (the ARCH term) = news about volatility from the previous period, measured as the

lag of the squared residual from the mean equation; and

2

1t (the GARCH term)= last periods forecast variance.

Asymmetric GARCH Models

The responsiveness of the conditional variance to rises and falls in asset return is an important

phenomenon in volatility estimation and the symmetric GARCH models fail in this regard.

Further, the symmetric GARCH model discussed above cannot provide explanation to the

leverage effects experienced in the exchange rate returns, therefore, a number of models have

been introduced to deal with this observable fact. These models are called asymmetric models.

In this paper, we use TGARCH, EGARCH and PGARCH models for tracing the asymmetric

phenomena.

The Threshold GARCH (TGARCH) Model:The generalized specification for the

conditional variance under TGARCH (Glosten et al., 1993; and Zakoian, 1994) is given by:

kt

r

k ktk

q

j jtj

p

i ititI 1

2

1

2

1

22 ...(4)

where It= 1 if

t< 0 and 0 otherwise.

In this model, good news, ti

> 0, and bad news, ti

< 0, have differential effects on the

conditional variance; good news has an impact of i, while bad news has an impact of

i+

i.

If i> 0, it means increased volatility due to bad news, and it is believed that there is a

leverage effect for the ithorder. If i

0, there is asymmetric effect.

The Exponential GARCH (EGARCH) Model: The EGARCH model was proposed by

Nelson (1991). In this model, the asymmetric responses of variance to shocks are captured

and at the same time the positivity of variance is also ensured. The specification for the

conditional variance is:

kt

ktr

k k

q

j jtjit

itp

i it

11

2

1

2 )log()log( ...(5)

Here it should be noted that the left-hand side takes the log of the conditional variance

over time which implies the exponential nature of the leverage effect. Further, the estimates

for the conditional variance are positive. The existence of leverage effects can be checked bythe hypothesis that i< 0. The impact is asymmetric if

i0. The sign of is anticipated to be

positive in most practical cases.

The Power GARCH (PGARCH) Model: Taylor (1986) and Schwert (1989) introduced

another class of asymmetric GARCH models, where instead of modeling for variance, the

standard deviation is modeled and is called the standard deviation GARCH model. The

power constraint of the standard deviation can be projected and the optional parameters

are added to capture irregularity of up to order r:


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)|(|11 itiit

p

i ijt

q

j jt ...(6)

where > 0, for 1|| i for i = 1, ,r, for all i > r, and pr .

The symmetric model puts i= 0 for all i. Here it is worth noting that if = 2 and

i= 0

for all i, the PGARCH model is simply a standard GARCH specification. As in the prior

models, the asymmetric effects are present if 0.

Results and Discussion

The descriptive statistics of the rupee value and its first log difference against pound sterling

(RPS, LRPS), 100 Japanese yen (RJY, LRJ), euro (RE, LRE) and US dollar (RD, LRD) are

depicted in Table 1. During the study period, the rupee value was minimum, i.e., 63.96, 32.69,

38.79 and 39.27, against Pound Sterling (PS), Japanese Yen (JY), Euro (E) and US Dollar (D),respectively, whereas the rupee value was maximum against PS, i.e., 106.03. The mean of

rupee exchange return series varied to a greater extent against euro with standard deviation

of 9.76. The skewness for the log difference of exchange return series, i.e., LRPS, LRJ, LRE and

LRD are 0.43, 0.2, 0.04 and 0.28, respectively. In a standard normal distribution, skewness

is zero. A positive and negative value of skewness in the log return series shows asymmetry.

So, in our result, LRPS and LRE show a negatively skewed distribution and LRJ and LRD

show a positively skewed distribution, i.e., asymmetric data series. In a standard normal

distribution, kurtosis is 3. A value lesser or greater than 3 kurtosis coefficients indicates

flatness and peakedness of the data series. The kurtosis of all the series is different from the

standard value, except for RPS (2.92) and RJY (2.9) which is very close to 3. The higher value

of kurtosis for LRPS, LRJ, LRE and LRD shows that the data series is peaked, moreover LRDdata series is highly peaked with kurtosis 10.94 as compared to normal distribution. Table 1

also shows that the Jarque-Bera (JB) test of normality for all the data series rejects the null

hypothesis of normality at 1% significant level.

