Structural Breaks and Long Memory Property in Korean Won Exchange Rates: Adaptive FIGARCH Model

This paper explores the issue of structural breaks and long memory property in the conditional variance process of the Korean exchange rates. To analyze the above in detail, this paper examines the dynamics of the structural breaks and the long memory in the conditional variance process of the Korean exchange returns by using the daily KRW-USD and KRW-JPY exchange rates for the period from 2000 through 2007. In particular, this paper employs the Adaptive FIGARCH model of Baillie and Morana (2009) which account for the structural breaks and the long memory property together. This paper also finds that the new Adaptive FIGARCH model outperforms the usual FIGARCH model of Baillie et al. (1996) when the structural breaks are present and that the long memory property in the conditional variance process of the Korean exchange returns is significantly reduced after the structural breaks are accounted for. Thus, these results suggest that the upward biased long memory property observed in the conditional variance process of the Korean exchange returns could partially have been imparted as a result of neglecting the structural breaks.


I. Introduction
This paper is concerned with a new approach to examining the long memory property and the structural breaks in the conditional variance process of Korea's daily exchange rates. In particular, this paper applies the Adaptive FIGARCH or A-FIGARCH model of Baillie and Morana (2009)  As pointed out by Baillie (1996), it has been well known that most daily and high frequency financial time series data including foreign exchange rates exhibit long memory properties intrinsic to their conditional variance process with persistent and hyperbolic decaying autocorrelations. Long memory properties have been presented in their squared returns (Ding et al. 1993;Granger and Ding 1996), power transformations of absolute returns (Lobato and Savin 1998), conditional variance  and other measures of volatility such as realized volatility (Ebens 1999;Andersen et al. 2003). Following Ding et al. (1993) and Dacorogna et al. (1993), several long memory volatility models have been developed to represent the long memory property in the conditional variance process of the financial time series data. Among them are the long memory stochastic volatility (LMSV) models of Breidt et al. (1998) and Harvey (1998), and the long memory autoregression conditional heteroskedasticity (LM-ARCH) models of Baillie et al. (1996), Bollerslev and Mikkelsen (1996) and Davidson (2004).
Even though these models have appeared to be useful in describing the long memory property in the conditional variance process, there has been more interest in finding the reasons and underlying causes for the empirical findings of the long memory property in the conditional variance process. For example, Granger and Ding (1996) have presented that the contemporaneous aggregation of the stable GARCH(1,1) process can result in very persistent autocorrelations which is the typical feature of the long memory property. And, Andersen and Bollerslev (1997) have shown that the contemporaneous aggregation of weakly dependent information flow process can produce the long memory property in the conditional variance process. In the high frequency perspectives, Müller et al. (1997) have presented that the long memory property in volatility can result from the reaction of short term dealers to the expected volatility trend, which cause persistence in the conditional variance process.
However, other papers have cast doubts on the validity of the empirical findings of the long memory property in the conditional variance process (Mikosch and Starica 1998;Diebold and Inoue 2001;Granger and Hyung 2004;Choi and Zivot 2007). Most of them have suggested that the observed long memory property in conditional variance process may be generated by the presence of various types of structural breaks or regime switches. Mikosch and Starica (1998), Granger and Hyung (2004) and Choi and Zivot (2007) have presented evidence that spurious long memory could be due to the presence of occasional structural breaks detected in the financial time series, and they have conjectured that the long memory persistence of the conditional variance process may be overstated due to the presence of those structural breaks. And, Starica and Granger (2004) have found that a non-stationary model with allowance for breaks in the unconditional variance can outperform a long memory model in forecasting. Similarly, Granger and Terasvirta (1999) and Diebold and Inoue (2001) have presented that a process that switches the regime or switches in sign could have the characteristics of a long memory property.
Thus, it could be necessary to consider both the structural breaks and the long memory property in the conditional variance process of financial time series data.
Following the paper of Diebold (1988) and Lamoreaux and Lastrapes (1990) which have initially suggested that the occurrences of the structural breaks in conditional variance process can generate extreme persistence, the subsequent papers by Lobato and Savin (1998), Beine and Laurent (2000), Morana and Beltratti (2004) and Martens et al. (2004) have presented that an appropriate model for the conditional Given the previous studies summarized above, this paper focuses on the possibility that both the structural breaks and the long memory property are likely to be present in the conditional variance process of the daily Korean exchange rate returns obtained from the KRW-USD and the KRW-JPY daily exchange rates.
First, this paper estimates the long memory property in the conditional variance process of the Korean exchange returns by using the long memory FIGARCH model of Baillie et al. (1996). This paper finds that there exists strong evidence for long memory property in the conditional variance process of the Korean exchange returns, which can be caused by some economic changes in the Korean economy during the sample period even without serious financial crises. In particular, the long memory property in the conditional variance process of the KRW-USD returns appears to be more persistent than that of the KRW-JPY returns.
Second, this paper examines both the structural breaks and the long memory

