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Generalised Asymmetric Power ARCH Modeling of Exchange Rate Volatility

Michael McKenzie and Heather Mitchell*

Department of Economics and Finance Royal Melbourne Institute of Technology GPO Box 2476V Melbourne 3001

____________________________________________________________ Abstract

This paper considers the ability of the Power ARCH model introduced by Ding, Granger and Engle (1993) to capture the stylised features of volatility in 17 heavily traded bilateral exchange rates. This Power ARCH model nests a number of models from the ARCH family. The relative merits of these nested ARCH models can be considered using the standard log likelihood ratio test. The results of this paper suggest that in the presence of symmetric responses to innovations in the market, the GARCH(1,1) model is preferred. Where asymmetry is present, than the inclusion of a leverage term is worthwhile as long as a power term is also included. KEYWORDS : Exchange Rate Volatility, Power GARCH. The authors would like to thank Robert Faff and Robert Brooks for their invaluable comments on an earlier version of this paper and Sveta Risman for her excellent research assistance.

*

Corresponding author : Email - [email protected]

1

INTRODUCTION

The behaviour of speculative price series has attracted the attention of researchers for nearly 100 years1. Mandelbrot (1963) and Fama (1965) provided the first generally accepted evidence which suggests that the distribution of such asset prices are characterised by a number of stylised ‘facts’ such as kurtosis and heteroskedasticity. Most importantly, asset returns are approximately uncorrelated but not independent through time as large (small) price changes tend to follow large (small) price changes. This temporal concentration of volatility is commonly referred to as ‘volatility clustering’ and it was not fully exploited for modeling purposes until the introduction of the ARCH model by Engle (1982). The ARCH model was unique in that it specified the variance of the error term in a regression equation as conditional on squared past errors. Hence, volatility in the ARCH model will exhibit periods of relative tranquillity and volatility effectively capturing this volatility clustering characteristic so common to economic and financial time series data.

The ARCH literature has developed so rapidly that it is hard to believe that it is only 15 years since the release of Engle’s seminal paper. Yet, there currently exists a veritable family of ARCH models incorporating the original ARCH model of Engle, the generalised ARCH (GARCH) model of Bollerslev (1986) as well as a host of other suitably acronymed models (see Bollerslev, Engle and Nelson (1994) or Bera and Higgins (1993) for a survey). Each of these subsequent contributions to the ARCH family have concentrated on refining both the mean and variance equations to better capture the stylised characteristics of the data.

One recent development in the ARCH literature, has focussed on the power term by which the data is to be transformed. The presence of volatility clustering is by no means unique to the squared returns of an asset’s price. In general, the absolute changes in an assets price will exhibit volatility clustering and the inclusion of a power term acts so as to emphasise the periods of relative tranquillity and volatility by magnifying the outliers in that series. It is possible to specify any power term to 1

Bachelier conducted a study of speculative asset prices in 1900 titled “Theory of Speculation” . An English translation of this work may be found in Cootner, P.H. Ed. (1964) “The Random Character of

2

complete this task from a myriad of options inclusive of any positive value. The common use of a squared term in this role is most likely a reflection of the normality assumption traditionally invoked (or at least seriously entertained) regarding the data. If a data series is normally distributed than we are able to completely characterise its distribution by its first two moments. As such, it is appropriate to focus on a squared term and hence a measure of the variance. However, if we accept that the data may have a non-normal error distribution, than one must transcend into the realm of the higher moments of skewness, kurtosis and beyond to adequately describe the data. In this instance, the superiority of a squared term is lost and other power transformations may be more appropriate. In fact, for non-normal data, by squaring the returns one effectively imposes a structure on the data which may potentially furnish sub-optimal modelling and forecasting performance relative to other power terms.

Recognising the possibility that a squared power term may not necessarily be optimal, Ding, Granger and Engle (1993) introduced a new class of ARCH model called the Power ARCH model2. Rather than imposing a structure on the data, the Power ARCH class of models estimates the optimal power term. Thus, this model permits a virtually infinite range of transformations inclusive of any positive value. This includes the standard class of ARCH model which specifies the use of a squared term and also the Taylor (1986) GARCH model which relates the conditional standard deviation as a function of past lagged absolute residuals and standard deviations. Ding, Granger and Engle (1993) specified a generalised asymmetric version of the Power ARCH model and applied it to U.S. stock returns data. The authors found that the model provided a good fit of the data and the optimal power term was 1.43. Hentschel (1995) proposed a more general class of Power ARCH model and also applied it to U.S. stock market data where the optimal value for the power term was found to be 1.524.

Stock Market Prices” Cambridge, Mass : M.I.T. Press. 2 Ding Granger and Engle (1993) also found that the absolute returns and their power transformations have a highly significant long-term memory property as the returns are highly correlated. For example, significant positive autocorrelations were found at over 2,700 lags in 17,054 daily observations of the S&P 500. As such, the long memory ARCH model introduced by Ding and Granger (1996) and the fractionally integrated GARCH (FIGARCH) model introduced by Ballie, Bollerslev and Mikkelsen (1996) may be viewed as closely related to the PARCH model.

3

The applicability of the Power ARCH class of model to stock market data has been well documented in papers such as Ding et al (1993) and Hentschel (1995). Yet, little is known about the applicability of this type of model to other types of equally as important financial asset price data such as exchange rates. One cannot immediately assume that the conclusions drawn for Power ARCH models fitted to stock market data also apply to exchange rate data. This is because the stylised characteristics of these two types of data differ in one very important respect. Stock market returns data commonly exhibits an asymmetry in that positive and negative shocks to the market do not bring forth equal responses.

This fact is most commonly attributed to the

‘leverage’ effect (see inter alia Black (1976)). However, such asymmetry is generally not present in exchange rate data and a valid theoretical reason for its existence is yet to be proposed. This difference in the innovations between the two markets may mean that whilst Power ARCH models suit the stock market data well, computationally less complex models from the ARCH family may provide a better fit for exchange rate data.