The visual inspection of the rupee value and log difference against the selected currencies

is depicted in Figures 1 to 4. From the figures, it is clearly visible that the high volatility or

frequent changes in LRPS, LRJ, LRE and LRD data series show the clustering.

Testing for Stationarity of the Series

To investigate whether the daily rupee value against PS, E, D and JY and their first log

difference are stationary series, the Augmented Dickey-Fuller (ADF) test (Dickey and Fuller,

1979), Philips-Perron (PP) test (Phillips and Perron, 1988) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test (KPSS, 1992) have been applied to confirm the results about the

stationarity of the series. The results of the unit root test are shown in Table 2. The ADF and

PP test statistics for LRD, LRPS, LRE and LRJ are significant at 1% level, thus rejecting the

null hypothesis of the presence of unit root in the data. Similar results are obtained with

KPSS test under the null hypothesis of absence of unit root in the series. ADF and PP tests

with null hypothesis of unit root and KPSS test with null hypothesis of no unit root in the

series confirm that LRD, LRPS, LRE and LRJ are stationary at levels itself.


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LRD 42.62 0.00 57.01 0.00 1.43 0.15

LRPS 42.82 0.00 56.94 0.00 0.98 0.33

LRE 58.76 0.00 58.76 0.00 1.64 0.10

LRJ 59.85 0.00 59.93 0.00 0.85 0.40

Table 2: Unit Root Test

Series

ADF Test

t-Statistics p-Value

PP test

Adj.

t-Statistics

p-Value

KPSS

Test

Statistics

p-Value

(Null: Unit Root) (Null: Unit Root) (Null: No Unit Root)

Testing for Heteroskedasticity

We cannot use homoskedastic model to estimate volatility. Thus, before modeling the

volatility of rupee exchange log return series against major currencies, testing for the

heteroskedasticity in residuals is necessary. At the beginning, we obtain the residuals from an

Autoregressive and Moving Average (ARMA) process as specified by Equation (7). This

model represents that the present value of a time series depends upon its past values, which

is the autoregressive component, and on the preceding residual values, which is the moving

average component (Murari, 2013). The ARMA(p, q) model can be presented in the following

general form:

11011 tttt YY ...(7)

where

Yt

is the dependent variable at time t;

is the constant term;

0and

1are residual coefficients;

Yt1

is the lagged dependent variable;

1is the regression coefficient;

tis the residual term; and

t1

is the previous values of the residual.

ARCH-LM Test

Once the residuals from ARMA(1, 1) are obtained, the existence of heteroskedasticity inresiduals of log exchange rate return series is checked using Engles Lagrange Multiplier (LM)

test for ARCH effects (Engle, 1982). This particular heteroskedasticity specification was

motivated by the observation that in many financial time series, the magnitude of residuals

appeared to be related to the magnitude of recent residuals (Chakrabarti and Sen, 2011).

However, ignoring ARCH effects may result in loss of efficiency.

The ARCH LM test-statistic is calculated from a supporting test regression. To test the null

hypothesis that there is no ARCH up to order qin the residuals, we use the following regression:


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tst

q

st

v 2

1 00

2 ...(8)

where is the residual.

In the above equation, the squared residuals are regressed on lagged squared residuals up

to order q. We use two (F-statistics and Engles LM test statistic) test statistics from this test

regression.

