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Long Memory and Nonlinearity in Conditional Variances: A Smooth Transition FIGARCH Model Rehim Kılıç∗ Georgia Institute of Technology This Version: February 2009 Under revision for re-submission. Comments are welcome. Abstract This paper introduces a new nonlinear long memory volatility process, denoted by Smooth Transition FIGARCH, or ST-FIGARCH, which is designed to account for both long memory and nonlinear dynamics in the conditional variance process. The nonlinearity is introduced via a logistic transition function which is characterized by a transition parameter and a variable. The model can capture smooth jumps in the altitude of the volatility clusters as well as asymmetric response to negative and positive shocks. A Monte Carlo study ﬁnds that the ST-FIGARCH model outperforms the standard FIGARCH model when nonlinearity is present, and performs at least as well without nonlinearity. Applications reported in the paper show both nonlinearity and long memory characterize the conditional volatility in exchange rate and stock returns and therefore presence of nonlinearity may not be the source of long memory found in the data. JEL Classification: C22, F31, G15. Keywords: FIGARCH, ST-FIGARCH, Volatility, Long memory, Nonlinearity, Asymmetry. ∗ School of Economics, Georgia Institute of Technology, 221 Bobby Dodd Way, Atlanta, GA 30332-0615, e-mail rehim.kilic@econ.gatech.edu. 1 1 Introduction There exists ample evidence that suggest daily and high frequency ﬁnancial returns exhibit persistent autocorrelation in their squared returns, power transformations of returns, and conditional variances as well as other measures of volatility. To model long memory in volatility process, Baillie et al. (1996), Bollerslev and Mikkelsen (1996) and Davidson (2004) suggested long memory ARCH models while Breidt et al., (1998), and Harvey (1998) proposed long memory stochastic volatility models (earlier studies on long memory in volatility measures include the seminal papers by Ding, et al. 1993 and Dacorogna et al. 1993). Among these, the Fractionally Integrated GARCH (FIGARCH) model of Baillie et al. (1996) has been used extensively in modeling volatility dynamics and long memory in commodities, equities and exchange rate returns. Besides Baillie et al. (1996), examples include Beltratti and Morana (1999), Baillie et al. (2000), Baillie and Osterberg (2000), Brunetti and Gilbert (2000) and Kılıç (2004, 2007 and 2008) among others. In the mean time, a literature has emerged with the objective to understand the underlying causes for the widespread empirical ﬁnding of long memory in volatility. Ding and Granger (1996) show that contemporaneous aggregation of stable GARCH(1, 1) process can generate aggregate processes that display hyperbolically decaying autocorrelations. Anderson and Bollerslev (1997) show that the contemporaneous aggregation of weakly dependent information ﬂow may produce the long memory in volatility. Muller et al. (1997) argue that long memory in volatility can arise from the reaction of short-term dealers to the dynamics of a proxy for the expected volatility trend (coarse volatility), which in turn is the cause of persistence in the higher frequency volatility, (or ﬁne volatility) process. A related literature discusses whether observed long memory property in volatility is real or spurious due to neglected level shifts and/or regime changes. Mikosch and Starica (1998), Beine and Laurent (2000), Bredit and Hsu (2002) and Granger and Hyung (2004) show that presence of occasional breaks in the data can cause slowly decaying autocorrelations and hence may lead to ﬁndings of long memory in the conditional volatility of exchange rate and stock returns. In 2 a related line of literature, Lamoreaux and Lastrapes (1990) argue level shifts in conditional variance process may cause extreme persistence of the Integrated GARCH (IGARCH) form. Hamilton and Susmel (1994) considered Markov regime switching models for volatility process with each regime characterized by strong persistence. In a recent paper, Amado and Teräsvirta (2008) propose a new time-varying parameter GARCH model and show that the long memory type behavior of the sample autocorrelations of absolute returns can also be explained by deterministic changes in the unconditional variance. On the other hand, Diebold and Inoue (2001) argue that long memory may be a useful description for forecasting purposes, even if the data generating process shows breaks and weak dependence. Morana and Belteratti (2004) provide supporting evidence on the existence of long memory in the variance process and argue that the presence of long memory in the volatility cannot be fully explained by unaccounted breaks. By incorporating a time-varying intercept term in the FIGARCH model, Baillie and Morona (2007) provide evidence of long memory in conditional volatility of S&P500 returns even after controlling for the structural change. In a recent paper, Ohanissian et al. (2008) suggest a new test for the long memory in volatility which allows for shifts in the volatility process and report evidence that supports the “true” long memory in volatility. There is also a separate line of volatility literature that provides notable evidence on the presence of asymmetric volatility dynamics. Studies by Nelson (1991), Ding et al. (1993), Glosten et al. (1993), Zakoian (1994), and Li and Li (1996) among others show that volatility dynamics diﬀers over negative and positive shocks. On the other hand, Hagerud (1997), Gonzalez-Rivera (1998), Lundbergh and Teräsvirta (1998), Lubrano (2001), Lanne and Saikkonen (2005) and Amado and Teräsvirta (2008) suggest nonlinear GARCH models and provide evidence of nonlinear volatility dynamics in conditional volatility process for stock and exchange rate returns. Given the extant literature on long memory, nonlinearity and asymmetric dynamics, this paper contributes to the literature by providing a new model, denoted by Smooth Transition Autoregressive FIGARCH, or ST − F IGARCH, which can jointly model long memory and nonlinearity in the conditional volatility process. The ST − F IGARCH model generalizes the 3 FIGARCH model to allow for nonlinear dynamics and asymmetry in variance by introducing a smooth transition speciﬁcation for the conditional variance. To our best knowledge, this is the ﬁrst study that explicitly introduces nonlinear dynamics into a long memory GARCH process namely the FIGARCH model which has been used extensively to investigate both GARCH eﬀects and long memory in volatility. The ST-FIGARCH model is capable of accommodating smooth changes both in the amplitude of volatility clusters as well as asymmetry in conditional volatility in a relatively parsimonious way. Such dynamics cannot be modeled by the standard FIGARCH model. Since the suggested model allows joint estimation of long memory and nonlinearity, it should provide useful insights into our understanding of nonlinear and asymmetric behavior as well as presence of long memory in the volatility of ﬁnancial and economic time series. The ST − F IGARCH model belongs to the family of smooth transition GARCH models developed in the previous literature (Hagerud 1997, Gonzàlez-Rivera 1998, Lundberg and Teräsvirta 1998 and Lanne and Saikkonen 2005). However, unlike the previous nonlinear GARCH models, ST − F IGARCH introduces nonlinearity and asymmetry in a long memory conditional volatility model. We should also mention that the adaptive FIGARCH model (A-FIGARCH) suggested by Baillie and Morona (2007) diﬀers from the ST-FIGARCH model. The A-FIGARCH is a time-varying intercept model that controls for deterministic changes in the constant term (and hence the structural change in the conditional volatility) by using Gallant’s (1984) ﬂexible functional form based on the Fourier decomposition. As discussed above, ST-FIGARCH generalizes the standard FIGARCH model to capture nonlinear and asymmetric dynamics and long memory in the conditional volatility process and hence diﬀers substantially from the A-FIGARCH model. Simulations and empirical applications reported in the paper show the usefulness of the proposed model. Simulations show that the Quasi-Maximum Likelihood Estimation (QMLE) works well in estimating the parameters of ST − F IGARCH model in ﬁnite samples. Simulations also indicate that ignoring the nonlinearity may lead to large standard errors and bias in parameters of standard FIGARCH model. Applications to several exchange rate and stock 4 market data show presence of statistically signiﬁcant long memory component even after controlling for nonlinearity in conditional volatility. Therefore, ﬁndings of the paper contribute to the discussion on the spuriousness of the observed long memory in volatility. The reported results suggest that smooth changes and asymmetry in the conditional volatility cannot be the cause of long memory observed in the data. The ﬁndings reported in the paper are in line with studies that provide support for the presence of “true” long memory in volatility. The rest of the paper is organized as follows. Section two discusses brieﬂy the FIGARCH model and introduces the ST − F IGARCH model. Section three provides our simulation results. In section four, we discuss our empirical ﬁndings on long memory and nonlinearity in exchange rate and stock markets. The last section concludes the paper. 