The purpose of this paper is to consider the applicability of the Power ARCH class of models to currency returns. The standard class of ARCH models have certainly been extensively applied to exchange rates data3 and it is worthwhile to consider whether allowing the data to select the term by which it is transformed, enhances the model. Tse and Tsui (1997) applied the APGARCH model of Ding et al (1993) to daily Malaysian - U.S. and Singapore - U.S. exchange rate data. The authors find that the model adequately describes the data and the optimal power term was found to be some 3

ARCH models have been extensively used to capture the dynamic properties by which exchange rate markets evolve (see inter alia Bollerslev (1987), Hsieh (1988, 1989), Baillie and Bollerslev (1989), Diebold and Nerlove (1989), Bollerslev (1990), Engle and Gonzalez-Rivera (1991), Mundaca (1991), Higgins and Bera (1992), Drost and Nijman (1993), Bollerslev and Engle (1993), and Byers and Peel (1995)). ARCH models have also been applied to exchange rate data in an attempt to gain some insight into a policy or theoretical issue. For example, the exchange rate volatility and trade flows issue has used ARCH models to generate estimates of volatility (see Asseery and Peel (1991), Pozo (1992b), Kroner and Lastrapes (1993), Gagnon (1993), Qian and Varangis (1994), Caporale and Doroodian (1994), McKenzie and Brooks (1997) and McKenzie (1998)). Engle, Ito and Lin (1990) used an ARCH model to investigate the efficiency of the foreign exchange market. Diebold and Pauly (1985), McCurdy and Morgan (1988) and Baillie and Bollerslev (1990) used ARCH models to investigate the stability of foreign exchange risk premium. Other macroeconomic issues such as the fixed verses floating debate (Pozo (1992a)), the impact of monetary policy regime changes on exchange rate volatility (Lastrapes (1989)) or the impact of joining the Exchange Rate Mechanism (ERM) of the European Monetary System on exchange rate volatility (Pesaran and Robinson (1993) and Neely (1993)) have also utilised ARCH models in their analysis.

4

value other than unity or two. Further, whilst no evidence of asymmetry is found for the Singapore dollar, it is present in the Malaysian currency.

This paper will consider the applicability of the Ding et al Power ARCH class of model to exchange rate data.

Unlike Tse and Tsui (1997) which only considered two

exchange rates, this paper will consider a wider range of more commonly traded exchange rates. Further, as the Power ARCH model nests within it a number of other models from the ARCH family, the relative merits of each of these nested models may be considered using the standard log likelihood test. The remainder of this paper proceeds as follows. In Section Two we detail the general model and discuss how various ARCH models are nested within this generalised asymmetric Power ARCH structure. Section Three describes the bilateral exchange rate data to be used in this study and presents the empirical results. Parameter estimates for the A-PARCH model are presented as are the results of the likelihood ratio testing procedure. Finally, Section Four contains some concluding remarks.

2

THE POWER ARCH MODEL

Nelson (1990a,b) and Gannon (1996a,b) have researched the relationship between the estimation of ARCH models and the specification of the mean equation. A general conclusion to be drawn for their work is that the specification of the mean equation bears little impact on the ARCH model when estimated in continuous time. McKenzie (1997) presents evidence which suggests that this result is also a characteristic in discrete time. In light of this evidence and keeping in mind that the purpose of this paper is to consider the impact of alternative variance equation specifications, the mean equation shall be specified as a naïve mean reverting equation in the form of rt = εt where rt is the returns to the market index and εt is the error term in period t. This error term may be decomposed into εt = σt et where et ∼ N (0,1) ie, is normally distributed with a zero mean and a variance of one. The general Power ARCH model introduced by Ding, Granger and Engle (1993) specifies σt as of the form:

5

σ

d t

= α0 +

∑ α (ε p

i =1

i

t −i

+ γ i εt −i

)

d

+

q

∑ βσ i =1

i

d t −i

(1)

where the αi and βi are the standard ARCH and GARCH parameters, γi is the leverage parameter and d is the parameter for the power term.

Following the approach of Ding, Granger and Engle (1993) and Hentschel (1995), it is possible to nest a number of the more standard ARCH and GARCH formulations within this Asymmetric Power GARCH model by specifying permissible values for α, β, γ, and d in Equation 1. Table 1 summarises the restrictions required to produce each of the models nested within this A-PGARCH model. From Table 1, where αi is free, d = 2 and both β and γ = 0, this model reduces to Engle’s (1983) ARCH model. Further, when we extend this model to allow both αi and βi to take on any value, we get Bollerslev’s (1986) GARCH model. Taylor (1986) and Schwert (1989) have suggested that it is the conditional standard deviation which should be the focus of an ARCH model. This model is also nested in the conditional variance equation where αi and βi are free, d = 1 and γ = 0.

Beyond the ARCH and GARCH models, it is also possible to nest other ARCH models in this equation which have been proposed by the literature.

The nonlinear ARCH

model (NARCH) of Higgins and Bera (1992) is obtained where d and αi are free (β = 0, γ =0). If we extend this NARCH model to allow βi to also be free, then a Power GARCH specification is the result.

The models nested so far have assumed a symmetrical response of volatility to innovations in the market. However, empirical evidence suggests that positive and negative returns to the market of equal magnitude will not generate the same response in volatility. Glosten, Jaganathan and Runkle (1993) provided one of the first attempts to model leverage effects using a model which utilises a GARCH type conditional variance specification. In this GJR-GARCH model, the power term and beta conform to the conventional GARCH restrictions (d = 2 and βi is free) however, αi is specified as αi(1+γi)2 and the leverage term is restricted to -4αiγi. Another leverage effect 6

GARCH model which is less restrictive then the GJR-GARCH model may also be specified.

A leverage GARCH model is obtained by extending Bollerslev’s original

GARCH model to allow γi to take on positive values (ie. γi ≤ 1) in the σtd equation . Taylor’s GARCH model may also be extended to include asymmetric effects by specifying γi ≤ 1 which describes a generalised TARCH model.

The original

TARCH model by Zakoian (1991) is defined if we restrict βi =0 (αi is free, d=1 and γi ≤ 1) in the generalised TARCH model. Full details and proofs of this nesting process may be found in Ding, Granger and Engle (1993).

