DEPARTAMENTO DE ECONOMÍA Pontificia Universidad Católica del Perú DEPARTAMENTO DE ECONOMÍA Pontificia Universidad Católica del Perú DEPARTAMENTO DE ECONOMÍA Pontificia Universidad Católica del Perú DEPARTAMENTO DE ECONOMÍA Pontificia Universidad Católica del Perú DEPARTAMENTO DE ECONOMÍA Pontificia Universidad Católica del Perú DEPARTAMENTO DE ECONOMÍA Pontificia Universidad Católica del Perú DEPARTAMENTO DE ECONOMÍA Pontificia Universidad Católica del Perú DEPARTAMENTO DE ECONOMÍA Pontificia Universidad Católica del Perú DEPARTAMENTO DE ECONOMÍA DT DECON DOCUMENTO DE TRABAJO APPROXIMATE BAYESIAN ESTIMATION OF STOCHASTIC VOLATILITY IN MEAN MODELS USING HIDDEN MARKOV MODELS: EMPIRICAL EVIDENCE FROM STOCK LATIN AMERICAN MARKETS Nº 502 Carlos A. Abanto-Valle, Gabriel Rodríguez, Luis M. Castro Cepero y Hernán B. Garrafa-Aragón DOCUMENTO DE TRABAJO N° 502 Approximate Bayesian Estimation of Stochastic Volatility in Mean Models using Hidden Markov Models: Empirical Evidence from Stock Latin American Markets Carlos A. Abanto-Valle, Gabriel Rodríguez, Luis M. Castro Cepero y Hernán B. Garrafa-Aragón Octubre, 2021 DOCUMENTO DE TRABAJO 502 http://doi.org/10.18800/2079-8474.0502 http://doi.org/10.18800/2079-8474.0502 Approximate Bayesian Estimation of Stochastic Volatility in Mean Models using Hidden Markov Models: Empirical Evidence from Stock Latin American Markets Documento de Trabajo 502 © Carlos A. Abanto-Valle, Gabriel Rodríguez, Luis M. Castro Cepero y Hernán B. Garrafa-Aragón Editado e Impreso: © Departamento de Economía – Pontificia Universidad Católica del Perú Av. Universitaria 1801, Lima 32 – Perú. Teléfono: (51-1) 626-2000 anexos 4950 - 4951 econo@pucp.edu.pe http://departamento.pucp.edu.pe/economia/publicaciones/documentos-de-trabajo/ Encargada de la Serie: Roxana Barrantes Cáceres Departamento de Economía – Pontificia Universidad Católica del Perú Barrantes.r@pucp.edu.pe Primera edición – Noviembre, 2021. ISSN 2079-8474 (En línea) mailto:econo@pucp.edu.pe mailto:Barrantes.r@pucp.edu.pe Approximate Bayesian Estimation of Stochastic Volatility in Mean Models using Hidden Markov Models: Empirical Evidence from Stock Latin American Markets Carlos A. Abanto-Valle†, Gabriel Rodŕıguez§, Luis M. Castro Cepero$,∗,+ and Hernán B. Garrafa-Aragón‡ †Department of Statistics, Federal University of Rio de Janeiro, Caixa Postal 68530, CEP: 21945-970, Rio de Janeiro, Brazil §Department of Economics, Pontificia Universidad Católica del Perú, 1801 Universitaria Avenue, Lima 32, Lima, Peru $Department of Statistics, Pontificia Universidad Católica de Chile, Santiago, Chile ∗Millennium Nucleus Center for the Discovery of Structures in Complex Data, Santiago, Chile +Centro de Riesgos y Seguros UC, Pontificia Universidad Católica de Chile, Santiago, Chile ‡Escuela de Ingenieŕıa Estad́ıstica de la Universidad Nacional de Ingenieria, Lima, Perú September 27, 2021 Abstract The stochastic volatility in mean (SVM) model proposed by Koopman and Uspensky (2002) is revisited. This paper has two goals. The first is to offer a methodology that requires less computational time in simulations and estimates compared with others proposed in the literature as in Abanto-Valle et al. (2021) and others. To achieve the first goal, we propose to approximate the likelihood function of the SVM model applying Hidden Markov Models (HMM) machinery to make possible Bayesian inference in real-time. We sample from the posterior distribution of parameters with a multivariate Normal distribution with mean and variance given by the posterior mode and the inverse of the Hessian matrix evaluated at this posterior mode using importance sampling (IS). The frequentist properties of estimators is anlyzed conducting a simulation study. The second goal is to provide empirical evidence estimating the SVM model using daily data for five Latin American stock markets. The results indicate that volatility negatively impacts returns, suggesting that the volatility feedback effect is stronger than the effect related to the expected volatility. This result is exact and opposite to the finding of Koopman and Uspensky (2002). We compare our methodology with the Hamiltonian Monte Carlo (HMC) and Riemannian HMC methods based on Abanto-Valle et al. (2021). Keywords: Stock Latin American Markets, Stochastic Volatility in Mean, Feed-Back Effect, Hamiltonian Monte Carlo, Hidden Markov Models, Riemannian Manifold Hamiltonian Monte Carlo, Non Linear State Space Models. JEL Classification: C11, C15, C22, C51, C52, C58, G12. Estimación Bayesiana Aproximada de Modelos de Volatilidad Estocástica en la Media usando Modelos Hidden Markov: Evidencia Emṕırica para Mercados Bursátiles de América Latina Carlos A. Abanto-Valle†, Gabriel Rodŕıguez§, Luis M. Castro Cepero$,∗,+ and Hernán B. Garrafa-Aragón‡ †Department of Statistics, Federal University of Rio de Janeiro, Caixa Postal 68530, CEP: 21945-970, Rio de Janeiro, Brazil §Department of Economics, Pontificia Universidad Católica del Perú, 1801 Universitaria Avenue, Lima 32, Lima, Peru $Department of Statistics, Pontificia Universidad Católica de Chile, Santiago, Chile ∗Millennium Nucleus Center for the Discovery of Structures in Complex Data, Santiago, Chile +Centro de Riesgos y Seguros UC, Pontificia Universidad Católica de Chile, Santiago, Chile ‡Escuela de Ingenieŕıa Estad́ıstica de la Universidad Nacional de Ingenieria, Lima, Perú Setiembre 27, 2021 Resumen El modelo de volatilidad estocástica en la media (SVM) propuesto por Koopman and Uspensky (2002) es re- visado. Este documento tiene dos objetivos. El primero es ofrecer una metodoloǵıa que requiere menos tiempo computacional en simulaciones y estimaciones comparadas con otras propuestas en la literatura como en Abanto- Valle et al. (2021) y otros. Para lograr el primer objetivo, proponemos aproximamos la función de verosimilitud del modelo SVM aplicando Hidden Markov Models (HMM) para hacer posible la inferencia bayesiana en tiempo real. Tomamos muestras de la distribución posterior de los parámetros usando una distribución Normal mul- tivariada con media y varianza dadas por la moda posterior y la inversa de la matriz Hessiana evaluada en esta moda posterior usando importance sampling (IS). Usando simulaciones, las propiedades frecuentistas de los estimadores son analizadas. El segundo objetivo es proporcionar evidencia emṕırica estimando el modelo SVM utilizando datos diarios para cinco mercados bursátiles de América Latina. Los resultados indican que la volatilidad impacta negativamente en los rendimientos sugiriendo que el efecto de retroalimentación de la volatilidad es más fuerte que el efecto relacionado con la volatilidad esperada. Este resultado es exacto y opuesto al hallazgo de Koopman and Uspensky (2002). Nosotros comparamos nuestra metodoloǵıa con los algoritmos de Hamiltoniano Montecarlo (HMC) y Riemannian HMC basados en Abanto-Valle et al. (2021). Palabras Claves: Mercado Bursátiles de América Latina, Volatilidad Estocástica en Media, Efecto Feed-Back, Hamiltonian Monte Carlo, Hidden Markov Models, Riemannian Manifold Hamiltonian Monte Carlo, Modelos Espacio Estado No Lineales. Clasificación JEL: C11, C15, C22, C51, C52, C58, G12. 