On the threshold hyperbolic GARCH models. Citation Statistics And Its Interface, 2011, v. 4 n. 2, p

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1 Title On the threshold hyperbolic GARCH models Authors) Kwan, W; Li, WK; Li, G Citation Statistics And Its Interface, 20, v. 4 n. 2, p Issued Date 20 URL Rights Statistics Its Interface. Copyright International Press.

2 Statistics Its Interface Volume 0 20) 8 On the threshold hyperbolic GARCH models Wilson Kwan, Wai Keung Li Guodong Li In the financial market, the volatility of financial assets plays a key role in the problem of measuring market risk in many investment decisions. Insights into economic forces that may contribute to or amplify volatility are thus important. The financial market is characterized by regime switching between phases of low volatility phases of high volatility. Nonlinearity long memory are two salient features of volatility. To jointly capture the features of long memory nonlinearity, a new threshold time series model with hyperbolic generalized autoregressive conditional heteroscedasticity is considered in this article. A goodness of fit test is derived to check the adequacy of the fitted model. Simulation empirical results provide further support to the proposed model. AMS 2000 subject classifications: Primary 9B84; secondary 62M0. Keywords phrases: hyperbolic GARCH model, long memory, threshold model, volatility.. INTRODUCTION The volatility of financial returns has been shown to exhibit long memory, its correlations remain positive for long lags decay slowly to zero, see Greene Fielitz 977), Ding, Granger Engle 993), Kokoszka Taqqu 996), Cont 200) among others. Although price changes appear to be unpredictable, the magnitude of those changes, measured either by the absolute values or the squares of the return series, appears to be predictable in the sense that large changes are likely to be followed by large changes of either sign whereas small changes are likely to be followed by small changes. Price fluctuations are then characterized by periods of low volatility irregularly interspersed by periods of high volatility. This phenomenon was first investigated by Melbrot 963) for commodity prices. The modeling of volatility has been presented in a wide variety of financial assets such as stocks, market indices exchange rates, the autoregressive conditional heteroscedasticity ARCH) model Engle, 982) or the generalized ARCH GARCH) model Bollerslev, 986) is one of the most popular tools. Dedicated to Professor Howell Tong in celebration of his 65th birthday. Corresponding author. The GARCH model can be rewritten as the following ARCH ) form, ) ) e t = ε t h /2 t, h t = γ + δ L) e 2 t, βl) where L is the back shift operator, δ ) β ) are polynomials, see Davidson 2004). The stard GARCH model was introduced originally to describe the dependence structure of volatility, failed to capture the long memory feature of volatility in the returns. Note that δ L) = L)δL) for the case of the integrated GARCH Engle Bollerslev, 986), i.e. x = is the root of δ x). Following the idea of the fractional integrated autoregressive moving average ARFIMA) model, Baillie, Bollerslev Mikkeslen 996) extended the common GARCH model to the fractional integrated GARCH FIGARCH) model, e t = ε t h /2 t, h t = γ + δl) βl) L)d F G ) e 2 t, where 0 < d F G <, the item L) in ) is replaced by L) d F G. However, the FIGARCH process has always infinite variance, there also exists a paradox for the definition of the long memory for ARCH ) processes. To overcome these problems, Davidson 2004) derived a new definition of the memory for ARCH ) processes based on the concept of the near-epoch dependence, proposed the hyperbolic GARCH HYGARCH) model, e t = ε t h /2 t, h t = γ+ δl) ) βl) + α L)d F G )] e 2 t. The main motivation of the above model is that it nests the FIGARCH integrated GARCH models. For more discussions about the HYGARCH models, we can refer to Kwan, Li Li 200a). In financial practice, the main feature for measuring risk in many investment decisions is the volatility of financial assets. Theoretical observational insights into economic forces that may contribute to or amplify volatility are thus important. Lo MacKinlay 990) suggested a possible relation with technical trading rules stock market overreaction. Shefrin 2000) Hirshleifer 200) emphasized the role of market psychology investor sentiment in financial markets. In the trading process, it is observed that price changes are driven by a combination of news about fundamentals evolutionary forces. Under different trading

