Parsimonious heterogeneous ARCH models for high frequency modeling

Teran, Juan Carlos Ruilova; Morettin, Pedro Alberto Article Parsimonious heterogeneous ARCH models for high frequency modeling Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Teran, Juan Carlos Ruilova; Morettin, Pedro Alberto (2020) : Parsimonious heterogeneous ARCH models for high frequency modeling, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 13, Iss. 2, pp. 1-19, https://doi.org/10.3390/jrfm13020038 This Version is available at: https://hdl.handle.net/10419/239105 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Risk and Financial Management Article Parsimonious Heterogeneous ARCH Models for High Frequency Modeling Juan Carlos Ruilova 1and Pedro Alberto Morettin 2,* 1Itaú Bank, São Paulo 04344-902, Brazil; [email protected] 2Department of Statistics, University of São Paulo, São Paulo 05508-090, Brazil *Correspondence: [email protected] Received: 6 January 2020; Accepted: 8 February 2020; Published: 20 February 2020   Abstract: In this work we study a variant of the GARCH model when we consider the arrival of heterogeneous information in high-frequency data. This model is known as HARCH(n). We modify the HARCH(n) model when taking into consideration some market components that we consider important to the modeling process. This model, called parsimonious HARCH(m,p), takes into account the heterogeneous information present in the financial market and the long memory of volatility. Some theoretical properties of this model are studied. We used maximum likelihood and Griddy-Gibbs sampling to estimate the parameters of the proposed model and apply it to model the Euro-Dollar exchange rate series. Keywords: GARCH model; HARCH model; PHARCH model; Griddy-Gibs; Euro-Dollar 1. Introduction High frequency data are those measured in small time intervals. This kind of data is important to study the micro structure of financial markets and also because their use is becoming feasible due to the increase of computational power and data storage. Perhaps the most popular model used to estimate the volatility in a financial time series is the GARCH(1,1) model; see Engle (1982), Bollerslev (1986): rt=σtεt,εt∼iid (0,1), σ2 t=α0+α1r2 t−1+β1σ2 t−1,(1) with α0>0, α1≥0, β1≥0, α1+β1<1. When we use high frequency data in conjunction with GARCH models, these need to be modified to incorporate the financial market micro structure. For example, we need to incorporate heterogeneous characteristics that appear when there are many traders working in a financial market trading with different time horizons. The HARCH(n) model was introduced by Müller et al. (1997) to try to solve this problem. In fact, this model incorporates heterogeneous characteristics of high frequency financial time series and it is given by rt=σtεt, σ2 t=c0+∑n j=1cj∑j i=1rt−i2,(2) where c0> 0, cn> 0, cj≥ 0 ∀j= 1, . . . , n− 1 and εt are identically and independent distributed (i.i.d.) random variables with zero expectation and unit variance. However, this model has a high computational cost to fit when compared with GARCH models, due to the long memory of volatility, so the number of parameters to be estimated is usually large. J. Risk Financial Manag. 