1 Introduction Stochastic volatility (SV) models initially proposed by Taylor (1982, 2008), compose a well-known class of models to estimate volatility. These models have gained attention in the financial econometrics literature because of their flexibility to capture the nonlinear behavior observed in financial time series returns1. According to Melino and Turnbull (1990) and Carnero et al. (2004), an appealing aspect of the SV model is its close association to financial economic theories and its ability to capture the stylized facts often observed in daily series of financial returns in a more appropriate way. The daily asymmetrical relation between equity market returns and volatility has received a lot of attention in the financial literature; see for instance, Black (1976), Campbell and Entchel (1992), and Bekaert and Wu (2000). Asymmetric equity market volatility is important for al least three reasons. First, it is an important characteristic of the market volatility dynamics, has asset pricing implications and is a feature of priced risk factors. Second, it plays an important role in risk prediction, hedging and option pricing. Finally, asymmetric volatility implies negatively skewed returns distributions, i.e. it may help explain some of the market’s chances of losing. On the other hand, the relation between expected returns and expected volatility have been extensively examined in recent years. Overall, there appears to be stronger evidence of a negative relationship between unexpected returns and innovations to the volatility process, which French et al. (1987) interpreted as indirect evidence of a positive correlation between the expected risk premium and ex ante volatility. If expected volatility and expected returns are positively related and future cash flows are unaffected, the current stock index price should fall. Conversely, small shocks to the return process lead to an increase in contemporaneous stock index prices. This theory, known as the volatility feedback theory hinges on two assumptions: first, the existence of a positive relation between the expected components of the return and volatility process and second, volatility persistence. An alternative explanation for asymmetric volatility where causality runs in the opposite direction is the leverage effect put forward by Black (1976), who asserted that a negative (positive) return shock leads to an increase (decrease) in the firm’s financial leverage ratio, which has an upward (downward) effect on the volatility of its stock returns. However, French et al. (1987) and Schwert (1989) argued that leverage alone cannot account for the magnitude of the negative relationship. For example, Campbell and Entchel (1992) found evidence of both volatility feedback and leverage effects, whereas Bekaert and Wu (2000) presented results suggesting that the volatility feedback effect dominates the leverage effect empirically. 1The other important branch of the literature is GARCH models where time-varying variance is modeled as a de- terministic function of past squared perturbations and lagged conditional variances. Details and explanations of the extensive GARCH literature may be found in Bollerslev et al. (1992, 1994) and Diebold and Lopes (1995). On the other hand, SV models are reviewed in Taylor (1994), Ghysels et al. (1994), and Shephard (1996); among others. 2 Frequently, the volatility of daily stock returns has been estimated with SV models, but the results have relied on an extensive pre-modeling of these series to avoid the problem of simultaneous estimation of the mean and variance. Koopman and Uspensky (2002) introduced the SV in mean (SVM) model to deal with this problem. In the SVM setup, the unobserved volatility is incorporated as an explanatory variable in the mean equation of the returns under the normality assumption of the innovations. Moreover, they derived an exact maximum likelihood estimation based on Monte Carlo simulation methods. Abanto-Valle et al. (2012) extended the SVM model to the class of scale mixture of Normal distributions and developed an Markov Chain Monte Carlo (MCMC) algorithm to sample parameters and the log-volatilities from a Bayesian perspective. Recently, Abanto-Valle et al. (2021) apply Hamiltonian Monte Carlo (HMC) and Riemann Manifold HMC (RMHMC) methods within the MCMC algorithm to update the log-volatilities and parameters of the SVM model, respectively. However, the resulting MCMC algorithms has some undesirable features. In particular, the procedure is quite involved, requiring a large amount of computer-intensive simulations. In addition, the computational cost increases rapidly with the sample size. This paper has two objectives. The first is to offer an algorithm that requires less computational time in simulations and estimates even when the sample increases as compared with MCMC algorithms as proposed in Abanto-Valle et al. (2021). For this, this article applies an alternative approximate Bayesian estimation method to the SVM model considered by Abanto-Valle et al. (2012) and Abanto-Valle et al. (2021). First, we approximate the likelihood function by integrating out the log-volatilities as suggested by Langrock (2011), Langrock et al. (2012) and Abanto-Valle et al. (2017). Second, we get the maximum a posteriori by using a numerical optimization routine, and third, we use importance sampling to sample from the posterior distribution of the parameters using a multivariate normal distribution where the mean and variance are given by the maximum a posteriori and the inverse of the Hessian matrix evaluated at the maximum a posteriori, respectively. The second objective is to provide empirical evidence estimating the SVM model using daily data for five Latin American stock markets. Time-varying volatility for developed economies’ financial variables have been studied extensively; see for instance, Liesenfeld and Jung (2000), Jacquier et al. (2004) and Abanto-Valle et al. (2010). However, empirical studies of the volatility