II. Long Memory Property and FIGARCH Model
This section is concerned with some of the intriguing features of the daily Korean exchange returns obtained from the KRW-USD and the KRW-JPY spot exchange rates. In particular, it explores some aspects of the long memory property in the conditional variance process of the Korean exchange which has been well documented in the papers of Lee (2000), Han (2003) and Jo and Lee (2004). It focuses on the long memory parameters estimated from the FIGARCH model of Baillie et al. (1996). The FIGARCH model has been shown to provide good descriptions of the conditional variance process for the daily Korean exchange rates return by several previous studies such as Jo and Lee (2004) and Han (2003).   And, the returns of the daily exchange rates are defined in a conventional manner as continuously compounded rates of return and calculated as the first difference of the natural logarithm of prices. The return (y t ) at day t is defined as where t = 1,.., 2075 and , t n P is the spot exchange rate at day t.
The details of the descriptive statistics for the daily returns of the KRW-USD and the KRW-JPY spot exchange rates are provided in Table 1 indicating the existence of highly persistent autocorrelations in the conditional variance process. The problem of the serial correlation seems to be more significant in the KRW-USD returns than in the KRW-JPY returns. Thus, the conditional variance process of the KRW-USD returns appears to be more persistent relative to the KRW-JPY returns. Also, there exists small and negative first order of autocorrelations in the Korean exchange returns. This may be due to the market microstructure effects Bollerslev 1997, 1998) or large outliers in the data (Goodhart and Figliuoli 1991;Goodhart and Giugale 1993;Ghosh 1997).  (b), both returns are centered on zero but there exist obvious volatility clustering. And, the extreme turbulences in the market are also seen to induce a heavy tailed, undefined variance of unconditional returns phenomenon, as studied by Koedijk et al. (1990). And, in Figure 3 (a) and (b) which present the autocorrelation function of the returns, squared returns and absolute returns of the daily KRW-USD and the KRW-JPY exchange rates, the first order autocorrelation in the returns is small and negative for the Korean exchange returns while higher order autocorrelations of the raw returns are not significant at conventional levels.
2) According to Jarque and Bera (1987), the standard errors of the sample skewness and the sample kurtosis in their corresponding normal distributions are (6/T) 1/2 and (24/T) 1/2 . functions of the absolute returns as presented by Granger and Ding (1996). And, the degree of the long memory property seems to be more significant in the KRW-USD returns than the KRW-JPY returns.  (1,d,1) process, 3)

對外經濟硏究 제15권 제2호 2011년 여름호
As presented by Baillie et al. (1996) and Baillie and Morana (2009), it is well known that for 0<d≤1, the FIGARCH process has an undefined unconditional variance. However, the process does posses a finite sum to its cumulative impulse response weights. This makes the FIGARCH model different from other possible forms of long memory ARCH models proposed by Karanassos et al. (2004). But the FIGARCH model appears to be strictly stationary and ergodic for 0<d≤1 .
In particular, the FIGARCH process has impulse response weights, σ 2 t = ω/(1 -β) + γ(L)ε 2 t , where for large lag k, γ k ≈ ck d-1 where c is a positive constant. Thus, the conditional variance can be expressed as a distributed lag of past squared unconditional disturbances with coefficients that decay at a slow, hyperbolic rate, which is essentially consistent with the long memory property, or "Hurst effect" of hyperbolic decay. Thus, the attraction of the FIGARCH process is that for 0 < d < 1, it is sufficiently flexible to allow for intermediate ranges of persistence.
3) The exact parametric specification of the model that best represents the degree of autocorrelation in the conditional mean and the conditional variance is chosen based on the Box-Pierce portmanteau statistics. The test statistics shows that the models specified for the Korean exchange returns do a good job of capturing the autocorrelations in the mean and volatility of the return series. In each case there is no evidence of additional autocorrelation in the standardized residuals or squared standardized residuals, indicating that the chosen model specification provides an adequate fit.