3 strategies, expectations about future prices dividends of a risky asset, a nonlinear phenomenon structure may occur in the price volatility caused by the interaction between traders, fundamentalists technical analysts. The financial market is thus characterized by an irregular regime switching between phases of low volatility phases of high volatility, some recent works can be referred to Schwert 989), Rabemanjara Zakoian 993) Li Li 996). This motivates us to propose a threshold HY- GARCH model in section 2, it allows for a different HYGARCH structure for each regime. In the process of modeling volatility with ARCH-type models, more more evidences have shown that many financial time series may be so heavy-tailed that the distribution of the innovation ε t is far from normality, see Mikosch Starica 2000) Hall Yao 2003). The commonly used Gaussian quasi-maximum likelihood estimation MLE) may be lack of efficiency, the Student s t MLE seems preferable in analyzing finance economic time series although it is harder to be extended to the quasi-mle in theory, see Bollerslev 987). Davidson 2004) also employed the Student s t MLE to fit the ARFIMA-HYGARCH models. Section 3 derives the asymptotic properties of its student s t MLE, a diagnostic tool based on the squared residual autocorrelations is considered to check whether the fitted threshold HYGARCH model is adequate in section 4. Two simulation experiments are performed in section 5, section 6 analyzes the daily exchange rates of Korean Won against US dollar. 2. THE THRESHOLD ARFIMA-HYGARCH MODELS Let {y t } be a stationary ergodic time series generated by the threshold ARFIMA-HYGARCH model with P regimes, 2) L) di) ARF φ i) L)y t = ψ i) L)e t, e t = ε t h /2 t 3) h t = γ i) + { α i) + α i) L) di) F G ] δ i) L) β i) L) }e2 t, as r i z t d < r i, where d is the delay parameter, = r 0 < < r P = +, L is the back-shift operator, φ i) x) = p i) k= φi) k xk, ψ i) x) = q i) k= ψi) k xk, δ i) x) = m i) k= δi) k xk, β i) x) = s i) k= βi) k xk, p i), q i), m i) s i) with i =,..., P are known positive integers, the innovation sequence {ε t } is identically independently distributed i.i.d.) with mean zero variance one, L) d = j= dγj d) Γ d)γj + ) Lj as 0 < d <. For the ith regime, denoting θ i) V = γ i), β i),..., βi), δ i) s i),..., δi), d i) m i) F G, αi) ), we can rewrite model 3) into the following ARCH ) form, 4) h t = γ i) + π i) L)e 2 t = γ i) + as r i z t d < r i, where the π i) j j= π i) j e2 t j s are functions of θ i) V. The threshold variable z t may be the observed time series y t or the innovation sequence e t, or even some exogenous variable, see Tong 990) Liu, Li Li 997). Assumption. For all i P, α i) 0, 0 < d i) F G π i) j 0 with j =, 2,...; the polynomials δ i) x) β i) x) have no common root. Davidson 2004) pointed out that one of the main motivations for using 3) is that it nests the FIGARCH integrated GARCH models. It is observed that when d i) F G = 0, the conditional variance model becomes an ordinary GARCH model. Thus, the focus of this article will be on the range 0 < d i) F G. As argued by Davidson 2004), there are two kinds of memory to be recognized: hyperbolic decaying memory geometric decaying memory, with the former being defined as long memory. For model 3), when 0 < d i) F G <, πi) j = Oj d ), i.e. the coefficients decay hyperbolically, the conditional variance h t in 4) or 3) will exhibit the long-memory effect. Conrad 200) discussed the non-negativity conditions for the conditional variance of a HYGARCH process, these constraints are required for each regime to make sure that the conditional variance of the threshold models 2) 3) is always