MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY WITH A FLEXIBLE COEFFICIENT GARCH MODEL

Transcription

1 MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY WITH A FLEXIBLE COEFFICIENT GARCH MODEL MARCELO C. MEDEIROS AND ALVARO VEIGA ABSTRACT. In this paper a flexible multiple regime GARCH-type model is developed with the aim of describing sign and size asymmetries in financial volatility as well as intermittent dynamics and excess of kurtosis. The proposed model nests some of previous specifications found in the literature and has the following advantages: First, contrary to most of the previous models, more than two limiting regimes can be modeled and the number of regimes is determined by a simple sequence of of tests that circumvents identification problems usually found in nonlinear time series models. The second advantage is that the stationarity restriction on the parameters is relatively weak, allowing for rich dynamics. It is shown that the model may have explosive regimes and still be strictly stationary and ergodic. A small simulation shows that the proposed model can generate series with high kurtosis, low first-order autocorrelation of the squared observations, and at the same time exhibiting the so called Taylor effect, even with Gaussian errors. Estimation of the parameters is addressed and the asymptotic properties of the quasi-maximum likelihood estimator is derived. A Monte-Carlo experiment is designed to evaluate the finite sample properties of the sequence of tests. Empirical examples are used to illustrate the use of the model in practical situations. KEYWORDS: Volatility, GARCH models, multiple regimes, nonlinear time series, smooth transition, finance, asymmetry, leverage effect, excess of kurtosis.. INTRODUCTION MODELING AND FORECASTING the conditional variance, or the volatility, of financial time series has been one of the major topics in financial econometrics. Forecasted conditional variances are used, for example, in portfolio selection, derivative pricing and hedging, risk management, market timing, and market making. Among solutions to tackle this problem, the ARCH Autoregressive Conditional Heteroscedasticity) model proposed by Engle 98) and the GARCH Generalized Autoregressive Conditional Heteroscedasticity) specification introduced by Bollerslev 986) are certainly among the most widely used and are now fully incorporated into econometric practice. One drawback of the GARCH model is the symmetry in the response of volatility to past shocks, failing to accommodate sign asymmetries. Researchers, beginning with Black 976), pointed out the asymmetric response of the conditional variance of the series to unexpected news, represented by shocks: Financial markets become more volatile in response to bad news negative shocks) than to good news positive shocks). Goetzmann, Ibbotson, and Peng 00) found evidence of asymmetric sign effects in volatility as far back as 857 for the

2 M. C. MEDEIROS AND A. VEIGA NYSE. They report that unexpected negative shocks in the monthly return of the NYSE from 857 to 95 increase volatility almost twice as much as equivalent positive shocks in returns. Similar results were also reported by Schwert 990). The above mentioned asymmetry has motivated a large number of different volatility models which were applied with relatively success in several situations. Nelson 99) proposed the Exponential GARCH E-GARCH) model. In his proposal, the natural logarithm of the conditional variance is modeled as a nonlinear ARMA model with a term that introduces asymmetry in the dynamics of the conditional variance according to the sign of lagged returns. Glosten, Jagannanthan, and Runkle 993) proposed the GJR-GARCH model, where the impact of the lagged squared returns on the current conditional variance changes according to the sign of the past return. A similar specification, known as Threshold GARCH T-GARCH) model was developed by Rabemananjara and Zakoian 993) and Zakoian 994). Ding, Granger, and Engle 993) proposed the Asymmetric Power ARCH which nests several GARCH specifications and Engle and Ng 993) popularized the news impact curve as a measure of how new information is incorporated into volatility estimates. The authors also developed formal statistical tests to check the presence of asymmetry in the volatility dynamics. More recently, Fornari and Mele 997) generalized the GJR-GARCH model by allowing all the parameters to change according to the sign of the past return. Their proposal is known as the Volatility-Switching GARCH VS- GARCH) model. Based on the Smooth Transition AutoRegressive STAR) model, Haregud 997) and Gonzalez-Rivera 998) proposed the Smooth Transition GARCH ST-GARCH) model. While the latter only considered the Logistic ST-GARCH LST-GARCH) model, the former discussed both the Logistic and the Exponential ST-GARCH EST-GARCH) alternatives. In the logistic ST-GARCH specification the dynamics of the volatility is very similar to those of the GJR-GARCH model and depends on the sign of the past returns. The difference is that the former allows for a smooth transition between regimes. In the EST-GARCH model, the sign of the past returns does not play any role in the dynamics of the conditional variance. It is the magnitude of the lagged squared return that is the source of asymmetry. Anderson, Nam, and Vahid 999) combined the ideas of Fornari and Mele 997), Haregud 997), and Gonzalez-Rivera 998) and proposed the Asymmetric Nonlinear Smooth Transition GARCH ANST-GARCH) model and found evidence in favor of their specification. Inspired by the Threshold Autoregressive TAR) model, Li and Li 996) proposed the Double Threshold ARCH DT-ARCH) model and Liu, Li, and Li 997) generalized it, proposing the Double Threshold GARCH DT-GARCH) process. The idea is to model both the conditional mean and the conditional variance as a threshold process. More recently, based on the regressiontree literature, Audrino and Bühlmann 00) put forward the Tree Structured GARCH in order to model multiple limiting regimes in volatility. See also Cai 994) and Hamilton and Susmel 994) for regime switching GARCH specifications based on the Markov-switching model.