2020,13, 38; doi:10.3390/jrfm13020038 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2020,13, 38 2 of 19 We propose a new model known as the parsimonious heterogeneous autoregressive conditional heteroscedastic model, in short-form PHARCH, as an extension of the HARCH model. Specifically, we call a PHARCH(m,p), with aggregations of different sizes a1 , . . . , am , where mis the number of the market components, the model given by rt=σtεt, σ2 t=C0+C1(rt−1+. . . +rt−a1)2+. . . + +Cm(rt−1+. . . +rt−am)2+b1σ2 t−1+. . . +bpσ2 t−p, (3) where εt∼i.i.d. (0,1),C0>0, Cj≥0, ∀j=1, . . . , m−1, Cm>0, bj≥0, j=1, . . . , p. HARCH models are important because they take account the natural behavior of the traders in the market. However they have some problems, mainly because they need to include several aggregations, so the number of parameters to estimate is large, because of the large memory feature of financial time series. Parsimonious HARCH includes only the most important aggregations in its structure, which makes the model more realistic. We can see some simulations in Figure 1, where the characteristics of clustering and volatility are better represented in PHARCH processes than in ARCH or HARCH processes. Sigma 0 2000 4000 6000 8000 10000 0 50 150 250 Returns 0 2000 4000 6000 8000 10000 -10 0 5 10 Sigma 0 2000 4000 6000 8000 10000 0 20 40 60 Returns 0 2000 4000 6000 8000 10000 -5 0 5 10 Sigma 0 2000 4000 6000 8000 10000 0 5 10 20 Returns 0 2000 4000 6000 8000 10000 -10 -5 0 5 ARCH(1) HARCH(5) PHARCH(5) with aggregations 1, 4, 96, 480 and 1920 Figure 1. Simulations of ARCH, HARCH and PHARCH processes. The organization of the paper is as follows. In Section 2we provide some background information on Markov chains and give the necessary and sufficient conditions for the PHARCH model to be stationary. In Section 3we obtain forecasts for the proposed model, and in Section 4we introduce the data that will be used for illustrative purposes. The actual application is given in Section 5, and we close the paper with some conclusions in Section 6. 2. Background In this section we provide briefly some background on Markov chains and results on stationarity of PHARCH models. 2.1. Markov Chains Suppose that X={Xn , n∈Z+} , Z+:={ 0,1,2, . . .} are random variables defined over (Ω , F , B(Ω)) , and assume that X is a Markov chain with transition probability P(x , A) , x∈Ω , A⊂Ω . Then we have the following definitions: J. Risk Financial Manag. 2020,13, 38 3 of 19 1. A function f:Ω→IR is called the smallest semi-continuous function if liminfy→xf(y)≥ f(x) , x∈Ω . If P(· , A) is the smallest semi-continuous function for any open set A∈ B(Ω) , we say that (the chain) Xis a weak Feller chain. 2. A chain X is called ϕ -irreducible if there exists a measure ϕ on B(Ω) such that, for all x , whenever ϕ(A)>0, we have, U(x,A) = ∑∞ n=1Pn(x,A)>0. 3. The measure ψ is called maximal with respect to ϕ , and we write ψ>ϕ , if ψ(A) = 0 implies ϕ(A) = 0, for all A∈ B(Ω) . If X is ϕ -irreducible, then there exists a probability measure ψ , maximal, such that Xis ψ-irreducible. 4. Let d={d(n)} a distribution or a probability measure on Z+ , and consider the Markov chain Xd with transition kernel Kd(x,A):= ∞ ∑ n=0 Pn(x,A)d(n). If there exits a transition kernel Tsatisfying Kd(x,A)≥T(x,A),x∈Ω,A∈ B(Ω), then Tis called the continuous component of Kd. 5. If X is a Markov chain for which there exits a (sample) distribution d such that Kd has a continuous component T, with T(x,Ω)>0, ∀x, then Xis called a T-chain. 6. A measure πover B(Ω),σ-finite, with the property π(A) = ZΩπ(dx)P(x,A),A∈ B(Ω), is called an invariant measure. The following two lemmas will be useful. See Meyn and Tweedie (1996) for the proofs and further details. We denote by IA(·)the indicator function of A. Lemma 1. Suppose that X is a weak Feller chain. Let C∈ B(Ω) be a compact set and V a positive function. If V(Xn)−E[V(Xn+1)|Xn]≥0, Xn∈Cc, then there exists an invariant measure, finite on compact sets of Ω. Lemma 2. Suppose that X is a weak Feller chain. Let C∈ B(Ω) be a compact set, and V a positive function that is finite at some x0∈Ω. If V(Xn)−E[V(Xn+1)|Xn]≥1−bIC(Xn),Xn∈Ω, with b a constant, b <∞, then there exists an invariant probability measure πon B(Ω). Lemma 3. Suppose that X is a ψ-irreducible aperiodic chain. Then the following conditions are equivalent: 1. There exists a function f:Ω→[1, ∞) , a set C∈ B(Ω) , a constant b<∞ and a function V:Ω→ [0, ∞), such that V(Xn)−E[V(Xn+1)|Xn]≥f(Xn)−bIC(Xn),Xn∈Ω. J. Risk Financial Manag. 