characteristics of the financial markets in Latin America are very scarce and are far from being thoroughly analyzed despite their growth in recent years, see Abanto-Valle et al. (2011), Rodŕıguez (2016), Rodŕıguez (2017), Rodŕıguez (2017), Lengua Lafosse and Rodŕıguez (2018), Alanya and Rodŕıguez (2019). Moreover, Abanto-Valle et al. (2021) use HMC amd RMHMC methods to analyse the SVM model using Latin American markets. For this reason, we perform a detailed empirical study of five Latin American indexes: MERVAL (Argentina), IBOVESPA (Brazil), IPSA (Chile), MEXBOL (Mexico) and IGBVL (Peru) in the context of the SVM model using the HMM approach. We also include the S&P 500 returns in order to perform some comparisons. All the results are in line with Abanto-Valle et al. (2021). The remainder of this paper is organized as follows. Section 2 describes the SVM model, the approximated 3 likelihood of the SVM model based on hiden Markov models (HMM) techniques and the Bayesian inference procedure. In section 3, we conduct a simultation study to verify the frequentist properties of estimators compared to the methods used in Abanto-Valle et al. (2021) including computational time intensity. Section 4 is devoted to the application of the proposed methodology to five indexes of Latin American countries and the S&P 500. Finally, some concluding remarks and suggestions for future developments are given in Section 5. 2 The Stochastic Volatility in Mean (SVM) Model The SVM model is defined by yt = β0 + β1yt−1 + β2e ht + e ht 2 ǫt, (1a) ht+1 = µ+ φ(ht − µ) + σηt, (1b) where yt and ht are, respectively, the compounded return and the log-volatility at time t, for t = 1, . . . , T . We assume that |β1| < 1, |φ| < 1, i.e., that the returns and the log-volatility process are stationary and that the initial value h1 ∼ N (µ, σ2 η 1−φ2 ). Furthermore, the innovations ǫt and ηt are assumed to be mutually independent and normally distributed with mean zero and unit variance. The SVM model incorporates the unobserved volatility as an explanatory variable in the mean equation. Therefore, the aim of the SVM model is to estimate the ex-ante relation between returns and volatility and the volatility feedback effect (parameter β2). 2.1 Likelihood evaluation by iterated numerical integration and fast evaluation of the approximate likelihood using HMM techniques To formulate the likelihood, we require the conditional pdfs of the random variables yt, given ht and yt−1 (t = 1, . . . , T ), and of the random variables ht, given ht−1 (t = 2, . . . , T ). We denote these by p(yt | yt−1, ht) and p(ht | ht−1), respectively. The likelihood of the SVM model defined by equations (1a) and (1b) can then be derived as L = ∫ . . . ∫ p(y1, . . . , yT , h1, . . . , hT | y0)dhT . . . dh1 = ∫ . . . ∫ p(y1, . . . , yT | y0, h1, . . . , hT )p(h1, . . . , hT )dhT . . . dh1 = ∫ . . . ∫ p(h1)p(y1 | y0, h1) T ∏ t=2 p(yt | yt−1, ht)p(ht | ht−1)dhT . . . dh1. Hence, the likelihood is a high-order multiple integral that cannot be evaluated analytically. Through numerical integration, using a simple rectangular rule based on m equidistant intervals, Bi = (bi−1, bi), i = 1, . . . ,m, with midpoints b∗i and length b, the likelihood can be approximated as follows: 4 L ≈ bT m ∑ i1=1 . . . m ∑ iT =1 p(h1 = b∗i1)p(y1 | y0, h1 = b∗i1) × T ∏ t=2 p(yt | yt−1, ht = b∗it)p(ht = b∗it | ht−1 = b∗it−1 ) = Lapprox . (2) This approximation can be made arbitrarily accurate by increasing m, provided that the interval (b0, bm) covers the essential range of the log-volatility process. We note that this simple midpoint quadrature is by no means the only way in which the integral can be approximated (cf. Langrock et al., 2012). The numerical evaluation of approximate likelihood given in (2) will usually be computationally intractable since it involves mT summands. However, it can be evaluated numerically using the tools well-established for HMMs, which are the models that have exactly the same dependence structure as the stochastic volatility in mean models, but with a finite and hence discrete state space (cf. Langrock, 2011; Langrock et al., 2012). In the given scenario, the discrete states correspond to the intervals Bi, i = 1, . . . ,m, in which the state space has been partitioned. A fundamental property of HMM, which we exploit here, is that the likelihood can be evaluated efficiently using the so-called forward algorithm, a recursive scheme which iteratively traverses forward along the time series, updating the likelihood and the state probabilities in each step (Zucchini et al., 2016). Under the HMM, to apply the forward algorithm results in a convenient closed-form matrix product expression for the likelihood. For the SVM model is given by: Lapprox = δP(y1)ΓP(y2)ΓP(y3) · · ·ΓP(yT−1)ΓP(yT )1 ′ . (3) Note that, in equation (3), the m×m-matrix Γ = ( γij ) plays the role of the transition probability matrix in case of an HMM, defined by γij = p(ht = b∗j | ht−1 = b∗i ) · b, which is an approximation of the probability of the log-volatility process changing from some value in the interval Bi to some value in the interval Bj ; this conditional probability is determined by (1b). The vector δ is the analogue to the Markov chain initial distribu- tion in case of an HMM. It is defined such that δi is the density of the N (µ, σ2 η 1−φ2 )-distribution –the stationary distribution of the log-volatility process– multiplied by b. Furthermore, P(yt) is an m×m diagonal matrix with the ith diagonal entry p(yt | yt−1, ht = b∗i ) determined by (1a). Finally, 1′ is a column vector of ones. Using the matrix product expression given in (3), the computational effort required to evaluate the approx- imate likelihood is linear in the number of observations, say T , and quadratic in the number of intervals used in the discretization, say m. In other words, the likelihood can be calculated in a fraction of seconds, even for high values of T and m. Furthermore, the approximation can be‘ Although we are using the HMM forward algorithm to evaluate the (approximate) likelihood, the specifi- cations of δ, Γ and P(xt) given above do not define precisely an HMM. In general, the row sums of Γ will be only approximately equal to one, and the vector components δ will only approximately sum to one. If desired, this can be remedied by scaling each row of Γ and the vector δ to total 1. 