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VOLATILITY MODELING OF DAILY EXCHANGE RATE BETWEEN UNITED STATES DOLLAR AND NIGERIA NAIRA

VOLATILITY MODELING OF DAILY EXCHANGE RATE BETWEEN UNITED STATES DOLLAR AND NIGERIA NAIRA

VOLATILITY MODELING OF DAILY EXCHANGE RATE BETWEEN UNITED STATES DOLLAR AND NIGERIA NAIRA

     


                            

VENUE: UG COMPUTER LABBOURATORY

Abstract


Exchange rates are important financial problem that is receiving attention globally. This study investigated the volatility modeling of daily exchange rate between United States dollar and Nigeria naira using GARCH (1, 1), GJR-GARCH (1, 1), TGRACH (1, 1) and TS-GARCH (1, 1) models by using daily data over the period June 2000 to July 2011. The aim of the study is to determine volatility modeling of daily exchange rate between United States dollar and Nigeria naira. The results from all the models show that volatility is persistent (i.e. exceed 1) indicating GARCH (1, 1), GJR-GARCH (1, 1), and TS-GARCH (1, 1) models variances are not stationary but for TGARCH (1, 1) model (i.e. below 1) indicating the variance is stationary. The GJR-GARCH (1, 1) and TGARCH (1, 1) models show the existence of statistically significant asymmetry effect. The TGARCH (1, 1) and TS-GARCH (1, 1) models are found to be the best models because they have maximum likelihood and lower Akaike information criteria and Schwarz information criteria.   



















                                              

1.0   Introduction

Exchange rates currency is important financial problem that has received a great deal of interest from statisticians, actuaries and financial economists. The collapse of the Breton-Woods arrangement coming out of the 1973 world crisis gave way to a period of floating exchange rates throughout the developed and the third worlds (Esquivel and Larrain, 2002). Since then, the volatility modeling and forecasting in spot foreign exchange rates have attracted much attention and fostered a growing body of theoretical and empirical works to investigate the behavior of currency movements. In Nigeria, like most developing countries, volatility modeling and forecasting have attracted little attention for the simple reason that the foreign exchange market and the capital market are largely undeveloped (Hamadu and Ibiwoye, 2010). The traditional measure of volatility as represented by variance or standard deviation is unconditional and does not recognize that there are interesting patterns in asset volatility; e.g., time-varying and clustering properties. Researchers have introduced various models to explain and predict these patterns in volatility. Engle (1982) modeled the heteroskedasticity by relating the conditional variance of the disturbance term to the linear combination of the squared disturbances in the recent past. Bollerslev (1986) generalized the ARCH model by modeling the conditional variance to depend on its lagged values as well as squared lagged values of disturbance, which is called generalized autoregressive conditiona l heteroskedasticity (GARCH). Little or no work has been done on modeling exchange rate volatility in Nigeria particularly using GARCH models. The exchange rate volatility has implications for many issues in the arena of finance and economics. Such issues include impact of foreign exchange rate volatility on derivative pricing, global trade patterns, countries balance of payments position, government policy making decisions and international capital budgeting.

 

1.1 Statement of the problem

The major reason for this research is that the country depends heavily on imports from various countries as most industries in Nigeria import their raw materials from foreign countries. Apart from raw materials, there were massive importations of finished goods with the adverse consequences for domestic production. The implications of these problems mentioned above are enormous on the exchange rate. This study is therefore designed to determine volatility modeling of daily exchange rate between United States dollar and Nigeria naira with different types of GARCH family models.


1.2 Objectives of the Study

The objectives of this study; (I) Is to determine volatility modeling of daily exchange rate between United Sates dollar and Nigeria naira with different types of GARCH family models.

 (II) And to observe some stylize fact characteristic volatility of exchange rate time series, such as persistence, volatility clustering, leverage effect etc.