2020,13, 38 4 of 19 2. The chain is positive recurrent with invariant probability measure π, and π(f)<∞. 2.2. Stationarity of PHARCH(m,p) Models We first give a necessary condition for Model (3) to be stationary. We know that Ert−irt−j=0, ∀i6=j, and if rtis stationary, we must have Er2 t=Eσ2 t=C0+Er2 tm ∑ i=1 aiCi+Eσ2 tp ∑ i=1 bi, so Ehr2 ti=C0 1−∑m i=1aiCi+∑p i=1bi. Therefore, ∑m i=1aiCi+∑p i=1bi<1. (4) To prove a sufficient condition it will be necessary to represent the PHARCH(m,p) as a Markov process. We use the definitions given in the previous section, so the process Xt=rt−1, . . . , rt−am+1,σt, . . . , σt−p+1,(5) whose elements follow Equation (3), is also a T-chain. The proofs of the following results are based on Dacorogna et al. (1996), and they are given in the Appendix A. Proposition 1. The Markov Chain Xtthat represents a PHARCH(m,p) process is a T-chain. Proposition 2. The Markov Chain that represents a PHARCH(m,p) process is recurrent with an invariant probability measure (stationary distribution), and its second moments are finite if the condition given in (4) is satisfied. Note that if εt∼t(v) (a Student’s tdistribution with ν degrees of freedom) in (3), then the necessary and sufficient condition becomes m ∑ i=1 v v−2aiCi+ p ∑ i=1 bi<1, for v>2. 3. Forecasting In this section we make some considerations about forecasting and validation of the proposed model. Usually two data bases are used for testing tha forecasting ability of a model: one (in-sample), used for estimation, and the other (out-of-sample) used for comparing forecasts with true values. There is an extra complication in the case of volatility models: there is no unique definition of volatility. Andersen and Bollerslev (1998) show that if wrong estimates of volatility are used, evaluation of forecasting accuracy is compromised. We could use the realized volatility as a basis for comparison, or use some trading system. We could, for example, have a model for hourly returns and use the realized volatility computed from 15 min returns for comparisons. In general, we can compute vh,t=∑ah i=1r2 t−i , where ah is the aggregation factor (4, in the case of 15 min returns). Then use some measure based on sh=˜ vh,t−vh,t , for example, mean squared error, where ˜ vh,t is the volatility predicted by the proposed model. See Taylor and Xu (1997), for example. J. Risk Financial Manag. 2020,13, 38 5 of 19 Now consider Model (3). The forecast of volatility at origin tand horizon `is given by ˆ σ2 t(l)= E(σ2 t+l|Xt) =E(C0+C1rt+l−1+. . . +rt+l−a12+. . . + +Cm(rt+l−1+. . . +rt+l−am)2+b1σ2 t+l−1+. . . +bpσ2 t+l−p|Xt), where Xt=(rt,σt,rt−1,σt−1,...), for l=1, 2, . . . Since a0=1<a1<a2<. . . <am<∞, then we have three cases: (i) If l=1, ˆ σ2 t(l) = E(C0+C1rt+l−1+. . . +rt+l−a12+. . . + +Cs(rt+l−1+. . . +rt+l−as)2+. . . + +Cm(rt+l−1+. . . +rt+l−am)2+ +b1σ2 t+l−1+. . . +bpσ2 t+l−p/Xt) =C0+C1rt+. . . +rt+1−a12+. . . + +Cs(rt+. . . +rt+1−as)2+. . . + +Cm(rt+. . . +rt+1−am)2+b1σ2 t+. . . +bpσ2 t+1−p. (ii) If lis such that as−1<l<as,s=1, 2, . . . , m, then we have, ˆ σ2 t(l) = E(C0+C1rt+l−1+. . . +rt+l−a12+. . . + +Cs(rt+l−1+. . . +rt+l−as)2+. . . + +Cm(rt+l−1+. . . +rt+l−am)2+ +b1σ2 t+l−1+. . . +bpσ2 t+l−p/Xt) =E(C0+∑s−1 i=1Ci∑ai j=1rt+l−j2+ +∑m i=sCi∑l−1 j=1rt+l−j2+∑m i=sCi∑ai j=lrt+l−j2+ +∑m i=sCi∑l−1 j=1rt+l−j∑ai j=lrt+l−j+ +b1σ2 t+l−1+. . . +bpσ2 t+l−p/Xt) =E(C0+∑s−1 i=1Ci∑ai j=1σt+l−jεt+l−j2+ +∑m