Unconditional skewness from asymmetry in the conditional mean and variance

Transcription

1 Unconditional skewness from asymmetry in the conditional mean and variance Annastiina Silvennoinen, Timo Teräsvirta, and Changli He Department of Economic Statistics, Stockholm School of Economics, P. O. Box 6501, SE Stockholm, Sweden March 2, 2005 Abstract In this paper we consider the third-moment structure of a class of nonlinear time series models. Empirically it is often found that financial time series exhibit unconditional skewness, and thus it is of importance to know what kind of models can accommodate for this. On the other hand, if one chooses to use a nonlinear model, one has to recognize its implications on the unconditional third-moment structure. We especially investigate the effect of having a nonlinear first or second conditional moment, and different conditions under which the unconditional distribution exhibits skewness as well as nonzero third-order autocovariance structure. The nonlinear specification of the conditional mean is found to be of greater importance than the specification of the conditional second moment. Several examples are discussed and, whenever possible, explicit analytical expressions are provided for all third-order moments and cross-moments. JEL classification: C22 Key words: Unconditional skewness; GARCH; Time series annastiina.silvennoinen@hhs.se timo.terasvirta@hhs.se changli.he@hhs.se Acknowledgements. This research has been supported by the Jan Wallander s and Tom Hedelius s Foundation, Grant No. J We would like to thank Pentti Saikkonen, Markku Lanne, Tony Hall, and Mika Meitz for useful discussions. Material from this paper has been presented at the University of Technology, Sydney, and at the workshop Econometrics and Computational Economics, Helsinki, November We would like to thank the participants in these occasions for their comments. The responsibility for any errors and shortcomings in this paper remains ours. 1

2 1 Introduction Empirically it is often found that both unconditional and conditional skewness are present in financial markets. The unconditional stock market return distributions in particular appear skewed. One approach to tackle the problem of modelling such phenomena is to allow the error distribution to be asymmetric. Lanne and Saikkonen (2004) introduced a GARCH-inmean model with a skewed error distribution to model the conditional skewness. A conceptual disadvantage of this approach is that it is difficult to explain why the genuine independent and identically distributed noise component should have a skewed distribution. Another feature of financial data is the nonlinear or asymmetric response to shocks both in the conditional mean and variance, see e.g. Brännäs and de Gooijer (2004). The modeller should be aware of the fact that using nonlinear models may have an effect on the unconditional skewness. This paper investigates ways of introducing unconditional skewness while assuming a symmetric error distribution. More specifically, we investigate how the unconditional third-moment structure of the variable of interest is affected by the presence of asymmetry (nonlinearity) in the conditional mean and/or in the conditional variance. We begin by introducing asymmetry in the conditional first moment. Explicitly, the conditional mean is first modelled as an asymmetric moving average (asma) proposed by Wecker (1981), and as another alternative we consider a GARCH-in-mean specification by letting the conditional standard deviation or variance enter the conditional mean equation. For the purpose of deriving analytical expressions for the third-order moments, GARCH models which parameterize the conditional standard deviation are more convenient than the ones where conditional variance is parameterized. In the latter case the calculations would involve terms for which analytical expressions are not available. With GARCH models where the conditional standard deviations are used this difficulty is circumvented. Our choice for the conditional standard deviation process is the asymmetric extension of the absolute value GARCH which allows for asymmetry also for the second unconditional moment. The model is, more or less, a reparameterized version of the threshold GARCH (TGARCH) model of Zakoïan (1994), and it nests the absolute value GARCH (AVGARCH) model of Taylor (1986) and Schwert (1989). These models are nested in the family of GARCH models of Hentschel (1995), which is useful in view of recent results on the conditions for ensuring strict stationarity and existence of moments. Recently, Brännäs and de Gooijer (2004) proposed a model that introduces asymmetry both in the conditional first and second moments. The variance process is an extension of the QGARCH model of Sentana (1995) allowing more flexibility in the way the asymmetric response to shocks is parameterized. The authors considered the first and second moments of their asma asqgarch model but did not consider the third-moment structure. Because they choose to parameterize the conditional variance, not the conditional standard deviation, finding analytical expressions for the third moments of the model is difficult. In fact, it seems that in their model even the lower-order moments lack explicit analytical expressions. The paper is organized as follows. In Section 2 the general model framework is presented. In Section 3 the first and second-moment structure as well as the third-order moment structure are derived. Specific examples are examined in detail in Section 4. Section 5 concludes. Technical derivations and detailed expressions for the moments considered can be found in the Appendix. 2