2 2.1 A Nonlinear Long Memory Conditional Volatility Model A digression on FIGARCH model Suppose that a discretely sampled time series process can be written as yt = μ + ut with ut = ζt σt , for t = 1, · · · , T. (1) where ζt is a zero-mean and unit variance process, σt is a time-varying measurable function with respect to the information set available at time t − 1 (Ωt−1 ). Therefore, σt2 is the time dependent conditional variance of yt . Baillie et al. (1996) introduce the F IGARCH(p, d, q) model by deﬁning u2t via the well-known “ARMA in squares” representation φ(L)(1 − L)d u2t = ω + β(L)vt (2) where vt = u2t − σt2 , for some ω ∈ R+ , 0 ≤ d ≤ 1 and lag polynomials are deﬁned as φ(L) = 1 − q φi L , β(L) = 1 − i i=1 p i=1 5 βi Li where L is the lag operator, φ(L) and β(L) are ﬁnite order lag polynomials with the roots assumed to lie outside the unit circle, 0 < d < 1 is the fractional diﬀerencing (long memory) parameter. As it can be seen from (2), F IGARCH(p, d, q) model nests the GARCH(p, q) and the integrated GARCH (IGARCH) models in the sense that when d = 0, FIGARCH model reduces to GARCH model while for d = 1 it becomes an IGARCH model. The conditional variance of ut , or inﬁnite ARCH representation of FIGARCH process, is given by σt2 = φ(L) ω + [1 − (1 − L)d ]u2t = ω/β(1) + λ(L)u2t , β(1) β(L) (3) where λ(L) = λ1 L + λ2 L2 + · · · . For any 0 < d < 1, the λj coeﬃcients will be characterized by a slow hyperbolic decay implying persistent impulse response weights (Conrad and Karanasos 2006). For the general F IGARCH(p, d, q) process to be well deﬁned and the conditional variance to be positive for all t, all the coeﬃcients in the inﬁnite ARCH representation need to be nonnegative, i.e. λj ≥ 0 for j = 1, 2, · · · . The conditions for nonnegativity of lag coeﬃcients in λ(L) are not easy to establish for general F IGARCH(p, d, q) models, but as illustrated in Baillie et al. (1996), it is possible to show suﬃcient conditions in a case by case basis. In a recent paper, Conrad and Haag (2006) derive necessary and suﬃcient conditions for the nonnegativity of the conditional variance in the F IGARCH(p, d, q) model of the order p ≤ 2 and suﬃcient conditions for the general model with p > 2. A key feature of the FIGARCH process is the distributed lag coeﬃcients (λj ) in inﬁnite ARCH representation are approximately equal to λj ∼ cj d−1 where c is a positive constant. Therefore the conditional variance can be expressed as the distributed lag of past squared innovations with coeﬃcients decaying at a hyperbolic rate, which is consistent with the long memory property observed in economic and ﬁnancial time series. It is well known that for 0 < d ≤ 1 the F IGARCH(p, d, q) process has an undeﬁned unconditional variance. However, the process does possess a ﬁnite sum to its cumulative impulse response weights. Moreover, following the arguments in Baillie et al. (1996) the F IGARCH process does appear to be 6 strictly stationary and ergodic for 0 ≤ d ≤ 1. Similar to GARCH(1, 1) model, a F IGARCH(1, d, 1) speciﬁcation has been by far the most commonly used long memory model for the conditional variance (Conrad and Haag 2006). Therefore the nonlinear model proposed in this paper will be discussed in terms of the F IGARCH(1, d, 1) representation. 2.2 The Smooth Transition FIGARCH Model Our proposed nonlinear FIGARCH model allows the conditional variance process to depend on the evolution of a variable, called transition variable. Depending on the sign and the magnitude of the transition variable, conditional variance can evolve smoothly between low and high volatility regimes. Although there are several ways to introduce asymmetry and nonlinearity into conditional variance process, in this paper we use ideas developed in the smooth transition autoregressive (STAR) models (see for example Granger and Teräsvirta 1993) and use a smoothly changing logistic function to characterize nonlinearity and asymmetry in the volatility process. Consider the following “Smooth Transition ARMA-in-squares” formulation, what we call Smooth Transition F IGARCH(1, d, 1) (ST − F IGARCH(1, d, 1)) model, (1 − φL)(1 − L)d u2t = ω + [1 − βL(1 − G(zt−s , γ)) − β ∗ LG(zt−s , γ)] vt , (4) where vt = u2t − σt2 , 0 < d < 1, β and β ∗ are the volatility dynamics parameters, G(zt−s , γ) = 1 1+exp(−γzt−s ) is the logistic transition function with transition variable z which is assumed to be covariance stationary. Note that the stationarity assumption ensures that the volatility process visits extreme regimes with positive probability. If the transition variable is not stationary, then the process may stay in one regime indeﬁnitely with positive probability. Note also that z can either be distributed continuously or discretely. This allows us to consider a wide number of variables to be considered as the transition variable. The parameter γ is the transition parameter which characterizes the speed of transition between regimes, s is refereed to be the 7 Figure 1: The logistic transition function for γ = ±{0.5, 1, 5, 10, 100} delay parameter and typically greater than 0, indicating that the transition variable is s− period lagged value of z. The transition function G(.) is bounded between 0 and 1. The typical shape of the transition function is illustrated in Figure 1 for γ > 0 and γ < 0. As the absolute value of γ increases so does the speed of transition between the regimes associated with G(.) = 0 and G(.) = 1 as a function of z, and the transition between the two extreme volatility regimes/states becomes abrupt as γ → ±∞. Typically γ is assumed to be either negative or positive in smooth transition models for identiﬁcation purposes. In this paper, we do not set the regimes a priori and hence do not restrict the parameter space for γ and let the data at hand label the regime. Apart from this, the dynamic characteristics of the model will be symmetric for γ > 0 and γ < 0 and hence in the following we will discuss interpretation of the regimes under the assumption γ > 0. Note that as zt−s → ∞, G(.) → 1, and hence the model in (4) becomes the F IGARCH(1, d, 1) process with volatility dynamics parameter β ∗ . and when zt−s → −∞, G(.) → 0 and hence ST − F IGARCH(1, d, 1) model reduces to the F IGARCH(1, d, 1) model with parameter β. When zt−s → 0 or when γ = 0, then G(.) → parameter β+β ∗ 2 . 1 2 and hence the model in (4) reduces to F IGARCH(1, d, 1) model with On the other hand, the regime associated with G(.) = 1 can be thought to be the upper regime and the regime where G(.) = 0 is the lower regime when γ > 0 and the names reversed when γ < 0. Depending on the sign of the γ, the transition variable z characterizes the conditional 8 volatility process. The choice of transition variable depends on the time series process modeled. A useful feature of the ST-FIGARCH model is that the researcher can choose z to ﬁt his or her research problem. In some cases, economic theory may provide guidance while in some others available empirical information may be used. Possible choices may include time, (which may be useful if one thinks that conditional volatility may have smoothly changing shifts) functions of past values of the returns series and past values of unobserved shocks as in Gonzàlez-Rivera (1998). Another choice would be to use a variable, which may relate to the return and hence volatility. One such example would be news that may cause smooth changes in the volatility dynamics in exchange rates and stock markets. Yet another such example may be changes in the key policy variables such as interest rates which may be linked to the occasional shifts in conditional variance process. Central Bank intervention or the amount of currencies purchased or sold during an intervention can also be considered relevant transition variables as changes in these factors may lead to diﬀerent volatility regimes. Re-arranging the terms in Equation (4) the ST − F IGARCH(1, d, 1) model can be written in the following alternative form, 2 2 + β ∗ G(zt−s , γ)σt−1 σt2 = ω + β(1 − G(zt−s , γ))σt−1 + [1 − βL(1 − G(zt−s , γ)) − β ∗ LG(zt−s , γ)] − (1 − φL)(1 − L)d u2t . (5) It can be observed from Equation (5), in the ST-FIGARCH model, for a given γ = 0, the amplitude of the volatility clusters and hence the dynamics of conditional volatility will be characterized by β and β ∗ . In other words, the amplitude of the volatility clusters will change between G(.) = 0 and G(.) = 1 with the degree of smoothness given by the slope and hence the speed of the transitions across regimes. Equation (5) implies that the conditional variance of ut is given by σt2 (1 − φL)(1 − L)d ω + 1− u2 , (6) = 1 − β(1 − G(zt−s , γ)) − β ∗ G(zt−s , γ) 1 − βL(1 − G(zt−s , γ)) − β ∗ LG(zt−s , γ) t which shows that the constant term also changes smoothly and takes