i=sCi∑l−1 j=1σt+l−jεt+l−j2+ +∑m i=sCi∑ai j=lrt+l−j2+ +∑m i=sCi∑l−1 j=1σt+l−jεt+l−j∑ai j=lrt+l−j+ +b1σ2 t+l−1+. . . +bpσ2 t+l−p/Xt), and given the independence of εtand E(εt) = 0, we have Ert−irt−j=0, ∀i6=j; hence, ˆ σ2 t(l) = E(C0+∑s−1 i=1Ci∑ai j=1σ2 t+l−j+∑m i=sCi∑l−1 j=1σ2 t+l−j+ +∑m i=sCi∑ai j=lrt+l−j2+b1σ2 t+l−1+. . . +bpσ2 t+l−p/Xt) =C0+∑s−1 i=1Ci∑ai j=1ˆ σ2 t+l−j+∑m i=sCi∑l−1 j=1ˆ σ2 t+l−j+ +∑m i=sCi∑ai j=lrt+l−j2+b1˜ σ2 t+l−1+. . . +bp˜ σ2 t+l−p, where, for i=1, . . . , p, we have that ˜ σ2 t+l−i=(σ2 t+l−i,i≥l ˆ σ2 t+l−i,i<l (iii) If lis such that l>am,s=1, 2, . . . , m, then it follows J. Risk Financial Manag. 2020,13, 38 6 of 19 ˆ σ2 t(l)= E C0+ m ∑ i=1 Ci ai ∑ j=1 σ2 t+l−j+b1σ2 t+l−1+. . . +bpσ2 t+l−p/Xt! =C0+ m ∑ i=1 Ci ai ∑ j=1 ˆ σ2 t+l−j+b1˜ σ2 t+l−1+. . . +bp˜ σ2 t+l−p, where for i=1, . . . , p, we have ˜ σ2 t+l−i=(σ2 t+l−i,i≥l ˆ σ2 t+l−i,i<l 4. High Frequency Data In this section we further elaborate on high frequency data and introduce the series that will be analyzed later. High frequency data are very important in the financial environment, mainly because there exist large movements in short intervals of time. This aspect represents an interesting opportunity for trading. Furthermore, it is well known that volatilities in different frequencies have significant cross-correlation. We can even say that coarse volatility predicts fine volatility better than the inverse, as shown in Dacorogna et al. (2001). As an example, take the tick by tick foreign exchange (FX) time series Euro-Dollar, from January First 1999 to December 31, 2002. Returns are calculated using bid and ask prices, as rt=ln pbid t+pask t/2−ln pbid t−1+pask t−1/2. (6) We discard Saturdays and Sundays, and we replace holidays with the means of the last ten observations of the returns for each respective hour and day. After cleaning the data (see Dacorogna et al. (2001), for details) we will consider equally spaced returns, with sampling interval ∆t= 15 min. This seems to be adequate, as many studies indicate. Figure 2shows Euro-Dollar returns calculated as above. The length of this time series is 95,317. The figure shows that the absolute returns present a seasonal pattern. This is due to the fact that physical time does not follow, necessarily, the same pattern as the business time. This is a typical behavior of a financial time series and we will use a seasonal adjustment procedure similar to that of Martens et al. (2002). However, we will use absolute returns instead of squared returns; that is, we will compute the seasonal pattern as Sd,s,h=1 s s ∑ j=1 |(rd,j,h|, (7) where rd,s,h is the return in the weekday d, week sand hour h, and sis the number of weeks from the beginning of the series. Therefore, Sd,Ns,h is the rolling window mean of the absolute returns with the beginning fixed. In Figure 3we have the autocorrelation function of these returns and of squared returns. The seasonality pattern is no longer present. FX data has some distinct characteristics, mainly because they are produced twenty four hours a day, seven days a week. In particular, Euro-Dollar is the most liquid FX in the world. However, there are periods where the activity is greater or smaller, causing seasonal patterns to occur, as seen above. Let us analyze some facts about these returns that we will denote simply by rt . We can see in Figure 4the histogram fitted with a non-parametric density kernel estimate, using unbiased cross-validation method to estimate the bandwidth. It shows fat tails and high kurtosis, namely, 121, while its skewness coefficient is − 0.079, showing almost symmetry. A normality test (Jarque-Bera) rejects the hypothesis that these returns are normal. J. Risk Financial Manag. 