5 2.2 Bayesian Inference for the SVM model Because we have some constraints in the original parametric space of the SVM model (|β1| < 1, |φ| < 1, ση > 0), we consider the transformations for the following parameters : γ = log ( 1+β1 1−β1 ) , ψ = log ( 1+φ 1−φ ) , and ω = log(σ). Let θ = (β0, γ, β2, µ, ψ, ω) ′ and p(θ) be the prior distribution of θ. As the likelihood function is invariant to 1:1 transformations, from equation (3), we obtain the posterior distribution up to a normalization constant, namely: p(θ | y0,yT ) ∝ p(θ)Lapprox(θ), (4) where yT = (y1, . . . , yT ) ′. Suposse we wish to calculate an expectation E p(θ|y0,yT ) [h(θ)], which can be calculated by using the importance density q(θ) as follows: E p(θ|y0,yT )[h(θ)] = ∫ h(θ)p(θ | y0,yT )dθ ∫ p(θ | y0,yT )dθ = ∫ h(θ)p(θ|y0,yT ) q(θ) q(θ)dθ ∫ p(θ|y0,yT ) q(θ) q(θ)dθ = E q(θ) [ h(θ)ω(θ) ] E q(θ) [ ω(θ) ] , (5) where ω(θ) = p(θ|y0,yT ) q(θ) and Eq[.] denotes an expected value with respect to the importance density q(θ). Therefore a sample of independent draws θ1, . . . , θm from q(θ) can be used to estimate Ep(θ|y0,yT )[h(θ)] by h̄ = ∑m i=1 h(θi)ω(θi) ∑m i=1 ω(θi) . (6) It is shown that using one sample θ′ is in estimating the ratio in (5) is more efficient than using two samples (one for the numerator and another for denominator); see Chen et al. (2008). It follows from the strong law of large numbers that h̄ → Ep(θ|y0,yT )[h(θ)] as m → ∞ almost surely; see Geweke (1989). In the same way, a variance of h̄(θ) can be consistenly estimated by ∑m i=1 ω(θi) 2[h(θi)− h̄]2/[ ∑m i=1 ω(θi)] 2. 3 Simulation Study To assess the performance of the methodology described in the previous Section, we conducted some sim- ulation experiments. All the calculations were performed using stand-alone code developed by the authors using the Rcpp interface inside the R package. First, we simulated a data set comprising T = 6000 ob- servations from the SVM model, specifying β = (0.14, 0.03,−0.10)′, µ = 0.3, φ = 0.98, ση = 0.2 and y0 = 0.2, which correspond to typical values found in daily series of returns; see for example Leão et al. (2017) and Abanto-Valle et al. (2017). The resulting transformed true parameter vector θ = (β0, γ, β2, µ, ψ, ω) ′ = 6 (0.14, 0.06,−0.10, 0.30, 4.5951,−1.6094)′. Figure 1 shows the resulting artificial data set. We set the priors distributions as follows: (β0, γ, β2) ∼ N3(03, 100I3), µ ∼ N (0, 100), ψ ∼ N (4.5, 100) and ω ∼ N (−1.5, 100), where Nr(., .) and N (., .) denote the r−variate and univariate normal distributions and 0r and Ir are the r× 1 vector of zeros and the r × r the identity matrix, respectively. In order to investigate the influence of the choice of m on the accuracy of the likelihood approximation in the posterior distribution, and of the sample size T on the computing time, we fitted the SVM model using m = 50, 100, 150, 200 (i.e., different levels of accuracy), bm = −b0 = 4, to subsamples of length T = 1500, 3000, 6000 of the original simulated series. Table 1 reports the results of the maximum to posteriori (MAP). In general, we observe that all MAP estimates approach their true values as we go from T = 1500 to T = 6000. This convergence is faster and clearer for ψ and ω. In the case of µ, the MAP estimates are different for T = 1500 observations but they converge to their true value rapidly when sample size increases. In particular, the MAP for β2 approaches its true value when T=6000. The log likelihood value is fairly stable for any value of m and sample size. Most of MAP estimates obtained by numerical maximization become stable for values of m around 100, for all the sample sizes considered here. However, when T = 6000, we observe that stabilization is reached for m = 150 or more. Therefore, we recommend to use m = 150, 200 in empirical applications. Next, the multivariate normal distribution with mean and covariace matrix being the MAP and the inverse of the hessian matrix evaluated at the MAP is used as an importance density. We draw a sample of size 1000 using sampling importance resampling. Based on it, we calculate the expected value (posterior mean) and the standar deviation in the original escale using equation (6). The results are reported in Table 2. In all the cases the posterior credibility intervals of 95% contain the true value of the parameters. Considering a time serie of size 6000, the proposal methodology spend 174.18 seconds to simulate and report the results, making useful our proposal in real time applications. We also investigated the influence of the choice of b0 and bm (the results are not presented here for space reasons). Overall, we observe that the estimator performance is not affected much. However, when these values are chosen either too small (not covering the support of the log-volatility process, e.g., bm = −b0 = 2) or too large (leading to a partition of the support into unnecessarily wide intervals and poor approximation of the likelihood, e.g. m = 50 and bm = −b0 = 15), the estimator performance could be affected. In practice, it can easily be checked post-hoc if the chosen range, specified by b0 and bm, is adequate, by investigating the stationary distribution of the fitted log-volatility process. The second simulation experiment studies the properties of the estimators of the SVM model parameters. We generated 300 datasets from the SVM model, specifying β = (0.14, 0.03,−0.10)′, φ = 0.98, σ = 0.2, µ = 0.30. For each generated data set, we fitted the SVM model using m = 50, 100, 150, 200 and bm = −b0 = 4, for T = 1500, T = 3000 and T = 6000, respectively. Tables 3, 4 and 5 report the sample mean, the mean relative bias (MRB), the mean relative absolute deviation (MRAD) and the mean squared error (MSE) of the 7 parameter estimates for T = 1500, 3000, 6000, respectively. For all the sample sizes, i.e., T = 1500, T = 3000, and T = 6000, higher values of MRAD are found for the estimator of β1 and µ, while none of the other estimators exhibited a notable bias. The bias found for µ does not substantially affect the resulting model and its performance in forecasting, since it merely indicates a minor shift of the volatility process. It is important to stress that the MSEs are smaller for the larger sample size, as presumed. The obtained results for m = 50 are similar to those using higher values of m. Consequently, a more acceptable approximation of the likelihood is achieved. Overall, it can be concluded that the use of the HMM machinery to maximize the approximate poste- rior distributions of SVM models numerically leads to a good estimator performance, considering a modest computational effort. 