on values between ψ = 9 ω/(1−β) and ψ ∗ = ω/(1−β ∗ ) depending on if the conditional volatility is in the regime dictated by G(.) = 0 or G(.) = 1 respectively. This shows that in the ST-FIGARCH model, since the constant term will change between extreme regimes, the level of the conditional volatility should be changing over diﬀerent regimes. Therefore, the ST-FIGARCH model may capture changes in the the conditional volatility which if not modeled adequately may spuriously lead to ﬁnding of long memory in the volatility as argued by some of the papers discussed in the Introduction. Conditional on a regime, one can derive the conditions for non-negativity of conditional volatility process as in Conrad and Haag (2006). In the limiting regimes, conditions for nonnegativity of conditional variance will be the same as the conditions given in Conrad and Haag (2006) with the autoregressive parameter replaced by β/β ∗ or β+β ∗ 2 for upper/lower and middle regimes respectively. For example, in the regime, G(+∞, γ > 0) = 1 or G(−∞, γ < 0) = 1, conditional volatility is well deﬁned and positive provided that ω > 0, λ∗1 = d + φ − β ∗ ≥ 0, and φ ≤ (1 − d)/2 for the case 0 < β ∗ < 1 and for the case −1 < β ∗ < 0, λ∗1 ≥ 0, λ∗2 ≥ 0, ∗ and φ ≤ ((1 − d)/2) ββ ∗+((2−d)/3) +(1−d)/2 as stated in Conrad and Haag (2006, pp. 421). For all other regimes, the conditions for non-negativity of conditional volatility process can be checked for conditional on the existing regime at date t. Note that since the transition function is bounded between 0 and 1 and β(1 − G) + β ∗ G is a convex combination of β and β ∗ provided that the nonnegativity conditions are satisﬁed in the extreme regimes, they should also be satisﬁed for all the intermediate regimes with probability one. Note that ARCH(∞) representation of ST − F IGARCH model given in Equation (6) shows that inﬁnite ARCH terms depend on the regime in a given date t. In the extreme regime G(+∞, γ > 0) = G(−∞, γ < 0) = 1, the inﬁnite ARCH representation is σt2 = ∞ ω + λ∗j u2t−j 1 − β∗ (7) j=0 and with λ∗0 = 1, λ∗1 = d + φ − β ∗ , λ∗j = β ∗ λ∗j−1 + (fj − φ)(−gj−1 ) for all j ≥ 2 and fj = j−1−d j j gj = fj qj−1 = i=1 fi for j = 1, 2, · · · (see Conrad and Haag 2006 for details). Similarly for 10 the regime where G(+∞, γ < 0) = G(−∞, γ > 0) = 0, the inﬁnite ARCH representation is σt2 = ∞ ω + λj u2t−j 1−β (8) j=0 with λ0 = 1, λ1 = d + φ − β and λj = βλj−1 + (fj − φ)(−gj−1 ) for all j ≥ 2. The ST-FIGARCH model can capture the asymmetric dynamics in volatility for negative and positive shocks when the transition variable is the lagged value of the error term. This can be seen by analyzing the conditional volatility in the extreme regimes. For γ > 0, in order for volatility to be higher for G(.) = 0 (the regime where zt−s = ut−s → −∞) than G(.) = 1 (the regime where zt−s = ut−s → +∞), we should have ∞ ∞ j=0 j=0 ω ω λj u2t−j > + λ∗j u2t−j . + ∗ 1−β 1−β By analyzing the terms of this inequality one can show that the above condition reduces to β ∗ > β. Similarly for γ < 0, in order for volatility to be higher for negative shocks (that is for the regime G(zt−s = ut−s → −∞ γ < 0) = 1) than the positive shocks (for the regime G(zt−s = ut−s → +∞ γ < 0) = 0) we should have β > β ∗ . For example, if the transition variable is the lagged error process ut−1 , then one can anticipate to see for big negative shocks (i.e. ut−1 < 0) volatility to be higher than the big positive shocks (ut−1 > 0) and hence β < β ∗ when γ > 0 and β ∗ < β when γ < 0. This way, ST-FIGARCH model captures both asymmetry and long memory as well as smooth changes in conditional variance in a parsimonious way. Since in practice, the asymmetry conditions typically depend on the value of the transition function at any given date t, we suggest to to use plots of estimated conditional volatility process over the transition variable together with the transition function, to investigate asymmetry and nonlinearity in conditional volatility. Further insights on some of the properties of ST-FIGARCH model can be obtained by considering its implied news impact curve (Pagan and Schwert 1990) which shows the relationship 2 , keeping between the current shock ut and the conditional volatility in the next period, σt+1 11 all other information constant. For the standard F IGARCH(1, d, 1) model the news impact curve (NIC) is thus deﬁned as N IC(ut |σt2 2 2 = σ ) = ω + βσ + (1 − β)u2t − (1 − φ) ∞ πi (d) u2t . i=0 Note that in contrast to the GARCH models NIC from FIGARCH models may depend upon the long memory parameter as well as dynamics parameters β and φ. Note also that the value of the conditional volatility σt2 moves the curve vertically and the moves are proportional to changes in σ 2 as in the GARCH model. However, the impact of a shock on next period’s conditional volatility depends on β, φ as well as the inﬁnite sum which is a function of the long memory parameter d. In a similar fashion, the news impact curve for the ST − F IGARCH(1, d, 1) model can be written as N IC(ut |σt2 = σ 2 ) = ω + [β + (β ∗ − β)G(zt−s , γ)] σ 2 + ∞ ∗ 2 πi (d) u2t , [1 − β − (β − β)G(zt−s , γ)] ut − (1 − φ) i=0 where πi = Γ(i−d) Γ(i+1)Γ(−d) and Γ(.) denotes the Gamma function.1 Note that the NIC for the ST − F IGARCH model depends on the transition function and the transition parameter and hence the prevailing regime. Diﬀerently from the F IGARCH model, the impact of a shock on next period’s conditional volatility depends on the regime and the impact moves smoothly between the extreme regimes where G(.) = 0 and G(.) = 1. Moreover, the eﬀect of an increase in σ 2 on the NIC depends on the value of the transition function and hence the regime. In this model a change in the value of σ 2 also moves the curve vertically but the size of these moves depends on the regime. Note that if the transition variable is the lagged shacks, the impact of a shock on the next period’s conditional volatility will also depend upon the sign and the size of the shock itself. Fixing ω = 0.1, φ = 0.1 and changing other parameters of the model, Figure (2) depicts 1 In practice, we truncate the inﬁnite sum in these equations by selecting a ﬁnite number. In computing NICs we have truncated at 1000 consistent with the suggestion of Baillie et al. (1996). 12 NICs of the ST − F IGARCH(1, d, 1) models as functions of u in the range [−4, 4]. The ﬁgure displays the NICs for various degrees of nonlinearity (γ ∈ {0, 0.5, 1, 5, 10, 25}) with two sets of parameter values for the long memory parameter (d ∈ {0.4, 0.7}) and the nonlinear dynamics parameters (β, β ∗ ) = ({0.3, 0.6}, {0.4, 0.5}). Displayed plots indicate the asymmetry in the response of conditional volatility to negative and positive shocks. Note that since STFIGARCH model approaches to the FIGARCH model as γ → 0 or as the diﬀerence between (β ∗ and β decreases, so does the observed asymmetric response. Therefore as the degree of nonlinearity increases so does the asymmetric response. We also note that as the shocks moves from negative to positive (i.e. in the neighborhood of zero) there is a distortion in the NICs as such the conditional volatility ﬁrst increases and then stays calm for some small positive values of shocks and then starts to increase after the shocks reaches a certain threshold level. The length of the initial increase in conditional volatility around u = 0 depends on the transition parameter as well as the diﬀerence between β ∗ and β which measures the diﬀerence in nonlinear dynamics across extreme regimes. 2.3 Estimation of ST − F IGARCH Model Estimation and inference for the parameters of ST − F IGARCH model can be carried out by the method of Quasi Maximum Likelihood (QMLE), where the Gaussian log likelihood (ζ, ut ) = −0.5T ln(2π) − 0.5 T i=1 [ln(σt2 ) + u2t ] σt2 is numerically maximized with respect to the vector of parameters ζ = (μ, d, ω, β, β ∗ φ, γ) . Therefore, the QMLE implements simultaneous estimation of all the model’s parameters, including the transition parameter γ. Under fairly general conditions, the asymptotic distribution of the QMLE is T 1/2 (ζ̂ − ζ0 ) −→ N 0, A(ζ0 )−1 B(ζ0 )A(ζ0 )−1 , where T is the sample size (adjusted for the initial values), ζ0 denotes the true value of the vector of parameters, A(ζ0 ) is the Hessian and B(ζ0 ) is the outer product of gradient evaluated 13 Figure 2: News Impact Curves for ST − F IGARCH Model d = 0.7, β = 0.3, β ∗ = 0.6 d = 0.4, β = 0.3, β ∗ = 0.6 d = 0.4, β = 0.4, β ∗ = 0.5 d = 0.7, β = 0.4, β ∗ = 0.5 Key: News Impact Curves for ST-FIGARCH model. 