3

DATA AND RESULTS

The data to be used in this study consists of daily bilateral spot exchange rate data taken over the period January 1986 to December 19974. Seventeen bilateral exchange rates are included in this study : German Mark / U.S. Dollar (DEM/USD); Japanese Yen / U.S. Dollar (YEN/USD); British Pound / U.S. Dollar (GBP/USD); Swiss Franc / U.S. Dollar (CHF/USD); Canadian Dollar / U.S. Dollar (CAD/USD); German Mark / French Franc (DEM/FRF); Australian Dollar / U.S. Dollar (AUD/USD); German Mark / Japanese Yen (DEM/JPY); and German Mark / British Pound (DEM/GBP). These exchange rates comprise nine of the top ten most heavily traded bilateral currencies in the global foreign exchange market (data was not available for the French Franc / U.S. Dollar)5. The eight remaining exchange rates included in this study were the Hong Kong Dollar / U.S. Dollar (HKD/USD); Singapore Dollar / U.S. Dollar (SGD/USD); Italian Lira / U.S. Dollar (ITL/USD); Austrian Schilling / U.S. Dollar (AUT/USD); Belgian Franc / U.S. Dollar (BEF/USD); Danish Krone / U.S. Dollar (DKK/USD); Spanish Peseta / U.S. Dollar (ESP/USD); and Netherlands Guilder / U.S. Dollar (NLG/USD). These additional currencies were selected based on the high volume of trading in the foreign exchange market in each bilateral rates home country.

3.1

ASYMMETRIC POWER GARCH MODEL ESTIMATION

4

Except for the Belgium data which started in April 1992. All data sourced from Datastream. Bank for International Settlements (1996) “Central Bank Survey of Foreign Exchange and Derivatives Market Activity 1995” BIS, Basle. 5

7

For each spot exchange rate series, the continuously compounded percentage return was estimated as rt = log (et / et-1) where et is the spot exchange rate on day t. As discussed in Section 2, each bilateral exchange rate return series was fitted with a model which specified a naïve no change mean equation and a generalised asymmetric Power ARCH (APGARCH) conditional variance equation. As the error term in this mean equation most likely contains outliers, the assumption of a standard Gaussian distribution is inappropriate. To accommodate such leptokurtosis, the models in this paper were estimated assuming the conditional errors were drawn from a conditional tdistribution (see Bollerslev (1987))6.

Table 2 presents the results of this estimation procedure and from this table one can see that all of the ARCH and GARCH coefficients are statistically significant at the 5% level. Further, the sum of the ARCH and GARCH coefficients for all of the models estimated was less than unity indicating that shocks to the model are transitory rather than permanent.

The estimated parameter for the conditional t-distribution was

significantly greater than one for all cases, indicating that the variance is finite. The power term (d) is also presented in Table 2 and ranges in value from 1.987 in the case of the German - Japanese exchange rate to 1.010 for the Hong Kong - U.S. exchange rate. The average power term across all of the models estimated was 1.371. For seven of the models estimated the power term was significantly different from two and for an additional five models the power term was significantly different from unity. This means that for 12 of the 17 models estimated, the optimal power term was some value other than unity or two which would seem to support the use of a model which allows the power term to be estimated. The power term provided by this model is typical of those generated by the other three models in which the power term is estimated (ie. NARCH, Power GARCH and APARCH).

The generalised Ding et al Power ARCH model includes and asymmetry term (γ) which allows positive and negative shocks of equal magnitude to elicit an unequal response from the market. Table 2 presents details of this asymmetry term and reveals that for eleven of seventeen models fitted, the estimated coefficient was negative. 6

To test the validity of this assumption an ARCH(1) model assuming normal and t-distributed errors

8

Further, for the CHF/USD, CAD/USD, AUD/USD, SGD/USD and DEM/FRF, the asymmetry term was statistically significant.

Such a result was expected since

response asymmetry is generally attributed solely to stock market data. We should perhaps be surprised that five models do exhibit evidence of unequal responses to market innovations.

No theoretical reason exists to justify the presence of such

asymmetry.

3.2

LIKELIHOOD RATIO TESTING

The parameters of the APGARCH model were allowed to vary in accordance with Table 1 to produce each of the nested models. These models were robust in a manner described for the APGARCH model in Section 3.1. To conserve space the authors decline to present individual model details although they are available upon request. One may test the significance of the restrictions required to nest these models using the standard log likelihood ratio testing procedure.

The results of each pairwise

comparison are presented in Table 3 (p value in parentheses) and insignificant test scores are bolded indicating a failure to reject the null.

3.2.1 TESTING OF NESTED ARCH MODELS

As can be seen from Table 3, the standard ARCH model of Engle (1982) is nested within a number of the more exotic ARCH specifications including NARCH in which the power term is free, Leverage ARCH in which an asymmetry term is included and the GJR-ARCH and Asymmetric PARCH model which both include a power and asymmetry term although the former is a much more restricted version of the model. The log likelihood tests give a clear signal that the augmentation of Engle’s basic ARCH model with a leverage and/or power term does not significantly enhance the model. The ARCH model is clearly preferred to the NARCH model in all 17 exchange rate series and the Leverage ARCH model in 13 cases. As one would expect given the previous results, the inclusion of both an asymmetry and a power term is also of no great benefit according to the likelihood tests as the standard ARCH specification was

was fitted to each return series. The log likelihood test unanimously preferred the latter.

9

favoured to the GJR-ARCH model in 13 instances and the APARCH model and in 14 cases.

The standard ARCH model is also nested within a number of GARCH parameterisations including : Bollerslev’s GARCH; Leverage GARCH; Glosten, Jagannathan and Runkle’s (1993) GJR-GARCH; Power GARCH; and Asymmetric Power GARCH. The log likelihood ratio test clearly dismisses the standard ARCH model in favour of any model specification which includes a GARCH term. For all seventeen exchange rate series under study, the calculated likelihood ratio test statistic is clearly significant indicating a preference for the various GARCH parameterisations.

The more exotic Leverage ARCH and GJR-ARCH model parameterisations introduce a leverage term to the ARCH model and are themselves nested in more general models. The Leverage ARCH model is nested within the APARCH,

Leverage

GARCH and APGARCH models. The likelihood test results in Table 3 indicate that the augmentation of the Leverage ARCH model with a power term (APARCH) does not appear to enhance the model as the simpler version was preferred in all 17 cases. Consistent previous results, the Leverage GARCH and APGARCH models which contain a GARCH term are again clearly preferred across all currencies to the Leverage ARCH model. The GJR-ARCH model is nested within the APARCH, GJRGARCH and APGARCH models. Interestingly, the GJR-ARCH is preferred to the less restrictive APARCH model in all cases except the Swiss-U.S. exchange rate whilst the GARCH term included in the latter two models is preferred in all of the currencies tested.