1.3 Scope and Limitation

The study is designed to make use of daily Exchange rate between United States dollar and Nigeria naira from 1st June, 2000 to 26th July, 2011. This range is chosen as the focus for this study. The data has gotten from the http://fxtop.com/en/historates.php-cached-Block all . The aim of which is to determine the volatility modeling of daily exchange rate between United States dollar and Nigeria naira with different types of GARCH family models.

                                            

2.0 Literature review 

Volatility modeling of exchange rate has many practical applications in statistics, economics and finance with wide discussion in the literature. The basic ARCH/GARCH models are frequently applied and quoted to describe the volatility in financial markets, such as, stock exchanges and foreign exchange markets. Hsieh (1989) used 10 years (1974 – 1983) of daily closing-bid prices, consisting of 2,510 observations, for five countries in comparison of US dollar to estimate the autoregressive conditionally heteroscedastic (ARCH) and generalized autoregressive conditionally heteroscedastic (GARCH) models along with the other modified/altered types of ARCH and GARCH. The findings of Hsieh (1989) proved that the standardized residuals from all the ARCH and GARCH models using the standard normal density were highly leptokurtic, and the EGARCH proved to fit the data, better than GARCH model. Mundaca (1991) modeled the NOK/US Dollar exchange rate through ARCH and GARCH models, the results of which supported that three out of four analyzed series fitted better through GARCH than the ARCH model. Alberg et al. (2006) investigated the forecasting performance of GARCH, EGARCH, GJR and APARCH models and found that the EGARCH model, which used a skewed Student-t distribution, produced significant results than any other model. 


 3.0 Methodology and the Source of data

The data to be used is classified under the secondary source of data having been obtained from http://fxtop.com/en/historates.php-cached-Block.all. The aim of which is to determine volatility modeling of daily exchange rate between United States dollar and Nigeria naira with different types of GARCH family models.


3.1 Fat Tails

Financial time series often exhibit leptokurtosis, meaning that the distribution of their returns is fat-tailed (i.e. the kurtosis exceed the kurtosis of a standard Gaussian distribution, see Mandelbrot, 1963, or Fama, 1965). When compared to the normal distribution, the empirical distribution of financial time series exhibits a fourth moment (kurtosis) higher than the normal value of 3 and therefore has fatter tails2.


3.2 Volatility Clustering

Volatility clustering is often observed (i.e. large changes tend to be followed by large changes and small changes tend to be followed by small changes; see Mandelbrot, 1963, for early evidence).  


3.3 Mean Reverting

Volatility clustering implies that volatility comes and goes. Thus a period of high volatility will eventually give way to more normal volatility and similarly, a period of low volatility will be followed by a rise Engle and Patton (2001). Mean reversion in volatility is generally interpreted as meaning there is a normal level of volatility to which volatility will eventually return.


3.4 Persistence of Volatility

. A shock in the volatility series seems to have very “long memory” and impact on future volatility over a long horizon. The Integrated GARCH (IGARCH) model of Engle and Bollerslev (1986) captures this effect but a shock in this model impacts upon future volatility over an infinite horizon and the unconditional variance does not exist for this model. Volatility is said to be persistence if today’s return has a large effect on the forecast variance many periods in the future. Engle et al, (2001).


3.5 Asymmetric and Leverage Effects 

However, stylized fact of financial volatility is that bad news (negative shocks) tends to have a larger impact on volatility than good news (positive shocks). In other words, volatility tends to be higher in a falling market than in a rising market. Based on this conjecture, the asymmetric news impact on volatility is commonly referred to as the leverage effect Engle (2001). Nelson (1991) proposed a GARCH-class model named Exponential GARCH that allows for asymmetric effects and therefore solves one of the important shortcomings of the symmetric models. While the GARCH model imposes the nonnegative constraints on the parameters3, EGARCH models the log of the conditional variance so that there are no restrictions on these parameters: Black (1976), chritie (1982), Nelson (1991), Glosten et al. (1993) and Engle an Ng (1993) all find evidence of volatility being negatively related to equity returns. 