14 at the true parameter values. Although, there is no formal results that show the asymptotic consistency and normality of QMLE of FIGARCH models, simulations reported in Baillie et al. (1996) suggest that it performs well for the sample sizes typically observed in high frequency ﬁnancial data. Lee and Hansen (1994), and Lumsdaine (1996) showed consistency and asymptotic normality of QMLE for the strictly stationary and ergodic GARCH(1, 1) process. Berkes et al. (2003) shows consistency and asymptotic normality of QMLE for the general strictly stationary and ergodic GARCH(p, q) model. Also, recently Jensen and Rahbek (2004) showed consistency and asymptotic normality of IGARCH(1, 1) process which is nonstationary and nonergodic. Although a formal proof of consistency and asymptotic normality of QMLE for F IGARCH and ST − F IGARCH models is beyond the scope of this paper, one may expect similar results to hold for the F IGARCH(1, d, 1) and ST − F IGARCH(1, d, 1) models following the arguments and simulation evidence reported in Baillie et al. (1996). We examine ﬁnite sample performance of QMLE of ST-FIGARCH model by Monte Carlo experiments in the next section. As discussed in Baillie et al. (1996) to carry out QMLE we need to condition on pre-sample values and truncate the inﬁnite lag polynomial in equation (6). Following Baillie et al. (1996), we set the truncation lag to 1000. 3 Simulation Results In this section we report and discuss Monte Carlo evidence on the impact of estimating ST − F IGARCH models under diﬀerent data generating scenarios. The objective of the simulations is to gain insights into the QMLE of parameters of ST − F IGARCH model in sample sizes that are typical of high frequency ﬁnancial and economic time series data. All the experiments are carried out by specifying an uncorrelated process yt for the mean with various forms of long memory and nonlinear dynamics for the conditional variance process. Speciﬁcally the data is generated from Eqs. (1) and (5) with ζt ∼ N (0, 1) and zt = ut−1 as the transition variable. In all experiments we set μ = 0, ω = 0.1, β = 0.3, β ∗ = 0.5 and φ = 0.1. 15 Table 1: Monte Carlo Results: Eﬀects of ignoring nonlinearity bias(β) d = 0.3 d = 0.45 d = 0.7 0.175 0.200 0.189 d = 0.3 d = 0.45 d = 0.7 0.210 0.215 0.228 RM SE(β) s.e(β) bias(d) RM SE(d) s.e(d) γ = 1, ST − F IGARCH(1, d, 1) Model 0.189 0.064 0.098 0.109 0.053 0.207 0.072 0.105 0.112 0.051 0.199 0.071 0.089 0.105 0.038 γ = 10, ST − F IGARCH(1, d, 1) Model 0.230 0.101 0.106 0.118 0.072 0.238 0.110 0.108 0.119 0.068 0.258 0.106 0.103 0.116 0.069 RM SF E(σ) 0.080 0.084 0.091 0.098 0.097 0.102 Notes: Table reports the simulation results on bias, RMSE, s.e. for d and β from QMLE of F IGARCH(1, d, 1) model. The true DGP is the ST − F IGARCH(1, d, 1) model and the estimated model is the F IGARCH(1, d, 1) model. The long memory parameter d and the transition parameter γ are varied to see the eﬀect of changes in long memory and the speed of transition on the key parameters of the model. The experiments were conducted for three diﬀerent values of long memory parameter (d ∈ {0.3, 0.45, 0.7}) and for ﬁve values of transition parameter (γ ∈ {−10, −1, 0, 1, 10}). Since the simulation results were similar for γ < 0, we discuss results for γ ≥ 0 only in the following (results for γ < 0 can be obtained upon request). Clearly, estimation of the ST-FIGARCH model should prove superﬂuous in the experiment for which γ = 0 as the true data generating process (DGP) is a martingale-FIGARCH model. The interest in experiments in which γ = (1, 10) centers on the performance of QMLE when the pure martingale-FIGARCH and the nonlinear martingale-ST-FIGARCH models are estimated in the presence of nonlinear dynamics in the conditional volatility process. We have generated 500 simulations with 10,000 observations in each replication and discarded the ﬁrst 7,000 simulated observations in each replication to minimize the impact of initialization. This left 3000 observations for each Monte Carlo replication. The F IGARCH(1, d, 1) and the ST − F IGARCH(1, d, 1) models were estimated for each replication. Tables 1 through 3 report the Monte Carlo bias (bias), root mean squared error (RM SE) and the standard error (s.e.) of the QMLE of the F IGARCH(1, d, 1) and ST − F IGARCH(1, d, 1) models. Following Baillie and Morona (2007), we also use root mean square forecast error (RM SF Eσ ) statistic to evaluate the ability of the models in ﬁtting the conditional variance process. The measure 16 is deﬁned as ⎞ ⎛ 500 T 1 2 − σ 2 )⎠ , ⎝ 1 (σ̂t,j RM SF Eσ = t,j 500 T t=1 j=1 2 and σ 2 are the estimated and actual variances for the simulation j. where σ̂t,j t,j Table 1 reports simulation results from the experiments where the true DGP is the ST − F IGARCH(1, d, 1) model while the estimated model is the F IGARCH(1, d, 1) model. In other words, we seek to explore potential impact of ignoring nonlinearity on the QMLE of the long memory parameter d and the volatility dynamics parameter β in FIGARCH model. Estimation of misspeciﬁed model, i.e. the F IGARCH(1, d, 1) model, produces positive bias in estimates of long memory and volatility dynamics parameter in the FIGARCH model. The size of the bias as well as the standard error and the RMSE of d tend to increase with the increase in the severeness of nonlinearity (i.e. as γ increases). As can be observed from the ﬁrst panel of Table 1, ignoring nonlinearity may severely bias estimates of β in F IGARCH model for any given d with rather large RMSE and standard errors. The bias, especially the RMSE and standard error of β increase with the increase in γ. Also, reported RM SF E(σ) tends to increase with the increase in γ suggesting pour predictive power for F IGARCH model. We should however note that in computing the bias and other measures for β we used β+β ∗ 2 = 0.4 as the true value of β when true DGP was ST − F IGARCH model while the estimated model was F IGARCH model in Table 1. Therefore caution should be exercised in interpreting the results for β in Table 1. Comparison of columns 5-7 of Table 1 with the results in columns 2-4 in the ﬁrst panel of Table 2 reveals that QMLE of d in ST-FIGARCH and FIGARCH models has roughly the same degree of bias (with slightly higher bias from the FIGARCH model than the STFIGARCH model). Interestingly enough, similar comparisons show that both standard errors and the RMSE of d are about the same or slightly lower from ST − F IGARCH model than the F IGARCH model. These results suggest there is no added cost of estimating ST −F IGARCH model as opposed to the F IGARCH model. Note that results in the last column of Table 1 and the last column of ﬁrst panel in Table 2 show ST − F IGARCH model performs well compared 17 with the F IGARCH model in terms of predictive power. These ﬁndings are consistent with the results reported in Baillie and Morona (2007) in that their simulations also suggest no cost of estimating their A-FIGARCH model when the true model is a FIGARCH process. Inspection of columns 5-7 in the ﬁrst panel of Table 2 reveals that when the true DGP is ST − F IGARCH model with γ = 0, estimating ST − F IGARCH model typically produces relatively large ﬁnite sample bias for γ with considerably large RMSE and standard errors. This makes sense as in the ST −F IGARCH model with γ = 0, the the slope of the transition function is zero and the transition function takes on value of 1/2 for all z and hence the process stays always in the middle regime. This suggests that there is no nonlinear/asymmetric dynamics in the conditional volatility process that can be identiﬁed accurately by QMLE. Careful inspection of second and third panels of Table 2 show that QMLE of d and γ have considerably low ﬁnite sample bias. Note the bias decreases with the degree of nonlinearity (increases in γ). We should also note that although QMLE has small bias for γ, it usually have larger RMSE and standard errors. Despite the decrease in the RMSE and standard error of estimated γ with increases in the d and γ, there is still notable uncertainty about the ﬁnite sample distribution of γ. This might be partly because of the behavior of the transition function G(zt−s , γ). In the ST − F IGARCH model the partial derivative, zt e−γzt ∂σt2 ∗ 2 ∗ 2 = (β − β)σt−1 − (β + β )ut−1 , ∂γ (1 + e−γzt )2 is part of the score. When the outer product of the score and the hessian are calculated, the partial derivative is squared, producing values that are close to zero for especially very small and very large positive and negative values of γ. This in turn may lead to large standard error for γ. Therefore caution should be exercised in evaluating the signiﬁcance of the estimated transition parameters in nonlinear smooth transition models (see also Gonzàlez-Rivera 1998 and Lundberg and Teräsvirta 1998 on diﬃculties involved in precision of transition parameters.) Table 3 displays the simulation results for the other key parameters of ST − F IGARCH model. Since simulations show no major impact on φ when true and estimated models are ST- 18 Table 2: Monte Carlo Results for d and γ bias(d) d = 0.3 d = 0.45 d = 0.7 0.070 0.075 0.073 d = 0.3 d = 0.45 d = 0.7 0.060 0.060 0.063 d = 0.3 d = 0.45 d = 0.7 0.040 0.045 0.032 RM SE(d) s.e(d) bias(γ) RM SE(γ) s.e(γ) γ = 0, F IGARCH(1, d, 1) Model 0.088 0.058 0.659 3.974 3.910 0.091 0.047 0.632 4.215 4.082 0.082 0.036 0.698 3.860 3.790 γ = 1, ST − F IGARCH(1, d, 1) Model 0.070 0.041 0.066 1.851 1.671 0.067 0.040 0.072 1.667 1.630 0.068 0.033 0.070 1.643 1.602 γ = 10, ST − F IGARCH(1, d, 1) Model 0.053 0.037 0.049 1.658 1.652 0.053 0.034 0.051 1.448 1.422 0.049 0.030 0.050 1.309 1.302 RM SF E(σ) 0.118 0.127 0.125 0.062 0.055 0.048 0.050 