Finally, a comparison of the APARCH model to the APGARCH model

produced highly significant test statistics in each case clearly preferring indicating a preference for the latter model.

To complete the likelihood testing procedure for the ARCH based models, the NARCH and TARCH models are also nested. The NARCH model is nested within the APARCH, APGARCH and Power GARCH models. The test results reveal that the addition of a leverage term to the NARCH model (APARCH) has little to offer (12 currencies preferred the NARCH model) and a unanimous preference is indicated for the models which contain a GARCH term. The TARCH model is nested in the 10

APARCH, Generalised TARCH and APGARCH models.

The data is undecided

whether the inclusion of a free power term to the TARCH model is worthwhile as the APARCH model is preferred in eight cases. However, the data clearly prefers the two models with a GARCH term over the TARCH model

In general, the results of the nested testing procedure for the ARCH based models would tend to suggest that the inclusion of a GARCH term significantly enhances the explanatory power of the model and that the addition of a power and/or a leverage term does little to enhance the standard ARCH specification provided by Engle (1982).

3.2.2

TESTING OF NESTED GARCH MODELS

The GARCH model of Bollerslev (1986) is nested within the Leverage GARCH, GJRGARCH, Power GARCH and APGARCH models. The likelihood testing procedure applied to each model pair produced the results presented in Table 3 and indicates that the simple GARCH specification is superior to the models augmented solely by an asymmetry term whether it be restricted as in the GJR-GARCH model or unrestricted as in the Leverage GARCH model. In both of these cases, 14 currencies supported the simple GARCH model and only the Canadian - U.S., Singapore - U.S. and German French exchange rates prefer the inclusion of an asymmetry term. Not surprisingly, these are the currencies which produced the three most significant t-statistics associated with the leverage term in the APGARCH model as discussed in Section 3.

These results suggest that the sole inclusion of an asymmetry term does not significantly enhance the explanatory power of the GARCH model. The inclusion of a power term may also be assessed using the likelihood testing procedure. Comparing the standard GARCH model to the Power GARCH model, 11 of 17 currencies tested generated an insignificant test score indicated a relative preference for Bollerslev’s model. As one would expect, these 11 currencies include the 10 currencies for which the power term generated in the APGARCH model was not statistically different from two. Each of the currencies which preferred the Power GARCH model (CHF/USD, CAD/USD, AUD/USD, HKD/USD, AUT/USD and DKK/USD), generated a power

11

term which was significantly different from two. Thus, specifying the GARCH model with a squared power term does not in general seem restrictive.

Whilst the inclusion of either a power term or an asymmetry term does not significantly enhance the GARCH model, it is worthwhile to consider the usefulness of including both terms. A comparison of the GARCH model and the APGARCH model reveals that, for nine currencies, the GARCH model is preferred whilst the remaining eight series prefer the APGARCH. Table 2 reveals that for these eight currencies, the APGARCH model contained either a significant asymmetry term or a power term which was significantly different from two. The relative performance of the Power GARCH model and the APGARCH model reveals that for 13 of 17 currencies tested the Power GARCH model was preferred as only in the case of the CAD/USD, AUD/USD, SGD/USD and DEM/FRF (which generated a significant leverage term) was a significant test score generated. Thus, where asymmetry is present in the data, it would appear that the inclusion of a leverage term is worthwhile but only where a power term is present.

To complete the log likelihood testing of GARCH based nested models, a comparison between the Leverage GARCH and the APGARCH model finds that 10 currencies favour the former model without the power term. Further, when we compare the GJR-GARCH to the APGARCH model the more restrictive version is again slightly preferred in the same 10 instances.

In general, the results of this testing procedure would seem to indicate that a standard GARCH(1,1) model is optimal when modeling exchange rate returns data.

The

addition of an asymmetry term to Bollerslev’s GARCH model, whether restricted as in the GJR-GARCH model, unrestricted as in the Leverage GARCH model or combined with a power term in the APGARCH model, does not significantly enhance the model. The inclusion of a free power term would also appear, in general, to be unnecessary. However, where leverage effects are present in the data, the inclusion of a leverage term is worthwhile but only where a power term is present.

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3.2.3 TESTING OF NESTED TAYLOR ARCH / GARCH MODELS

The Taylor family of ARCH models chooses unity as its power term and so specifies the conditional standard deviation of a series as a function of past absolute errors (and conditional standard deviations in the case of GARCH). The Taylor ARCH model is nested within a wide range of other ARCH models, including the TARCH, NARCH, Leverage Taylor ARCH, APARCH, Taylor GARCH, Power GARCH, Generalised TARCH, and APGARCH models. The pairwise comparisons of the Taylor ARCH model to the other ARCH specifications, clearly favour the simple Taylor ARCH model. In almost all cases, we fail to reject the Taylor ARCH model against more complex alternatives. Comparing the Taylor ARCH model to its nested counterparts which include a GARCH term (Taylor GARCH, Power GARCH, Generalised TARCH, and APGARCH), the latter are unambiguously preferred in all instances with a statistically significant test statistic in each case.

The Taylor GARCH model is nested within the Generalised TARCH, Power GARCH and APGARCH models. Restricting the power term to unity does appear to hamper the Taylor GARCH model as for 10 currencies the Power GARCH model is preferred. However, the exclusion of a leverage term does not appear to limit the Taylor GARCH model as only 6 of the 17 currencies tested preferred the Generalised TARCH model. Five of these six cases were the currencies which exhibited a significant leverage term in the APGARCH model (the Yen/USD is the other). Finally, when comparing the Taylor GARCH and the Generalised TARCH model to the APGARCH model, 11 and 10 currencies respectively preferred the APGARCH model.

In general, the results for the Taylor ARCH family of models exhibit some similarities to the results obtained for the standard ARCH model family. Firstly, the inclusion of a GARCH term clearly augments the model in a worthwhile fashion.

Further, the

addition of a leverage term to the model does not appear to bring with it any substantial gains nor does allowing both a free power term and an asymmetry term in the model. However, the use of unity as the power term in the Taylor model does appear restrictive and the Power GARCH model was generally preferred.

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4

CONCLUSION

The literature suggets that the Power ARCH class of models are generally applicable to stock market returns data.