3.6 ARCH (p) Model and its Properties   

Engle (2001) specifies that a good volatility model should be reflect and capture the stylized facts of asset returns. The simplest model for studying volatility in univariate time series is the Autoregressive Conditionally Heteroskedastic model of order p, denoted ARCH (p). The model was originally introduced by Engle (1982). For the time series {rt }, the ARCH(p) model specification is:

                                                                                                                          3.1

                                                                                                3.2

     ,   ut ~ IIN(0,1)                                                                                             3.3

                                                                                      3.4

where, = is the innovation /shock at day t and follows heteroskedastic error process

= Asset returns at day t

= Conditional mean of {}

= Volatility at day t i.e. Conditional variance

 = Squared innovation at day 

the time varying conditional variance is postulated to be a linear function of the past squared innovations. A sufficient condition for the conditional variance to be positive is that the parameters of the model satisfy the following constraint; >0, >0, >0…>0.


3.7 GARCH (p, q) Model and its Properties

In practice, it is often found that a large number of lags p, and a large number of parameters, are required to obtain a good model fit of ARCH (p) model. To circumvent this problem bollerslev (1986) proposed the generalized ARCH or GARCH (p, q) model, with the following formulation: 

                                                                          

                                                             3.5

where,  is the volatility at day t-I  


 for i= 1,…, P

 for i=1,…,q

 and  are as previously defined.

Under the GARCH (p, q) model, the conditional variance of,, depends on the squared innovations in the previous p periods, and the conditional variance in the previous q periods. The GARCH models are adequate to obtain a good volatility model fit for financial time series. Rearranging the GARCH (p,q) model by defining , it follows that 

                                                                         3.6

where, L is a backshift operator and 



which defines an ARMA (Max(q, p), q) model for  . By standard argument, the model is covariance stationary if and only if all the roots of   lie outside the unit circle. The ARMA representation in 3.6 allows for the use of time series techniques in the identification of the orders of p and q. For the sake of simplicity however, we are going to examine the GARCH (1, 1) model and investigate all the features of stylized facts exhibited by the model.


3.8 GARCH Model Extensions

In most 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 characteristics, and dynamics of a particular time series. Since bad news (negative shocks) tends to have a large impact on volatility than good news (positive shocks), hence here is need to talk about the GARCH extensions model and we have restricted our analysis to the more popular models of asymmetric volatility, Such as  TGARCH, GJR-GARCH, and TS-GARCH. 


3.8.1 TGARCH Model

Another GARCH Variant that is capable of modeling leverage effects is the threshold GARCH (TGARCH) model, which has the following form:

                                                                     3.7

where, 

= leverage effects coefficient. (if >0 it indicates presence of leverage effect.) That is depending on whether   is above or below the threshold value of zero,  has different effects on conditional variance: when  is positive, the total effects are given by  ; when  is negative, the total effects are given by  . So one would expect  to be positive for bad news to have larger impacts. This model also known as the GJR model (Glosten, Jagannathan and Runkle, 1993) proposed essentially the same model.

3.8.2 GJR-GARCH Model

The GJR-GARCH model is another volatility model that allows asymmetric effects. This was introduced by Glosten, Jagannathan and Runkle (1993). The general specification of this model is of the form:

                        3.8

where  is a dummy variable which takes the value of 1 when  is negative  and 0 when  it is positive. In this GJR-GARCH model, it is supposed that the impact of  on the conditional variance  differs when  is positive or negative. A nice aspect of the GJR model is that it is easy to test the null hypothesis of no leverage effects. In fact,  means that the news impact curve is symmetric, i.e. past negative shocks have the same impact on today’s volatility as positive shocks.