3 MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY 3 In this paper we aim to contribute to the literature by proposing a new flexible nonlinear GARCH model with multiple limiting regimes, called the Flexible Coefficient GARCH FC- GARCH) model, that nests several of the models mentioned previously. Our proposal has the following advantages: First, contrary to most of the previous models in the literature, more than two limiting regimes can be modeled and the number of regimes is determined by a rigorous, simple, and easily implemented sequence of tests that circumvents the identification problem usual in nonlinear time series models. To our knowledge, the only two exceptions that explicitly model more than two limiting regimes in the volatility are the DT-GARCH and the Tree-Structure GARCH models. However, in the former the authors did not discuss how to determine the number of regimes and in the latter the proposed procedure, based on the use of information criteria, may suffer from identification problems which will be discussed later. The second advantage is that the stationarity restriction on the FC-GARCH model parameters is relatively weak, allowing for rich dynamics. For example, the model may have explosive regimes and still be strictly stationary and ergodic, being capable of describing intermittent dynamics; the system spends a large fraction of time in a bounded region, but sporadically develops an instability that grows exponentially for some time and then suddenly collapses. Furthermore, data with very high kurtosis can be easily generated even with Gaussian errors. This allows for a better description of the large absolute returns of financial time series that standard GARCH models fail to describe satisfactorily. A small simulation shows that the FC-GARCH model is able to generate time-series with high kurtosis and, at the same time, positive but low first-order autocorrelations of squared observations. Consequently, the FC-GARCH model seems to be able to reproduce the so-called Taylor effect Granger and Ding 995). Other concurrent models such as the standard GARCH and the E-GARCH models are not able to adequately reproduce the above mentioned stylized facts of financial time series; See Malmsten and Teräsvirta 004) for a comprehensive discussion. We discuss the theoretical aspects of the FC-GARCH model: Conditions for the existence of the second- and fourth-order moments, model identifiability, existence, consistency, and asymptotic normality of the quasi-maximum likelihood estimators. Consistency and asymptotic normality are proved, respectively, under second- and fourth-order moment conditions. Furthermore, a sequence of simple Lagrange multiplier LM) tests is developed in order to determine the number of limiting regimes. Although the test is derived under the assumption that the errors are Gaussian, a robust version against non-gaussian errors is also discussed. A Monte-Carlo experiment is designed to evaluate the finite sample properties of the proposed sequence of tests with simulated data. The main finding is that the robust version of the test is well-sized and has a reasonable power. An empirical example with eight stock indexes shows that the FC-GARCH model produces normalized residuals with lower kurtosis than the GARCH and GJR-GARCH models. Moreover, the results show evidence of two limiting regimes for four series and three limiting

4 4 M. C. MEDEIROS AND A. VEIGA regimes for three of them. Only for one stock index there is no evidence of more than one regime. Furthermore, for all the series with three limiting regimes, the first limiting extreme) regime is associated with very negative shocks, representing great losses. The middle regime is related to tranquil periods and the third and extreme regime represents large positive shocks. Thus we found strong evidence of both size and sign asymmetries. The first limiting regime for seven of the series is extremely explosive indicating that bad news may induce very high volatility. In addition, the GARCH effect seems to be much less significant when the lagged return is positive. The structure of the paper is as follows. Section presents the model and its probabilistic properties are analyzed in Section 3. Estimation of the FC-GARCH model is considered in Section 4. Section 5 discusses the test for an additional regime and the modeling cycle procedure is summarized in Section 6. A Monte-Carlo simulation is carried out in Section 7 and empirical examples are considered in Section 8. Finally, Section 9 concludes. All technical proofs are consigned to the Appendix.. THE MODEL In this paper, we generalize the GARCH,) formulation, introducing a general regime switching scheme. The proposed model is defined as follows. DEFINITION. A time series {y t } follows a first-order Flexible Coefficient GARCH model with m = H + limiting regimes, FC-GARCHm,, ), if ) y t = h / t ε t, h t = G w t ; ψ) = α 0 + β 0 h t + λ 0 y t + αi + β i h t + λ i yt f st ; γ i, c i ), t =,..., T, where {ε t } is a sequence of identical and independently distributed zero mean and unit variance random variables, ε t IID0, ), G w t ; ψ) is a nonlinear function of the vector of variables w t = y t, h t, s t and is indexed by the vector of parameters ψ = α 0, β 0, λ 0, α,..., α H, β,..., β H, λ,..., λ H, γ,..., γ H, c,..., c H, R 3+5H, and f s t ; γ i, c i ), i =,..., H, is the logistic function defined as ) f s t ; γ i, c i ) = + e γ is t c i ). The parameter γ i, i =,..., H is called the slope parameter. When γ i, the logistic function becomes a step function. The variable s t is known as the transition variable. In this paper, we consider s t = y t. Hence, we model the differences in the dynamics of the conditional variance according to the sign and size of shocks in past returns.

5 MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY 5 The number of limiting regimes is defined by the hyper-parameter H. For example, suppose that in ) H =, c is very negative and c is very positive, than the resulting FC-GARCH model will have 3 limiting regimes that can be interpreted as follows: The first regime may be related to very low negative shocks very bad news ), the middle regime represents low absolute returns tranquil periods ), and, finally, the third one is related to very high positive shocks very good news ). The FC-GARCH model can be also interpreted as a time-varying GARCH,) model where all the parameters vary across regimes. Letting all the parameters to change turns the model very flexible and allows for very rich dynamics. It is important to notice that model ) nests several well-known GARCH specifications, such as, for example: The GARCH,) model if γ i = 0 or α i = β i = λ i = 0, i =..., h. The LST-GARCH,) model if α i = β i = 0, i =..., h and h =. The GJR-GARCH,) model if H =, γ, α = β = 0, and c = 0. The VS-GARCH,) model if H = and γ, c = 0. The ANST-GARCH,) model if H =, and c = 0. The variance component of the DT-ARCH,) if γ i and α i = β i = 0, i =..., h. The variance component of the DT-GARCH,) if γ i and s t = h t. The FC-GARCH model is a special case of the nonlinear GARCH model proposed in Lanne and Saikkonen 00) if β i = 0, i =,..., H or if s t = h t, and is a special case of the general GARCH specification presented in He and Teräsvirta 999), Ling and McAleer 00), and Carrasco and Chen 00) if s t = ε t. As the FC-GARCH is nonnested without additional restrictions in the general specifications mentioned above, the application of the results of He and Teräsvirta 999), Ling and McAleer 00), Lanne and Saikkonen 00), and Carrasco and Chen 00) is not straightforward. 3. MAIN ASSUMPTIONS AND PROBABILISTIC PROPERTIES OF THE FC-GARCH MODEL At this point, we need to make the following set of assumptions: ASSUMPTION. The true parameter vector ψ 0 Ψ R 3+5H is in the interior of Ψ, a compact and convex parameter space. ASSUMPTION. The sequence {ε t } of IID0, ) random variables is drawn from a continuous with respect to Lebesgue measure on the real line), symmetric, unimodal, positive everywhere density, and bounded in a neighborhood of 0. In addition, assume that h t is independent of {ε t, ε t+,...}. ASSUMPTION 3. The parameters c i, and γ i, i =,..., H, satisfy the conditions: R.) < M < c <... < c H < M < ;