4 Empirical Application We consider the daily closing prices of five Latin American stock markets: MERVAL (Argentina), IBOVESPA (Brazil), IPSA (Chile), MEXBOL (Mexico) and IGBVL (Peru). We use the S&P 500 in order to compare the results with Latin American stock markets because the U.S. stock market could be considered as a good benchmark. The data sets were obtained from the Yahoo finance web site available to download at http: //finance.yahoo.com. The period of analysis is from January 6, 1998, until December 30, 2016. Stock returns are computed as yt = 100× (logPt − logPt−1), where Pt is the (adjusted) closing price on day t. Table 6 shows the number of observations and summary descriptive statistics. The sample size differs between countries due to holidays and stock market non-trading days. According to Table 6, the IGBVL and S&P 500 returns are negatively skewed whereas the rest are positively skewed. The IGBVL returns are the most negatively skewed with -0.3915 and the IBOVESPA returns the most positively skewed with 0.5313. Regarding the kurtosis, all the daily returns of the five Latin American returns and the S&P 500 are leptokurtic (all kurtosis coefficients are higher than 3). Brazil, Peru, and Chile are the markets with the highest degree of kurtosis with the USA near Chile’s value. Although there are high differences between the minimum and maximum values, the most outstanding values correspond to Argentina and Brazil. We further observe that the IGBVL and IPSA returns show the highest level of first-order autocorrelation. These values decrease fast for the other orders of autocorrelation. In the case of returns, high first-order autocorrelation reflects the effects of non-synchronous or thin trading. The squared returns show high level of autocorrelation of order one, which can be seen as an indication of volatility clustering. We further observe that high-order autocorrelations for squared returns are still high and decrease slowly2. The Q(12) test statistic, 2This behavior has suggested that the literature considers that there is long memory in the volatility of returns, as well as the possibility that infrequent level shifts cause such behavior. For a discussion on this, see Diebold and Inoue 8 which is a joint test for the hypothesis that the first twelve autocorrelation coefficients are equal to zero, indicates that this hypothesis has to be rejected at the 5% significance level for all return and squared return series. We set the priors distributions as follows: (β0, γ, β2) ∼ N3(03, 100I3), µ ∼ N (0, 100), ψ ∼ N (4.5, 100) and ω ∼ N (−1.5, 100), where Nr(., .) and N (., .) denote the r−variate and univariate normal distributions and 0r and Ir are the r × 1 vector of zeros and the r × r the identity matrix, respectively. We use the procedure described in Section 2 by using bm = −b0 = 4 and m = 200, to ensure numerical stability in the results. We use the multivariate normal distribution with mean and variance being the MAP and the inverse of the hessian matrix evaluated at the MAP as importance density. To compare our methodology, we use the MCMC procedure based on HMC and RMHMC algorithms as described in Abanto-Valle et al. (2021) based on 30000 iterations and discarding the first 10000, only every 10-th values of the chain are stored. Table 7 summarizes these results for the five Latin American stock market and the S&P 500 returns using our proposal and MCMC methods based on Abanto-Valle et al. (2021). It is important to note tha all the estimates are similar for all markets considered here. The value of φ is very similar among all markets, suggesting similar degrees of persistence (ranging from 0.9523 for Argentina to 0.9851 for Mexico) with the HMM aproach. The MEXBOL, IBOVESPA and S&P 500 are more persistent than the other markets. The main difference between the results in in the case of Brazil, where the posterior mean of φ is 0.9841 with the HMM approach and 0.9730 with MCMC. The posterior mean estimates of σ show that all returns have similar estimates in the range from 0.1330 to 0.2757. The higest value is 0.2757 for the IGBVL jointly with the estimate of φ indicates that IGBVL is the most volatile stock market index in the region. Regarding the posterior mean of µ, we found that the estimates are statistically significant for the MERVAL, IBOVESPA and IPSA markets. For the MEXBOL, IGBVL and S&P 500 markets, the parameter µ is not significant because the credibility interval contains the null value. We observe that the posterior mean parameter β0 is always positive and statistically significant for all series. The value of β1 that measures the correlation of returns is as expected, small and very similar to the first-order autocorrelation coefficients reported in Table 6. The estimates of β1 are statistically significant for all series with the exception of Brazil and, although in the cases of Chile and Peru these values are 0.1876 and 0.1873, respectively, these values indicate a weak persistence with a rapid mean reversion. Regarding the parameter of interest (β2), this is more negative in the cases of USA, Brazil and Chile. Intermediate values are observed in Argentina and Mexico, while Peru presents the smallest value in absolute terms. Moreover, while all countries have a credibility interval that excludes the zero value, this does not happen in the case of Peru, so it is difficult to argue for an uncertainty effect in this market. It is important to note that the right side of the credibility interval is very close to zero in all markets except the U.S.. Therefore, the (2001) and Perron and Qu (2010), among others. For applications to different financial markets in Latin America, see Rodŕıguez (2017) and the references mentioned therein. 9 posterior mean of β2 parameter, which measures both the ex ante relationship between returns and volatility and the volatility feedback effect, is negative for all series and statistically significant for all the series with the exception of Peru. These findings are very similar and consistent with those found by Abanto-Valle et al. (2021). Following Koopman and Uspensky (2002), the volatility feedback effect (negative) dominates the positive effect which links the returns with the expected volatility. Our estimates are more negative compared to those of Koopman and Uspensky (2002) where the hypothesis that β2 = 0 can never be rejected at the conventional 5% siginificance level. Therefore, the volatility feedback effect is clearly dominant in our results (except for Peru) in comparison to those of Koopman and Uspensky (2002). These results confirm the hypothesis that investors require higher expected returns when unanticipated increases in future volatility are highly persistent. It is important to stress that these findings are consistent with higher values of