3.8.3 TS-GARCH Model

The TS-GARCH model developed by Taylor (1986) and Schwert (1990) is another popular model used to capture the information content in the thick tails, which is common in the return distribution of speculative prices. The specification of this model is based on standard deviations and is as follows:

                        3.9

3.9 Estimating GARCH Models

There are numbers methods uses in estimating the parameters of volatility models, among these popular methods include:

Maximum likelihood Estimation with Gaussian errors

 Kalman Filter Method

 Efficient Method of Moment

 Quasi-Newton Method

we implement the maximum likelihood method in the course of our research, when we assume Gaussian errors and Quasi maximum likelihood when we assume non Gaussian errors. This helps us to provide a general framework for the issue of estimating GARCH-type models and requires the need for the regression model with conditionally heteroskedastic error terms to be investigated. The model is given as:

                                                                                        3.10

In 3.10 above, contains a set of explanatory variables which determines the conditional mean of    i.e.  in principle  may contain fixed regressor variables or even stochastic terms, e.g. lagged dependent variables or lagged error terms or even both.  is an error term the distribution of which is specified conditional on an information set . The conditional distribution is assumed to be Gaussian in Order to make maximum likelihood estimation of the unknown parameters in 3.10 feasible.


3.9.1   Maximum Likelihood Estimation

 Suppose that the parameters of the GARCH process are  and the parameters of the regression function, which for simplicity is assumed to be linear, are. Then, under the assumption of normality, the model we want to estimate can be written as 

,                                                                                                 3.11

The density for observation t is therefore,

                                                                                   3.12 

Where, 

                                                                      3.13

3.13 is the standard normal density. 


3.10 Statistical Tests

Therefore, for all the models that have been stated earlier, there is need to test for heteroskedasticity and more importantly the covariance stationary of each of the series involved in order to obtain valid and accurate results. These tests and all others that will be implemented in the course of the research are introduced and explained as follows:


3.10.1 Augumented Dickey Fuller

This test was first introduced by Dickey and Fuller (1979) to test for presence of unit root(s). The regression model for the test is given as:

 

3.10.2 Hypothesis Testing:

  (The series contains unit root (s))

 (The series is stationary)


3.10.3    KPSS Test

The integration properties of a series   may also be investigated by testing:


That is, the null hypothesis that the data generating process (DGP) is stationary is tested against a unit root. Kwiatkowski, Philips, Schmidt and Shin (1992) have derived a test for this pair of hypotheses. If there is no linear trend term, they start from a DGP

    where,

is a random walk i.e.    and is a stationary process.


3.10.4   Portmanteau Test

A portmanteau test is a test used for investigating the presence of autocorrelation in time series. Number of lags u and h are predetermined.

Hypothesis:      i.e. all lags correlations are zero

 i.e. for at least one i= 1,…,h is is tested i.e. at least one lag with non zero 

correlations.




3.10.5   ARCH-LM Test

Before estimating a full GARCH model for a financial time series, it is usually good practice to test for the presence of ARCH effects in the residuals. This test is used to check if the error  (in ARCH (p) model) is truly a skedastic function. The regression is given thus:


where, 

are the coefficients of the regression and  is the intercept.

   There are no ARCH effects in the residuals under the null the LM statistic is distributed asymptotically as  statistic.


3.10.6   Test Statistic:  

The test is from Engle (1982) who derive a LM statistic for testing the null as , where n is the number of observation and   is from the regression. Reject the null if the p-value is less than critical value 


3.10.7 Structural Breaks Test

It has been shown by Diebold (1996) that breaks in the variance, which are not taken into account by the econometrician, will look as ARCH effects when the whole sample is used. In other words, it might be that for a sub sample the unconditional variance changes from say to and then back to the previous level. In this case to model the conditional variance as an ARCH model will be wrong thing to do. In this case, it is recommended to divide the sample and test for ARCH for the sub periods, if no ARCH effects are found for any of the sub periods but are found for the whole sample that is a clear indication of a break in the unconditional variance and not of ARCH effects. 


3.10.8   Jarque-Bera Test

Jarque and Bera (1987) have proposed a test for non-normality based on skewness and kurtosis of a distribution. The test checks the pair of hypothesis:

 and  the distribution is symmetry and hence normal.

  and the distribution is asymmetry and hence non-normal


3.11    Software’s for the Analysis

The software Gretl.1.9.7 and JMulTi 4.23 were used for the analysis of the research and they are free software’s designed to carter for multiple time series analysis. The package series was used to test for the stationarity and unit root in the data set. They also used to plot the graph of the actual exchange rate returns and autocorrelation function.