6 6 M. C. MEDEIROS AND A. VEIGA R.) γ i > 0. ASSUMPTION 4. The parameters γ i, c i, i =,..., H, are such that the logistic functions satisfy the following restrictions: f s t ; γ, c ) f s t ; γ, c )... f s t ; γ H, c H ), t 0, T. ASSUMPTION 5. The parameters α j, β j, and λ j, j = 0,..., H, satisfy the following restrictions: R.3) K j=0 α j > 0, K = 0,..., H. R.3) K j=0 β j 0, and K j=0 λ j 0, K = 0,..., H. Assumption is standard. Assumption is important for the mathematical derivations in this section and in Section 5. Assumption 3 guarantees the identifiability of the model see Section 4. for details) and the restrictions stated in Assumptions 4 and 5 ensure strictly positive conditional variances. Specifically, Assumption 4 ensures that the conditions in Assumption 5 are sufficient for the strict positivity of the conditional variance. We follow Liu, Li, and Li 997) and apply the results in Liu and Susko 99) to derive a sufficient condition for strict stationarity and ergodicity of the FC-GARCHm,, ) model, m = H +. Furthermore, we state a sufficient condition for the existence of the fourth-order moment of y t. First, defining s t = y t and u t = h t, ε t, it is easy to show that {u t } is a Markov chain with homogenous transition probability expressed as 3) u t = G u t ; ψ) + ɛ t, where G u t ; ψ) = with f i,t = f α0 + H α if i,t + β 0 + λ 0 ε t + H βi + λ i ε ) t fi,t h / t ε t ; γ i, c i ), and ɛ t = 0, ε t. THEOREM. Suppose that y t R follows a FC-GARCHm,,) process as in ) with s t = y t, and the bivariate stochastic process u t = h t, ε t R + R follows the Markov chain defined in 3). Under Assumptions 5, u t is strictly stationary and ergodic if 4) β 0 + λ 0 ) + 0 β i + λ i ) <. i=0 COROLLARY. Define αy t ) = α 0 + H α ify t ; γ i, c i ), βy t ) = β 0 + H β ify t ; γ i, c i ), and λy t ) = λ 0 + H λ ify t ; γ i, c i ). Under the Assumptions 5 and the restriction of Theorem T 5) plim βy t ) = 0. T h t,

7 MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY 7 Moreover, define αy t k ) if k = ; 6) φy t k ) = k αy t k ) otherwise, j= βy t j) and λy t k ) if k = ; 7) φyt k ) = k λy t k ) otherwise. j= βy t j) Then, 8) h t = φy t k ) + φy t k )yt k exists with probability one and is unique. k= k= Deriving a general primitive sufficient condition for the existence of the moments of y t is rather complicated. However, the moment condition stated in the following theorem can be used to find a sufficient condition for the existence of low-order moments of y t. In the subsequent corollary we derive a sufficient conditions for the existence of the second- and forth-order moments of y t. THEOREM. Suppose that y t R follows a FC-GARCHm,,) process as in ) with s t = y t and E ε k t = µk <, for k =,, 3,.... Under Assumptions 5 and assuming that the moments of order up to n = k exist, E yt n <, the kth-order moment of yt exists if 9) E β 0 + λ 0 ε t + βi + λ i ε ) k t fi,t F t <, where F t is filtration given by all the information up to time t. COROLLARY. Suppose that y t R follows a FC-GARCHm,,) process as in ) with s t = y t. Then, a sufficient condition for the existence of the second-order moment of y t is 4). Furthermore, define β U H β i and λ U H λ i. Under Assumptions 5, the fourth-order moment of y t exists if E ε 4 t = µ4 <, 4) holds, and 0) β0 + β 0 β U + β U + µ 4 λ 0 + λ 0 λ U + λ U + λ 0 β 0 + β 0 λ U + λ 0 β U + λ U β U <. REMARK. When H = 0, conditions 4) and 0) turn to be the usual conditions for the existence of the second- and fourth-order moments of GARCH models and, when H =, γ, α = 0, and β = 0, conditions 4) and 0) become the usual ones for the case of the GJR-GARCH model; see He and Teräsvirta 999).