2020,13, 38 7 of 19 Euro Dollar Returns Returns Date Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 1999 2000 2001 2002 2003 -0.010 0.010 0.030 Lag ACF 0 50 100 150 200 -0.06 -0.04 -0.02 0.0 Returns ACF Lag ACF 0 50 100 150 200 -0.05 0.05 0.15 0.25 Absolut Returns ACF Lag ACF 0 50 100 150 200 0.0 0.04 0.08 Squared Returns ACF Figure 2. Euro-Dollar returns: acf of returns, acf of absolute returns and acf of squared returns. The seasonally adjusted returns are then given by ˜ rt=˜ rd,s,h=rd,s,h Sd,s,h . (8) We may assume for example that the errors of a GARCH model fitted to these returns follow a Student’s tdistribution or a generalized error distribution, which represents better the fat tails of the distribution. Often the optimization of the likelihood function can be a very difficult task, due mainly to the flat behavior of likelihood function, as can be seen in Zumbach (2000). Bayesian methods are an alternative, and in the next section we will use the Griddy-Gibbs sampling to estimate the parameters of a PHARCH model. Figure 4also shows that the Euro-Dollar series has some clusters of volatility. This is a typical behavior of financial time series. A problem is that we do not know how many clusters there are and what their sizes are. The reason for this is that the information arriving is different for each sampling frequency. We can look these clusters as market components and they depend on the heterogeneity of the market. These market components are considered in our PHARCH model, as seen in Equation (3). Differently from GARCH-type models, PHARCH models have a variance equation with returns over intervals of different sizes. Therefore PHARCH models take into account the sign of the returns and not only their absolute value as GARCH models do. Two subsequent returns with similar sizes in the same direction will cause a higher impact on the variance than two subsequent returns with similar sizes but opposite signs. Now we need to determine the number and the size of the market components for the Euro-Dollar FX series. Ruilova (2007) proposed some technical rules to determine these market components, and Dacorogna et al. (2001) proposed some empirical rules. To help us to determine if the component sizes chosen are correct we can use the impact of the component. J. Risk Financial Manag. 2020,13, 38 8 of 19 We define the impact Iiof the ith component as, Ii=aiCi,∀i. (9) Note that the stationary condition to PHARCH(m) models can be written in terms of these impacts; namely, m ∑ i=1 Ii<1. We also notice that if we consider the Student’s tdistribution with ν degrees of freedom, the impact should be defined as Ii=v v−2aiCi,∀i≥1. (10) As remarked above, the number of components in a financial series can vary depending how the returns are being traded in this market. That is, liquid series can have a structure with more components than a non-liquid series. 5. Application Due to the complexity of the proposed model, the likelihod function may be flat in the neighbourhood of the maxima, so the optimization procedure using traditional procedures may fail. An alternative is to use Bayesian methods. Some references on the use of Bayesian procedures for the family of ARCH processes are Geweke (1989), Kleibergen and Dijk (1993), Geweke (1994) and Bauwens and Lubrano (1998). Lag ACF 0 50 100 150 200 -0.08 -0.06 -0.04 -0.02 0.0 ACF of Returns Clearing Seasonality Lag ACF 0 50 100 150 200 0.0 0.02 0.04 0.06 0.08 ACF of Squared Returns Clearing Seasonality Figure 3. Autocorrelation functions of the Euro-Dollar returns and squared returns after seasonal adjustment. J. Risk Financial Manag. 