Structural Breaks and Long Memory Property in Korean Won Exchange Rates 45
The above model is estimated for the daily Korean exchange returns of interest by maximizing the Gaussian log likelihood function, where Θ is a vector containing the unknown parameters to be estimated.
The consistency and asymptotic normality of the QMLE for the conditional variance process can be established on the basis of available results from the estimation of GARCH processes as pointed out by Baillie and Morana (2009).
Thus, the inference is usually based on the QMLE of Bollerslev and Wooldridge (1992), which is valid when zt is non-Gaussian. Denoting the vector of parameter estimates obtained from maximizing (5)    One of the quite powerful approaches to account for the structural breaks is to allow the intercept to be time varying as suggested by Baillie and Morana (2009).
They have provided that the A-FIGARCH model can be derived from the usual FIGARCH model of Baillie et al. (1996) by directly allowing the intercept in the conditional variance equation to be time varying according to the flexible functional form of Gallant (1984) and Andersen et al. (1997). The flexible function form model can allow for a very efficient modeling of structural breaks since it does not require any pretests to determine the actual location of break points nor add any estimation complexity. Hence, the A-FIGARCH process is formed from two basic components of a long memory volatility process and a deterministic time-varying intercept which allows for the breaks. Although the deterministic process modeled by the flexible functional form is smooth, it has been shown to be able to approximate accurately quite abrupt structural breaks.
The general A-FIGARCH (p,d,q,k) model becomes, There are also GARCH models allowing to model time varying unconditional moments such as the flexible coefficient GARCH model of Medeiros and Veiga (2004), the spine GARCH model of Engle and Rangel (2008) and the smooth transition model of Terasvirta and Gonzalez (2006). But, this paper focus on the FIGARCH model since the FIGARCH model is essentially a generalization of the GARCH model that allows the differencing parameter to be fractional with the hyperbolic memory and imposes fewer burdens in computation (Baillie and Morana 2009).
In order for the conditional variance to be positive with nearly certainty at each point in time, restrictions similar to those holding for the standard FIGARCH  Table 3. In particular, the parameters of the long memory property in the conditional variance process are estimated to be 0.14 and 0.06 for the KRW-USD and the KRW-JPY returns, respectively, and they are all statistically significant. It could be seen that the long memory parameters are significantly reduced compared to the estimated parameters under the basic FIGARCH model without considering the structural breaks, indicating that the degree of the long memory property in the conditional variance process can be upward biased and overstated when the structural breaks caused by economic changes in the Korean economy are neglected.
Also, the Loglikelihood Ratio (LR) test statistics, denoted by LR, for testing the 6) One interesting issue concerns the interpretation of the structural breaks whether or not they correspond to certain important economic changes like the changes in the interest rates, inflation, and economic growth by the economic policies. However, without more detailed information, it is difficult to distinguish these breaks. Furthermore, no formal test is available for detecting multiple structural breaks in the long memory I(d) process with unknown number of breaks (Granger and Hyung 2004).

IV. Conclusion
This paper considers the daily KRW-USD and KRW-JPY exchange rates over the period from 2000 through 2007. Special attention is devoted to account for both the structural breaks caused by economic changes in the Korean economy and the long memory property in the conditional variance process of the Korean exchange returns during the sample period.
Initially, this paper uses the usual FIGARCH model of Baillie et al. (1996) to figure out the long memory property in the conditional variance process of the daily Korean exchange returns series. This paper finds strong evidence for hyperbolic decay and significant persistence of autocorrelations in the conditional variance process of the daily Korean returns, a feature typical of the long memory property. Thus, the long memory property is found to be the characteristic feature in the conditional variance process of the daily Korean exchange returns. And, the usual FIGARCH model of Baillie et al. (1996) is found to provide an adequate fit and match the dynamics of the daily Korean exchange returns. In particular, the long memory property in the conditional variance process, the KRW-USD returns are more significant than that in the KRW-JPY returns.
Following many previous studies which have allowed for the possibility of structural breaks in the conditional variance process of financial time series data including foreign exchange rates, this paper applies the A-FIGARCH model of Baillie and Morana (2009) which is designed to provide a model for both structural breaks and the long memory property in the conditional variance process of the daily Korean exchange returns. In particular the structural breaks can be molded by allowing the intercept in the conditional variance equation to follow a slow time-varying function specified by the flexible functional form of Gallant (1984) and Andersen et al. (1997). Consequently, the results of this paper show the usefulness and the superiority of the A-FIGARCH model relative to other available modeling strategies for its value in terms of risk estimates, especially if the timing and the size of the breaks are shown to be predictable when the breaks are endogenous and need to be explored further.