8 8 M. C. MEDEIROS AND A. VEIGA It is important to notice that even with explosive regimes the FC-GARCHm,, ) may be still strictly stationary, ergodic, and with finite fourth-order moment. Furthermore, some the parameters of the limiting GARCH models may be greater than one. This flexibility generates models with higher kurtosis than the standard GARCH,), even with Gaussian errors. The following examples illustrate some interesting situations. Consider 3000 replications of the following FC-GARCH3,, ) models with Gaussian errors, each of which with 5000 observations. ) Example : y t = h / t ε t, ε t NID0, ) h t = h t + 0.8yt h t 0.0yt ) f 5000 yt )) + ) Example : 3) Example 3: h t yt ) f 5000 yt 0.0)). y t = h / t ε t, ε t NID0, ) h t = h t + 0.0yt h t 0.09yt ) f 3000 yt )) h t yt ) f 3000 yt 0.005)). y t = h / t ε t, ε t NID0, ) h t = h t + 0.0yt h t 0.0yt ) f 000 yt )) f 000 y t 0.0)). The models in Examples 3 have three extreme regimes each with the first regime being explosive as β 0 + λ 0 >. However, even with an explosive regime, the generated time-series is still stationary provided that β 0 + λ 0 ) + i=0 β i + λ i ) < for all the tree models. Furthermore, the fourth-order moment exists provided that condition 0) is also satisfied. Note also that β 0 > in Examples and 3. The model in Example 3 has the interesting property that the GARCH effect is only present in the extreme regimes. The regime associated with tranquil periods is homoskedastic. Figure shows the scatter plot of the estimated kurtosis and first-order autocorrelation of squared observations. The dots indicates the cases where the first-order autocorrelation of y t is greater than the first-order autocorrelation of y t. The crosses indicate the opposite. As can be seen by inspection of graph, the simulated FC-GARCH models seem to be able to reproduce some of the stylized facts usually observed in financial time-series. Table summarizes some

9 MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY 9 statistics about the estimated kurtosis and autocorrelations. As can be seen, the minimum value of the estimated kurtosis is over 3. In addition the mean values of estimated first-order autocorrelations are in accordance with typical numbers that are usually found in practical applications With Taylor effect Without Taylor effect With Taylor effect Without Taylor effect Estimated kurtosis of y t Estimated kurtosis of y t Estimated first order autocorrelation of y t Estimated first order autocorrelation of y t a) b) 450 With Taylor effect Without Taylor effect Estimated kurtosis of y t Estimated first order autocorrelation of y t c) FIGURE. Scatter plot of the estimated kurtosis and first-order autocorrelation of squared observations. The dots indicates the cases where the Taylor effect is satisfied and the crosses indicate the opposite. Panel a) concerns Example ). Panel b) concerns Example ). Panel c) concerns Example 3. TABLE. SIMULATED MODELS: DESCRIPTIVE STATISTICS. The table shows descriptive statistics for the estimated kurtosis and first-order autocorrelation of the squared observations over For each of the 3000 replications of Models ) 3). Kurtosis Autocorrelation Examples Min. Max. Mean Std. Dev. Min. Max. Mean Std. Dev. Example Example Example

10 0 M. C. MEDEIROS AND A. VEIGA 4. PARAMETER ESTIMATION As the distribution of ε t is unknown, the parameters of the FC-GARCH model should be estimated by quasi-maximum likelihood QML). Even for the GARCH model, the properties of the quasi-maximum likelihood estimator QMLE) are not completely clear. For example, Lee and Hansen 994) proved that the local QMLE is consistent and asymptotic normal if all the conditional expectations of ε +κ t < uniformly with κ > 0. Lumsdaine 996) required that E ε 3 t <. Jeantheau 998) discussed consistency of the QMLE under weaker conditions. More recently, Ling and McAleer 003) proved the consistency of the global QMLE under only second-order moment condition. The authors also proved the asymptotic normality of the global QMLE under the sixth-order moment condition. Compte and Lieberman 003) and Berkes, Horváth, and Kokoszka 003) proved consistency and asymptotic normality of the QMLE of the parameters of the GARCHp,q) model under second- and fourth-order moment conditions. In this paper we also prove consistency and asymptotic normality of the QMLE of the FC- GARCHm,, ) under second- and fourth-order moment conditions, respectively. Extending the results in Jensen and Rahbek 004) for non-stationary ARCH models to the case of the FC-GARCH model is not straightforward and is beyond the scope of this paper. However, this is an interesting topic for future research. The quasi-loglikelihood function of the FC-GARCH model is given by ) L T ψ) = T T l t ψ), where l t ψ) = lnπ) lnh t) y t h t. Note that the processes y t and h t, t 0, are unobserved and hence they are only arbitrary constants. Thus, L T ψ) is a quasi-loglikelihood function that is not conditional on the true y 0, h 0 ) making it suitable for practical applications. However, to prove the asymptotic properties of the QMLE is more convenient to work with the unobserved process {y u,t, h u,t ) : t = 0, ±, ±,...}, which satisfies ) h u,t = α 0 + β 0 h u,t + λ 0 y u,t + αi + β i h u,t + λ i yu,t f yu,t ; γ i, c i ). The unobserved quasi-loglikelihood function conditional on F 0 = y 0, y, y,...) is 3) L u,t ψ) = T T l u,t ψ), with l u,t ψ) = lnπ) lnh u,t) y u,t h u,t. The primary difference between L T ψ) and L u,t ψ) is that the former is conditional on any initial values, whereas the latter is conditional on an infinite series of past observations. In practical situations, the use of 3) is not possible.