3 2 The model framework Let y t be generated by y t = µ t + ε t, (1) ε t = z t h t (2) where µ t is the conditional mean of y t given F t 1 (the sigma-field generated by the available information until time t 1), h 2 t is the conditional variance of y t given F t 1, and z t } iid(0,1) and symmetric around zero. The symmetry assumption is made because the main point of the present work is to investigate possibilities of parameterizing the skewness in the unconditional marginal distribution of y t. A variety of model specifications for the conditional mean will be considered. For simplicity, we focus on first-order models which are also empirically often found to be adequate. Defining µ t = φε t 1 + φ + (ε + t 1 Eε+ t ) (3) where ε + t = max(0,ε t ), one obtains a first-order asymmetric moving average (asma) process of Wecker (1981). If µ t is a function of h t, µ t = φ(h δ t Ehδ t ), δ = 1 or 2 (4) one arrives at the GARCH-in-mean (GARCH M) model of Engle, Lilien, and Robins (1987). The error process ε t } is assumed to be a conditionally heteroskedastic white noise sequence with h d t = ω + c t 1h d t 1, d = 1 or 2 (5) where c t (z t ) is a well-defined function and h d t > 0 for all t. To ensure hd t > 0, suitable parameter restrictions must be imposed. The moment properties of the family of GARCH models defined by (5) are investigated in He and Teräsvirta (1999). It nests many but not all of the models in the family of GARCH models of Hentschel (1995). For instance, setting d = 2 in (5) and c t = αz 2 t + β yields the standard GARCH model of Bollerslev (1986). Setting d = 1 and c t = α z t + β + α z t (6) equation (5) defines the threshold GARCH (TGARCH) model of Zakoïan (1994). By setting α = 0, the model collapses into the AVGARCH model of Taylor (1986) and Schwert (1989). Furthermore, the GJR GARCH model of Glosten, Jaganathan, and Runkle (1993) and the nonlinear GARCH (NLGARCH) model of Engle (1990) are nested withing the framework. Note that any GARCH model defined by equation (5) is symmetric in its response to shocks if and only if c t (z t ) in (5) is an even function of z t. The following assumptions are made to ensure strict stationarity and the existence of moments for the process y t : Assumption 1 The random variable z t has a probability density function that is positive and continuous on the whole real axis. Assumption 2 The processes µ t and h t are measurable with respect to F t 1. 3

4 Assumption 3 The processes y t, µ t, and h d t are strictly stationary. For a given m 1, Ey m t <, Eµ m t <, and Eh m t <. The Assumptions 1 3 are very general and the conditions are not stated in their most strict form as it is not the main purpose of the paper. We note that the required positivity and continuity of the probability density function are not at all restrictive for any practical purposes. 1 General conditions for the Assumption 3 to hold for models that are nested in the Hentschel family can be found in Meitz and Saikkonen (2004). For the models (1), (2) and (5) with µ t = 0 necessary and sufficient conditions for the existence of third moments when d = 1 and sufficient conditions when d = 2 can be found in He and Teräsvirta (1999). 2 In Proposition 2 of the Appendix the moment conditions are stated explicitly for certain common models. 3 Moments We begin by considering the moment structure of the general model (1) and (2). Assume that the process y t is strictly stationary with finite third-order moments. Set γ i = E(y t Ey t ) i and γ ij (k) = Cov(yt,y i j t k ), i,j 1, for ease of notation. The unconditional mean and variance are Ey t = Eµ t and γ 2 = V arµ t + Eε 2 t, respectively. The autocovariances are γ 11 (k) = Cov(µ t,µ t k ) + Eµ t ε t k where k 1. Assuming µ t is an asma in (3) and φ + 0 renders the autocovariances nonzero for k 1, see Lemma 1 in the Appendix. The same holds if µ t is a function of h δ t, δ = 1 or 2, see Lemma 3. The third moment and third-order cross-moments of y t are given by γ 3 = E(µ t Eµ t ) 3 + 3Cov(µ t,ε 2 t ) γ 21 (k) = Cov(µ 2 t,µ t k) + Cov(µ t k,ε 2 t ) + Eµ2 t ε t k + Eε 2 t ε t k, k 1 γ 12 (k) = Cov(µ t,µ 2 t k ) + Cov(µ t,ε 2 t k ) + 2Eµ tµ t k ε t k, k 1. Define the unconditional skewness of y t as κ 3 = γ 3 /(γ 2 ) 3/2. The following proposition gives general conditions that yield zero skewness. Proposition 1 Under the Assumptions 1 3 with m = 3, E(µ t Eµ t ) 3 = 0 and Cov(µ t,ε 2 t ) = 0 are sufficient for κ 3 = 0. When µ t 0 in (1), the conditions in Proposition 1 are satisfied and γ 3 = 0. In this case the only nonzero cross-moments are γ 21 (k) = Eε 2 t ε t k, k 1. Therefore, only assuming that the conditional second moment is time-varying does not imply nonzero unconditional skewness. A time-varying conditional mean is required for that. 4 Examples In this section we consider two nonlinear specifications of the conditional mean in detail. The first one is an asymmetric moving average model of Wecker (1981). The second one introduces the 1 The assumptions of positivity and continuity of the density function of the error process are needed to operationalize Theorem 3 in Meitz and Saikkonen (2004), which could be used to prove stationarity at least in some cases. 2 If a µ t = 0 in (1), analytical expressions for the fourth-order moments (note that Ey 3 t = 0) can be found in He and Teräsvirta (1999). 4