Table 3 presents the results of heteroskedasticity test LM to check for the presence of

ARCH effect in the residual series of LRD, LRPS, LRE and LRJ at lag 1. From the table, we

infer that for all the log rupee exchange return series, both F-statistics and LM statistics are

significant at 1% level in the first lags. The zero p-value indicates the presence of ARCH

effect. Based on these results, we reject the null hypothesis of absence of ARCH effects

(homoskedasticity) in residual series of log rupee exchange return series. These results suggest

that the log return series of Indian rupee-against US dollar, pound sterling, euro and Japanese

yen have the presence of ARCH. This observation directs us to estimate the exchange rate

volatility using different classes of GARCH models.

Table 3: ARCH-LM Test Results

SeriesARCH

F-Statistics

Prob.

F(1,3336)LM-Statistics

Prob.

2 (1)

LRD 227.74 0.00 213.31 0.00

LRPS 49.19 0.00 48.50 0.00

LRE 64.87 0.00 63.67 0.00

LRJ 325.87 0.00 297.05 0.00

Symmetric Model: GARCH(p, q)

Table 4 shows the results of GARCH(p, q) model used for estimating the daily foreign exchange

rate volatility of Indian rupee against four major currencies of the world for the sample period

ranging from January 3, 2000 to September 30, 2013.

From the table we can see that all the coefficients of rupee against different currencies,

(Constant), (ARCH effect) and (GARCH effect), in the sample period are statistically

significant at 1% level. GARCH(1, 1) model for series LRD and LRJ showed the presence of

further ARCH in residuals. Thus, we estimated GARCH(2, 1) model for LRD and LRJ whichshows the significant ARCH-LM test statistics (Table 4). The lagged conditional and squared

variances have impact on the volatility and it is supported by both ARCH Term () and

GARCH Term() which is significant.

The highly significant (ARCH effect) in the sample period evidenced the presence of

volatility clustering in GARCH(1, 1) model in LRPS and LRE series. It also indicates that

the past squared residual term (ARCH term) is significantly affected by the volatility risk in

Indian rupee against pound sterling and euro. The coefficient of (GARCH effect) also


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shows highly statistical significance for rupee exchange rate against major world currencies.

It indicates that the past volatility of Indian foreign exchange rate is significantly influencing

the current rupee volatility.

The sum of coefficients of ARCH term and GARCH termand(persistent coefficients)

in GARCH(p, q) model reported in Table 4 are near to one for all the series, suggesting that

shocks to the conditional variance are highly persistent, i.e., the conditional variance process

is volatile. This shows that the volatility clustering phenomenon is implied in exchange rate

return series.

Asymmetric GARCH Models

Threshold GARCH/TGARCH(p, q)

The TGARCH model used to test leverage effect or asymmetry in the daily foreign exchange

rate volatility of Indian rupee against US dollar, pound sterling, euro and Japanese yen is

shown in Table 5. The estimated results of coefficients in TGARCH(p, q) model for the series

LRD, LRPS, LRE and LRJ are statistically significant at 1% and 5% levels of significance.

In the case of asymmetric term or leverage effect (), a statistically significant value suggests

that there exists the leverage effect and asymmetric behavior in daily Indian rupee exchange

rate against US dollar, pound sterling, euro and Japanese yen. Further for all the series, theleverage effect term shows a negative sign, indicating that positive shocks (good news) have

large effect on next period volatility than negative shocks (bad news) of the same sign or

magnitude. TGARCH(1, 1) modeling for LRD and LRJ shows the presence of further ARCH

in the residuals, thus we use TGARCH (2, 1) model in order to achieve better results. The

ARCH-LM test statistics for LRD and LRJ at TGARCH(2, 1) and for LRPS and LRJ at

TGARCH(1, 1) did not exhibit additional ARCH effect. This shows that the variance

equations are well precise.