φ combined with larger negative values for the in-mean parameter. We have indirect evidence of a positive intertemporal relation between expected excess market returns and its volatility as this is one of the assumptions underlying the volatility feedback hypothesis. Figure 2 shows the estimates of e ht 2 using our proposal and the smoothed mean of e ht 2 obtained via the MCMC output for all the series considered here. The dotted red line is the value obtained using our approach and the full black line via MCMC. We observe a great similarity between the estimating results for both methods. From a practitioners’ viewpoint, implementing the MCMC procedure developed in Abanto-Valle et al. (2021) requires to write about 800 lines of C++ code. In stark contrast, the approach discussed in the present paper is easily implemented using Rcpp inside R, writing less than 200 lines of code. In all the applications considered here the MCMC procedure takes about 1 hour and our HMM procedure takes about 20 minutes, for all the countries. Therefore, from the practical viewpoint, there is substantial merit in considering the HMM approach as an alternative to MCMC schemes. 5 Discussion This paper has two objectives. The first is to show gains in computational time using HMM methods versus MCMC methods developed and used, for example, in Abanto-Valle et al. (2021). The second objective is to empirically estimate the effect of the volatility in the mean (SVM model) using five Latin American stock markets comparing with the results obtained by Koopman and Uspensky (2002) and Abanto-Valle et al. (2021). Regarding, the first goal, this article introduces an approximate Bayesian inference via importance sampling, of the SVM model proposed by Koopman and Uspensky (2002). The likelihood function of the SVM model is approximated using HMMs machinery. The empirical application reveals similar results between our proposal methodology and the MCMC methods used by Abanto-Valle et al. (2021). However, our proposal is less time- 10 consuming, which is very important in real time applications. The SVM model allows us to investigate the dynamic relationship between returns and their time-varying volatility. Therefore, concerning the second goal, we illustrated our methods through an empirical application of five Latin American return series and the S&P 500 return. The β2 estimate, which measures both the ex- ante relationship between returns and volatility and the volatility feedback effect, was negative and significant for all the indexes considered here except for the IGBVL. The results are in line with those of French et al. (1987), who found a similar relationship between unexpected volatility dynamics and returns and confirm the hypothesis that investors require higher expected returns when unanticipated increases in future volatility are highly persistent. This fact is consistent with our findings of higher values of φ combined with larger negative values for the in-mean parameter. Future research considers extending the model and algorithm to include time-varying parameters, including the in-mean parameter. This fact would allow us a comparison with other algorithms, such as the one proposed in Chan (2017). Another extension is to incorporate heavy-tails as in Abanto-Valle et al. (2012) or skewness and heavy-tails simultaneously, as in Leão et al. (2017). References Abanto-Valle, C. A., Bandyopadhyay, D., Lachos, V. H., and Enriquez, I. (2010). Robust bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions. Computational Statistics and Data Analysis, 54:2883–2898. doi: https://doi.org/10.1016/j.csda.2009.06.011. Abanto-Valle, C. A., Langrock, R., Chen, M.-H., and Cardoso, M. V. (2017). Maximum likelihood estimation for stochastic volatility in mean models with heavy-tailed distributions. Applied Stochastic Models in Business and Industry, 33:394–408. doi: https://doi.org/10.1002/asmb.2246. Abanto-Valle, C. A., Migon, H. S., and Lachos, V. H. (2011). Stochastic volatility in mean models with scale mixtures of normal distributions and correlated errors: A bayesian approach. Journal of Statistical Planning and Inference, 141:1875–1887. doi: https://doi.org/10.1016/j.jspi.2010.11.039. Abanto-Valle, C. A., Migon, H. S., and Lachos, V. H. (2012). Stochastic volatility in mean models with heavy- tailed distributions. Brazilian Journal of Probability and Statistics, 26:402–422. doi: https://doi.org/10. 1214/11-BJPS169. Abanto-Valle, C. A., Rodŕıguez, G., and Garrafa-Aragón, H. B. (2021). Stochastic volatility in mean: Empir- ical evidence from latin-american stock markets using hamiltonian monte carlo and riemann manifold hmc 11 methods. The Quarterly Review of Economics and Finance, 80:272–286. doi: https://doi.org/10.1016/ j.qref.2021.02.005. Alanya, W. and Rodŕıguez, G. (2019). Asymmetries in volatility: An empirical study for the peruvian stock and forex returns. Review of Pacific Basin Financial Markets and Policies, 22:1–18. doi: https://doi.org/ 10.1142/S0219091519500036. Bekaert, G. and Wu, G. (2000). Asymmetric volatility and risk in equity markets. Review of Financial Studies, 13:1–42. doi: https://doi.org/10.1093/rfs/13.1.1. Black, F. (1976). Studies of stock price volatility changes. Proceedings of the 1976 Meetings of the American Statistical Association, Business and Economical Statistics Section, pages 177–181. Bollerslev, T., Chou, R. Y., and Kroner, K. F. (1992). ARCH modeling in finance: a review of the theory and empirical evidence. Journal of Econometrics, 52:5–59. doi: https://doi.org/10.1016/0304-4076(92) 90064-X. Bollerslev, T., Engle, R. F., and Nelson, D. B. (1994). ARCH models. In Engle, R. F. and MCFadden, D. L., editors, Handbook of Econometrics. Vol. 4., pages 2959–3038. Elsevier Science, Amsterdam. doi: https://doi.org/10.1057/9780230280830_2. Campbell, J. Y. and Entchel, L. H. (1992). No news is good news: An asymmetric model of changing volatility in stock returns. Journal of Financial Economics, 31:281–318. doi: https://doi.org/10.1016/0304-405X(92) 90037-X. Carnero, M. A., Peña, D., and Ruiz, E. (2004). Persistence and kurtosis in GARCH and Stochastic volatility models. Journal of Financial Econometrics, 2:319–342. doi: https://doi.org/10.1093/jjfinec/nbh012. Chan, J. C. (2017). The stochastic volatility in mean model with time-varying parameters: An application to inflation modeling. Journal of Business & Economic Statistics, 35:17–28. doi: https://doi.org/10.1080/ 07350015.2015.1052459. Chen, M.-H., Huan, L., Ibrahim, J., and Kim, S. (2008). Bayesian variable selection and computation for generalized linear models with conjugate priors. Bayesian Analysis, 3:585–614. doi: https://doi.org/10. 