2020,13, 38 15 of 19 Lag ACF 0 50 100 150 200 -0.010 0.005 ACF of Harch Residuals Lag Partial ACF 0 50 100 150 200 -0.010 0.005 PACF of Harch Residuals Lag ACF 0 50 100 150 200 -0.01 0.03 ACF of Absolute Harch Residuals Lag Partial ACF 0 50 100 150 200 -0.01 0.03 PACF of Absolute Harch Residuals Lag ACF 0 50 100 150 200 -0.005 0.010 ACF of Squared Garch Residuals Lag Partial ACF 0 50 100 150 200 -0.005 0.010 PACF of Squared Garch Residuals Figure 11. Autocorrelation and partial autocorrelation functions of the residuals, absolute residuals and squared residuals after PHARCH fitting. QQ-Plot of Euro Dollar Residuals Quantiles of t distribution Residuals -20 0 20 40 -60 -40 -20 0 20 40 Figure 12. QQ-plot of PHARCH residuals. 6. Conclusions PHARCH models are good models for the analysis of high frequency data, since the financial market agents behave differently, incorporating heterogeneous information to the J. Risk Financial Manag. 2020,13, 38 16 of 19 market microstructure. Nevertheless, their use still depends on solving some issues, mainly computational ones. One big challenge in the analysis of high frequency data is dealing with large amounts of observations, and complex models bring computational difficulties, even with the recent technological breakthroughs in computing technology. Therefore, the first issue here is to develop techniques that help us to improve the computational algorithms. Maximum likelihood estimation may collapse, as we have described earlier. Techniques such as genetic algorithms and neural networks are viable optimization alternatives. Another possibility is to use Bayesian techniques, such as the Griddy-Gibbs samples that we have used. The disadvantage of the Griddy-Gibbs sampler lies in its high computational load. From another viewpoint, more sophisticated volatility models might be developed, taking into account the arrival of information, for example, a stochastic volatility model or stochastic duration model; or we could adapt existing models such as CHARN (conditional heteroskedasticity nonlinear autoregressive) models to heterogeneity of information characteristics. Finally, extensions similar to those proposed to GARCH models could be studied for HARCH models. A feature of the HARCH models is that the market components are chosen in a subjective way. In the analysis of the Euro-Dollar series, we considered five components, with different aggregations. A different number of components could be proposed, depending of the degree of information one has. This is clearly a matter for further studies. One last remark is that the performance of the different estimation methods should be evaluated. This evaluation could be done using prediction capabilities, for example. Other possibility is calculating some measure of risk. Volatility models are often established with the purpose of computing the VaR (value at risk) or other risk measure or for establishing trading strategies. In this context, an evaluation of the performance of the proposed model and several estimation procedures should be interesting. A comparison of returns of different trading systems that use a proposed model will be of fundamental importance. Further details on these aspects can be found in Acar and Satchell (2002), Dunis et al. (2003), Ghysels and Jasiak (1994) and Park and Irwin (2005). Other models for high frequency data use the realized volatility as a basis, instead of models such as ours and models of the ARCH family, which assume that volatility is a letent variable. Among the former, we mention the autoregressive fractionally integrated moving average (ARFIMA) models, the heterogeneous autoregressive model of realized volatilidade (HAR–RV) of Corsi (2009) and the mixed data sampling regression (MIDAS) proposed by Ghysels Santa-Clara and Valkanov (2002). A comparison of the PHARCH models with HAR and MIDAS would be useful, but due to the length of the present paper, this will be the object of future research. Author Contributions: The authors study some theoretical properties of the PHARCH models and illustrate the theory with an application to real data. All authors have read and agreed to the published version of the manuscript. Funding: This work was partially funded by Fapesp grant 2013/00506-1. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Proof of Proposition 1. Let Xt following (3) and (5). Assume that the innovations distribution has a non-zero absolutely continuous component, with a positive density on a Borel set with a non-empty interior. Examples of this are the normal and Student’s tdistribution. Then Xt can be written as Xt=H(Xt−1 , εt) , where H is a non-linear continuous function for each εt fixed. Then, using the Continuous Mapping Theorem we obtain the weak convergence of Xt , namely, the conditional distribution Xt given Xt−1=yk converges to the conditional distribution of Xt given Xt−1=y if yk→y . So, the Markov Chain Xt that represents the PHARCH(m,p) process is also aT-chain. J. Risk Financial Manag. 2020,13, 38 17 of 19 Proof of Proposition 2. Define the function Vas: V(Xt):= am−1 ∑ i=1 αir2 t−i+2am−2 ∑ i=1 am−1 ∑ j=i+1 βjrt−irt−j+ p−1 ∑ i=0 γiσ2 t−i. A simple algebraic computation gives E"am−1 ∑ i=1 αir2 t+1−i|Xt#=α1σ2 t+ am−2 ∑ i=1 αi+1r2 t−i, E"2am−2 ∑ i=1 am−1 ∑ j=i+1 βjrt+1−irt+1−j|Xt#=2am−3 ∑ i=1 am−2 ∑ j=i+1 βj+1rt−irt−j, and using (4) we have, E"p−1 ∑ i=0 γiσ2 t+1−i|Xt#=γ0  C0+ m ∑ i=1 Ci  σ2 t+  ai−1 ∑ j=1 rt−j  2  + p ∑ i=1 biσ2 t+1−i  + + p−1 ∑ i=1 γiσ2 t+1−i. Therefore, taking αam=βam=γp=0 and a0=1 and grouping we have V(Xt)−E[V(Xt+1)|Xt]= m ∑ k=1 ak−1 ∑ j=ak−1 αj−αj+1−γ0 m ∑ i=k Ci!r2 t−j+ + m ∑ l=1 al−2 ∑ j=1 al−1 ∑ k=al−1+j βk−βk+1−γ0 m ∑ i=l Ci!rt−jrt−k+ + p−1 ∑ i=1 (γi−γi+1−γ0bi+1)σ2 t−i+ + γ0−γ1−γ0b1−α1−γ0 m ∑ i=1 Ci!σ2 t−γ0C0. We choose k∈Z+,al−1<k<al,l∈{1, . . . , m}, and βk=βk+1+∑m i=lCi. If we take αj>βj, for all j, then we have V(Xt)≥0, and we can take γ0=1, so V(Xt)−E[V(Xt+1)|Xt]= m ∑ k=1 ak−1 ∑ j=ak−1 αj−αj+1− m ∑ i=k Ci!r2 t−j + p−1 ∑ i=1 (γi−γi+1−bi+1)σ2 t−i(A1) + 1−γ1−b1−α1− m ∑ i=1 Ci!σ2 t−C0. Similarly, we can choose k∈ Z + , al−1<k<al , l∈{1, . . . , m} , αk>αk+1+∑m i=lCi and γi>γi+1+bi+1. If ∑m i=1aiCi+∑p i=1bi<1, then we have for the expressions in Equation (A1): ξi=αi−αi+1−∑m j=kCj>0, i=1, . . . , am−1 and kis chosen such that i∈(ak−1,ak); ξam−1+i=(γi−γi+1−bi+1)>0, i=1, . . . , p−1; and J. Risk Financial Manag. 2020,13, 38 18 of 19 ξam+p−1= 1−γ1−b1−α1− m ∑ i=1 Ci!>0. So, V(Xt)−E[V(Xt+1)|Xt]can be as large as we want if Xt∈Cc. Then, using Lemma 1we have that there exists an invariant measure, finite on compact sets of Ω . Choosing vmin =min(ξi), for i=1, . . . , am+p−1, we have, V(Xt)−E[V(Xt+1)|Xt]≥vmin am−1 ∑ i=1 r2 t−j+vmin p ∑ i=1 σ2 t−i−C0 ≥1−(C0+1)IB0,rC0+1 vmin , where B(c,r)is the ball with center cand radius r. Therefore, the Markov chain Xt=rt−1, . . . , rt−am+1,σt, . . . , σt−p+1 that represents the PHARCH(m,p) process is recurrent, with an invariant probability measure (stationary distribution). Now, if we consider fx1, . . . , xam+p−1=x2 1+. . . +x2 am+p−1, then, V(Xt)−E[V(Xt+1)|Xt]≥vmin 2frt−1, . . . , rt−am+1,σt, . . . , σt−p+1+ +C01B0,r2C0 vmin . We conclude, using Lemmas 2and 3, that the process Xt is a Tchain having a stationary distribution with finite second order moments. References Acar, Emmanuel, and Stephen Satchell. 2002. Adavanced Trading Rules. Oxford: Butterworth Heinemann. Andersen, Torben G., and Tim Bollerslev. 1998. Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39: 885–905. [CrossRef] Bauwens, Luc, and Michel Lubrano. 