4.0 Data Analysis

This chapter intends to use volatility modeling of daily exchange rate between dollar/naira for the period June, 2000 to July, 2011 to (I) determine the volatility modeling of daily exchange rate between dollar/naira with difference type of GARCH family models. (11) to observe some characteristic of volatility modeling of exchange rate time series, such as persistence, volatility clustering, leverage effect etc. The data has gotten from the http://fxtop.com/en/historates.php.cached-Block .   

   

Fig 4.1 Graphical Representations of Daily Exchange Rate US Dollar /NG Naira


From figure 4.1 above indicate that the series contains a trend component which can be remove before modeling. To remove the trend we take the first difference of the logarithms of the distribution and the series are preferred in analysis of financial time series because they have attractive statistical property which is stationarity.  


Fig. 4.2 Daily Exchange Rate USD/NGN from 1st June, 200 to 26th July, 2011


Fig. 4.2 reveals that some periods are riskier than others. Also the risky times are scattered randomly and there is some degree of autocorrelation in the riskiness of financial returns. (i.e large changes tend to be followed by large changes and small changes tend to be followed by small changes is volatility clustering (Mendelbret, 1963) and is one of the stylized facts of volatility of financial time series. We also observed that the clustering of periods of volatility that is large movements followed by further large movements; the variance of exchange rate returns is not constant over time. This is an indication of shock persistence. (Engle and Patton, 2001) which is another stylized fact of volatility of financial time series.


Figure 4.3 Frequency Distribution of Exchange rate Between US dollar and NG Naira


The figure 4.3 clearly shows that the error may not likely normally distribute, at the same time are farther apart. And they are not spread and its can result in to higher kurtosis (kurtosis greater than 3 of normal distribution) which is another characteristics. In the other hand, if the distribution is not normal, the skewness will be different from zero and the distribution is asymmetrical ( i.e negative skewness or lepkurtic). But are times, the leverage effect of financial time series result from asymmetrical effect produced from the skewness of the distribution. In this context, negative shock increase predictable volatility in asset markets more than positive. A shock which increases the volatility of the market, increase the risk of holding the currency (Rabinson and Longmore, 2004).


4.1 Unit Root Test for the Exchange Rate 

The ADF statistic test the null hypothesis of unit root against the alternative of no unit root and the decision rule is to reject the null hypothesis when the value of the test statistic is less than the critical value. The KPSS statistic tests the null hypothesis of stationarity against the alternative of non stationarity and the decision rule is to accept the null hypothesis when the value of the test s  tatistic is less than the critical value. The results of the ADF and KPSS tests are in Table 4.1 below.

Table 4.1 Results of the Unit Root test for the Exchange Rate.

Critical Values



ADF Test Statistics:

            -45.6949

KPSS Test Statistics:

0.0284


1%

-3.96

0.216



5%

-3.41

0.146



10%

-2.57

0.119




Table 4.1 the ADF test statistic is greater than all the critical values in absolute value so the hypothesis of non-stationarity is rejected. And for KPSS test statistic is less than the critical value hypothesis is accept.



4.2    Testing for ARCH Effects

We have shown that the series are stationary, our aims here is to test for the ARCH effects in the residuals of the exchange rate and to establish if the ARCH effects are due to structural breaks or not.

Table 4.2 Results of ARCH LM test, assuming without structural breaks.

Test Statistic

P-Value (chi^2)



217.0407

0.0000



 

Table 4.3 Results of ARCH-LM Test for Structural Breaks


Sub period 1

01/06/2000-01/06/2003

Sub period 2

02/06/2003-31/12/2007

Sub period 3

01/01/2008-26/07/2011


Test Statistic

77.7535

22.6693

16.9394



P-Value (chi^2)

0.0000

0.0000

0.0002





From Table 4.2-4.3 result show that whether or not we assume structural breaks in exchange rate. There is evidence of ARCH effects for all the exchange rate across the groups and they can be modeled as conditional heteroscedastic model. The ARCH effects are not due to structural breaks  as there are ARCH effects across the sub groups.