11 and Let MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY ψ T = argmaxl T ψ) = argmax ψ Ψ ψ Ψ T ψ u,t = argmaxl u,t ψ) = argmax ψ Ψ ψ Ψ T ) T l t ψ), ) T l u,t ψ). Define Lψ) = E l u,t ψ). In the following two subsections, we discuss the existence of Lψ) and the identifiability of the FC-GARCH model. Then, in Subsection 4.3, we prove con- B we show that sup L u,t ψ) L T ψ) ψ Ψ sistency of ψ T and ψ u,t. We first prove the consistency of ψ u,t. Using Lemma 3 in Appendix p 0, and the consistency of ψ T follows. Asymptotic normality of both estimators is considered in Subsection 4.4. We start proving asymptotic normality of ψ u,t. Then, using the results of Lemma 5, the proof for ψ T is straightforward. 4.. Existence of the QMLE. The following theorem proves the existence of Lψ). It is based on Theorem. in White 994), which establishes that under certain conditions of continuity and measurability on quasi-loglikelihood function, Lψ) exists. THEOREM 3. If the stationarity condition in Theorem is satisfied, then, under Assumptions 5, Lψ) exists and is finite. 4.. Identifiability of the Model. A fundamental problem for statistical inference with nonlinear time series models is the unidentifiability of the model parameters. To guarantee unique identifiability of the quasi-loglikelihood function, the sources of uniqueness of the model must be studied. Here, the main concepts and results will be briefly discussed. In particular, the conditions that guarantee that the FC-GARCH model is identifiable and minimal will be established and proven. First, two related concepts will be discussed: The concept of minimality of the model, established in Sussman 99), also called non-redundancy in Hwang and Ding 997); and the concept of reducibility of the model. DEFINITION. The FC-GARCHm,, ) model is minimal or non-redundant), if its inputoutput map cannot be obtained from a FC-GARCHn,, ) model, where n < m. One source of unidentifiability comes from the fact that a model may contain irrelevant limiting regimes. A limiting regime is represented by the functions µ i = α i + β i h t + λ i y t f yt ; γ i, c i ), i =,..., H. This means that there are cases where the model can then be reduced, eliminating some regimes without changing the input-output map. Thus, the minimality condition can only hold for irreducible models. DEFINITION 3. Define θ i = γ i, c i and let ϕ y t ; θ i ) = γ i y t c i ), i =,..., H. The FC-GARCH model defined in ) is reducible if one of the following three conditions holds:

12 M. C. MEDEIROS AND A. VEIGA ) One of the triples α i, β i, λ i ) vanishes jointly for some i, H. ) γ i = 0 for some i, H. 3) There is at least one pair i, j), i j, i =,..., H, j =,..., H, such that ϕ y t ;θ i ) and ϕ y t ;θ j ) are sign-equivalent. That is, ϕ y t ;θ i ) = ϕ y t ;θ j ), y t R, t =,..., T. DEFINITION 4. The FC-GARCH model is identifiable if there are no two sets of parameters such that the corresponding distributions of the population variable y are identical. Three properties of the FC-GARCH model cause unidentifiability of the models: P.) The property of interchangeability of the regimes. The value of the likelihood function of the model does not change if the regimes are permuted. This results in H! different models that are indistinct among themselves related to the input-output map). As a consequence, in the estimation of parameters, we will have H! equal local maxima for the quasi-loglikelihood function. P.) The fact that fy t ; γ i, c i ) = fy t ; γ i, c i ). P.3) The presence of irrelevant regimes. Conditions ) ) in the definition of reducibility give information about the presence of irrelevant regimes, which translate into identifiability sources. If the model contains a regime such that α i = 0, β i = 0, and λ i = 0, then parameters γ i and c i remain unidentified, for some i, H. On the other hand, if γ i = 0, then parameters α i, β i, λ i, and c i may take on any value without affecting the value of the quasi-loglikelihood function. Property P.3) is related to the concept of reducibility. In the same spirit of the results stated in Sussman 99) and Hwang and Ding 997) we show that, if the model is irreducible, properties P.) and P.) are the only forms of modifying the parameters without affecting the loglikelihood. Hence, establishing restrictions on the parameters of ) that simultaneously avoid model reducibility, any permutation of regimes, and symmetries in the logistic function, we guarantee the identifiability of the model. The problem of interchangeability property P.)) can be prevented by imposing the Restrictions R.) in Assumption 3. Now the consequences due to the symmetry of the logistic function property P.)) can be resolved if we consider Restrictions R.) in Assumption 3. The presence of irrelevant regimes, property P.3), can be circumvented by applying a specific-to-general model building strategy as suggested in Section 5. Corollaries. in Sussman 99) and.4 in Hwang and Ding 997) guarantee that an irreducible model is minimal. The fact that irreducibility and minimality are equivalent implies that there are no mechanisms, other than the ones listed in the definition of irreducibility, that can be used to reduce the complexity of the model without changing the functional input-output relation. Then, restrictions in Assumption 3 guarantee that if irrelevant regimes do not exist the model is identifiable and minimal.

13 MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY 3 We need an additional assumption before establishing sufficient conditions under which the FC-GARCH model is globally identifiable. ASSUMPTION 6. The parameters α i, β i, and λ i do not vanish jointly for some i, H. Assumption 6 guarantees that there are no irrelevant regimes as described in property P.3) above. THEOREM 4. Under Assumptions 3 and 6 the FC-GARCHm,, ) model is globally identifiable. Furthermore, Lψ) is uniquely maximized at ψ Consistency. The proof of consistency of the QMLE for the FC-GARCH model follows the same reasoning of Ling and McAleer 003). The following theorem states and proves the main consistency result. THEOREM 5. If the restriction of Theorem is satisfied, then, under Assumptions 5 and the additional assumption that E ε t <, ψu,t p ψ0 and ψ T p ψ Asymptotic Normality. To prove asymptotic normality we use the results in Ling and McAleer 003). First, we introduce the following matrices: Aψ 0 ) = E l u,t ψ) ψ ψ and ψ0 Bψ 0 ) = E T L u,t ψ) L u,t ψ) ψ ψ T E l u,tψ) l u,t ψ) T ψ ψ. ψ0 ψ0 Consider the additional matrices: A T ψ) = T { T 4) B T ψ) = T T h t ψ0 ψ0 ) ) )} h t h t y t y ψ ψ t h t h t h t ψ, and h t ψ l t ψ) l t ψ) ψ ψ = 4T T h t The following theorem states the asymptotic normality result. ) y t ht h t h t ψ ψ. THEOREM 6. If the restrictions of Theorems and are satisfied, then, under the Assumptions 5, 5) T / ψ T ψ 0 ) D N 0, Aψ 0 ) Bψ 0 )Aψ 0 ) ), where Aψ 0 ) and Bψ 0 ) are consistently estimated by A T ψ) and B T ψ), respectively. REMARK. Under Assumption, it is clear that Bψ 0 ) = reduces to the information matrix equality when E ε 4 t = 3. E ε 4 t ) Aψ0 ), which