5 conditional standard deviation or variance into the conditional mean. We choose the TGARCH model for the error process. This choice is dictated by our aim at obtaining analytical expressions for all unconditional third-order moments and cross-moments. Such expressions give an idea of how asymmetries in conditional first and second moments contribute to the unconditional third moments. Since the TGARCH model is asymmetric in its response to shocks and nests the symmetric AVGARCH model, it is possible to study the effect of this asymmetry on the unconditional skewness. The analytical expressions for the moments can be found in Lemma 2 and the subsequent Corollaries and in Lemma 4 in the Appendix. The expressions are rather involved but yield quite straightforward conclusions. Other GARCH models are likely to be similar in this respect but because most of them lack analytical expressions for third-order moments, one has to rely on simulations to obtain numerical values for them. 3 Whenever possible, we try to take examples of models such as the symmetric GARCH model of Bollerslev (1986), or the asymmetric QGARCH one of Sentana (1995), or the GJR GARCH model of Glosten, Jaganathan, and Runkle (1993). In fact, all results in the Appendix apply to the general family of GARCH models in (5). However, fully explicit expressions are provided only for the TGARCH or the AVGARCH model and not for models for which d = 2 in (5). Note that the QGARCH model is not a member of the family defined by (5). We begin by considering the complete asma TGARCH model and the effect of restricting the conditional standard deviation specification to be the symmetric AVGARCH model. Subsequently, we consider the third-moment structure when the conditional mean is simplified to only contain either the asymmetric component (φ = 0), the symmetric MA component (φ + = 0), or neither (φ = φ + = 0). In all these cases we consider both the TGARCH and the AVGARCH specifications for the conditional second moment. As a final example, the conditional mean is defined to be a function of the conditional second moment, which leads us to GARCH M models. 4.1 First-order asma with TGARCH or AVGARCH conditional standard deviation The third-moment structure of the asma TGARCH model is given in Lemma 2 and Lemma 5. It is apparent from the rather involved expressions that the first-order asma TGARCH model accommodates a rich variety of third-order moment structures. When the conditional standard deviation is restricted to follow the AVGARCH model, some of the expressions in Lemma 2 simplify, but not substantially. The resulting moment structure is given in Corollary 1. In this case the cross-moments Eε 2 t ε t k, Eε t k ε + t, Eε t kε +2 t, Eε t k ε t k 1 ε + t, and Eε t kε + t ε+ t k 1, k 1, equal zero, which can be seen from the expressions in Lemma 5 and Corollary 5. Thus the third-moment structure is still rich as long as the conditional mean is defined by an asma model. In particular, all third-order moments are nonzero whether or not the conditional second moment exhibits asymmetry. Since the expressions for the third-order moments are quite involved we illustrate the situation numerically. Figure 1 shows the amount of unconditional skewness that can be obtained from the asma TGARCH process for certain parameter values of the conditional mean. Each curve represents the level of unconditional skewness for a fixed value of φ + and a suitable range of values for φ. The parameter space is restricted to contain only parameter values for which 3 The possibility of using quantile measures as in Kim and White (2004) for unconditional skewness is yet to be explored. 5

6 the conditional mean process is invertible. Invertibility of the asma process has to be checked using simulations, see e.g. Brännäs and de Gooijer (1995). It is seen that the closer φ is to zero, the wider is the range of skewness implied by the model: from κ 3 = 1 for φ + = 3 to κ 3 = 1 for φ + = 3. When φ is large in absolute value, however, the invertibility condition restricts the asymmetry parameter φ + and thus limits the achievable amount of skewness. Next consider the case where φ = 0. It can be seen from the expressions in Corollary 2 that the third-moment structure is still rather rich and complex. It is, however, considerably simpler than in the case of φ 0. When the errors are restricted to follow a symmetric AVGARCH model, the expressions simplify somewhat, see Corollary 3, but again not very much. It can be seen from the expressions in Lemma 5 and Corollary 5 that Eε 2 t ε t k = Eε t k ε + t = Eε t k ε +2 t = Eε t k ε + t ε+ t k 1 = 0, k 1. Thus, regardless of the conditional variance, an asymmetric (nonlinear) conditional mean leads to a skew marginal distribution for y t and nonzero third-order cross-moments. From Figure 1 it is seen that the widest spread of skewness is achieved exactly when φ = 0. An obvious conclusion is that the asymmetry parameter φ + plays a very influential role in determining both the sign and the amount of skewness. 4.2 First-order MA with TGARCH or AVGARCH conditional standard deviation A much simpler third-moment structure follows when φ + = 0 in (3) while the conditional standard deviation follows the TGARCH model. This is evident from the results in Corollary 4. In this case we still have Ey 3 t 0. An interesting feature is that Ey2 t y t k 0, k 1, whereas Ey t y 2 t k = 0 for k > 1. It should also be noted that the only nonzero cross-moment of ε t is now Eε 2 t ε t k. Furthermore, also assuming φ = 0 in (3), i.e. having µ t = 0, forces the unconditional skewness to zero regardless of the asymmetry in the conditional second moment. In this case the only nonzero cross-moments are Ey 2 t y t k = Eε 2 tε t k, k 1. These results may be useful in specifying asma TGARCH models. Note that in Figure 1 the thick curve represents the skewness as a function of φ for φ + = 0, in which case the amount of skewness is small. At φ = 0 this curve intersects the x-axis in accordance with the results mentioned after Proposition 1. As a sidestep consider a model whose conditional mean specification is a first-order AR process: µ t = φy t 1 in (1). In this case κ 3 = 3 i=1 φi Eε t i h 2 t/(1 φ 3 ) (Eh 2 t /(1 φ2 )) 3/2. The unconditional skewness emerging from this model is very similar to that of the MA TGARCH model already discussed. In Figure 2 it is plotted as a function of the mean parameter φ for a range of values for β and keeping the other TGARCH parameter values fixed. It can be concluded that the amount of skewness obtained from a model with a linear mean is not large and the effect of increasing persistence in the GARCH process is negligible. This also holds for the asma TGARCH model, in which increasing persistence has almost no effect on the level of skewness. We now turn to the analytic form of the cross-moment Eε 2 tε t k in Corollary 4. When the conditional stardard deviation is defined as a TGARCH process, it follows that k 1 Eε 2 t ε t k = 2ωα Eh 2 t (Ec t ) k 1 j (Ec 2 t )j + 2α (Ec 2 t )k 1 Ec t z t 2 Eh 3 t, k 1, j=0 6