1998. Bayesian inference on GARCH models using the gibbs sampler. Econometrics Journal 1: 23–46. [CrossRef] Bélisle, Claude J. 1992. Convergence theorems for a class of simulated annealing algorithms on Rd .Journal of Applied Probability 29: 885–95. [CrossRef] Bollerslev, Tim. 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31: 307–27. [CrossRef] Corsi, Fulvio. 2009. A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics 7: 174–96 [CrossRef] Dacorogna, Michel M., Ramazan Gençay, Ulrich Muller, Richard B. Olsen, and Olivier V. Pictet. 2001. An Introduction to High-Frequency Finance. Amsterdam: Elsevier. Dacorogna, Michel M., Ulrich A. Müller, Paul Embrechts, and Gennady Samorodnitsky. 1996. How heavy are the tails of a stationary HARCH(k) process? A study of the moments. In Stochastic Processes and Related Topics. Boston: Birkäuser. Davis, Philip J., and Philip Rabinowitz. 1975. Methods Of Numerical Integration. New York: Academic Press. Dunis, Christian L., Jason Laws, and Patrick Naïm. 2003. Applied Quantitative Methods For Trading And Investment. Chichester: John Wiley & Sons Ltd. Engle, Robert F. 1982. Autoregressive conditional heteroskedasticity with estimates of the variance of u. k. inflation. Econometrica 50: 987–1008. [CrossRef] Geweke, John. 1989. Exact predictive densities in linear models with ARCH disturbances. Journal of Econometrics 40: 63–86. [CrossRef] J. Risk Financial Manag. 2020,13, 38 19 of 19 Geweke, John. 1994. Bayesian Comparison of Econometric Models. Technical Report Working Paper 532, Research Department, Federal Reserve Bank of Minneapolis, Minneapolis, MN, USA. Ghysels, Eric, and Joanna Jasiak. 1994. Stochastic Volatility and Time Deformation: An Apllication to Trading Volume and Leverage Effects. Technical report. Santa Fé: Western Finance Association Meeting. Ghysels, Eric, Pedro Santa-Clara, and Rossen Valkanov. 2002. The MIDAS Touch: Mixed Data Sampling Regression Models. Working paper, UNC and UCLA, Chapel Hill, NC, USA. Kleibergen, Frank, and Herman K. Van Dijk. 1993. Non-stationarity in GARCH models: A bayesian analysis. Journal of Applied Econometrics 8 (Suppl. 1): 41–61. Martens, Martin, Yuan-Chen Chang, and Stephen J. Taylor. 2002. A comparison of seasonal adjustment methods when forecasting intraday volatility. Journal of Financial Research 25: 283–99. [CrossRef] Meyn, Sean P., and Richard L. Tweedie. 1996. Markov Chains and Stochastic Stability. Heidelberg: Springer. Müller, Ulricn, Michel M. Dacorogna, Rakhal D. Davé, Richard B. Olsen, Olivier V. Pictet, and Jacob E. von Weizsäcker. 1997. Volatilities of different time resolutions—Analyzing the dynamics of market components. Journal of Empirical Finance 4: 213–89. [CrossRef] Park, Cheol-Ho, and Scott H. Irwin. 2005. The Profitability of Technical Trading Rules in US Futures Markets: A Data Snooping Free Test. Technical report, AgMAS Project Research Report. Urbana-Champaign: University of Illinois. Ritter, Christian, and Martin A. Tanner. 1992. Facilitating the gibbs sampler: The gibbs stopper and the griddy-gibbs sampler. Journal of the American Statistical Association 87: 861–68. [CrossRef] Ruilova, Juan Carlos. 2007. Modelos Arch Heterogêneos E Aplicações À Análise De Dados De Alta Freqüência. Ph.D. thesis, Institute of Mathematic and Statistics, University of São Paulo, São Paulo, Brazil. Taylor, Stephen J., and Xinzhong Xu. 1997. The incremental volatility information in one million foreign exchange quotations. Journal of Empirical Finance 4: 317–40. [CrossRef] Zumbach, Gilles. 2000. The pitfalls in fitting GARCH(1,1) processes. In Advances in Quantitative Asset Management. Technical report. Boston: Springer. c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).