0.045 0.037 Notes: Table reports the simulation results on bias, RMSE, s.e. for d and γ from QMLE of ST − F IGARCH(1, d, 1) model. The true DGP is for the ﬁrst panel is the F IGARCH while in second and third panels the ST − F IGARCH(1, d, 1) model and the estimated model is the ST − F IGARCH(1, d, 1) model. FIGARCH, for the sake of conserving space, we do not report results for φ. However, we should emphasize that when the true DGP is ST-FIGARCH and the estimated model is FIGARCH, we observe some positive bias and relatively large standard errors for φ (results for φ can be obtained upon request). Inspection of the results show that when γ = 0, estimates for β and β ∗ show some bias (β typically have negative bias while β ∗ have positive bias). Both estimates also have considerably large RMSE and large standard errors. This suggests that when γ is close to zero, since σt2 is in the middle regime where it follows roughly F IGARCH(1, d, 1) model with parameter (β + β ∗ )/2, QMLE underestimates β and overestimates β ∗ with large standard errors. Therefore QMLE has diﬃculty identifying nonlinear dynamics which should be expected as nonlinear eﬀects diminishes. On the other hand, when γ is far from zero, QMLE produces estimates for β and β ∗ that has negligible bias with rather small RMSE and standard errors. Both RMSE and standard errors of estimates fall signiﬁcantly as γ becomes larger. Displayed results on the last three columns of Table 3, show that bias, RMSE and standard error of ω are low and as the persistence and speed of transition increases they all tend to fall. 19 Table 3: Monte Carlo Results bias(β) RM SE(β) d = 0.3 d = 0.45 d = 0.7 -0.251 -0.265 -0.273 0.421 0.490 0.511 d = 0.3 d = 0.45 d = 0.7 -0.029 -0.025 -0.020 0.170 0.194 0.166 d = 0.3 d = 0.45 d = 0.7 -0.015 -0.012 -0.011 0.098 0.094 0.085 4 bias(β ∗ ) RM SE(β ∗ ) s.e(β ∗ ) bias(ω) γ = 0, F IGARCH(1, d, 1) Model 0.319 0.259 0.225 0.193 0.044 0.364 0.332 0.256 0.207 0.040 0.347 0.298 0.212 0.210 0.040 γ = 1, ST − F IGARCH(1, d, 1) Model 0.102 0.046 0.150 0.121 0.029 0.129 0.037 0.151 0.109 0.025 0.130 0.034 0.139 0.102 0.025 γ = 10, ST − F IGARCH(1, d, 1) Model 0.049 0.049 0.071 0.066 0.029 0.045 0.044 0.055 0.032 0.020 0.041 0.045 0.060 0.024 0.019 s.e(β) RM SE(ω) s.e(ω) 0.069 0.060 0.048 0.058 0.048 0.033 0.062 0.053 0.040 0.055 0.042 0.030 0.057 0.042 0.040 0.051 0.029 0.026 Applications to Exchange Rate and Stock Market Volatilities In this section, we report and discuss estimation of ST − F IGARCH and F IGARCH models for four daily major exchange rates and three S&P 500 composite stocks traded in the New York Stock Exchange as well as the S&P 500 index itself. The exchange rates are Canadian Dollar, Japanese Yen and Swiss Francs per US Dollar and US Dollar per British Pound. Exchange rate series are the noon buying rates in New York City and provided by Federal Reserve Economic Data delivery system (FRED). The sample period for exchange rates is March 1, 1973 to February 21, 2008. The sample size for all exchange rate series is 8785. The stocks are General Motors (GM), International Business Machines (IBM) and Intel (INT). Individual stock market data are obtained from Reuters. The observation period begins July 29, 1980 for GM, January 3, 1968 for IBM, and November 14, 1982 for INT. The index series covers the period January 3, 1950 to May 6, 2008 and obtained from CRSP. The sample for stocks ends on April 2, 2008 with sample sizes 6984, 10128, 6405 and 14679 for GM, IBM, INT and S%P 500 respectively. Table 4 reports summary statistics with the Ljung-Box portmanteau tests for up to 20thorder serial correlation in the returns and squared returns. Inspection of the Table reveals that except for Pound, daily exchange rates has a median return of 0% with low average returns (except for Canadian Dollar for which average returns is 0%). Daily stock and index returns are characterized by positive average returns with higher variation. Minimum and maximum values 20 21 sample 8784 8784 8790 8784 6983 10128 6404 14678 mean 0.000 -0.010 -0.012 -0.003 0.002 0.020 0.062 0.030 med 0.000 0.000 0.000 0.006 0.000 0.000 0.000 0.044 summary statistics min max var -2.073 2.670 0.106 -5.630 6.256 0.414 -4.408 5.827 0.534 -3.843 4.589 0.351 -23.601 16.647 3.988 -26.088 12.366 2.755 -24.889 23.400 7.540 -22.900 8.709 0.812 skew 0.078 -0.315 0.005 -0.139 -0.081 -0.281 -0.315 -1.274 kurt 6.113 8.347 6.113 6.719 9.258 15.567 8.494 36.555 Ljung-Box Q(rt ) Q(rt2 ) 33.1 5328.6 37.8 1037.4 27.5 1216.8 63.9 1982.9 31.2 703.4 43.8 700.7 89.2 1123.9 121.2 9629.0 |rt | L AN NM L T LGM 3, 0.289 0.221 0.108 0.087 0.077 0.041 0.048 0.053 0.052 0.050 0.034 0.045 0.025 0.040 0.005 0.008 AN NM L 0.197 0.093 0.052 0.040 0.032 0.036 0.041 0.015 rt L T LGM 3, 0.228 0.064 0.048 0.036 0.037 0.034 0.053 0.012 Key: The sample period for exchange rates is 03/01/1973-02/21/2008 producing 8784 daily returns for Canadian Dollar, Yen, and Pound and 8790 retruns for the Swiss Francs (observations for weekends and holidays were not available). The sample periods for the stock are as follows; 07/29/1980-04/02/2008 for GM, 01/03/1968-04/02/2008 for IBM, 11/14/1982-04/02/2008 form INT and 01/03/1950-05/06/2008 for the S&P 500 index. Q(rt ), and Q(rt2 ), denote the Ljung Box portmanteau tests for up to 20th-order serial correlation in the standardized daily returns and squared L returns respectively. AN NM L and T LGM 3, are the p-values for the tests based on neural networks and Taylor series expansion discussed in Baillie and Kapetanios (2007). series Canadian Dollar Japanese Yen Swiss Franc UK Pound GM IBM INT S&P 500 Table 4: Summary statistics for daily exchange rate and stock returns and tests for nonlinearity on absolute and squared daily returns 2 Figure 3: Autocorrelation functions for daily exchange rate and S&P 500 index returns UK Pound S&P 500 Index Key: Autocorrelations and 95% conﬁdence intervals for daily US Dollar-British Pound and S& P 500 index log price and absolute returns for lags 1 through 60 days. 22 as well as reported variance statistics show considerable variation in stock returns especially for the individual stocks. Both exchange rate and stock returns have nonzero skewness with notable excess kurtosis. Stock returns show higher Kurtosis than the exchange rates. Reported in last two columns of Table 4, Ljung-Box statistics suggest exchange rate and stock returns have serial correlation and squared returns show signiﬁcant dependence throughout time. This is also supported by the plots of autocorrelation functions displayed for UK Pound and S&P 500 index returns. Plots for all other series are similar and can be obtained upon request. Inspection of plots in Figure 2 shows autocorrelations for squared and absolute returns start of around 0.2-0.3 and slowly decay over time. Even after 60 days, autocorrelations are outside the 95% conﬁdence bands, suggesting presence of long memory in the conditional volatility process. As a preliminary to the following analysis, in the last four columns of Table 4, we report p-values from several tests that test the null of linearity against the alternative of nonlinearity in absolute and squared daily log exchange rate and equity returns. We use tests based on neural networks and Taylor series expansions as suggested recently by Baillie and Kapetanios (2007). Baillie and Kapetanios (2007) construct tests for the presence of nonlinearity of unknown form when the time series has a long-memory component. We refer reader to Baillie and Kapetanios (2007) for a discussion of these tests to conserve space and report results based on the neural network approximations of logistic nonlinear form (i.e., AN NM L in Baillie and Kapetanios 2007’s notation) and tests based on third order Taylor series approximation of logistic form L of nonlinearity (i.e. T LGM 3, ). The reported results clearly indicates presence of nonlinearity in various measures of volatility including the daily log absolute and squared returns for all equities and for all exchange rates except for the Canadian Dollar. The presence of nonlinearity as well as long memory suggests the possibility of formulating comparable GARCH models, such as FIGARCH with nonlinear terms. We present estimation results and several diagnostic statistics for the ﬁnal F IGARCH and ST − F IGARCH models in Tables 5 and 6 for daily exchange rate and stock returns respectively. We used BHHH algorithm with numerical derivatives in Gauss to maximize the likelihood functions. In each case, diﬀerent starting values for the parameters are used to check 23 the global maximum. The results were robust to diﬀerent initial values. Following Conrad and Haag (2006), we have also checked the necessary and suﬃcient conditions for the nonnegativity of conditional variance process for the FIGARCH and ST-FIGARCH models (for the extreme regimes). In this paper, we consider lagged values of error term as the transition variable. We leave future research other possible candidate transition variables. We have estimated STFIGARCH models with transition variable zt = ut−s where s ∈ {1, 2, · · · , 