1214/08-BA323. Diebold, F. X. and Inoue, A. (2001). Long memory and regime switching. Journal of Econometrics, 105:131–159. doi: https://doi.org/10.1016/S0304-4076(01)00073-2. 12 Diebold, F. X. and Lopes, J. A. (1995). Modeling volatility dynamics. In K., H., editor, In Macreoconomet- rics: Developments, Tensions and Prospects, pages 427–472. Kluwer Academic Publishers, Amsterdam. doi: https://doi.org/10.1007/978-94-011-0669-6_11. French, K. R., Schert, W. G., and Stambugh, R. F. (1987). Expected stock return and volatility. Journal of Financial Economics, 19:3–29. doi: https://doi.org/10.1016/0304-405X(87)90026-2. Geweke, J. (1989). Bayesian inference in econometric models using monte carlo integration. Econometrica, 57:1317–1339. doi: https://doi.org/10.2307/1913710. Ghysels, E., Harvey, A. C., and Renault, E. (1994). Stochastic volatility. In Madala, G. S. and Rao, C. R., editors, Handbook of Statistics Vol. 14, Statistical methods in Finance, pages 128–198. Elsevier Science, Amsterdam. doi: http://doi.org/10.1016/S0169-7161(96)14007-4. Jacquier, E., Polson, N. G., and Rossi, P. E. (2004). Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. Journal of Econometrics, 122(1):185–212. doi: https://doi.org/10.1016/ j.jeconom.2003.09.001. Koopman, S. J. and Uspensky, E. H. (2002). The stochastic volatility in mean model: empirical evidence from international tock markets. Journal of Applied Econometrics, 17:667–689. doi: https://doi.org/10.1002/ jae.652. Langrock, R. (2011). Some applications of nonlinear and non-gaussian state-space modelling by means of hidden markov models. Journal of Applied Statistics, 38:2955–2970. doi: https://doi.org/10.1080/02664763. 2011.573543. Langrock, R., MacDonald, I. L., and Zucchini, W. (2012). Some nonstandard stochastic volatility models and their estimation usingstructured hidden markov models. Journal of Empirical Finance, 19:147–161. doi: https://doi.org/10.1016/j.jempfin.2011.09.003. Leão, W. L., Abanto-Valle, C. A., and Chen, M.-H. (2017). Stochastic volatility in mean model with leverage and asymmetrically heavy-tailed error using generalized hyperbolic skew student’s t distribution. Statistics and its Interface, 10:29–541. doi: https://doi.org/10.4310/SII.2017.v10.n4.a1. Lengua Lafosse, P. and Rodŕıguez, G. (2018). An empirical application of a stochastic volatility model with gh skew student’s t-distribution to the volatility of latin-american stock returns. The Quarterly Review of Economics and Finance, 69:155–173. doi: https://doi.org/10.1016/j.qref.2018.01.002. 13 Liesenfeld, R. and Jung, C. R. (2000). Stochastic volatility models: conditional normality versus heavy-tailed distributions. Journal of Applied Econometrics, 15(2):137–160. doi: https://doi.org/10.1002/(SICI) 1099-1255(200003/04)15:2<137::AID-JAE546>3.0.CO;2-M. Melino, A. and Turnbull, S. M. (1990). Pricing foreign currency options with stochastic volatility. Journal of Econometrics, 45:239–265. doi: https://doi.org/10.1016/0304-4076(90)90100-8. Perron, P. and Qu, Z. (2010). Long-memory and level shifts in the volatility of stock market return indices. Journal of Business and Economic Statistics, 28:275–290. doi: https://doi.org/10.1198/jbes.2009. 06171. Rodŕıguez, G. (2016). Modeling latin-american stock markets volatility: Varying probabilities and mean rever- sion in a random level shifts model. Review of Development Finance, 6:26–45. doi: https://doi.org/10. 1016/j.rdf.2015.11.002. Rodŕıguez, G. (2017). Modeling latin-american stock and forex markets volatility: Empirical application of a model with random level shifts and genuine long memory. The North American Journal of Economics and Finance, 42:393–420. doi: https://doi.org/10.1016/j.najef.2017.07.016. Rodŕıguez, G. (2017). Selecting between autoregressive conditional heterocedasticidity models: An empirical application to the volatility of stock returns in peru. Economic Analysis Review, 32:69–94. doi: http: //doi.org/10.4067/S0718-88702017000100069. Schwert, G. W. (1989). Why does stock market volatility change over time? Journal of Finance, 44:1115–1153. doi: https://doi.org/10.1111/j.1540-6261.1989.tb02647.x. Shephard, N. G. (1996). Statistical aspects of ARCH and stochastic volatility. In Cox, D. R., Hinkley, D. V., and Barndorff-Nielsen, O. E., editors, Time Series Models in Econometrics, Finance and Other Fields, Monographs on Statistics and Applied Probability 65, pages 1–67. Chapman and Hall, London. doi: https://doi.org/10.1007/978-1-4899-2879-5_1. Taylor, S. (1982). Financial returns modelled by the product of two stochastic processes-a study of the daily sugar prices 1961-75. In Anderson, O., editor, Time Series Analysis: Theory and Practice, Vol 1, pages 203–226. North-Holland, Amsterdam. Taylor, S. (2008). Modeling Financial Time Series. Wiley, Chichester, 2nd edition. doi: https://doi.org/10. 1142/6578. Taylor, S. J. (1994). Modelling stochastic volatility: a review and comparative study. Mathematical Finance, 4:183–204. doi: https://doi.org/10.1111/j.1467-9965.1994.tb00057.x. 14 Zucchini, W., MacDonald, I. L., and Langrock, R. (2016). Hidden Markov Models for Time Series: An Intro- duction Using R. Chapman & Hal, Boca Raton, FL, 2nd edition. doi: https://doi.org/10.1201/b20790. 15 Table 1: SVM Model, simulated data set: maximum a posteriori (MAP) of the parameters and computing times in seconds for the HMM method (bm = −b0 = 4). True values of the parameters: θ = (β0, γ, β2, µ, ψ, ω) ′ = (0.1400, 0.0600,−0.1000, 0.3000, 4.5951,−1.6094)′ size m log p(θ | yT ) β0 γ β2 µ ψ ω time 50 -2339.82 0.1037 0.0322 -0.0248 -0.0135 4.4144 -1.4568 3.75 1500 100 -2339.82 0.1038 0.0325 -0.0249 -0.0139 4.4141 -1.4562 8.76 150 -2339.82 0.1038 0.0325 -0.0249 -0.0140 4.4141 -1.4562 15.33 200 -2339.82 0.1038 0.0325 -0.0249 -0.0140 4.4141 -1.44562 23.02 50 -5197.48 0.1196 0.0485 -0.0710 0.4390 4.4783 -1.4421 8.01 3000 100 -5197.49 0.1210 0.0544 -0.0718 0.4201 4.4496 -1.4439 18.40 150 -5197.49 0.1210 0.0544 -0.0718 0.4201 4.4496 -1.4439 35.65 200 -5197.49 0.1210 0.0544 -0.0718 0.4201 4.4496 -1.4439 65.06 50 -9934.81 0.1231 0.0592 -0.0800 0.3116 4.6130 -1.5051 14.75 6000 100 -9934.80 0.1232 0.0589 -0.0802 0.3133 4.6103 -1.5050 31.43 150 -9934.80 0.1230 0.0596 -0.0800 0.3082 4.6034 -1.5028 62.41 200 -9934.80 0.1230 0.0596 -0.0800 0.3082 4.6034 -1.5028 89.45 16 Table 2: SVM Model estimation results using HMM approach, simulated data set. First line: posterior mean and the standar deviation and the computing time in seconds. Second line: 95% posterior credibility