Volatility Dynamics in Foreign Exchange Rates: Further Evidence from Malaysian Ringgit and Singapore Dollar 1Kin-Yip Ho and 2Albert K. Tsui 1Department of Economics, Cornell University Ithaca, NY 14853, USA, 2Department of Economics, National University of Singapore, Singapore 117570 Email: [email protected] Keywords: Exchange rate volatility; Fractional integration; Long memory; Bivariate asymmetric GARCH; Varying conditional correlations EXTENDED ABSTRACT also examine the robustness of the volatility dynamics of the two currencies against the Japanese The volatility dynamics of foreign exchanges have yen as the alternative numeraire currency besides the been the focus of research since Bollerslev’s (1986) dollar. This may provide bearing for the market seminal work on the generalized autoregressive practitioners to formulate their currency hedging conditional heteroscedasticity (GARCH) modelling. strategies. Several well-established empirical regularities may be highlighted as follows: [a] evidence of volatility Consistent with results of Tse and Tsui (1997) and clustering is detected in the exchange rates returns; Tsui and Ho (2004), we do not find support of [b] asymmetric effects in exchange rate volatility are asymmetric volatility when these currencies are not common; and [c] exchange rate volatility may measured against the dollar. But we find strong display significant persistence and dependence evidence of negative asymmetric effects for the between observations. Among others, Franses and Singapore dollar when it is measured against the yen. van Dijk (2000) provide an in-depth review of this Additionally, we detect significant evidence of subject and illustrate the importance of capturing extreme persistence in the impacts of foreign conditional variance using GARCH-type models in exchange shocks, regardless of the choice of the the empirical finance research. numeraire currency and the volatility structure. It seems that the impacts of exchange rate shocks In this paper we follow up the study of the Malaysian display much longer persistence than the standard ringgit and the Singapore dollar in the Asia-Pacific exponential decay. markets by Tse and Tsui (1997). A family of bivariate GARCH-type models with time-varying Moreover, the likelihood ratio tests indicate that the correlations is proposed to analyse the volatility bivariate fractionally integrated models generally dynamics of the Malaysian ringgit and the Singapore outperform those models without the long-memory dollar, respectively. The proposed models can structure. Furthermore, we find relatively weaker capture stylized features of long-memory, evidence of time-varying correlations when the asymmetric conditional volatility, and time-varying Malaysian ringgit and the Singapore dollar are correlations simultaneously in returns of the two measured against the dollar. However, we detect currencies. They not only retain the flexibility and significant support of time-varying correlations when intuition of the univariate GARCH structure but also these currencies are measured against the yen. It satisfy the positive-definite condition of the maps out an interesting time path of the conditional conditional variance and covariance matrix. correlations between the Malaysian ringgit and the Singapore dollar. The fractionally integrated models are able to distinguish between long persistence and exponential decay in the impact of exchange rate volatilities. We 828 1. Introduction used and the estimation results. Section 4 provides some concluding remarks. The volatility dynamics of foreign exchanges have been the focus of research since Bollerslev’s (1986) 2. Methodology seminal work on the generalized autoregressive conditional heteroscedasticity (GARCH) modelling. We highlight the gist of the bivariate GARCH(1,1) Several well-established empirical regularities may model with time-varying conditional correlations be highlighted as follows: [a] evidence of volatility (VC-GARCH) proposed by Tse and Tsui (2002). clustering is detected in the exchange rates returns; Then we incorporate two structures of asymmetric [b] asymmetric effects in exchange rate volatility are volatility and long memory into the conditional not common; and [c] exchange rate volatility may variance equations so as to synthesize the bivariate display significant persistence and dependence GARCH-type models. between observations. Among others, Franses and van Dijk (2000) provide an in-depth review of this Let yt = (y1t, y2t)’ be the bivariate vector of variables subject and illustrate the importance of capturing with time-varying variance-covariance matrix Ht, conditional variance using GARCH-type models in and let μit(ξi) be the arbitrary conditional mean the empirical finance research. functions which depend on ξi, a column vector of parameters. A typical bivariate GARCH(1,1) model In this paper we follow up the study of the Malaysian can be specified as follows: ringgit and