The purpose of this paper was to consider the

applicability of the Power ARCH model to exchange rate data which does not commonly exhibit the asymmetry characterstic prevalent in stock market returns. Seventeen heavily traded bilateral exchange rate return series were fitted with the generalised asymmetric Power ARCH model proposed by Ding, Granger and Engle (1993). The results of this estimation procedure found that 12 of the currencies tested did not exhibit evidence of asymmetry. For these 12 currencies, log likelihood testing of nested ARCH models indicated that a simple GARCH(1,1) model was generally preferable and the inclusion of a power and/or leverage term did little to enhance the model.

For the five currencies in which evidence of asymmetry was found, the

inclusion of an asymmetry term was a worthwhile addition to the model as long as the power term by which the data was transformed was estimated within the model.

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5

TABLES

TABLE 1 TAXONOMY OF NESTED ARCH MODEL SPECIFICATIONS This table shows the restrictions required on the asymmetric Power ARCH model of Ding, Granger and Engle (1993) to nest particular special cases of other ARCH/GARCH models. σ td = α 0 +

ρ

∑α (ε i =1

i

t −i

+ γ i ε t −i

)

d

q

+

∑βσ i =1

i

d t −i

d

αi

βi

γI

ARCH

2

free

0

0

GARCH

2

free

free

0

Leverage ARCH

2

free

0

| γi | ≤ 1

Leverage GARCH

2

free

free

| γi | ≤ 1

GJR-ARCH

2

αi(1 + γi)2

0

-4 αiγi

GJR-GARCH

2

αi(1 + γi)2

free

-4 αiγi

Taylor ARCH

1

free

0

0

Taylor GARCH

1

free

free

0

TARCH

1

free

0

| γi | ≤ 1

Generalised TARCH

1

free

free

| γi | ≤ 1

NARCH

free

free

0

0

Power GARCH

free

free

free

0

Asymmetric PARCH

free

free

0

| γi | ≤ 1

Asymmetric PGARCH

free

free

free

| γi | ≤ 1

Model

15

TABLE 2 ASYMMETRIC POWER GARCH [APGARCH(1,1)] MODEL SUMMARY The following table reports the results of the estimation of an asymmetric power GARCH model with t-distributed errors fitted to 17 different exchange rate returns series sampled at a daily frequency over the period January 1986 to December 1997. The ARCH and GARCH coefficients are presented as are the power and leverage effect term (t-statistics in parentheses). The sum of the ARCH and GARCH coefficients should sum to be less than one. The final column presents the estimated parameter and standard error for the t-distribution assumed for the data.

Exchange Rate JPY/USD

α1

β1

0.037 0.913 (3.706) (54.017) 0.021 0.960 GBP/USD (2.971) (150.237) 0.028 0.957 CHF/USD (4.193) (95.144) 0.047 0.934 CAD/USD (5.364) (92.847) 0.054 0.893 AUD/USD (5.010) (53.150) 0.139 0.803 HKD/USD (12.707) (59.836) 0.093 0.820 SGD/USD (5.811) (43.740) 0.051 0.926 ITL/USD (4.981) (98.456) 0.046 0.922 AUT/USD (5.127) (70.401) 0.046 0.925 BEF/USD (3.690) (51.144) 0.040 0.944 DKK/USD (5.067) (99.419) 0.034 0.941 ESP/USD (3.668) (99.812) 0.042 0.936 NLG/USD (4.741) (86.163) 0.037 0.939 DEM/USD (3.459) (75.689) 0.071 0.896 DEM/FRF (5.253) (80.260) 0.053 0.916 DEM/JPY (3.663) (61.210) 0.069 0.913 DEM/GBP (5.599) (82.377) Note : a - significantly different from unity b -significantly different from two

d

γ

Σ(α+β)

1.492 (4.751) 1.846 a (4.970) 1.074 b (3.204) 1.148 b (5.949) 1.299 b (5.269) 1.010 b (11.541) 1.729 a (6.750) 1.697 a (5.856) 1.293 b (5.067) 1.192 b (4.099) 1.207 b (4.566) 1.675 (4.662) 1.461 (5.125) 1.669 (4.307) 1.658 a (6.359) 1.987 (3.978) 1.596 a (6.041)

-0.123 (1.417) 0.100 (1.269) -0.245 (1.948) 0.273 (2.805) 0.212 (2.164) -0.028 (0.484) 0.149 (3.454) -0.030 (0.544) -0.062 (0.785) -0.182 (1.307) -0.067 (0.837) 0.039 (0.556) -0.068 (0.980) -0.065 (0.764) -0.325 (4.600) 0.007 (0.101) -0.008 (0.127)

0.950

16

0.981 0.985 0.981 0.947 0.942 0.913 0.977 0.968 0.971 0.984 0.975 0.978 0.976 0.967 0.969 0.982

t-distribution parameter 3.568 (0.277) 3.725 (0.334) 4.772 (0.511) 4.425 (0.347) 3.589 (0.299) 2.028 (0.112) 3.289 (0.243) 5.123 (0.516) 3.835 (0.342) 3.974 (0.465) 4.108 (0.366) 4.194 (0.402) 4.814 (0.506) 5.116 (0.639) 4.185 (0.441) 7.942 (1.294) 5.296 (0.527)

TABLE 3 Likelihood Ratio Tests for Nested ARCH Models Fitted to Bilateral Exchange Rate Data Nested Test ARCH vs GARCH

YEN/USD GBP/USD CHF/USD CAD/USD AUD/USD HKD/USD SGD/USD ITL/USD

33.885 (0.000) ARCH vs Leverage GARCH 34.272 (0.000) ARCH vs GJR-GARCH 34.269 (0.000) ARCH vs NARCH 0.647 (0.255) ARCH vs Power GARCH 34.352 (0.000) ARCH vs Asymmetric Power 35.322 GARCH (0.000) ARCH vs GJR-ARCH 0.232 (0.496) ARCH vs Leverage ARCH 0.200 (0.527) ARCH vs Asymmetric Power 0.880 ARCH (0.414) Leverage ARCH vs 0.681 Asymmetric Power ARCH (0.243) Leverage ARCH vs Leverage 34.073 GARCH (0.000) Leverage ARCH vs 35.122 Asymmetric Power GARCH (0.000) GJR-ARCH vs Asymmetric 0.649 Power ARCH (0.254) GJR-ARCH vs GJR 34.038 GARCH (0.000) GJR-ARCH vs Asymmetric 35.090 Power GARCH (0.000) GARCH vs Leverage GARCH 0.387 (0.378) GARCH vs GJR-GARCH 0.385 (0.380) GARCH vs Power GARCH 0.467 (0.334) GARCH vs Asymmetric 1.437 Power GARCH (0.237) Leverage GARCH vs 1.050 Asymmetric Power GARCH (0.147) GJR-GARCH vs Asymmetric 1.052 Power GARCH (0.146) NARCH vs Power GARCH 33.705 (0.000)