4.3 Autocorrelation Function (ACF)


Having discovered, that the exchange rate could be modeled as conditional heteroskedasticity variance, the next is to examine autocorrelation function (ACF) to see the degree of correlation in the data points of the series. The one with higher degree of correlation will be the right candidate to model with. 


Fig. 4.4 ACF of the daily Exchange rate returns 


Fig 4.5 ACF of the Squared of daily Exchange rate returns

Fig 4.4-4.5 shows that all the exchange rates are serially correlated. However, compared to the autocorrelation in the exchange rate, autocorrelation is stronger in the squared of the exchange  rate. This indicates that there is substantial dependence in the volatility of the exchange rate. We formally confirm the presence of autocorrelation in the exchange rate and squared exchange rate series by portmanteau test.  

4.4    Jarque Bera Test for Normality.

To achieve the first objective of the research, we examine the characteristics of the unconditional distribution of the exchange rate. This will enable us to explore and explain some stylized facts embedded in the financial time series. Jarque Bera normality test is used to demonstrate this and the results are given in Table 4.5 below

Table 4.5 Jarque – Bera Test for Normality

Mean

0.000098319


Median

0.00000


Minimum

-0.12180


Maximum

0.12991


Std.dev.

0.009954


Skewness

0.3795


Kurtosis

42.9911


Jakue Bera Test

271377


P- value

0.0000



 

Table 4.5 the results indicate the positive mean of daily exchange rate between USD/NGN, and standard deviation appear to be higher which follow the introduction of market determine exchange rate. The skewness is positively skews relative to the normal distribution (0 for the normal distribution). This is an indication of a non symmetric series. The kurtosis is very much larger than 3, the kurtosis for a normal distribution. Skewness indicates non-normality, while relatively large kurtosis suggests that distribution of the exchange rate return series is leptokurtic ( i.e exhibit fat tail ), Jarkue-Bera normality test statistics, indicating that neither return series has normal distribution. 


4.7   Model Checking

Before the interpretation and use of the model, we are to look at some tests/plots to check whether this volatility models have adequately captured all of the persistence in the variance of returns. And in as much as the model is adequate, then standard residuals are uncorrelated as specified in the results below. 


  Fig. 4.6 GARCH (1, 1) Residuals^2 of the exchange rate returns.  


Figure 4.7 Show that there is no serial correlation observed in the residuals of the returns..

 

 Table 4.9 Results for no Remaining ARCH Effects in Residual

F- Test

P- value


0.00308

0.9511



From the table 4.9 above, the null hypothesis that there is no ARCH effects remaining at every lag is accepted since all the p- values are greater than 0.05 therefore the models are adequate.


Table 4.10 Results of ARCH LM Test for GARCH (1, 1) Residuals

            Lag

Test Statistics

P- value



1

1.2885

0.2568



2

1.4737

0.4786



3

1.5147

0.6789



4

1.5777

0.8128




The null hypothesis of no ARCH effects should be accepted as in table 4.10 and the conclusion is that the models are adequate.


Fig 4.7. Estimated conditional volatility of the exchange rate using GARCH (1, 1)


Fig 4.7 shows that the time series reverts quite quickly to its mean and is uniformly distributed.



Fig. 4.8. Estimated conditional volatility of the exchange rate using GARCH (1, 1)


Fig 4.8 clearly Show that, the estimated volatility function is skewed and thus reveals asymmetry as display a U-shaped structure. We can observe that downward movements in the daily Dollar/Naira exchange rate return are follows by higher volatilities than upward movements of the same size of changes. That is the abnormal return will have more volatility especially for the negative abnormal.


