5}. Reported models are selected on the basis of extensive diagnostic statistics. In all series, zt = ut−1 is found to provide the best model (complete estimation results can be obtained upon request). In each table, the ﬁrst panel reports parameter estimates and QMLE standard errors. Second panels in each table report the estimated constant term from the FIGARCH model (ψ = ω/(1 − β)) and the ST-FIGARCH model in the extreme regimes (i.e. ψ = ω/(1 − β) and ψ ∗ = ω/(1 − β ∗ )) together with the percentage diﬀerence across low and high volatility regimes. In the third panels, we report summary diagnostic statistics for the estimated models (including the Ljung-Box statistics and skewness and kurtosis values from the standardized residuals) as well as log-likelihood values and Akaike and Schwartz Information criteria. We also report p-values from positive sign bias, negative sign bias and both sign and size bias tests suggested by Engle and Ng (1993). Finally the last panel in each table gives the robust Wald statistic for testing the diﬀerential volatility dynamics across extreme regimes (that is, the null β ∗ = β against the alternative β ∗ > β when γ > 0 and β ∗ < β when γ < 0). Parameter estimates indicate that for all daily exchange rates and individual as well as S&P 500 Index, a small and positive MA component characterizes the conditional mean of daily returns. For all return series estimates of long memory parameter d are signiﬁcant and greater than 0 but less than unity (robust Wald tests, not reported, can be obtained upon request). Estimated long memory parameters are around 0.3-0.4 for most of the series. Two exceptions are the Pound and Canadian Dollar for which the estimated value for d are 0.550 and 0.602 from F IGARCH model respectively. However, introduction of nonlinearity reduces the estimates for d to 0.386 for the Pound and to 0.515 for the Canadian Dollar. Overall, estimated values for d for all series suggest that ignoring nonlinear dynamics may induce an upward bias 24 Table 5: Estimated FIGARCH and ST-FIGARCH Models for Daily US Dollar Exchange rate returns μ θ1 d ω β β∗ φ γ ψ, ψ ∗ %Δ AIC SIC m3 m4 Q(20) Q(10) Q2 (20) Q2 (10) ppsb pnsb pssb W ald Swiss Francs -0.007 -0.007 (0.007) (0.007) 0.022 0.023 (0.012) (0.011) 0.437 0.422 (0.027) (0.025) 0.015 0.015 (0.002) (0.002) 0.616 0.510 (0.025) (0.038) . 0.665 . (0.037) 0.248 0.246 (0.017) (0.018) . 2.347 . (1.422) . [0.072] 0.039 0.031, 0.045 . 45.2 -9121.4 -9108.3 18254.1 18232.5 18296.6 18289.2 -0.166 -0.178 4.525 4.455 34.022 33.098 19.740 19.223 15.466 13.955 7.412 6.429 0.116 0.176 0.063 0.148 0.033 0.127 . 7.124 . {0.008} . [0.044] Japanese Yen 0.000 -0.003 (0.006) (0.006) 0.036 0.042 (0.012) (0.012) 0.410 0.389 (0.019) (0.020) 0.014 0.012 (0.001) (0.001) 0.536 0.540 (0.022) (0.024) . 0.642 . (0.023) 0.284 0.280 (0.022) (0.021) . 13.072 . (6.896) . [0.067] 0.030 0.026, 0.034 . 30.8 -7815.1 -7802.9 15642.1 15621.8 15684.6 15678.5 -0.181 -0.204 9.026 8.563 36.941 36.949 26.789 27.307 16.792 18.937 9.229 10.845 0.094 0.247 0.043 0.112 0.002 0.114 . 40.196 . {0.000} . [0.010] Canadian Dollar 0.003 0.004 (0.002) (0.002) 0.043 0.024 (0.011) (0.010) 0.602 0.515 (0.041) (0.033) 0.001 0.001 (0.0001) (0.000) 0.738 0.516 (0.259) (0.026) . 0.718 . (0.027) 0.189 0.211 (0.023) (0.021) . 1.215 . (1.142) . [0.127] 0.003 0.002, 0.006 . 50.0 -1118.1 -1111.9 2248.2 2239.9 2290.7 2296.5 0.226 0.217 5.045 5.001 27.517 40.265 13.693 33.589 13.795 12.219 6.850 6.120 0.116 0.403 0.113 0.129 0.093 0.103 . 1.761 . {0.185} . [0.116] UK Pound 0.008 -0.002 (0.004) (0.006) 0.053 0.047 (0.011) (0.012) 0.550 0.386 (0.021) (0.019) 0.000 0.014 (0.002) (0.001) 0.721 0.581 (0.010) (0.023) . 0.640 . (0.020) 0.221 0.301 (0.010) (0.019) . 11.573 . (14.965) . [0.118] 0.001 0.033, 0.039 . 18.2 -7261.3 -7117.5 14534.6 14250.9 14577.1 14307.5 0.488 -0.004 19.282 8.210 37.690 34.287 26.453 25.703 8.861 10.593 2.540 3.950 0.107 0.413 0.030 0.240 0.042 0.171 . 17.704 . {0.000} . [0.030] Key:The numbers in parenthesis are Quasi-Likelihood Standard Errors. is the maximized log-likelihood value. AIC and SIC are the Akike and the Schwartz Information Criteria. %Δ stands for the % diﬀerence in the estimated constant term between extreme regimes in the ST-FIGARCH model. m3 and m4 are the estimated skewness and kurtosis of residuals respectively. Q(20), Q(10) and Q2 (20) and Q2 (10) are the Ljung-Box statistics for testing presence of serial correlation up to order 20 and 10 in standardized residuals and squared residuals respectively. ppsb , pnsb and pssb are the p-values for positive sign bias, negative sign bias and the both sign and size bias test respectively as suggested by Engle and Ng (1993). The values in square brackets corresponding to rows for γ are the simulated p-value for the one-sided t-statistic for testing γ = 0 against the alternative γ > 0 or γ < 0 depending on if the estimated γ is positive or negative respectively. W ald is the robust Wald test for testing β = β ∗ versus β ∗ > β or β > β ∗ in the ST-FIGARCH models with γ > 0 and γ < 0 respectively. The values in curly brackets are the asymptotic p−values obtained from χ2 distribution with one degree of freedom, while the values in square brackets are the simulated p-values as explained in the text. 25 in the long memory parameter. This ﬁnding is consistent with the ﬁndings from simulations reported in Table 1 in that, ignoring nonlinearity and estimating the F IGARCH model may produce some positive bias in the estimates of long memory parameter. Estimated volatility dynamics parameters (β, β ∗ and φ) are statistically signiﬁcant at conventional signiﬁcance levels. Estimated parameters satisfy the nonnegativity constraints in both F IGARCH and ST − F IGRCH models. Note that if the nonlinear dynamics in the conditional volatility process is not pronounced, β and β ∗ statistically should not be diﬀerent from one other. To test the null hypothesis that β = β ∗ , we utilize a robust Wald test which presumably should have an asymptotic χ2 distribution under the null. However, under this null, the transition parameter γ is not identiﬁed and the null model becomes the F IGARCH model. In other words, one test for linear F IGARCH against the alternative of ST − F IGARCH model is then to test the equality of β and β ∗ . If γ were known, the test would be distributed as χ2 with one degree of freedom. However the dependence on the unknown parameter γ may make the test not behave in the standard fashion (Davies 1987). A similar issue arises when testing for the slope parameter γ = 0 as under the null, the model reduces to a F IGARCH(1, d, 1) speciﬁcation with parameter β+β ∗ 2 . Therefore, β and β ∗ are not uniquely identiﬁed. One ap- proach is to use tests that are based on auxiliary regressions where nonlinearity is approximated around γ = 0. See for example tests suggested by Luukonen et al. (1988) and Teräsvirta (1994) and the tests suggested in the context of Smooth Transition GARCH models by Harvey (1998) and Gonzàlez-Rivera (1998). Since the behavior of tests that are based on auxiliary models under the null of linearity are not well-known in the context of long memory models, we use Wald and t tests and calculate the p−values by simulations.2 First, we have estimated the model under the null hypothesis (F IGARCH(1, d, 1) for all the series studied) and saved the residuals. Then we generated data by calibrating on the parameters of the estimated null models with errors drawn randomly 2 In the case of Smooth Transition Models for the conditional mean process Kılıç (2004) shows that the conventional linearity tests based on Taylor series approximations may suggest spurious nonlinearity if the data generating process has persistence. To our best knowledge properties of linearity tests with persistence dynamics are not well-known. 26 Table 6: Estimated FIGARCH and ST-FIGARCH Models for Daily Stock returns INT μ θ1 θ2 d ω β β∗ φ γ ψ, ψ ∗ %Δ AIC SIC m3 m4 Q(20) Q(10) Q2 (20) Q2 (10) ppsb pnsb pssb W ald 0.105 (0.032) 0.032 (0.013) 0.040 (0.013) 0.340 (0.019) 0.559 (0.077) 0.404 (0.057) . . 0.163 (0.047) . . . 0.938 . -15067.5 30149.1 30196.4 -0.376 6.325 35.786 13.105 12.796 15.870 0.122 0.013 0.022 . . . 0.097 (0.032) 0.033 (0.013) 0.040 (0.013) 0.304 (0.021) 0.539 (0.077) 0.401 (0.054) 0.498 (0.056) 0.214 (0.042) 1.723 (1.010) [0.059] 0.901, 1.073 19.1 -15064.0 30146.1 30207.0 -0.376 6.277 36.754 12.957 12.994 9.730 0.316 0.224 0.188 9.564 {0.002} [0.041] GM 0.027 (0.022) . . . . 0.315 (0.022) 0.219 (0.021) 0.605 (0.011) . . 0.335 (0.011) . . . 0.554 . -14276.4 28562.9 28597.1 -0.088 7.099 23.928 10.974 7.234 2.890 0.087 0.002 0.019 . . . IBM 0.018 (0.014) . . . . 0.305 (0.034) 0.225 (0.050) 0.429 (0.056) 0.650 (0.049) 0.330 (0.040) 1.744 (1.231) [0.063] 0.394, 0.662 63.2 -14260.7 28535.4 28583.4 -0.030 6.708 23.273 10.662 8.151 3.189 0.247 0.211 0.121 27.795 {0.000} [0.011] Key: See Table 5. 27 0.039 (0.013) . . . . 0.425 (0.018) 0.107 (0.010) 0.584 (0.025) . . 0.295 (0.021) . . . 0.220 . -18421.2 36852.4 36888.6 -0.162 8.275 19.457 4.283 13.853 8.442 0.093 0.043 0.039 . . . 