14 4 M. C. MEDEIROS AND A. VEIGA 5. DETERMINING THE NUMBER OF REGIMES The number of regimes in the FC-GARCH model, represented by the number of transition functions in ), is not known in advance and should be determined from the data. One possibility is to begin with a small model a GARCH,) model or just a white noise) and sequentially add regimes to it. The decision of adding another regime may be based on the use of model selection criteria MSC) or cross-validation. For example, Audrino and Bühlmann 00) used the Akaike s Information Criterium AIC) to select the number of regimes in their Tree-Structured GARCH model. However, this has the following drawback. Suppose the data have been generated by a FC-GARCH model with m regimes m transition functions). Applying an MSC to decide whether or not another regime should be added to the model requires estimation of a model with m logistic functions. In this situation the larger model is not identified and its parameters cannot be estimated consistently. This is likely to cause numerical problems in quasi maximum likelihood estimation. Besides, even when convergence is achieved, lack of identification causes a severe problem in interpreting the MSC. The FC- GARCH model with m regimes is nested in the model with m + regimes. A typical MSC comparison of the two models is then equivalent to a likelihood ratio test of m regimes against m + ones; see Teräsvirta and Mellin 986) for discussion. The choice of MSC determines the asymptotic) significance level of the test. But then, when the larger model is not identified under the null hypothesis, the likelihood ratio statistic does not have its customary asymptotic χ distribution when the null holds. In this paper we tackle the problem of determining the number of regimes of the FC-GARCH model with a specific-to-general strategy but we circumvent the problem of identification in a way that enables us to control the significance level of the tests in the sequence and thus compute an upper bound to the overall significance level of the procedure. The following developments will be based on the assumption that the errors ε t are Gaussian. However, this does not pose a problem as far as the results will be robustfied against nonnormal errors. Consider a FC-GARCH with H limiting regimes defined as 6) y t = h / t ε t, h t =α 0 + β 0 h t + λ 0 y t + H αh + β H h t + λ H yt f yt ; γ H, c H ). αi + β i h t + λ i yt f yt ; γ i, c i ) + The idea is to test the presence of the additional regime represented by the term 7) αh + β H h t + λ H yt f yt ; γ H, c H ).

15 A convenient null hypothesis is MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY 5 8) H 0 : γ H = 0, against the alternative H a : γ H > 0. Note that model 6) is not identified under the null hypothesis. To remedy this problem, we follow Lundbergh and Teräsvirta 00) and expand the logistic function f y t ; γ H, c H ) into a first order Taylor expansion around the null hypothesis γ H = 0. After merging terms, the resulting model for h t is 9) h t = α 0 + β 0 h t + λ 0 y t + H πy t + δh t y t + ρy 3 t + R, αi + β i h t + λ i yt f yt ; γ i, c i ) + where R is the remainder, α 0 = α 0 α Hγ H c H 4, β 0 = β 0 β Hγ H c H 4, λ 0 = λ 0 λ Hγ H c H 4, π = γ H 4, δ = β Hγ H 4, and ρ = λ Hγ H 4. Define f i,t fy t ; γ i, c i ), i =,..., H. Under H 0, R = 0 and the quasi maximum likelihood approach enables us to state the following result: THEOREM 7. If the stationarity condition in Theorem is satisfied, then, under Assumptions 5 and the additional assumption that E y 6 t < under the null, the LM statistic { 0) LM = T T ) } yt T { T ) } yt d t d t d t d t, ĥ 0,t ĥ 0,t where ĥ0,t is the estimated conditional variance of the process under the null, d t = ẑ t, û t, ) ẑ t = h t ĥ 0,t ψ = t x t H t + β 0 + β i f i,j x k, H0 x t = ĥ 0,t û t = h t ĥ 0,t θ = H0 ĥ 0,t k= v t t +, ĥ0,t, y t, f,t,..., f H,t, k= j=k+ t j=k+ H β 0 + f,t ĥ 0,t,..., f H,t ĥ 0,t, f,t y t,..., f H,t y t, β i f i,j ) v k, α + β h 0,t + λ yt ) f,t,..., α H + γ β H ĥ 0,t + λ H yt ) f H,t, γ H α + β h 0,t + λ yt ) f,t,..., α H + c β H h 0,t + λ H yt ) f H,t c H, The idea of circumvent the identification problem by approximating the nonlinear contribution by a low-order Taylor expansion under the null was originally proposed by Luukkonen, Saikkonen, and Teräsvirta 988).