Table 4: Estimation Results of GARCH(p, q) Model

LRD LRPS LRE LRJ

GARCH

(1, 1)

GARCH

(2, 1)

GARCH

(1, 1)

GARCH

(1, 1)

GARCH

(1, 1)

GARCH

(2, 1)

Constant () 0.000342 0.00015 0.00505 0.00400 0.019517 0.01295

ARCH Effect (1) 0.273045* 0.39708* 0.05143* 0.04942* 0.100899* 0.17630*

ARCH Effect (2) 0.23775* 0.10819*

GARCH Effect () 0.7846* 0.87002* 0.93588* 0.94305* 0.871744* 0.91288*

i+

j1.057645 1.02935 0.98731 0.99247 0.972643 1.08918

Residual Diagnostics: ARCH-LM Test

F-Statistic 12.35045 0.603694 0.602994 0.00669 17.3168 2.598767Prob.F(1,3336) 0.0004 0.4372 0.4375 0.9348 0 0.107

Note: * indicates that the coefficients are significant at 1% level.


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Exponential GARCH/EGARCH(p, q) Model

Table 6 presents the estimated results of EGARCH(p, q) model to test the asymmetric behavior

of Indian foreign exchange rate against major world currencies.

Note: * and ** indicate that the coefficients are significant at 1% and 5% levels, respectively.

Table 5: Estimation Results of TGARCH(p, q) Model

LRD LRPS LRE LRJ

TGARCH

(1, 1)

TGARCH

(2, 1)

TGARCH

(1, 1)

TGARCH

(1, 1)

TGARCH

(1, 1)

TGARCH

(2, 1)

Constant () 0.000342 0.000147 0.004964 0.004177 0.021446 0.014193

ARCH Effect (1) 0.306522* 0.39711* 0.05690* 0.06106* 0.120594* 0.18210*

ARCH Effect (2) 0.00301** 0.02535**

Leverage Effect () 0.06619* 0.23511* 0.01110* 0.02503* 0.04014* 0.10183*

GARCH Effect () 0.783613* 0.86903* 0.93635* 0.943543* 0.867794* 0.910441*

i+

j1.090135 1.26614 0.99325 1.00460 0.988388 1.09254

Residual Diagnostics: ARCH-LM TestF-Statistic 11.66325 0.633212 0.652476 0.013669 22.14605 3.376216

Prob. F(1,3336) 0.0006 0.4262 0.4193 0.9069 0 0.0662


All the parameters presented in the table are statistically significant at 1% and 5% levels.

The significance of EGARCH term () indicates the presence of asymmetric behavior of

volatility of Indian rupee against US dollar, pound sterling, euro and Japanese yen. The positive

coefficients of EGARCH term suggest that the positive shocks (good news) have more effect

on volatility than that of negative shocks. The null hypothesis of no heteroskedasticity in

Table 6: Estimation Results of EGARCH(p, q) Model

LRD LRPS LRE LRJ

EGARCH(1, 1)

EGARCH(2, 1)

EGARCH(1, 1)

EGARCH(1, 1)

EGARCH(1, 1)

EGARCH(2, 1)

Constant () 0.3515 0.26373 0.10255 0.09471 0.12811 0.10544

ARCH Effect (1) 0.440692* 0.57298* 0.11638* 0.10868* 0.158255* 0.30817*

ARCH Effect (2) 0.23452** 0.17716**

Leverage Effect () 0.03715* 0.027829* 0.009054* 0.021398* 0.014956* 0.017479*

GARCH Effect () 0.987516* 0.993237* 0.98770* 0.98836* 0.986088* 0.990482*

i+

j1.428208 1.56622 1.10408 1.09704 1.144343 1.29865

Residual Diagnostics: ARCH-LM Test

F-Statistic 26.94159 1.483617 0.239194 0.023062 110.6067 1.706964

Prob. F(1,3336) 0 0.2233 0.6248 0.8793 0 0.1915


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the residuals is accepted for LRPS and LRE, but not for LRD and LRJ in EGARCH(1, 1)

model.

Power GARCH/PGARCH(p, q) Model

The results of modeling the standard deviation (PGARCH) rather than modeling of variance

as in most of the GARCH family models are presented in Table 7.