0.028 (0.013) . . . . 0.402 (0.031) 0.101 (0.014) 0.502 (0.037) 0.701 (0.028) 0.290 (0.028) 0.934 (0.279) [0.005] 0.203, 0.338 66.5 -18387.1 36788.1 36838.7 -0.090 7.424 17.653 3.793 17.932 12.415 0.250 0.134 0.117 52.945 {0.000} [0.001] S&P 500 0.048 0.049 (0.006) (0.006) 0.123 0.120 (0.009) (0.009) . . . . 0.468 0.447 (0.019) (0.020) 0.017 0.018 (0.002) (0.002) 0.661 0.715 (0.019) (0.025) . 0.604 . (0.024) 0.269 0.260 (0.014) (0.015) . -1.707 . (0.970) . [0.044] 0.051 0.045, 0.063 . 40.0 -17040.2 -17030.8 34092.5 34077.5 34138.0 34138.3 -0.458 -0.435 7.140 6.940 23.895 24.692 15.116 15.723 10.935 11.884 8.828 9.819 0.111 0.201 0.039 0.120 0.044 0.151 . 13.068 . {0.000} . [0.015] from the residuals of the estimated null model.3 We ran 1,000 simulations with sample size of 5000 + T where T is the sample size for each series. Then we dropped the ﬁrst 5000 data points and estimated alternative models (ST − F IGARCH(1, d, 1) and computed the corresponding Wald statistic for testing β = β ∗ and the t−statistic for testing γ = 0 in each run. The reported p−values, in brackets, are the frequency of times the absolute value of simulated tests are greater than the absolute value of the actual tests reported in Tables 5 and 6. The reported asymptotic and simulated p−values for the W ald test suggests that for exchange rate and stock market conditional volatilities, there is considerable evidence against the null that volatility dynamics is the same across diﬀerent regimes. In other words, we reject the null β = β ∗ at conventional signiﬁcance levels for all series except for the Canadian Dollar at conventional signiﬁcance levels. The estimated values for β and β ∗ show the existence of asymmetry in conditional volatility as whenever γ > 0, estimates for parameters satisfy, β < β ∗ and when γ < 0, β ∗ < β for all cases. Consistent with parameter estimates, results in the second panels of each table show there is considerable diﬀerence between the constant term across low and high volatility regimes contrary to what the FIGARCH model would suggest. Except for S&P 500 returns, estimated transition parameters are all positive. As discussed in Section 2, the sign of the transition parameter labels the regimes rather than the dynamics across diﬀerent regimes. Consistent with the simulation results, estimated standard errors for the transition parameter γ are fairly large. Despite the large standard errors, estimated transition parameters for Swiss Franc, Yen, Intel, and S&P 500 returns are signiﬁcant at 10% level by using the asymptotic normal critical values. For IBM returns, the signiﬁcance level is 1%. Note also that simulated p−values are consistent with the ﬁndings from the asymptotic critical values. Only exceptions are Canadian Dollar and the UK Pound for which the marginal signiﬁcance levels are 0.127 and 0.118 respectively. These simulated values are much smaller than the p−value one would get by using the standard critical values, suggesting marginal evidence against the null of γ = 0 for these currencies. We note also that estimates of γ are more precise for stock returns than the exchange rate returns. Simulated p−values suggest that 3 We have also drawn errors from a Normal distribution as well in simulating the p-values. Results are found to be very similar. These can be obtained upon request. 28 Figure 4: Conditional Standard Deviations from F IGARCH and ST − F IGARCH models and Transition Function over the transition variable IBM FIGARCH ST FIGARCH Transition Fun 15 1 Transition Fun .8 10 .6 .4 5 .2 0 −30 −20 0 −10 0 transition variable... 10 ST FIGARCH Transition Fun 8 S&P 500 FIGARCH 1 Transition Fun .8 6 .6 4 .4 2 .2 0 −20 −10 0 transition variable... 29 0 10 for all individual stocks, S&P 500 and two out of four exchange rates the estimated transition function has statistically signiﬁcant slope. Comparison of F IGARCH and ST − F IGARCH in terms of the diagnostics and information criteria, reveals that ST −F IGARCH models perform better than the F IGARCH models in several dimensions. First, for all series the log-likelihood values from ST − F IGARCH are higher than the values from F IGARCH models. Both AIC and SIC select ST − F IGARCH models over the F IGARCH model for all series except for the Canadian Dollar and Intel for which SIC selects the F IGARCH model. Estimated skewness and kurtosis values for the residuals are mostly lower for the ST − F IGARCH model than the F IGARCH model. Reported Ljung-Box statistics for the serial correlation in residuals and squared residuals are also favorable for the ST − F IGARCH model. Reported p-values for the sign bias and sign and size bias tests due to Engle and Ng (1993) also show that FIGARCH models may have statistically signiﬁcant sign and size biases in their residuals. Results strongly rejects the null of sign and both sign and size eﬀects in the residuals of ST-FIGARCH models for all the series except for the Canadian Dollar for which there is only marginal evidence of nonlinearity and asymmetry. To gain further insights into the properties of estimated ST −F IGARCH models, in Figures 3 and 4, we display estimated conditional standard deviations from F IGARCH and ST − F IGARCH models with the estimated transition functions for four series, IBM, S&P 500, Swiss Francs and Japanese Yen over the transition variable (plots for other series are qualitatively similar and can be obtained upon request). Estimated transition functions have the expected shapes. Careful inspection of the plots reveal that for large negative values of shocks, ST − F IGARCH model implies larger conditional standard deviations than for the positive news. Compared with the F IGARCH model, estimated conditional standard deviations from STFIGARCH models are typically higher for large negative values of transition variable than those from the FIGARCH model. Moreover the estimated standard deviations from the FIGARCH model are higher than those from the ST-FIGARCH model for large positive shocks. Overall, the estimated standard deviations from the FIGARCH models are symmetric over the large 30 Figure 5: Conditional Standard Deviations from F IGARCH and ST − F IGARCH models and Transition Function over the transition variable Japanese Yen FIGARCH ST FIGARCH Transition Fun 1 3 Transition Fun .8 2 .6 .4 1 .2 0 −5 0 transition variable... 0 5 Swiss Francs FIGARCH ST FIGARCH Transition Fun 1 2.5 Transition Fun .8 2 .6 1.5 .4 1 .2 .5 0 −4 −2 0 2 transition variable... 31 4 6 negative and positive news while asymmetric from the ST-FIGARCH model. The plots suggest considerable evidence of asymmetric volatility in both stock and exchange rate markets. The degree of asymmetric volatility dynamics seems more pronounced for the stock returns than the exchange rate returns. This is consistent with the earlier ﬁndings reported in the literature. Our ﬁndings show considerable asymmetric dynamics in conditional volatility in exchange rate markets as well. In this sense, ﬁndings here lends support to Lanne and Saikkonen (2005) which also report asymmetric and nonlinear dynamics in German Mark-US Dollar returns by using a nonlinear GARCH model. The diﬀerence in estimated conditional standard deviations between two models increases in the extreme regimes (lower and upper regimes) and declines in the middle regime where ST − F IGARCH model approaches to the standard F IGARCH model with parameter 5 β+β ∗ 2 . Conclusions In this paper, we have introduced a new nonlinear F IGARCH model, namely the ST − F IGARCH model which allows both nonlinearity and long memory in the conditional variance process. Nonlinear dynamics in volatility is modeled by employing a smoothly changing Logistic function which is considerably ﬂexible and captures smooth jumps as well as asymmetric dynamics in conditional volatility. In the ST-FIGARCH model, the transition between low and high volatility regimes is characterized by the slope of the transition function and a transition variable which can be identiﬁed by economic theory, available information or certain policy variables of interest. Properties of estimation of the new model compared with the FIGARCH model by simulations. Simulations show the ST − F IGARCH model outperforms the standard F IGARCH model when there is nonlinearity. Results suggest that ignoring nonlinearity may cause some bias and loss in eﬃciency in estimates of long memory parameter and dynamic parameters in FIGARCH models. Empirical applications to exchange rate and stock returns show notable evidence on the ST − F IGARCH model. Results show eﬀects of shocks strongly depend on 32 both the level of the conditional variance as well as the sign and the size of the shock itself. Findings reveal that for large negative shocks, conditional volatility is larger than for small and especially positive shocks. The ﬁndings of the paper show that existence of nonlinearity and asymmetry may not be the source of true long memory in volatility and both long memory as well as asymmetry and nonlinearity can exist in economic and ﬁnancial data. In this sense, our ﬁndings are in line with the literature which provide evidence on the existence of true long memory in the volatility process. Estimation of the model suggested in this paper shows presence of substantial long memory and nonlinearity in conditional volatility. However, the concept of realized volatility has now generally become as a more desirable measure of daily volatility when high-frequency ﬁnancial data are available (see Andersen et al., 2001, 2003). It would be interesting to extend the ideas developed in this paper other measures of volatility such as realized volatility to examine jointly long memory and nonlinearity. This topic is currently under investigation by the author. 