110.430 (0.000) 111.393 (0.000) 111.392 (0.000) 0.745 (0.222) 110.437 (0.000) 111.340 (0.000) 4.012 (0.005) 4.015 (0.005) 3.747 (0.024) 0.013 (0.870) 107.378 (0.000) 107.325 (0.000) 0.266 (0.466) 107.380 (0.000) 107.328 (0.000) 0.963 (0.165) 0.962 (0.165) 0.007 (0.904) 0.910 (0.403) 0.053 (0.744) 0.052 (0.747) 111.183 (0.000)

33.165 (0.000) 33.827 (0.000) 33.822 (0.000) 0.061 (0.726) 36.109 (0.000) 37.850 (0.000) 1.192 (0.123) 1.174 (0.125) 1.509 (0.221) 0.134 (0.604) 32.653 (0.000) 36.676 (0.000) 2.701 (0.020) 32.630 (0.000) 36.658 (0.000) 0.662 (0.250) 0.657 (0.252) 2.944 (0.015) 4.685 (0.009) 4.023 (0.005) 4.028 (0.005) 42.235 (0.000)

66.294 (0.000) 69.024 (0.000) 69.057 (0.000) 0.618 (0.266) 72.809 (0.000) 76.432 (0.000) 4.911 (0.002) 4.955 (0.002) 3.967 (0.019) 0.049 (0.753) 64.069 (0.000) 71.477 (0.000) 0.945 (0.169) 64.145 (0.000) 71.521 (0.000) 2.731 (0.019) 2.763 (0.019) 6.516 (0.000) 10.139 (0.000) 7.408 (0.000) 7.376 (0.000) 73.427 (0.000)

47.787 (0.000) 49.473 (0.000) 49.417 (0.000) 0.258 (0.472) 50.278 (0.000) 52.658 (0.000) 1.983 (0.046) 1.939 (0.049) 1.886 (0.152) 0.003 (0.942) 47.533 (0.000) 50.718 (0.000) 0.097 (0.660) 47.434 (0.000) 50.675 (0.000) 1.686 (0.066) 1.630 (0.071) 2.491 (0.026) 4.871 (0.008) 3.185 (0.016) 3.241 (0.011) 50.020 (0.000)

179.116 (0.000) 179.112 (0.000) 179.112 (0.000) 0.027 (0.815) 196.108 (0.000) 196.198 (0.000) 0.617 (0.267) 0.108 (0.643) 0.149 (0.861) 0.752 (0.220) 179.005 (0.000) 196.090 (0.000) 0.389 (0.378) 178.495 (0.000) 195.581 (0.000) 0.004 (0.929) 0.004 (0.929) 16.991 (0.000) 17.081 (0.000) 17.085 (0.000) 17.085 (0.000) 198.843 (0.000)

132.984 (0.000) 138.595 (0.000) 138.591 (0.000) 1.505 (0.083) 133.507 (0.000) 139.134 (0.000) 7.819 (0.000) 7.777 (0.000) 8.666 (0.000) 0.889 (0.182) 130.818 (0.000) 131.357 (0.000) 0.847 (0.193) 130.773 (0.000) 131.315 (0.000) 5.611 (0.001) 5.607 (0.001) 0.523 (0.307) 6.150 (0.002) 0.540 (0.299) 0.543 (0.298) 132.002 (0.000)

131.912 (0.000) 132.391 (0.000) 132.391 (0.000) 0.450 (0.343) 132.756 (0.000) 132.909 (0.000) 0.566 (0.287) 0.583 (0.280) 0.430 (0.650) 0.153 (0.580) 131.808 (0.000) 132.326 (0.000) 0.136 (0.602) 131.825 (0.000) 132.343 (0.000) 0.478 (0.328) 0.479 (0.328) 0.843 (0.194) 0.997 (0.369) 0.518 (0.309) 0.518 (0.309) 133.206 (0.000)

17

AUT/USD BEF/USD DKK/USD ESP/USD NLG/USD DEM/USD DEM/FRF DEM/JPY DEM/GBP

46.807 (0.000) 46.833 (0.000) 46.833 (0.000) 0.859 (0.190) 50.352 (0.000) 50.765 (0.000) 1.048 (0.148) 1.001 (0.157) 0.424 (0.655) 0.577 (0.283) 45.832 (0.000) 49.764 (0.000) 0.624 (0.264) 45.785 (0.000) 49.717 (0.000) 0.026 (0.819) 0.026 (0.820) 3.545 (0.008) 3.958 (0.019) 0.393 (0.005) 3.932 (0.005) 51.211 (0.000)

19.962 (0.000) 20.154 (0.000) 20.133 (0.000) 0.079 (0.690) 21.383 (0.000) 22.273 (0.000) 0.002 (0.954) 0.015 (0.863) 0.060 (0.942) 0.045 (0.764) 20.139 (0.000) 22.258 (0.000) 0.058 (0.732) 20.131 (0.000) 22.271 (0.000) 0.192 (0.536) 0.171 (0.559) 1.420 (0.092) 2.311 (0.099) 2.119 (0.040) 2.140 (0.039) 21.303 (0.000)

61.270 (0.000) 61.345 (0.000) 61.344 (0.000) 0.704 (0.235) 65.699 (0.000) 65.999 (0.000) 0.113 (0.635) 0.142 (0.594) 0.552 (0.576) 0.069 (0.710) 61.203 (0.000) 65.857 (0.000) 0.665 (0.249) 61.232 (0.000) 65.887 (0.000) 0.076 (0.697) 0.075 (0.699) 4.429 (0.003) 4.730 (0.009) 4.654 (0.002) 4.655 (0.002) 66.403 (0.000)