7 Figure 1: asma TGARCH: Unconditional skewness of y t as a function of φ for the following values of φ + and the TGARCH parameters: x-axis: φ, y-axis: unconditional skewness, lines: φ + = 3.0, 2.5,..., 3.0 (top to bottom), thick line corresponds to φ + = 0; TGARCH parameters: ω = 0.005, α = 0.05, β = 0.90, and α = where the expressions for the moments of c t and h t are given in Lemma 5. Hence Eε 2 tε t k 0 for α 0. Assuming α = 0 yields Eε 2 t ε t k = 0, k 1, which implies that the third moment and all the third-order cross-moments in Corollary 4 are zero. In fact, any parameterization of h t or h 2 t that has the property Eε2 t ε t k = 0 for k 1 gives the same result. As stated in Corollary 5 all symmetric GARCH models within the family (5) have that property. The linear GARCH model is an example of such a case. As further examples, consider the nonlinear cases where the errors are governed by a first-order QGARCH process h 2 t = ω +αε 2 t 1 +βh2 t 1 +α ε t 1 (Sentana (1995)) or by a GJR GARCH process h 2 t = ω + αε 2 t 1 + βh2 t 1 + α ε +2 t 1 (Glosten, Jaganathan, and Runkle (1993)). If the errors follow a QGARCH process then the expression for Eε 2 tε t k is given by Eε 2 t ε t k = α (α + β) k 1 Eh 2 t, k 1, where Eh 2 t = ω 1 (α + β). If the errors follow a GJR GARCH process, d = 2 and c t = αzt 2 +β+α z t +2 for Eε 2 tε t k is formally given by Eε 2 t ε t k = α (α + β + α Ez +2 t ) k 1 Ez +3 t Eh 3 t, k 1, in (5), the expression but an explicit expression for Eh 3 t is not available. Also in these cases, Eε 2 tε t k 0 if and only if α 0. Thus, if both the conditional mean and the conditional second moment are symmetric, 7

8 Figure 2: AR TGARCH: Unconditional skewness of y t as a function of φ for the following values of β, δ = 1, and the TGARCH parameters: x-axis: φ, y-axis: unconditional skewness, lines: β = 0.90, 0.91,..., 0.94 (see legend); TGARCH parameters: ω = 0.005, α = 0.05, and α = beta 0.90 beta 0.91 beta 0.92 beta 0.93 beta the unconditional marginal distribution for y t is symmetric as well. It is emphasized that the results just discussed are only obtained if φ 0. Thus, at least some linear dependence in y t } is necessary for skewness in the unconditional distribution of y t. We may also note that in the case of the first-order asqgarch model h 2 t = ω + αε2 t 1 + βh 2 t 1 + α ε t 1 + α ε + t 1 (Brännäs and de Gooijer (2004)), the expression for Eε2 t ε t k becomes very complicated, and it does not seem possible to derive an analytical form for it. However, it seems to be nonzero for k 1 as its components are not trivially zero GARCH-in-mean model It is well known that when the conditional standard deviation, conditional variance or any other nontrivial function of these, enters the conditional mean, γ 11 (k) 0 for k 1 if γ 2 < ; see Hong (1991). It may be less well known that in this case γ 3 0. As an example, consider the following TGARCH M model (1),(2), and (4) (6) so that Ey t = 0. Since h δ t is a positivevalued variable and its distribution is asymmetric, it follows that Ey 3 t 0. Assume now that y t is strictly stationary and has a finite third moment. 5 Then the third moment and third-order cross-moments of the TGARCH M process are given by Lemma 4. In Figure 3 the unconditional skewness from a TGARCH M model with δ = 1 is plotted as a function of φ. One can see that 4 In fact, it seems that there is no explicit expression for any moment Eε m, m > 1, for this model at least it seems that there does not exist an analytic form for Eh 2 t. 5 The fact that TGARCH M process is strictly stationary with finite third moment follows from Assumptions 1 and the assumptions in Proposition 2 with m = 3 as well as from the assumption of strict stationarity of the process h d t. 8