16 6 M. C. MEDEIROS AND A. VEIGA and v t = y t, ĥ0,t y t, y 3 t, f i,t γ i = f i,t f i,t ) y t c i ), i =,..., H, f i,t c i = f i,t f i,t ) γ i, i =,..., H, has a χ distribution with 3 degrees of freedom under the null hypothesis. REMARK 3. The sixth-order moment condition is necessary for the existence of E v t v t. Under the normality assumption, the test can be carried out in stages as follows. ) Estimate model ) under the null and call the estimated variance ĥ0,t. Compute SSR 0 = T. yt /ĥ0,t ) ) ) Regress yt /ĥ0,t on ẑ t and û t and compute the sum of the squared residuals SSR. 3) Compute the LM statistic ) LM = T SSR 0 SSR SSR 0, or the F statistic ) F = SSR 0 SSR )/3 SSR /T 5H + ). Under H 0, LM is approximately distributed as a χ with p degrees of freedom and F has approximately an F distribution with 3 and T 5H + degrees of freedom. Although the test statistic is constructed under the assumption of normality, we may follow Lundbergh and Teräsvirta 00) and consider a robust version of the LM test against nonnormal errors. The robust version of the test can be constructed following Wooldridge 990, Procedure 4.). The test is carried out as follows: ) Same as before. ) Regress û t on ẑ t and compute ) the residual vectors r t, t =,..., T. y 3) Regress on t r t, and compute the residual sum of squares SSR from that ĥ 0,t regression. The test statistic 3) LM R = T SSR has an asymptotic χ distribution with 3 degrees of freedom under the null hypothesis. As pointed out in Lundbergh and Teräsvirta 00) the robust version of the LM test should be always preferred to nonrobust ones. At relevant sample sizes when the errors are normal, they are roughly as powerful as normality-based LM tests.

17 MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY 7 6. MODELING CYCLE At this point we are ready to combine the above statistical ingredients into a coherent modeling strategy. We begin testing linearity against an ARCHq) model at significance level δ 3. The model under the null hypothesis is simply a homoskedastic model. If the null hypothesis is not rejected, the homoskedastic model is considered as the data generating process. In case of a rejection, a GARCH,) model is estimated and tested against a FC-GARCH,,) model with two regimes at the significance level δϱ, 0 < ϱ <. Another rejection leads to estimating a model with two regimes and testing it against a model with three at the significance level δϱ. The sequence is terminated at the first non-rejection of the null hypothesis. The significance level is reduced at each step of the sequence and converges to zero. This way we avoid excessively large models and control the overall significance level of the procedure. An upper bound for the overall significance level may be obtained using the Bonferroni bound Gourieroux and Monfort 995, p. 03). The selection of the parameter ϱ is ad hoc. To avoid selecting small models few regimes), is a good practice to carry the modeling cycle with different values of ϱ. Evaluation following the estimation of the final model is carried out by subjecting the model to the misspecification tests discussed in Lundbergh and Teräsvirta 00). 7. MONTE-CARLO EXPERIMENT The purpose of this section is to check the performance of test described in Section 5. We make use of the following four data generating processes DGPs). ) Model A: GARCH,): α =.0 0 5, β = 0.85, λ = ) Model B: GARCH,): α =.0 0 5, β = 0.90, λ = ) Model C: FC-GARCH3,, ): α 0 = 0 4, β 0 = 0.96, λ 0 = 0.8, α = , β = 0.60, λ = 0.0, α = 0 4, β = 0.0, λ = 0.05, γ = 5000, γ = 5000, c = 0.005, and c = ) Model D: FC-GARCH3,, ): α 0 = 6 0 5, β 0 =.0, λ 0 = 0.0, α = 5 0 5, β = 0.65, λ = 0.09, α = 0 5, β = 0.0, λ = 0.04, γ = 3000, γ = 3000, c = 0.005, and c = In all DGPs the error term has a probability function either Gaussian or a standardized t with 0 degrees of freedom. Model A has theoretical kurtosis 3.08 when the error distribution is Gaussian and 4.6 when the errors are t-distributed. Model B has a higher kurtosis: 8.55 with normality of the errors and 5.9 when the distribution of the errors is a t. Furthermore, 3 Bollerslev 986) pointed out that under the null of no heteroskedasticity, there is no general Lagrange Multiplier test for GARCHp,q). This is due to the fact that the Hessian is singular if both p > 0 and q > 0.

18 8 M. C. MEDEIROS AND A. VEIGA model A has a well defined sixth-order moment even with t-distributed errors, while model B does not. We include model B in our simulation in order to evaluate the effect of the violation of the sixth-order moment assumption in the behavior of the test. Models C and D are different specifications of a FC-GARCH3,, ) model and were previously analyzed in the examples in Section 3. Using the result of Theorem, it can be shown that Models C and D satisfy the sixthorder moment condition. All the simulations are based on series with 000 observations and the first 500 observations of each generated series are always discarded to avoid any initialization effect; see Lundbergh and Teräsvirta 00). For each experiment, a total of 000 replications have been generated. Only the results concerning the robust version of the tests are shown in order to save space. Figures shows the size-discrepancy plot for models A and B for both Gaussian and t- distributed errors. The empirical size is relatively close to the nominal size and the discrepancies are extremely low. For model B, the discrepancies are higher than for model A. One possible reason for that is the violation of the sixth-order moment assumption Nominal size empirical size Nominal size empirical size Nominal size Nominal size a) b) FIGURE. Size-discrepancy plots. Simulations with 000 observations. Lines with circles refer to Gaussian errors and lines with triangles refer to t-distributed errors. Panel a) refers to model A. Panel b) refers to model B. The power-size curve is shown in Figure 3. Observing the graphs, we can see that the power of the test is extremely high. This is also expected since the simulated FC-GARCH models have dynamics very different from standard GARCH models Empirical power 0.6 Empirical power Nominal size Nominal size a) b) FIGURE 3. Power-size plots. Simulations with 000 observations. Lines with circles refer to Gaussian errors and lines with triangles refer to t-distributed errors. Panel a) refers to model C. Panel b) refers to model D.