83.902 (0.000) 84.052 (0.000) 84.051 (0.000) 1.165 (0.127) 84.161 (0.000) 84.377 (0.000) 0.950 (0.168) 0.935 (0.171) 0.059 (0.943) 0.099 (0.656) 83.116 (0.000) 83.442 (0.000) 1.009 (0.155) 83.102 (0.000) 83.427 (0.000) 0.149 (0.585) 0.149 (0.585) 0.258 (0.473) 0.474 (0.622) 0.325 (0.420) 0.326 (0.420) 85.326 (0.000)

65.318 (0.000) 65.584 (0.000) 65.580 (0.000) 0.742 (0.223) 66.250 (0.000) 66.590 (0.000) 0.618 (0.266) 0.615 (0.267) 0.102 (0.903) 0.051 (0.749) 64.969 (0.000) 65.974 (0.000) 0.516 (0.310) 64.961 (0.000) 65.971 (0.000) 0.266 (0.466) 0.261 (0.470) 0.932 (0.172) 1.271 (0.280) 1.005 (0.156) 1.010 (0.155) 66.992 (0.000)

48.989 (0.000) 49.137 (0.000) 49.136 (0.000) 0.656 (0.252) 49.079 (0.000) 49.316 (0.000) 0.614 (0.268) 0.630 (0.262) 0.375 (0.687) 0.100 (0.654) 48.507 (0.000) 48.686 (0.000) 0.989 (0.160) 48.522 (0.000) 48.702 (0.000) 0.147 (0.587) 0.147 (0.588) 0.090 (0.672) 0.326 (0.722) 0.179 (0.550) 0.179 (0.549) 49.735 (0.000)

261.993 (0.000) 231.649 (0.000) 231.648 (0.000) 0.031 (0.802) 218.131 (0.000) 232.478 (0.000) 1.879 (0.053) 1.901 (0.051) 2.047 (0.129) 0.146 (0.589) 229.748 (0.000) 230.578 (0.000) 0.168 (0.562) 229.769 (0.000) 230.600 (0.000) 14.656 (0.000) 14.655 (0.000) 1.138 (0.131) 15.485 (0.000) 0.829 (0.198) 0.831 (0.197) 218.099 (0.000)

66.159 (0.000) 66.108 (0.000) 66.102 (0.000) 0.224 (0.503) 66.090 (0.000) 66.152 (0.000) 1.050 (0.147) 1.115 (0.135) 0.919 (0.399) 0.196 (0.532) 64.993 (0.000) 65.037 (0.000) 0.131 (0.609) 65.052 (0.000) 65.102 (0.000) 0.051 (0.750) 0.057 (0.736) 0.069 (0.709) 0.007 (0.993) 0.044 (0.767) 0.050 (0.752) 66.314 (0.000)

135.463 (0.000) 135.467 (0.000) 135.467 (0.000) 0.331 (0.416) 136.690 (0.000) 136.659 (0.000) 1.090 (0.140) 1.083 (0.141) 1.177 (0.308) 0.094 (0.665) 134.384 (0.000) 135.576 (0.000) 0.087 (0.676) 134.377 (0.000) 135.569 (0.000) 0.004 (0.926) 0.004 (0.925) 1.227 (0.117) 1.196 (0.302) 1.192 (0.123) 1.192 (0.123) 136.358 (0.000)

TABLE 3 (Continued) Likelihood Ratio Tests for Nested ARCH Models Fitted to Bilateral Exchange Rate Data Nested Test NARCH vs Asymmetric Power GARCH NARCH vs Asymmetric Power ARCH Power GARCH vs Asymmetric Power GARCH Asymmetric Power ARCH vs Asymmetric Power GARCH Taylor ARCH vs Leverage Taylor ARCH Taylor ARCH vs TARCH Taylor ARCH vs NARCH Taylor ARCH vs Taylor GARCH Taylor ARCH vs Power GARCH Taylor ARCH vs Generalised TARCH Taylor ARCH vs Asymmetric Power ARCH Taylor ARCH vs Asymmetric Power GARCH Taylor GARCH vs Generalised TARCH Taylor GARCH vs Power GARCH Taylor GARCH vs Asymmetric Power GARCH TARCH vs Asymmetric Power ARCH TARCH vs Generalised TARCH TARCH vs Asymmetric Power GARCH Generalised TARCH vs Asymmetric Power GARCH

YEN/USD GBP/USD CHF/USD CAD/USD AUD/USD HKD/USD SGD/USD ITL/USD

34.675 (0.000) 0.233 (0.495) 0.970 (0.163) 34.441 (0.000) 0.003 (0.941) 0.286 (0.449) 0.529 (0.304) 29.934 (0.000) 33.464 (0.000) 32.138 (0.000) 0.526 (0.305) 34.435 (0.000) 2.204 (0.035) 3.530 (0.007) 4.500 (0.011) 0.293 (0.444) 31.852 (0.000) 34.149 (0.000) 2.296 (0.032)

112.085 (0.000) 4.492 (0.003) 0.903 (0.179) 107.593 (0.000) 0.506 (0.314) 4.706 (0.002) 3.564 (0.008) 108.325 (0.000) 111.723 (0.000) 108.732 (0.000) 0.417 (0.361) 112.625 (0.000) 0.407 (0.367) 3.398 (0.009) 4.301 (0.014) 0.326 (0.419) 104.026 (0.000) 107.920 (0.000) 3.893 (0.005)

43.977 (0.000) 4.617 (0.002) 1.741 (0.062) 39.359 (0.000) 1.636 (0.070) 0.702 (0.236) 3.144 (0.012) 1527.987 (0.000) 1528.550 (0.000) 1530.254 (0.000) 1837.351 (0.000) 1530.291 (0.000) 2.267 (0.033) 0.563 (0.288) 2.305 (0.100) 1841.969 (0.000) 1881.290 (0.000) 1881.328 (0.000) 0.038 (0.783)

77.050 (0.000) 4.585 (0.002) 3.623 (0.007) 72.466 (0.000) 0.885 (0.183) 6.950 (0.000) 1.758 (0.061) 77.552 (0.000) 78.218 (0.000) 81.658 (0.000) 0.216 (0.511) 81.841 (0.000) 4.106 (0.004) 0.666 (0.248) 4.289 (0.014) 2.426 (0.028) 74.708 (0.000) 74.892 (0.000) 0.183 (0.545)