9 Figure 3: TGARCH M: Unconditional skewness of y t as a function of φ for the following values of β, δ = 1, and the TGARCH parameters: x-axis: φ, y-axis: unconditional skewness, lines: β = 0.90, 0.91,..., 0.94 (see legend); TGARCH parameters: ω = 0.005, α = 0.05, and α = beta 0.90 beta 0.91 beta 0.92 beta 0.93 beta the range of possible skewness increases with the persistence of the GARCH process. Assuming φ = 0 implies γ 3 = 0 regardless of the asymmetry in the second conditional moment. This is also seen from Figure 3 where all the lines intersect the x-axis at φ = 0. In this case the only nonzero cross-moments are Ey 2 t y t k = Eh 2 t ε t k; see the discussion in the previous subsection. Assuming that h t is defined by the standard AVGARCH model, the expressions for the moments simplify somewhat. Then Eh 2 t ε t k = Eh t ε t k = Eh t h t k ε t k = 0, k 1, as can be seen from the expressions in Lemma 5 and Corollary 5. However, provided that φ 0, all third-order moments are nonzero regardless of whether the conditional standard deviation is symmetric or asymmetric. This example demonstrates that the third-moment structure in the case of the TGARCH M or AVGARCH M model is richer than it is in MA TGARCH and MA AVGARCH models. However, assuming an asma process for the conditional mean gives even more room for skewness in the marginal density of y t. 5 Conclusions In this paper we show how different parameterizations of the conditional mean and/or variance contribute to the asymmetry in the unconditional distribution of y t. This is important because marginal distributions of return series often appear skewed. It is thus useful to know the structure of the unconditional distribution implied by a model that has asymmetries/nonlinearities in the first and the second conditional moment. The models we have considered in detail are the asma TGARCH and the TGARCH M 9

10 model, which contain nonlinearities both in the conditional mean and the conditional standard deviation. We derive the analytic expressions for the third-order moment structure of these models and consider various special cases of the asma TGARCH model where either the mean and/or the standard deviation specification is restricted to be symmetric. Similar considerations are made in the case of the TGARCH M model. In general, we find that asymmetries (nonlinearities) in the conditional mean are of greater importance than they are in the conditional second moment when it comes to generating skewed marginal distributions. If the conditional mean is symmetric (linear), then the unconditional skewness can only follow from the asymmetry of the conditional second moment. However, in that case the third-moment structure of the variable of interest is no longer particularly flexible. But then, if the conditional mean is nonlinear, the distribution of y t can be even strongly skewed regardless of whether the conditional second moment is symmetric or asymmetric. It would be interesting to consider a wider variety of specifications for the conditional mean and variance and derive the corresponding expressions in these cases. However, for many specifications analytical expressions are not available. It would be useful to fit the asma TGARCH model as well as other GARCH specifications to empirical series to see how well they actually manage to capture the asymmetry of the returns present in the data. 6 Of course, any comparison of the unconditional moments estimated from the data with the moments implied by the fitted model (plug-in estimation) is dependent on simulations whenever analytical expressions for the moments of interest are not available. This topic will be taken up in future work. 6 See Brännäs and de Gooijer (2004) for related work. 10

11 Appendix Proposition 2 Consider first the asma model (1) (3), and (5) where d = 1 and c t = α z t +β+ α z t (TGARCH), or d = 2 and c t = αzt 2 +β (GARCH) or c t = αzt 2 +β+α z t +2 (GJR GARCH), and the assumptions (i) E z t m <, (ii) Ec m/d t < 1. Consider then the asma model (1) (3) where h 2 t = ω + αε 2 t 1 + βh2 t 1 + α ε t 1 (QGARCH), and the assumptions (i) and (ii ) E(αz 2 t + β + α z 2 t ) m/d < 1. If the processes y t, µ t, and ε t are strictly stationary, then the assumptions (i) and (ii), or (ii ), are sufficient for the existence of the mth-order moments of the process y t. Proof. The conditions (i) and (ii), or (ii ), imply that Eh m t < and E ε t m <, see Meitz and Saikkonen (2004) (Theorem 3 and the related discussion). These in turn ensure the existence and finiteness of the mth-order moments of the mean process µ t and the process y t. Lemma 1 (First and second-moment structure of asma) Consider an asymmetric MA process (1) (3) and (5) that is strictly stationary with finite second moment. The unconditional first and second-order moments and cross-moments of y t are given by where Ey t = 0, (7) V ar y t = Eµ 2 t + Eε2 t, (8) γ 11 (k) = Eµ t µ t k + Eµ t ε t k, k 1, (9) Eµ 2 t = φ 2 Eε 2 t + (2φφ + + φ +2 )Eε +2 t φ +2 (Eε + t )2 Eµ t µ t k = + Eε t k ε + t + φ +2 Eε + t ε+ t k φ+2 (Eε + t )2 φφ Eµ t ε t k = φeε 2 t + φ + Eε +2 t, k = 1 φ + Eε t k ε + t 1, k > 1. The moments of ε t and ε + t can be found in Lemma 5. Proof. The results are readily obtained by straightforward algebra. Lemma 2 (Third-moment structure of asma) Consider an asymmetric MA process (1) (3) and (5) that is strictly stationary with finite third moment. The unconditional third-order moments and cross-moments of y t are given by γ 3 = Eµ 3 t + 3Eµ tε 2 t, (10) γ 21 (k) = Eµ 2 tµ t k + Eµ t k ε 2 t + Eµ 2 tε t k + Eε 2 tε t k, k 1 (11) γ 12 (k) = Eµ t µ 2 t k + Eµ tε 2 t k + 2Eµ tµ t k ε t k, k 1 (12) 11