non-negative. Assumption 2. For all i P, d i) > 0.5; q i) ARF k= ψi) k < ; the polynomials φi) x) ψ i) x) have no common root. The parameter d i) ARF is usually used to describe the extent of long memory in the ARFIMA processes. The ARFIMA process is short memory or long memory respectively when d i) ARF 0.5, 0) or 0, ), is stationary or nonstationary respectively when d i) ARF 0.5, 0.5) or 0.5, ), see Ling Li 997). However, it may be difficult to discuss the memory properties the stationarity of the models 2) 3) since the effects are mixed for the threshold models. We leave it for possible future research. Note that the definition of long memory here is different from that in the conditional variance in 3) or 4). The condition max i < is sufficient to make sure q i) that k= ψi) k the threshold model given by 2) 3) is invertible, see Ling Tong 2005) Ling, Tong Li 2007). In the following sections, we will concentrate our discussion on two-regime threshold models, i.e. P = 2 in models 2) 3), the notation r is replaced by r for simplicity. 2 W. Kwan, W.K. Li G. Li

4 3. ESTIMATION OF THRESHOLD ARFIMA-HYGARCH MODELS As in Bollerslev 987) Davidson 2004), this section will consider a Student s t MLE for the threshold ARFIMA- HYGARCH models proposed in the previous section. Let θ i) M = φi),..., φi), ψ i) p i),..., ψi) q i), d i) ARF ) with i = 2, θ M = θ ) M, θ2) M ), θ V = θ ) V, θ2) V ) θ = θ M, θ V ). Denote by Θ a suitable compact set in R l with l = 2 i= pi) + q i) + r i) + s i) ) + 8. Suppose that the delay parameter d the threshold value r are known, y,..., y n are generated by the threshold models 2) 3) with the true parameter vector θ 0, an interior point of Θ. Define functions q ) e t θ) = ψ ) k e t kθ) + L) d) ARF φ ) L)y t ]Iz t d < r) k= q 2) + ψ 2) k e t kθ) + L) d2) ARF φ 2) L)y t ] k= Iz t d r) conditional on y 0, y,..., is L n θ, ν) ) ] e t θ) = log f ν ht θ) 2 log h tθ) = n log 2 Γν + )/2) πν 2)Γν/2) log h t θ) + ν + ) log + the MLE can be defined as τ n = θ n, ν n ) = argmax L n θ, ν), θ Θ,ν V e 2 )] t θ), ν 2)h t θ) where V 2, ) is a compact set, the true degrees of freedom ν 0 is an interior point of V. Define c x) = ν log f ν 0 x), c 2 x) = x log f ν 0 x) c 3 x) = xc 2 x)]. It holds that Ec ε t )] = 0, Ec 2 ε t )] = 0, Ec 3 ε t )] = 0, Ec ε t )c 2 ε t )] = 0, Ec 2 ε t )c 3 ε t )] = 0, h t θ) =γ ) + j= + γ 2) + π ) j e 2 t jθ)]iz t d < r) j= π 2) j e 2 t jθ)]iz t d r). Ec 2 2ε t )] = ν 0 ν 0 + ) ν 0 2)ν 0 + 3) Ec 2 3ε t )] = ν 0 2ν 0 + 3), see Kwan, Li Li 200a). For quantities Ec 2 ε t )] Ec ε t )c 3 ε t )], it is difficult to get their explicit forms, we may calculate them using sample averages. Consider the derivative functions of the likelihood function L n θ, ν), Note that the above two functions are all dependent on past observations infinitely far away, however, there are only n values available in real applications. Hence, some initial values are needed, we may simply assume y s = e s θ) = 0 for s 0. The effect of the initial values can be shown to be negligible asymptotically, see Weiss 986) Ling Li 997). Without loss of generality, we assume that the values y s for s 0 are all observable in deriving the asymptotic results in this the next sections. Suppose the innovation ε t follows t ν0, where t ν is the normalized Student s t distribution with mean zero, variance one, ν degrees of freedom, its density is given by f ν x) = ) ν+)/2 Γν + )/2) + x2, πν 2)Γν/2) ν 2) where ν > 2. Then the log likelihood function of y,..., y n, L n θ 0, ν 0 ) θ = L n θ 0, ν 0 ) ν = c ε t ) c 2 ε t ) e t ht θ + c 3ε t ) ] h t. h t θ Then, the Fisher information matrix is denoted by ) Iθ J I τ = J Ec 2, ε t )] where I θ = Ec 2 2ε t )]E e t h t θ ] e t θ + Ec 2 h t 3ε t )]E h 2 t θ ] h t J = Ec ε t )c 3 ε t )]E. h t θ ] h t θ, Threshold HYGARCH model 3