the Singapore dollar in the Asia-Pacific markets by Tse and Tsui (1997). To ensure = μ ξ + ε = consistency in comparison, we confine our yit it ( i ) it , i 1,2 (1) investigation to the GARCH-type models. Instead of using univariate APARCH models by Tse and Tsui ε ε (1997), we propose a family of bivariate MGARCH where ( 1t , 2t )' | Φ t −1 ~ (O , H t ) (2) models to concurrently capture the stylized features Note that Φt is the σ-algebra generated by all the of volatility asymmetry, long-range persistence in available information up to time t. The random volatility, and time-varying correlations. The disturbance terms ε and the conditional variance proposed models automatically ensure the positive it equations h are modelled as follows: definiteness of the conditional variance-covariance iit matrix once convergence is obtained. The parameter ε = h e , where e ~ N(0,1) (3) estimates are relatively easy to interpret, as the it iit it it univariate GARCH-type equations are retained. =η + α ε 2 + β Unlike Bollerslev’s (1990) constant correlation hiit i i it −1 ihiit −1 (4) MGARCH model, the time-varying conditional where (4) is the standard Bollerslev’s (1986) correlations models are able to map out the time-path symmetric GARCH(1,1) model. of conditional correlations between the currencies. In addition, we investigate the behaviour of long- Denote the ij-th element (i, j = 1, 2) in H by h . The memory persistence in volatility of the Malaysian t ijt conditional correlation coefficients can be defined ringgit and the Sinagapore dollar using fractionally h integrated GARCH-type models. These models help ρ = ijt as ijt . Tse and Tsui (2002) assume to distinguish between long persistence and h h exponential decay in the impacts of exchange rate iit jjt volatilities. Moreover, we study the robustness of that the time-varying conditional correlation matrix the volatility dynamics of the two currencies against = {ρ } Γ t ijt is generated by the following recursion the Japanese yen as the alternative numeraire currency besides the dollar. This may help the ρ =(1−π −π )ρ +π ρ +π ψ (5) market practitioners to formulate their currency 12t 1 2 12 1 12,t−1 2 12,t−1 hedging strategies. 2 e1,t −a e2,t −a ψ = ∑ a =1 The rest of the paper is organized as follows. Section 12 ,t −1 (6) 2 2 2 2 2 presents the methodology of synthesizing features ( e − )( e − ) ∑∑aa==1 1,t a 1 2,t a of volatility asymmetry, long-memory and time- varying correlations in a bivariate GARCH-type framework. Section 3 briefly describes the data sets 829 2 e2 + e2 − 2ρ e e GARCH-type models obtained by the BBM θ = − 1 − 1 − ρ2 − 1t 2t 12t 1t 2t lt ( ) loghiit log(1 ) approach. 2 ∑i=1 2 12t 2(1− ρ2 ) 12t [a]Fractionally integrated GARCH(1,1) model (7) (FIGARCH(1,d,1)) η The conditional correlation in (5) inherits the 2 h = i + λ (L)ε (10) prototype properties of GARCH(1,1) structure. iit 1− β i it When π and π are zeros, the conditional correlation i 1 2 where equation in (5) is reduced to Bollerslev’s (1990) ∞ − λ (L) = λ La = 1 − (1 − β L) 1 (1 − φ L)(1 − L) d i . constant-correlation structure (CC-GARCH). The i ∑ a =1 ia i i likelihood ratio test can be readily applied to [b] Fractionally integrated asymmetric GARCH(1,1) compare the performance of both models. model ((FIAGARCH)(1,d,)) model ω The structures of asymmetric volatility and long h = i + λ ( L )( ε − γ ) 2 (11) iit 1 − β i it i memory dynamics are to be incorporated into the i VC-GARCH model by modifying the symmetric where λ is defined as in (10). conditional variance equation in (4). To maintain i (L) consistency in comparison, we choose two well- Note that (11) is similar to the FIGARCH(1,d,1) established asymmetric structures among the model in (10), except that it allows past return GARCH-type models. They include: the asymmetric shocks to have asymmetric effects on the conditional GARCH (1,1) (AGARCH (1,1)) model proposed by volatility. Engle (1990) and the asymmetric power ARCH (1,1) [c] Fractionally integrated APARCH(1,1) model (APARCH (1,1)) model of Ding, Granger, and Engle (FIAPARCH(1,d,1)) (1993), respectively. Indeed, Tse and Tsui (1997) δ η δ use the APARCH (1,1) model to capture the possibly h i 2 = i + λ (L)(| ε | −γ ε ) i (12) iit − β i it i it asymmetric effects of exchange shocks on future 1 i volatilities. In addition, these asymmetric GARCH- λ type models are less restrictive in assumptions and where i (L) is defined as in (10). Similar to the are more flexible to accommodate alternative FIAGARCH(1,d,1) model in (11), (12) allows past variations. Their main features are briefly shocks to have asymmetric effects on the conditional summarized as follows: volatility. Details of the derivations are given in Tsui [a] Engle’s (1990) asymmetric GARCH(1,1) and Ho (2004). (AGARCH(1,1)) model: = ω + α ε − γ 2 + β hiit i i ( it −1 i ) i hiit −1 (8) All parameters of GARCH-type models can be estimated using Bollerslev-Wooldridge’s (1992) γ where i is the asymmetric coefficient.