12 where Eµ 3 t = (3φ 2 φ + + 3φφ +2 + φ +3 )Eε +3 t (6φφ φ +3 )Eε +2 t Eε + t 3φ 2 φ + Eε 2 t Eε+ t +2φ +3 (Eε + t )3 Eµ 2 tµ t k = φ 3 Eε 2 t ε t k + (2φ 2 φ + + φφ +2 )Eε t k ε +2 t +φ 2 φ + (Eε 2 t ε + t k Eε2 t Eε + t ) + (2φφ+2 + φ +3 )(Eε +2 t 2φφ +2 Eε t k ε + t Eε+ t 2φ +3 (Eε + t ε+ t k Eε+ t (Eε + t )3 ) ε + t k Eε+2 t Eε + t ) Eµ t µ 2 t k = φ 2 φ + (Eε 2 t k ε+ t Eε 2 teε + t ) + (2φφ+2 + φ +3 )(Eε + t ε+2 t k Eε+2 t Eε + t ) 2φφ +2 Eε t k ε + t Eε+ t 2φ +3 (Eε + t ε+ t k Eε+ t (Eε + t )3 ) Eµ t ε 2 t = φeε 2 t ε t 1 + φ + (Eε 2 t ε+ t 1 Eε2 t Eε+ t ) Eµ t k ε 2 t = φeε 2 t ε t k 1 + φ + (Eε 2 t ε+ t k 1 Eε2 t Eε+ t ) Eµ t ε 2 t k = Eµ 2 t ε t k = Eµ t µ t k ε t k = φ + (Eε +3 t Eε 2 t Eε+ t ), k = 1 φ + (Eε 2 t (k 1) ε+ t Eε 2 teε + t ), k > 1 (2φφ + + φ +2 )Eε +3 t 2φφ + Eε 2 teε + t 2φ +2 Eε +2 t Eε + t, k = 1 (2φφ + + φ +2 )Eε t (k 1) ε +2 t 2φ +2 Eε t (k 1) ε + t Eε+ t +φ 2 Eε 2 tε t (k 1), k > 1 φφ + (Eε 2 tε + t 1 Eε2 teε + t + Eε t 1 ε +2 t ) + φ +2 (Eε +2 t ε + t 1 Eε+2 t Eε + t ) +φ 2 Eε 2 t ε t 1, k = 1 φφ + Eε t (k 1) ε t k ε + t + φ +2 (Eε t (k 1) ε + t ε+ t k Eε t (k 1)ε + t Eε+ t ), k > 1. The moments of ε t and ε + t can be found in Lemma 5. Proof. The results are readily obtained by tedious but straightforward algebra. Corollary 1 (Third-moment structure of asma with symmetric GARCH) Consider an asymmetric MA process (1) (3) and (5) that is strictly stationary with finite third moment. Furthermore, assume that c t in (5) is even with respect to z t. The unconditional third-order moments and cross-moments of y t are given by (10)-(12) in Lemma 2 where Eµ 3 t = (3φ 2 φ + + 3φφ +2 + φ +3 )Eε +3 t (6φφ φ +3 )Eε +2 t 3φ 2 φ + Eε 2 t Eε+ t + 2φ +3 (Eε + t )3 Eµ 2 t µ t k = φ 2 φ + (Eε 2 t ε+ t k Eε2 t Eε+ t ) + (2φφ+2 + φ +3 )(Eε +2 t ε + t k Eε+2 t Eε + t ) 2φ +3 (Eε + t ε+ t k Eε+ t (Eε + t )3 ) Eε + t 12