5 Suppose that the process {y t } generated by models 2) 3) is strictly stationary ergodic with finite fourth moment, Assumptions 2 are satisfied. Then it holds that I τ <. By a method similar to Ling Li 997) Straumann 2005), we can show that the information matrix I τ is also positive definite. By the Taylor expansion the routine arguments for MLEs, we have the following asymptotic results. Theorem 3.. Suppose {y t } is strictly stationary ergodic with finite fourth moment. If Assumptions 2 hold, then n τn τ 0 ) N0, I τ ) in distribution as n, where τ 0 = θ 0, ν 0 ). When the delay parameter d the threshold value r are unknown, we may estimate them by d, r) = argmax L n θ n, ν n ), d D,r a,b] where D is a set including all possible values of d, a, b] is a predetermined interval, say quantiles of the observed sequence, see Tong 990). In the Fisher information matrix I τ, the quantities Ec 2 ε t )], Ec 2 2ε t )], Ec 2 3ε t )] Ec ε t )c 3 ε t )] are all functions of ν 0, can be estimated in practice by replacing ν 0 by ν n. Let ε t = e t θ n )/h /2 t θ n ). We may alternatively estimate these four quantities by their respective sample averages, e.g. using n n c2 ε t ) n n c ε t )c 3 ε t )] respectively to estimate Ec 2 ε t )] Ec ε t )c 3 ε t )], where ν 0 is replaced by ν n. Furthermore, n n h t θ n ) h 2 t θ n ) n ] e t θ n ) e t θ n ) e t e t θ θ = E h t θ θ + o p ), ] h t θ n ) h t θ n ) h t h t θ θ = E h 2 t θ θ + o p ), h t θ n ) h t θ n ) θ ] h t = E + o p ). h t θ Correspondingly, we can consistently estimate I τ by Îτ. 4. DIAGNOSTIC CHECKING Residual autocorrelations from traditional autoregressive moving average models have been found useful in model diagnostic checking, see Box Jenkins 976). Li Mak 994) derived the asymptotic distribution of the square residual autocorrelations, constructed a diagnostic tool for the adequacy of the conditional heteroscedastic models, see also Li 2004). Following their ideas, we derive a portmanteau test to check whether the fitted threshold ARFIMA-HYGARCH model is adequate. Without confusion, we use the notations e t τ) h t τ) in this section, denote e t τ n ) h t τ n ) respectively by ê t ĥt for simplicity, where τ = θ, ν) τ n is the Student s t MLE in section 3. It is observed that e t τ) h t τ) do not depend on ν further e t τ) does not depend on θ V. Hence, e t τ)/ ν = 0, e t τ)/ θ V = 0 h t τ)/ ν = 0. Note that {ê t /ĥ/2 t } is the residual sequence for models 2) 3), n n êt/ĥ/2 t ) = o p ) n n ê2 t /ĥt) = + o p ). Then, for a positive integer k, the lag-k squared residual autocorrelation is r k = n t=k+ ê2 t /ĥt )ê 2 t k /ĥt k ) n ê2 t /ĥt ) 2. Let R = r,..., r K ), we next consider the asymptotic distributions of R. Let Ĉ = Ĉ,..., ĈK) C = C,..., C K ), where Ĉ k = n t=k+ ) ê2 t ê 2 t k ĥ t ĥ t k C k = n n t=k+ ε2 t )ε 2 t k ) is the corresponding values when τ n in Ĉk is replaced by the true parameter vector τ 0. Note that n n ê2 t /ĥt ) 2 = κ + o p ), where κ = Eε 4 t ) = 6ν 0 4) + 2. Then it is sufficient to derive the asymptotic distribution of Ĉ. Taking the Taylor expansion, we can obtain that 5) Ĉ = C + X τ n τ 0 ) + o p n /2 ), where X = X,..., X K ), X k = Eh t e 2 t k /h t k ) h t / τ)]. It holds that, by Theorem 3., 6) n τn τ 0 ) = Iτ n + o p ), n ) c ε t ) + c 2ε t ) e t τ + c 3ε t ) ht, h t ] h t τ where c i ε t ) with i =, 2 3 are defined as in section 3. Note that Eε 2 t )ε 2 t k )c ε t )] = 0 for k > 0, Eε 2 t )c 2 ε t )] = 0 Eε 2 t )c 3 ε t )] =. By 5), 6), the central limit theorem the Cramér-Wold device, under the conditions of Theorem 3., we can show that n R N0, Σ) in distribution as n, where Σ = I κ 2 X Iτ X. 4 W. Kwan, W.K. Li G. Li

6 Table. Estimation result from 500 simulated series of model regime γ β α d F G δ ν n = 000 Bias MSE Bias MSE n = 2000 Bias MSE Bias MSE n = 4000 Bias MSE Bias MSE Table 2. Estimation result from 500 simulated series of model 2 regime γ β α d F G δ ν n = 000 Bias MSE Bias MSE n = 2000 Bias MSE Bias MSE n = 4000 Bias MSE Bias MSE Denote X = X,..., X K ), where ) X k = ê2 t k n ĥ t k t=k+ ĥ t h t τ n ). τ It can be shown that X = X + o p ). Let κ = 6 ν n 4) + 2, then we can construct consistent estimators of Σ, denoted by Σ. Based on the asymptotic normality of R, we consider the portmanteau test, Q R K) = n R Σ R. Under the conditions of Theorem 3., if the threshold ARFIMA-HYGARCH model proposed in this paper is correctly specified, the quantity Q R K) will be asymptotically distributed as χ 2 K, the