33 References Amado, C. and T. Teräsvirta (2008), “Modeling Conditional and Unconditional Heteroscedasticity with Smoothly Time-Varying Structure,” Stockholm School of Economics, SSE/EFI Working Paper Series in Economics and Finance, No. 691. Andersen, T. G. and T. Bollerslev (1997), “Heterogeneous Information Arrivals and the Return Volatility Dynamics: Uncovering the Long-Run in High Frequency Returns,” Journal of Finance, 52, 975-1005. Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P. (2001), “The Distribution of Realized Exchange Rate Volatility,” Journal of the American Statistical Association, 96, 42-55. Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P. (2003), “Modeling and Forecasting Realized Volatility,” Econometrica, 72, 569-614. Baillie, R.T., T. Bollerslev, and H.O. Mikkelsen (1996), “Fractionally integrated Generalized Autoregressive Conditional Heteroscedasticity,” Journal of Econometrics, 74, 3-30. Baillie, R.T. and W.P. Osterberg (2000), “Deviations from Daily Uncovered Interest Rate Parity and the Role of Intervention,”Journal of International Financial Markets, Institutions and Money, 10, 363-379. Baillie, R. T., and C. Morana (2007), “Modeling Long Memory and Structural Breaks in Conditional Variances: An Adaptive FIGARCH Approach”,.Working Paper No. 593, Queen Mary University of London, Department of Economics. Baillie, R. T.,A. A. Çeçen and Y.-W. Han (2000), “High Frequency Deutsche Mark-US Dollar Returns: FIGARCH Representations and Non Linearities,” Multinational Finance Journal, 4, 247-267. Beine, M., Laurent, S. (2000), “Structural Change and Long Memory in Volatility: New Evidence from Daily Exchange Rates,” University of Liège, manuscript, Liège. Beltratti, A., Morana, C., (1999), “Computing value at risk with high frequency data”, Journal of Empirical Finance 6, 431 455. Berkes, I., E. Gombay, L. Horvth, and P. Kokoszka (2004), “Sequential Change-Point Detection in GARCH(p,q) Models,” Econometric Theory, 20, 1140.1167. Bollerslev, T., and H. O. Mikkelsen. (1996), “Modelling and Pricing Long Memory in Stock Market Volatility.” Journal of Econometrics, 73, 151184. Bredit, F. J., N. Crato and P. de Lima (1998), “The Detection and Estimation of Long Memory in Stochastic Volatility,” Journal of Econometrics, 83, 325-348. 34 Bredit, F. and Hsu, N. (2002), “A class of nearly long-memory time series models,” International Journal of Forecasting 18, 265-281. Brunetti, C. and C. L. Gilbert (2000), “Bivariate FIGARCH and fractional cointegration,” Journal of Empirical Finance, 7, 509-530. Conrad, C., and M. Karanasos (2006), “The Impulse Response Function of the Long Memory GARCH Process,” Economics Letters, 90, 3441. Conrad, C. and B. R. Haag (2006), “Inequality Constraints in the Fractionally Integrated GARCH Model,” Journal of Financial Econometrics, 3, 413449. Dacorogna, M. M., U. A. Muller, T. J. Nagler, R. B. Olsen and O. V. Pictet (1993), “A geographical model for the daily and weekly seasonal volatility in the foreign exchange market,” Journal of International Money and Finance, 12, 413-438. Davidson, J. (2004), “Moment and memory properties of linear conditional heteroscedasticity models, and a new model,” Journal of Business and Economics Statistics, 22, 16-29. Davies, R. B. (1987), “Hypothesis Testing when a nuisance parameter is present under the alternative,” Biometrika 74, 33-43. Diebold, F.X., and A. Inoue (2001), “Long Memory and Regime Switching,” Journal of Econometrics 105, 131-159. Ding, Z., and C. W. J. Granger. (1996), “Modeling Volatility Persistence of Speculative Returns: A New Approach,” Journal of Econometrics, 73, 185215. Ding, Z., C. W. J. Granger, and R. F. Engle (1993), “A Long Memory Property of Stock Market Returns and a New Model,” Journal of Empirical Finance, 1, 83106. Engle, R. F., and V. K. Ng (1993) “Measuring and testing the impact of news on volatility,” Journal of Finance, 48, 1749-1777. Gallant, R. (1984), “The Fourier Flexible Form,” American Journal of Agricultural Economics, 66, 204-208. Glosten, L., R. Jagannathan, and D. Runkle (1993), “On the Rekation Between Expected Value and the Volatility of the Nominal Excess Return on Stocks,” Journal of Finance, 48, 1779-1801. Gonzàlez-Rivera, G. (1998), “Smooth Transition GARCH Models”, Studies in Non-linear Dynamics and Econometrics, 3, 61.78. Granger, C.W.J. and T. Teräsvirta (1993), Modelling nonlinear economic relationships. Ox35 ford: Oxford University Press. Granger, C. and Hyung, N. (2004), “Occasional structural breaks and long memory with an application to the S&P500 absolute stock returns,” Journal of Empirical Finance 11, 399-421. Hagerud, G. E. (1997), “A New Non-Linear GARCH Model”, Ph.D. thesis, EFI, Stock-holm School of Economics. Hamilton, J. D., and R. Susmel (1994), “Autoregressive Conditional Heteroskedasticity and Changes in Regime”, Journal of Econometrics, 64, 307.333. Harvey, A. C., (1998), “Long Memory in Stochastic Volatility,” in Forecasting Volatility in Financial Markets, J. Knight and S. Satchell (editors), 307-320, Oxford: Butterworth-Heineman. Jensen, S. T., and A. Rahbek (2004), “Asymptotic Inference for Nonstationary GARCH,” Econometric Theory, 20, 1203.1226. Kılıç R., (2004), “On the long memory properties of emerging capital markets: evidence from Istanbul stock exchange” Applied Financial Economics, 14, 915-922. Kılıç R., (2007), “Conditional Volatility and Distribution of Exchange Rates: GARCH and FIGARCH Models with NIG Distribution,” Studies in Nonlinear Dynamics and Econometrics, 11, Issue 3, Article 1. Kılıç R., (2008), “A conditional variance tale from an emerging economy’s freely ﬂoating exchange rate”, Manuscript, School of Economics, Georgia Institute of Technology. Kim, S., N. Shephard and S. Chib (1998), “Stochastic volatility: likelihood inference and comparison with ARCH models,” Review of Economic Studies 65, 36193. Lamoureux, C. G., and W. D. Lastrapes (1990), “Persistence in Variance, Structural Change, and the GARCH Model,” Journal of Business & Economic Statistics, 8, 225-234. Li, C. W. and Li, W. K. (1996), “On a double threshold autoregressive heteroscedastic time series model,” Journal of Applied Econometrics, 11, 253-274. Lanne, M., and P. Saikkonen (2005), “Nonlinear GARCH Models for Highly Persistent Volatility”, Econometrics Journal, 8, 251-276. Lee, S-W. and B.E. Hansen (1994), “Asymptotic Theory for the GARCH(1,1): Quasi-Maximum Likelihood Estimator,” Econometric Theory, 10, 29-52. Lubrano, M. (2001), “Smooth Transition GARCH models: a Bayesian approach,” Recherches Economiques de Louvain, 67, 257-287. 36 Lumsdaine, R. L. (1996), “Consistency and Asymptotic Normality of the Quasi-Maximum Likelihood Estimator in IGARCH(1,1) and Covariance Stationary GARCH(1,1) Models,” Econometrica, 64, 575-596. Lundberg, S., and T. Teräsvirta (1998), “Modeling Economic High-Frequency Time Series with STAR-STGARCH Models,” Department of Economics, Stockholm School of Economics. Luukkonen, R., P. Saikkonen, T. Terasvirta (1988), “Testing Linearity Against Smooth Transition Autoregressive Models,” Biometrica, 75, 491-499. Mikosch, T. and Starica, C. (1998), “Change of structure in ﬁnancial time series, long range dependence and the GARCH model,” mimeo. University of Groningen, Department of Mathematics. Morana, C. and A. Beltratti, (2004), “Structural change and long-range dependence in volatility of exchange rates: either, neither or both?” Journal of Empirical Finance 11, 629-658. Muller, U. A., M. M. Dacorogna, R. D. Davé, R. B. Olsen, O. V. Pictet, and J. E. von Weizsacker, (1997), “Volatilities of Diﬀerent Time Resolutions-Analyzing the Dynamics of Market Components,” Journal of Empirical Fiance, 4, 213-239. Nelson, D. (1991), “Conditional heteroscedasticity in asset returns: A new approach”, Econometrica, 59, 347-370. Ohanissian, A., J. R. Russell, and R. S. Tsay (2008), “True or Spurious Long Memory? A New Test” Journal of Business & Economic Statistics, 26, pp.161-175 Pagan, A. R. and G. W. Schwert (1990), “Alternative modes for conditional stock volatility,” Journal of Econometrics, 45, 267-290. Teräsvirta, T. (1994), “Speciﬁcation, Estimation and Evaluation of Smooth Transition Autoregressive Models,” Journal of the American Statistical Association, 89, 208-218. Zaﬀaroni, P. (2004), “Stationarity and memory of ARCH(∞) models.” Econometric Theory, 147-160. Zakoıan, J. M. (1994), “Threshold Heteroskedastic Models,” Journal of Economic Dynamics and Control, 18, 931-955. 37