13 Eµ t µ 2 t k = φ 2 φ + (Eε 2 t k ε+ t Eε 2 teε + t ) + (2φφ+2 + φ +3 )(Eε + t ε+2 t k Eε+2 t Eε + t ) 2φ +3 (Eε + t ε+ t k Eε+ t (Eε + t )3 ) Eµ t ε 2 t = φ + (Eε 2 tε + t 1 Eε2 teε + t ) Eµ t k ε 2 t = φ + (Eε 2 tε + t k 1 Eε2 teε + t ) Eµ t ε 2 t k = Eµ 2 tε t k = Eµ t µ t k ε t k = φ + (Eε +3 t Eε 2 teε + t ), k = 1 φ + (Eε 2 t k ε+ t 1 Eε2 teε + t ), k > 1 (2φφ + + φ +2 )Eε +3 t 2φ 1 φ + 1 Eε2 t Eε+ t 2φ +2 1 Eε+2 t Eε + t, k = 1 0, k > 1 φφ + (Eε 2 t ε + t 1 Eε2 teε + t + Eε t 1 ε +2 t ) + φ +2 (Eε +2 t ε + t 1 Eε+2 t Eε + t ), k = 1 0, k > 1. The moments of ε t and ε + t can be found in Lemma 5. Corollary 2 (Third-moment structure of asma with φ = 0) Consider an asymmetric MA process (1) (3) and (5) with φ = 0 that is strictly stationary with finite third moment. The unconditional third-order moments and cross-moments of y t are given by (10)-(12) in Lemma 2 where Eµ 3 t = φ +3 (Eε +3 t 3Eε +2 t Eε + t + 2(Eε + t )3 ) Eµ 2 tµ t k = φ +3 (Eε +2 t ε + t k Eε+2 t Eε + t ) 2φ+3 (Eε + t ε+ t k Eε+ t (Eε + t )3 ) Eµ t µ 2 t k = φ +3 (Eε + t ε+2 t k Eε+2 t Eε + t ) 2φ+3 (Eε + t ε+ t k Eε+ t (Eε + t )3 ) Eµ t ε 2 t = φ + (Eε 2 tε + t 1 Eε2 t Eε + t ) Eµ t k ε 2 t = φ + (Eε 2 tε + t k 1 Eε2 teε + t ) Eµ t ε 2 t k = Eµ 2 tε t k = Eµ t µ t k ε t k = φ + (Eε +3 t Eε 2 t Eε + t ), k = 1 φ + (Eε 2 t k ε+ t 1 Eε2 t Eε+ t ), k > 1 φ +2 (Eε +3 t 2Eε +2 t Eε + t ), k = 1 φ +2 (Eε t k ε +2 t 1 2Eε t kε + t 1 Eε+ t ), k > 1 φ +2 (Eε +2 t ε + t 1 Eε+2 t Eε + t ), k = 1 φ +2 (Eε t k ε + t 1 ε+ t k 1 Eε t kε + t 1 Eε+ t ), k > 1. The moments of ε t and ε + t can be found in Lemma 5. 13

14 Corollary 3 (Third-moment structure of asma with φ = 0 and symmetric GARCH) Consider an asymmetric MA process (1) (3) and (5) with φ = 0 that is strictly stationary with finite third moment. Furthermore, assume that c t in (5) is even with respect to z t. The unconditional third-order moments and cross-moments of y t are given by (10)-(12) in Lemma 2 where Eµ 3 t = φ +3 (Eε +3 t 3Eε +2 t Eε + t + 2(Eε + t )3 ) Eµ 2 tµ t k = φ +3 Eε +2 t ε + t k φ+3 Eε +2 t Eε + t 2φ +3 (Eε + t ε+ t k Eε+ t (Eε + t )3 ) Eµ t µ 2 t k = φ +3 Eε + t ε+2 t k φ+3 Eε +2 t Eε + t 2φ +3 (Eε + t ε+ t k Eε+ t (Eε + t )3 ) Eµ t ε 2 t = φ + (Eε 2 tε + t 1 Eε2 teε + t ) Eµ t k ε 2 t = φ + (Eε 2 tε + t k 1 Eε2 teε + t ) Eµ t ε 2 t k = Eµ 2 tε t k = Eµ t µ t k ε t k = φ + (Eε +3 t Eε 2 t Eε+ t ), k = 1 φ + (Eε 2 t k ε+ t 1 Eε2 t Eε + t ), k > 1 φ +2 (Eε +3 t 2Eε +2 t Eε + t ), k = 1 0, k > 1 φ +2 (Eε +2 t ε + t 1 Eε+2 t Eε + t ), k = 1 0, k > 1. The moments of ε t and ε + t can be found in Lemma 5. Corollary 4 (Third-moment structure of MA) Consider an asymmetric MA process (1) (3) and (5) with φ + = 0 that is strictly stationary with finite third moment. The unconditional third-order moments and cross-moments of y t are given by (10)-(12) in Lemma 2 where and Eµ 3 t = Eµ tµ 2 t k = Eµ tε 2 t k = 0 Eµ t ε 2 t = φeε 2 t ε t 1 Eµ 2 tµ t k = φ 3 Eε 2 tε t k Eµ t k ε 2 t = φeε 2 t ε t (k+1) Eµ 2 t ε t k = Eµ t µ t k ε t k = 0, k = 1 φ 2 Eε 2 t ε t (k 1), k > 1 φ 2 Eε 2 tε t 1, k = 1 The moments of ε t and ε + t can be found in Lemma 5. 0, k > 1. 14