chi-square distribution with K degrees of freedom. Similarly, we can consider the portmanteau test based on the residual autocorrelations, however, it is usually less powerful in detecting departures of the conditional variance specifications, see Li Li 2008). Hence, we concentrate on Q R K) only. 5. SIMULATION RESULTS In this section, we conduct two simulation experiments to demonstrate the usefulness of the asymptotic results. The following threshold HYGARCH,d F G,) model is considered in both experiments, y t = ε t h /2 t, h t = γ ) + δ) L α ) + α ) L) d) β ) F G) ) e 2 t, L if y t 0, h t = γ 2) + δ2) L α 2) + α 2) L) d2) β 2) F G) ) e 2 t, L if y t > 0, where ε t follows Student s t distribution with mean zero, variance one ν degrees of freedom. Note that the conditional variance can be rewritten as h t = γ ) + k= π) k e2 t k if y t d 0, or γ 2) + k= π2) k e2 t k if y t d > 0. In practice, some truncations are often applied to π i) L) = k= πi) k Lk, see Chung 999) Lombardi Gallo 2002). It is remarked that 50 terms of π i) k are used. Table 3. Estimation result from 500 simulated series of model 3 regime γ β α d F G δ ν n = 000 Bias MSE Bias MSE n = 2000 Bias MSE Bias MSE n = 4000 Bias MSE Bias MSE In the first experiment, three sub-models of the threshold HYGARCH, d F G, ) are considered, the values of the parameter vector θ i) = γ i), β i), α i), d i) F G, δi) ) θ = Threshold HYGARCH model 5

7 θ ) ; θ 2) ; ν) are listed as follows, Model : 0., 0., 0.80, 0.45, 0.4; 0., 0.3, 0.80, 0.45, 0.6; 0), Model 2: 0., 0., 0.80, 0.20, 0.4; 0., 0.3, 0.80, 0.45, 0.6; 0), Model 3: 0., 0., 0.80, 0.45, 0.4; 0., 0.3, 0.65, 0.45, 0.6; 0). We consider three different sample sizes, 000, , there are 500 replications for each sample size. The Student s t likelihood-based MLE in section 3 were calculated, the biases Bias) the empirical root mean squared errors MSE) are summarized in Tables -3. It is observed that the biases are generally small. As sample size increases, all biases empirical root mean squared errors decrease. The second experiment is conducted to check the empirical sizes powers of the test statistic Q R K) in section 5. The generating processes are Pair : θ = 0.,, 0.80, 0.25, ; 0.,, 0.80, 0.45, ; 0), or 0., 0., 0.80, 0.45, 0.4; 0., 0.3, 0.80, 0.45, 0.6; 0), Pair 2: θ = 0., 0., 0.80, 0.20, ; 0., 0.3, 0.80, 0.45, ; 0), or 0., 0., 0.80, 0.20, 0.4; 0., 0.3, 0.80, 0.45, 0.6; 0), Pair 3: θ = 0., 0.,,, 0.4; 0., 0.3,,, 0.6; 0), or 0., 0., 0.80, 0.45, 0.4; 0., 0.3, 0.65, 0.45, 0.6; 0), where, for each pair, the first model is called the true model, the second one is the misspecified model. We generated the samples by both the true model the misspecified model, however the generated samples were always estimated by the true model. For samples generated by the misspecified models, the estimated results are expected to be worse since a sub-model was considered to fit the samples, hence they are used to assess the powers. Note that the first two pairs correspond to the misspecification of δ, while the last pair correspond to the misspecification of the α d F G. As in the first experiment, the sample sizes are set to be 500, 000, or 2000, there are 500 replications for each sample size. We consider five different values for K: 3, 6, 0, The empirical sizes powers are presented in Table 4 for Q R K), they are based on the upper fifth percentile of the chi-squared distribution with the corresponding degrees of freedom. It can be seen that all empirical sizes are close to the nominal value 0.05, the test is more powerful as the sample size n increases. As is well known, the proposed test statistic can be regarded as a pure significance test hence its power would not be very high with general departures from the null Table 4. Empirical size power of Q R K) Size Power n Pair Q R3) Q R 6) Q R0) Q R 5) Q R25) Pair 2 Q R 3) Q R6) Q R 0) Q R5) Q R 25) Pair 3 Q R3) Q R 6) Q R0) Q R 5) Q R 25) model. Like many of the tests proposed for testing linearity Luukkonen, Saikkonen Terasvirta, 988; Saikkonen Luukkonen, 988), its power might be substantially improved if it can be shown to be equivalent to some classical tests under certain alternative hypotheses. 