19 MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY Nominal size empirical size Nominal size empirical size Nominal size Nominal size a) b) FIGURE 4. Size-discrepancy plots. Simulations with 000 observations. Lines with circles refer to Gaussian errors and lines with triangles refer to t-distributed errors. Panel a) refers to model C. Panel b) refers to model D. To check the performance of the proposed methodology to detect the correct number of regimes we conduct the following experiment. First, we estimate Models C and D with three regimes H = in )). Then, we test for an additional regime, conditionally on the estimation of the previous ones. Figure 4 shows size-discrepancy plots. Of course those graphs are not true size-discrepancy because we do not know the true nominal size of the sequence of tests. However, conditionally on the estimation of a FC-GARCH,,) model, those graphs can give a glimpse of the performance of the tests in detecting the additional regime. As can be observed the test for an additional regime is well-sized. For the nominal sizes over 0.4 the discrepancies are larger when Model D is considered. However, for the significance levels typically used in applications, the distortions are very low. To conduct a power experiment, we estimate Models C and D with two regimes and test for an additional one. As shown in Figure 5 the power is rather low, specially for Model C. For example, considering Model C and a 5% significance level, the rejection frequency is 0.9 for Gaussian errors and 0. with t errors. For a 0% significance level, the rejection frequencies become 0.30 and 0. for Gaussian and t distributions, respectively. Considering Model D and a 5% level, the frequencies of rejection of the null hypothesis are 0.3 and 0.3 for Normal and t distributions, respectively. For a 0% level, the numbers become 0.44 and One possible explanation is that there are only few observations in the extreme regimes and a larger number of them are necessary to detect those regimes. When series with 3000 observations are considered, the power increases dramatically. 8. EMPIRICAL EXAMPLES The data that are used in this paper consist of eight indexes: Amsterdam EOE), Frankfurt DAX), Hong Kong Hang Seng), London FTSE00), New York, S&P 500), Paris CAC40), Singapore Singapore All Shares) and Tokyo Nikkei). For all series, except for the CAC40 index, the sample begins in January, and ends in December, The number of observations is 38. The CAC40 index begins in July, and ends in December, 3 987, a total of 736 observations.

20 0 M. C. MEDEIROS AND A. VEIGA Empirical power Empirical power Nominal size a) b) FIGURE 5. Power-size plots. Simulations with 000 observations. Lines with circles refer to Gaussian errors and lines with triangles refer to t-distributed errors. Panel a) refers to model C. Panel b) refers to model D. As our interest is on the conditional variance and not in the conditional mean, we focus on the unpredictable part of the returns and follow Engle and Ng 993). The procedure involves a day-of-the-week effect adjustment and an autoregression which removes the linear predictable part of the daily returns. Let y t be the daily return at day t. We start regressing y t on a constant and four variables: Mon t, Tue t, Wed t, and Thu t, which are dummy variables for Monday, Tuesdays, Wednesdays, and Thursdays, respectively. The residual from the regression, u t, is therefore regressed on a constant and on u t,..., u t 7. We choose seven lags in order to remove any remaining day-of-the-week effect not captured by the dummy variables. The residual from the autoregression, r t, is the unpredictable return. Table shows the adjustment results. Table 3 shows descriptive statistics and diagnostics, where σ is the standard deviation, SK is the skewness, K is the kurtosis, and Q0) and QS0) are, respectively, the p-values of the Ljung-Box statistic for tenth-order serial correlation in the unpredictable returns and squared returns. Sb, Nsb, P sb, and Jsb are, respectively, the p-values of the sign bias, negative sign bias, positive sign bias, and joint tests for asymmetry proposed by Engle and Ng 993). ARCH4) is the p-value of the fourth-order ARCH LM test described in Engle 98). From the Ljung-Box test statistic at the % significance level we find no significant serial correlation left in the series after our adjustment procedure. The coefficients of skewness and kurtosis both indicate that the series have a distribution that is fat-tailed and skewed to the left, with the exception of the Singapore All Shares which is skewed to the right. Furthermore, the Ljung-Box statistic in the squares and the ARCH LM test strongly suggest the presence of time-varying volatility. Moreover, there are evidence of asymmetries in the conditional variance of all the series. The negative sign bias and joint tests reject the null hypothesis of no asymmetric effect for all the eight indexes. The positive sign bias test strongly rejects the null hypothesis for the EOE, FTSE00, and CAC40 indexes. For the Singapore All Shares series, the test rejects the null hypothesis at a 8% level. The sign bias test rejects the null for the DAX, Hang Seng, and Nikkei indexes. The overall evidence is that the size of negative past returns strongly affects the current volatility: large negative unpredictable returns cause more volatility than small ones.

21 MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY TABLE. DAILY RETURNS: MEAN ADJUSTMENT REGRESSIONS. The table reports the results of an adjustment procedure to remove the day-of-the-week effects and the predictable part of the daily return series. The procedure is similar to the one in Engle and Ng 993). If Pt is the level of the index at day t, then the daily return is given by yt = lnpt) lnpt ). First, yt is regressed on a constant and four dummies representing the days-of-the week. The residual from this regression is regressed on a constant and seven lags to obtain the residual rt, which is the unpredictable return. Mont, Tuet, Wedt, and Thut are dummy variables for Monday, Tuesdays, Wednesdays, and Thursdays, respectively. Each of these day-of-the-week dummies takes a value of on the corresponding weekday and a value of 0 otherwise. ut is the residual of the day-of-the-week adjustment regression. The figures between parentheses bellow the estimates are the Newey-West HAC robust standard errors. The sample period is from January, to December, 3 997, a totalizing 38 observations. The only exception is the CAC40 index, for which the sample begins in July, 9 987, a total of 736 observations. Day-of-the-week adjustment yt = θ0 + θmont + θtuet + θ3wedt + θ4thut + ut Series θ0 θ θ θ3 θ4 EOE ) DAX ) Hang Seng ) FTSE ) S&P ) CAC ) All Shares ) Nikkei ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Continued on next page.