15 If the conditional second moment is parameterized such that Eε t ε t k = 0 for all k 1 (for instance c t in (5) is symmetric with respect to z t ), then the third-order moments and crossmoments are zero. Lemma 3 (First and second-moment structure of GARCH M) Consider a GARCH M process (1), (2), (4), and (5) that is strictly stationary with finite second moment. The unconditional first and second-order moments and cross-moments of y t are given by (7) (9) in Lemma 1 where Eµ 2 t = φ 2 (Eh 2δ t (Eh δ t) 2 ) Eµ t µ t k = φ 2 (Eh δ th δ t k (Ehδ t) 2 ) Eµ t ε t k = φeh δ tε t k. The moments of ε t and h t can be found in Lemma 5. Proof. The results are readily obtained by straightforward algebra. Lemma 4 (Third-moment structure of GARCH M) Consider a GARCH M process (1), (2), (4), and (5) that is strictly stationary with finite third moment. The unconditional thirdorder moments and cross-moments of y t are given by (10) (12) in Lemma 2 where Eµ 3 t = φ 3 (Eh 3δ t 3Eh 2δ t Eh δ t + 2(Eh δ t) 3 ) Eµ 2 t µ t k = φ 3 (Eh 2δ t hδ t k 2Ehδ t hδ t k Ehδ t Eh2δ t Ehδ t + 2(Ehδ t )3 ) Eµ t µ 2 t k = φ 3 (Eh δ th 2δ t k 2Ehδ th δ t k Ehδ t Eh 2δ t Eh δ t + 2(Eh δ t) 3 ) Eµ 2 tε t k = φ 2 (Eh 2δ t ε t k 2Eh δ tε t k Eh δ t) Eµ t µ t k ε t k = φ 2 (Eh δ t hδ t k ε t k Eh δ t ε t keh δ t ) Eµ t ε 2 t = φ(eh δ+2 t Eh δ teh 2 t) Eµ t ε 2 t k = φ(eh δ tε 2 t k Ehδ teh 2 t) Eµ t k ε 2 t = φ(eh 2 t h δ t k Ehδ teh 2 t). The moments of ε t and ε + t can be found in Lemma 5. If the c t in (5) is even with respect to z t, then Eµ 2 tε t k = Eµ t µ t k ε t k = 0. Proof. The results are readily obtained by straightforward algebra. For the results for the symmetric GARCH we make use of Corollary 5. 15

16 Lemma 5 Consider the GARCH model (2) and (5). Suppose that ε t is strictly stationary. Provided that the moments exist, the moments of ε t, ε + t, and h t are given by Eε m d 0m Eh m t t =, m even 0, m odd where and Eε +m t Eε m t ε(+)n t k = Eε +m t ε (+)n t k = d + 0m Ehm t d 0m Eh m t ε (+)n t k, m even 0, m odd = d + 0m Ehm t ε(+)n t k Eε t k ε + t ε(+) t k 1 = d + 01 Eh tε t k ε (+) t k 1 Eh dm t = Eh dn t ε (+)m t k = Eh d t ε t k ε (+) t k 1 Eh dn t h dm t k = Eh d t hd t k ε t k with k 1 and where 1 m ( ) m ω j d 1 d m0 j (m j)0 Eh d(m j) t j=1 n ( n j=0 j) ω j d (+) (n j)m Ehd(n j)+m t, k = 1 n ( n j=0 j) ω j d (n j)0 Eh d(n j) t ε (+)m t (k 1), k > 1 = d k 1 10 d 11Eh d+1 t ε (+) t 1 n ( n j=0 j) ω j d (n j)0 Eh d(n j+m) n j=0 = d k 1 10 d 11Eh 2d+1 t ( n j) ω j d (n j)0 Eh d(n j) t d ij = Ec i t zj t, i 0, j 0 d + ij = Ec i t z+j t, i 0, j > 0 t, k = 1 h dm t (k 1), k > 1 In the expressions above the notation (+) means that + is either included or excluded from the equation in question. When d = 1 the recursions above yield analytically explicit expressions whereas for d = 2 some of them involve moments that have to be calculated numerically through simulations. If the conditional standard deviation follows the TGARCH process (5) and (6) then d ij = d + ij = 0 h 1,h 2,h 3 i h 1 +h 2 +h 3 =i h 3 +j even 0 h 1,h 2,h 3 i h 1 +h 2 +h 3 =i i! h 1!h 2!h 3! αh 1 β h 2 α h 3 E z t i+j h 2, i 1, j 0 i! h 1!h 2!h 3! αh 1 β h 2 α h 3 Ez + i+j h 2 t, i 1, j 1 For the AVGARCH process the expressions for d ij and d + ij are obtained by setting α = 0, restricting the index h 3 = 0, and defining 0 0 = 1. Furthermore, if z t nid(0,1), then the first four moments of the censored variable z + t and z t are given in Lemma 6. 16

17 Proof. The results are readily obtained by straightforward but tedious algebra. Corollary 5 Consider the GARCH model (2) and (5) in Lemma 5. If the process for the conditional second moment is symmetric in its response to shocks, then d nm = 0 whenever m is odd. Hence, of the moments in Lemma 5, and where m is odd. Eh dn t ε m t k = Ehd t hd t k ε t k = Eh d t ε t kε (+) t k 1 = 0 Eε n t ε m t k = Eε+dn t ε m t k = Eε t kε + t ε(+) t k 1 = 0 Proof. If c t (z t ) in (5) is an even function of z t then the function c n t (z t)zt m z t for any odd m and therefore d nm = 0. is an odd function of Lemma 6 Assume z t nid(0,1). Then the first four moments of z t, z t, and z + t are given by m Ezt m E z t m Ez t +m π 1 2π π 2 2π Proof. Straightforward by noticing that the censored variable z + t can be expressed as z + t = max(0,z t ) = 1 2 ( z t + z t ) and hence Ez +m t = 1 2 E z t m. 17

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