It is evident from Table 7 that the estimated coefficients are significant and negative for

all the exchange rate return series in PGARCH(1, 1) model, indicating that negative shocks

are associated with higher volatility than positive shocks. The ARCH-LM test statistics did

not exhibit additional ARCH effect for LRPS and LRE under PGARCH(1, 1) model, but

LRD and LRJ witnessed the presence of further ARCH in the residuals of the model.

Thus, PGARCH (2, 1) model is estimated to eliminate the presence of ARCH effect where

null hypothesis of no ARCH is accepted. This shows that the variance equations are well

specified.

Conclusion

Exchange rate volatility estimation is considered as an important concept in many economic

and financial applications like currency rate risk management, asset pricing, and portfolio

allocation. This paper attempts to explore the comparative ability of different statistical and

econometric volatility forecasting models in the context of Indian rupee against US dollar,

pound sterling, euro and Japanese yen. Four different models were considered in this study.

The volatility of the rupee exchange rate returns has been modeled by using univariate

GARCH models. The study includes both symmetric and asymmetric models that capture

the most common stylized facts about currency returns such as volatility clustering and

Table 7: Estimation Results Using PGARCH(p, q)

LRD LRPS LRE LRJ

PGARCH

(1, 1)

PGARCH

(2, 1)

PGARCH

(1, 1)

PGARCH

(1, 1)

PGARCH

(1, 1)

PGARCH

(2, 1)

Constant 0.000342 0.000141 0.002933 0.005614 0.025614 0.016315

ARCH Effect (1) 0.272314* 0.395929* 0.030942* 0.055991* 0.114605* 0.173314*

ARCH Effect (2) 0.00175* 0.04097**

Leverage Effect () 0.06084* 0.23595* 0.04739** 0.15821* 0.09243* 0.09735*

GARCH Effect () 0.783649* 0.869056* 0.935482* 0.944103* 0.870101* 0.913422*

Power Parameter 2.000885* 2.013997* 3.030564* 1.400777* 1.457562* 1.498009*

i+

j1.055963 1.264985 0.966424 1.000094 0.984706 1.086736

Residual Diagnostics: ARCH LM Test

F-Statistic 11.66511 0.605952 0.463149 0.003725 41.81468 9.529872

Prob. F(1,3336) 0.0006 0.4364 0.4962 0.9513 0 0.002



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leverage effect. These models are GARCH(1, 1), EGARCH(1, 1), TGARCH(1, 1) and

PGARCH(1, 1) for log difference of rupee exchange rate return series against pound sterling

and euro and GARCH(2, 1), EGARCH(2, 1), TGARCH(2, 1) and PGARCH(2, 1) for log

difference of rupee exchange rate return series against US dollar and Japanese yen.

GARCH(1, 1) and GARCH(2, 1) models are used for capturing the symmetric effect, whereas

the other group of models for capturing the asymmetric effect. The paper finds strong evidence

that daily rupee exchange returns volatility could be characterized by the above-mentioned

models. For all series, the empirical analysis was supportive of the symmetric volatility

hypothesis, which means rupee exchange rate returns are volatile and that positive and

negative shocks (good and bad news) of the same magnitude have the same impact and effect

on the future volatility level. The parameter estimates of the GARCH(p, q) models indicate

a high degree of persistence in the conditional volatility of exchange rate returns of rupee

against world major currencies which means an explosive volatility.

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Reference # 01J-2015-01-02-01


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C o p y r i g h t o f I U P J o u r n a l o f A p p l i e d F i n a n c e i s t h e p r o p e r t y o f I U P P u b l i c a t i o n s a n d i t s

c o n t e n t m a y n o t b e c o p i e d o r e m a i l e d t o m u l t i p l e s i t e s o r p o s t e d t o a l i s t s e r v w i t h o u t t h e

c o p y r i g h t h o l d e r ' s e x p r e s s w r i t t e n p e r m i s s i o n . H o w e v e r , u s e r s m a y p r i n t , d o w n l o a d , o r e m a i l

a r t i c l e s f o r i n d i v i d u a l u s e .

currency volatility

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