4.8 Analysis of some Volatility Models

Table 4.11: Parameter Estimates of GARCH models for the Period, June 2000 – July 2011 

   

GARCH(1,1)

GJR-GARCH(1,1)

TGARCH(1,1)

TS-GARCH(1,1)








 


 











Persistence


  

()



  0.5576

(1.2791)







  0.9755

(0.0114)






1.5331


  

()



  1.1485

(8.7710)



-0.3312

(6.5431)



  0.9732

(0.0159)






1.9561

  

()



  0.2427

(0.1568)



-0.7970

(0.3044)



  0.9681

(0.1187)






0.81238

  

()



  0.2725

(0.1919)  



 




0.9647

(0.0149)





 

1.2372



AIC

-33337.26213

-33347.01232

-33392.98844

-33370.04158



SI C

-33305.70391

-33309.14245

-33355.11858

-33338.48336



LL

16673.63106

16679.50616

16702.49422

16690.02079



From table 4.9 show that coefficient is not statistically significant in the GARCH and GJR-GARCH models but significant at the 5% level in TGARCH and TS-GARCH models. This appears to show the presence of volatility clustering in TGARCH and TS-GARCH models. Conditional volatility for these models tends to rise (fall) when the absolute value of the standardized residuals is large (smaller). The coefficients of (a determinant of the degree of persistence) are statistically significant in all the models. The sum of    and in the GARCH model exceed 1. This appears to show that shocks to volatility are very high. The GJR-GARCH models that is + + (/2) exceed 1. This also appears to show that shocks to volatility are very high and the variances are not stationary under The GJR-GARCH model. But under TGARCH model + + (/2) is less than 1 showing persistent volatility in the TGARCH model. The sum of and  in the TS-GARCH model exceeds 1. This appears to show that shocks to volatility are very high and will remain forever as the variances are not stationary under TS-GARCH model. So however, in sum, the Nigeria exchange rate market is characterized by high volatility persistence. The   coefficient is asymmetry and leverage effects, are negative and statistically significant at the 5% level in the GJR-GARCH and TGARCH models. However, leverage effect will only exist >0 in the GJR-GARCH and TGARCH models and negative values of  in the GJR-GARCH and TGARCH models , the hypothesis of leverage effect is rejected for all models but   asymmetry effect is accepted for the GJR-GARCH and TGARCH models. The results from the asymmetry models rejected the hypothesis of leverage effect. That is the GJR-GARCH and TGARCH models show the existence of statistically significant asymmetry effect. The TS-GARCH and TGARCH models are found to be the best models. 








5.0 Summary and Conclusion

We use daily Exchange rate volatility model between United States dollar and Nigeria naira for the period 1st June, 2000 to 26th July, 2011. (I) is to determine volatility modeling of daily exchange rate between United States dollar and Nigeria naira with different types of GARCH family models. (II) And to observe some stylize fact characteristics volatility of    exchange rate, such as persistence, volatility clustering, leverage effect etc. The data has gotten from the http://fxtop.com/en/historates.php-cached-Block all . And we investigated that, the basic volatility of dollar/naira exchange rate using, GARCH (1, 1), GJR-GARCH (1, 1), TGARCH (1, 1) and TS-GARCH (1, 1) models. The results from all the models show that volatility is persistent (i.e. exceed 1) indicating GARCH (1, 1), GJR-GARCH (1, 1), and TS-GARCH (1, 1) models variances are not stationary but for TGARCH (1, 1) model (i.e. below 1) indicating the variance is stationary. The results from all the asymmetry models rejected the hypothesis of leverage effect. The GJR-GARCH (1, 1) and TGARCH (1, 1) models show the existence of statistically significant asymmetry effect. The TGARCH (1, 1) and TS-GARCH models are found to be the best models because they have maximum likelihood and lower Akaike and Schwarz information criteria.


 5.1 Recommendation 

From the results of the analysis, TGARCH (1, 1) and TS-GARCH (1, 1) models are recommended to be the best models because they have all the parameters of the variance being significant and they have  maximum likelihood, lower Akaike  and Schwarz information criteria. The research can serve as a step to observe the volatility modeling of the Nigeria exchange rate. Specified periods data can be tested by future researchers by developing new and more models to capture the effect and predictions of the volatility behavior of Nigeria exchange rate. 






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