52.399 (0.000) 1.627 (0.071) 2.380 (0.029) 50.772 (0.000) 0.314 (0.428) 3.225 (0.011) 1.412 (0.093) 56.367 (0.000) 57.533 (0.000) 58.797 (0.000) 4.288 (0.003) 59.912 (0.000) 2.430 (0.027) 1.165 (0.127) 3.545 (0.029) 5.915 (0.001) 55.572 (0.000) 56.687 (0.000) 1.115 (0.135)

198.866 (0.000) 0.305 (0.435) 0.090 (0.671) 211.126 (0.000) 3.378 (0.009) 0.005 (0.920) 2.043 (0.043) 200.919 (0.000) 200.968 (0.000) 201.021 (0.000) 4.645 (0.002) 201.058 (0.000) 0.102 (0.651) 0.049 (0.754) 0.139 (0.870) 0.189 (0.539) 203.541 (0.000) 203.578 (0.000) 0.037 (0.786)

137.629 (0.000) 7.161 (0.000) 5.627 (0.001) 130.468 (0.000) 1.395 (0.095) 6.584 (0.000) 1.231 (0.117) 125.671 (0.000) 133.923 (0.000) 130.344 (0.000) 0.466 (0.334) 139.551 (0.000) 4.673 (0.002) 8.252 (0.000) 13.880 (0.000) 2.498 (0.025) 123.760 (0.000) 132.966 (0.000) 9.207 (0.000)

133.359 (0.000) 0.880 (0.185) 0.153 (0.580) 132.479 (0.000) 1.022 (0.153) 1.624 (0.071) 3.711 (0.006) 129.191 (0.000) 134.250 (0.000) 129.113 (0.000) 0.580 (0.281) 134.403 (0.000) 0.077 (0.695) 5.059 (0.001) 5.213 (0.005) 0.300 (0.438) 127.489 (0.000) 132.779 (0.000) 5.290 (0.001)

18

AUT/USD BEF/USD DKK/USD ESP/USD NLG/USD DEM/USD DEM/FRF DEM/JPY DEM/GBP

51.624 (0.000) 1.283 (0.109) 0.413 (0.364) 50.341 (0.000) 0.511 (0.312) 0.496 (0.319) 0.868 (0.188) 52.420 (0.000) 53.142 (0.000) 53.369 (0.000) 1.435 (0.090) 53.555 (0.000) 0.950 (0.168) 0.723 (0.229) 1.135 (0.321) 2.718 (0.020) 52.873 (0.000) 53.059 (0.000) 0.186 (0.542)

22.194 (0.000) 0.019 (0.845) 0.891 (0.182) 22.213 (0.000) 2.460 (0.027) 0.036 (0.789) 1.967 (0.047) 23.946 (0.000) 24.571 (0.000) 25.279 (0.000) 3.232 (0.011) 25.462 (0.000) 1.333 (0.102) 0.625 (0.264) 1.515 (0.220) 3.213 (0.011) 25.243 (0.000) 25.426 (0.000) 0.182 (0.546)

66.703 (0.000) 0.152 (0.581) 0.301 (0.438) 66.551 (0.000) 0.933 (0.172) 0.073 (0.703) 0.589 (0.278) 65.143 (0.000) 65.637 (0.000) 65.515 (0.000) 0.693 (0.239) 65.937 (0.000) 0.372 (0.388) 0.494 (0.320) 0.795 (0.452) 0.541 (0.298) 65.588 (0.000) 66.010 (0.000) 0.423 (0.358)

85.542 (0.000) 1.106 (0.137) 0.216 (0.511) 84.436 (0.000) 0.049 (0.753) 0.748 (0.221) 1.042 (0.149) 81.854 (0.000) 85.101 (0.000) 82.185 (0.000) 0.973 (0.163) 85.317 (0.000) 0.331 (0.416) 3.246 (0.011) 3.462 (0.031) 0.133 (0.606) 81.437 (0.000) 84.569 (0.000) 3.132 (0.012)

67.332 (0.000) 0.844 (0.194) 0.339 (0.410) 66.487 (0.000) 0.005 (0.921) 0.803 (0.205) 0.316 (0.426) 64.088 (0.000) 66.673 (0.000) 64.706 (0.000) 0.112 (0.636) 67.012 (0.000) 0.617 (0.267) 2.584 (0.023) 2.924 (0.054) 0.278 (0.456) 63.903 (0.000) 66.210 (0.000) 2.307 (0.032)

49.972 (0.000) 0.281 (0.453) 0.237 (0.491) 49.691 (0.000) 0.234 (0.494) 15.409 (0.000) 1.565 (0.077) 3692.112 (0.000) 3694.966 (0.000) 3693.113 (0.000) 3629.821 (0.000) 3695.203 (0.000) 1.002 (0.157) 2.854 (0.017) 3.091 (0.045) 3630.103 (0.000) 3677.704 (0.000) 3679.793 (0.000) 2.089 (0.041)

232.447 (0.000) 2.015 (0.045) 14.348 (0.000) 230.432 (0.000) 1.095 (0.139) 2.946 (0.015) 3.279 (0.010) 216.722 (0.000) 221.144 (0.000) 230.090 (0.000) 0.098 (0.657) 235.492 (0.000) 13.367 (0.000) 4.422 (0.003) 18.770 (0.000) 2.114 (0.040) 227.143 (0.000) 232.545 (0.000) 5.402 (0.001)

66.376 (0.000) 1.143 (0.130) 0.062 (0.724) 65.233 (0.000) 1.659 (0.069) 1.383 (0.096) 2.898 (0.016) 61.121 (0.000) 66.311 (0.000) 62.005 (0.000) 0.139 (0.598) 66.373 (0.000) 0.884 (0.184) 5.190 (0.001) 5.253 (0.005) 0.243 (0.486) 60.621 (0.000) 64.990 (0.000) 4.369 (0.003)

136.328 (0.000) 0.846 (0.193) 0.030 (0.805) 135.482 (0.000) 0.112 (0.636) 0.662 (0.250) 0.009 (0.894) 132.146 (0.000) 136.172 (0.000) 132.246 (0.000) 0.849 (0.193) 136.142 (0.000) 0.010 (0.655) 4.026 (0.005) 3.996 (0.018) 0.002 (0.944) 131.584 (0.000) 135.479 (0.000) 3.896 (0.005)

6.

REFERENCES

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