6. EMPIRICAL RESULTS In the asset pricing model, temporary bubbles with prices deviating from the fundamental may arise when the fractions of traders believing in those bubbles is sufficiently large. Driven by evolutionary competition between different trading strategies, the model generates excess volatility, see Timmermann 993, 996), Routledge 999) Farmer Joshi 2002). Traders may believe that, in a heterogeneous world, prices will deviate from their fundamental value. Traders can be recognized into two main types: fundamentalists trend followers. Fundamentalists are Efficient Market Hypothesis EMH) believers. The EMH states that it is impossible, except through luck, to consistently outperform the market by using any information that the market already knows. Trend followers are chartists. They extrapolate the latest observed price change. Fundamentalists believe that prices will move towards its fundamental rational expectations RE) value. In contrast, the trend followers are not entirely unaware of the fundamental price. When prices move far away from the fundamental value, they start believing that a price correction towards the fundamental price is about to occur. The fractions of the two different trader types change over time upon price deviations from the RE fundamental value. The financial market is thus characterized by an irregular regime switching. 6 W. Kwan, W.K. Li G. Li

8 0.05 log return 8 conditional variance Figure. Log-returns of the daily exchange rates of the Korean Won again the US dollar from 2 January, 999 to 3 January, Figure 2. Fitted conditional variance, multiplied by 0 4, of the daily exchange rates of the Korean Won again the US dollar from 2 January, 999 to 3 January, In this section, we apply the proposed threshold HY- GARCH model to the daily exchange rates of the Korean Won against the US dollar. There are 2089 observed values for the Won/USD exchange rates from 2 January, 999 to 3 January, We consider a self-excited threshold model, the threshold value is assumed to be zero. In Kwan, Li Ng 200b), it is found that the value of the delay parameter, d, equals to one is a natural choice. The purpose here is to measure the asymmetric behavior of positive returns negative returns. In practice, the threshold value is not too far from zero. As in Kwan, Li Li 200a), the long memory characteristic is found in the volatility. The fitted threshold model is y t = ε t h /2 t, where h t = { L L if y t 0, or L) ])} y 2 t h t = { L L L) ])} y 2 t if y t > 0, the fitted degrees of freedom is the stard errors are given in the corresponding subscripts. The fitted conditional variances ĥt are plotted in Figure 2. The diagnostic tool Q R K) was also performed here, the calculated p-values for K = 3, 6, 0, 5 25 are respectively 0.89, 0.97, 0.99, This result shows that the exchange rates possess an hyperbolic memory. The log likelihood has the value of for the fitted threshold model, while it is for the HYGARCH model as stated in Kwan, Li Li 200a). A likelihood ratio test was performed with the null hypothesis of no threshold, the calculated p-value is This demonstrates that the threshold HYGARCH model is better than the model without threshold in this application. 7. CONCLUSION In this article, a threshold HYGARCH model is proposed, some properties, estimation a diagnostic procedure were presented. The application of this model to the exchange rates of Korean Won against US dollar seems to suggest that the long-memory phenomenon with a threshold structure may exist in the volatility of some financial time series. In other words, the volatility exhibits a nonlinear long memory feature. This phenomenon may also exist in many other time series. The proposed threshold HYGARCH model should be useful in modeling time series that exhibit nonlinear long memory feature. ACKNOWLEDGMENTS The authors are most grateful to the editor two anonymous referees for suggestions valuable comments that led to the improvement of this paper. W.K. Li thanks Threshold HYGARCH model 7

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