interval. True values of the parameters: β = (0.14, 0.03,−0.10)′ , µ = 0.3, φ = 0.98 and ση = 0.2 size m β0 β1 β2 µ φ ση time 50 0.1053 (0.0291) 0.0169 (0.0267) -0.0248 (0.0253) -0.0229 (0.2552) 0.9789 (0.0072) 0.2321 (0.0274) 13.41 (0.0505,0.1567) (-0.0406,0.0683) (-0.1059,0.0228) (-0.5081,0.4594) (0.9556,0.9876) (0.1816,0.2871) 100 0.1035 (0.0288) 0.0162 (0.0266) -0.0247 (0.0262) -0.0683 (0.3247) 0.9789 (0.0074) 0.2291 (0.0279) 15.29 1500 (0.0506,0.1565) (-0.0380,0.0718) (-0.1037,0.0222) (-0.4863,0.4206) (0.9573,0.9878) (0.1850,0.2937) 150 0.1039 (0.0283) 0.0172 (0.0260) -0.0244 (0.0253) -0.0006 (0.2589) 0.9780 (0.0072) 0.2330 (0.0289) 28.39 (0.0539,0.1603) (-0.0369,0.0694) (-0.1030,0.0243) (-0.5030,0.5158) (0.9544,0.9869) (0.1850,0.3005) 200 0.1037 (0.0278) 0.0149 (0.0279) -0.0236 (0.0248) -0.0229 (0.2590) 0.9792 (0.0071) 0.2306 (0.0279) 43.22 (0.0495,0.1598) (-0.0378,0.0697) (-0.1033,0.0254) (-0.5113,0.5124) (0.9558,0.9873) (0.1861,0.2922) 50 0.1181 (0.0230) 0.0245 (0.0189) -0.0705 (0.0144) 0.4325 (0.1974) 0.9792 (0.0047) 0.2305 (0.0199) 13.27 (0.0774,0.1635) (-0.0115,0.0620) (-0.1188,-0.0438) (0.0586,0.7896) (0.9651,0.9857) (0.1986,0.2811) 100 0.1173 (0.0226) 0.0240 (0.0196) -0.0709 (0.0141) 0.4209 (0.1932) 0.9787 0.2312 (0.0194) 30.69 3000 (0.0746,0.1678) (-0.0130,0.0609) (-0.1077,-0.0446) (0.0522,0.7749) (0.9662,0.9850) (0.1995,0.2758) 150 0.1179 (0.0234) 0.0266 (0.0190) -0.0715 (0.0140) 0.4447 (0.1899) 0.9788 (0.0048) 0.2314 (0.0179) 55.68 (0.0780,0.1663) (-0.0070,0.0663) (-0.1099,-0.0444) (0.0859,0.7933) (0.9651,0.9848) (0.1992,0.2751) 200 0.1171 (0.0230) 0.0239 (0.0193) -0.0709 (0.0145) 0.4377 (0.1966) 0.9791 (0.0049) 0.2319 (0.0189) 85.8 (0.0761,0.1656) (-0.0081,0.0645) (-0.1119,-0.0434) (0.0665,0.7899) (0.9658,0.9850) (0.1991,0.2799) 50 0.1237 (0.0152) 0.0297 (0.0134) -0.0800 (0.0103) 0.313 (0.1513) 0.9809 (0.0032) 0.2213 (0.0128) 28.21 (0.0946,0.1537) (0.0029,0.0567) (-0.1112,-0.0609) (0.0423,0.5759) (0.9731,0.9850) (0.1988,0.2485) 100 0.1233 (0.0153) 0.0297 (0.0132) -0.0796 (0.0108) 0.3101 (0.1539) 0.9811 (0.0033) 0.2213 (0.0129) 59.90 6000 (0.0949,0.1531) (0.0044,0.0567) (-0.1107,-0.0588) (0.0219,0.6027) (0.9727,0.9859) (0.1957,0.2478) 150 0.1230 (0.0153) 0.0298 (0.0137) -0.0801 (0.0103) 0.3123 (0.1421) 0.9808 (0.0032) 0.2220 (0.0131) 124.07 (0.0974,0.1002) (0.0321,0.1255) (-0.1106,-0.0589) (0.0471,0.3372) (0.0197, 0.1177) (0.1977,0.2497) 200 0.1229 (0.0155) 0.0293 (0.0302) -0.0800 (0.0107) 0.3092 (0.1495) 0.9805 (0.0031) 0.2220 (0.0130) 174.18 (0.0950,0.1509) (0.0048,0.1509) (-0.1099,-0.0589) (0.0200,0.5939) (0.9728, 0.9854) (0.1988,0.2500) 17 Table 3: SVM Model: Simulaton study results based on 300 replicates using the HMM method (bmax = −bmin = 4 and T = 1500) Parameter True value mean MRB MARD MSE m = 50 β0 0.14 0.1340 -0.0428 0.2208 0.0015 β1 0.03 0.0288 -0.0387 0.7322 0.0008 β2 -0.10 -0.0983 -0.0172 0.1976 0.0007 µ 0.30 0.3079 0.0263 0.6619 0.0647 φ 0.98 0.9732 -0.0069 0.0083 0.0001 σ 0.20 0.2395 0.1974 0.2018 0.0022 m = 100 β0 0.14 0.1344 -0.0397 0.2193 0.0015 β1 0.03 0.0285 -0.0492 0.7219 0.0008 β2 -0.10 -0.0984 -0.0158 0.1974 0.0007 µ 0.30 0.3094 0.0312 0.6582 0.0632 φ 0.98 0.9732 -0.0069 0.0084 0.0001 σ 0.20 0.2391 0.1958 0.2013 0.0022 m = 150 β0 0.14 0.1348 -0.0401 0.2210 0.0015 β1 0.03 0.0286 -0.0409 0.7316 0.0008 β2 -0.10 -0.0988 -0.0182 0.1975 0.0007 µ 0.30 0.3205 0.0305 0.6703 0.0662 φ 0.98 0.9735 -0.0068 0.0083 0.0001 σ 0.20 0.2389 0.1956 0.2006 0.0022 m = 200 β0 0.14 0.1340 -0.0372 0.1436 0.0015 β1 0.03 0.0288 -0.0443 0.7313 0.0008 β2 -0.10 -0.0983 -0.0124 0.1965 0.0007 µ 0.30 0.3079 0.0685 0.6948 0.0695 φ 0.98 0.9732 -0.0067 0.0083 0.0001 σ 0.20 0.2395 0.1943 0.1987 0.0022 18 Table 4: SVM Model: Simulaton study results based on 300 replicates using the HMM method (bmax = −bmin = 4 and T = 3000) Parameter True value mean MRB MARD MSE m = 50 β0 0.14 0.1368 -0.0225 0.1436 0.0006 β1 0.03 0.0296 -0.0134 0.5113 0.0003 β2 -0.10 -0.0990 -0.0104 0.1349 0.0003 µ 0.30 0.3097 0.0326 0.4465 0.0293 φ 0.98 0.9772 -0.0028 0.0046 0.00004 σ 0.20 0.2201 0.1006 0.1077 0.0007 m = 100 β0 0.14 0.1369 -0.0221 0.1446 0.0006 β1 0.03 0.0293 -0.0232 0.5047 0.0003 β2 -0.10 -0.0990 -0.0103 0.1349 0.0003 µ 0.30 0.3097 0.0324 0.4463 0.0281 φ 0.98 0.9772 -0.0028 0.0047 0.00004 σ 0.20 0.2200 0.0998 0.1074 0.0007 m = 150 β0 0.14 0.1371 -0.0206 0.1439 0.0006 β1 0.03 0.0294 -0.0216 0.5014 0.0003 β2 -0.10 -0.0990 -0.0094 0.1348 0.0003 µ 0.30 0.3205 0.0333 0.4520 0.0297 φ 0.98 0.9772 -0.0028 0.0047 0.00004 σ 0.20 0.2199 0.0997 0.1069 0.0007 m = 200 β0 0.14 0.1369 -0.0218 0.1436 0.0006 β1 0.03 0.0294 -0.0185 0.5074 0.0003 β2 -0.10 -0.0990 -0.0095 0.1350 0.0003 µ 0.30 0.3053 0.0179 0.4571 0.0303 φ 0.98 0.9772 -0.0028 0.0046 0.00004 σ 0.20 0.2199 0.0994 0.1067 0.0007 19 Table 5: SVM Model: Simulaton study results based on 300 replicates using the HMM method (bmax = −bmin = 4 and T = 6000) Parameter True value mean MRB MARD MSE m = 50 β0 0.14 0.1396 -0.0038 0.1432 0.0003 β1 0.03 0.0293 -0.0211 0.3412 0.0002 β2 -0.10 -0.0994 -0.0063 0.0959 0.0001 µ 0.30 0.3101 0.0336 0.3382 0.0156 φ 0.98 0.9787 -0.0013 0.0027 0.00001 σ 0.20 0.2098 0.0488 0.0637 0.0002 m = 100 β0 0.14 0.1396 -0.0025 0.0940 0.0003 β1 0.03 0.0294 -0.0181 0.3407 0.0002 β2 -0.10 -0.0993 -0.0062 0.0970 0.0001 µ 0.30 0.3110 0.0366 0.3431 0.0160 φ 0.98 0.9786 -0.0013 0.0027 0.00001 σ 0.20 0.2097 0.0485 0.0637 0.0002 m = 150 β0 0.14 0.1396 -0.0031 0.0935 0.0003 β1 0.03 0.0295 -0.0166 0.3398 0.0002 β2 -0.10 -0.0993 -0.0067 0.0958 0.0001 µ 0.30 0.3113 0.0378 0.3401 0.0159 φ 0.98 0.9787 -0.0013 0.0027 0.00001 σ 0.20 0.2097 0.0484 0.06346 0.0002 m = 200 β0 0.14 0.1396 -0.0027 0.0930 0.0003 β1 0.03 0.0294 -0.0196 0.3400 0.0002 β2 -0.10 -0.0993 -0.0066 0.0960 0.0001 µ 0.30 0.3107 0.0358 0.3391 0.0156 φ 0.98 0.9782 -0.0012 0.0027 0.00001 σ 0.20 0.2097 0.0485 0.0637 0.0002 20 Table 6: Summary Statistics for daily stock returns data INDEX MERVAL IBOVESPA IPSA MEXBOL IGBVL S&P 500 Size 4651 4698 4737 4759 4597 4777 Mean 0.0701 0.0376 0.0296 0.0464 0.0478 0.0177 S. D. 2.2125 2.0262 1.0695 1.4276 1.4111 1.2418 Minimum -14.2896 -17.2082 -7.6381 -10.3410 -13.2908 -9.4695 Maximum 16.1165 28.8325 11.8034 12.1536 12.8156 10.9572 Skewness 2.2091 0.5313 0.1372 0.1458 -0.3915 -0.2086 Kurtosis 7.3418 16.8094 11.6866 8.7449 13.5715 10.6576 Returns ρ̂1 0.0550 0.0130 0.1840 0.0910 0.1890 -0.0700 ρ̂2 0.0020 -0.0180 0.0220 -0.0300 0.0080 -0.0450 ρ̂3 0.0240 -0.0390 -0.0190 -0.0301 0.0680 0.0100 ρ̂4 0.0070 -0.0320 0.0250 -0.0030 0.0640 -0.0080 ρ̂5 -0.0090 -0.0170 0.0270 -0.0150 0.0250 -0.0460 Q(12) 33.37 44.51 190.52 54.57 240.53 66.95 Squared Returns ρ̂1 0.2580 0.1990 0.2320 0.1430 0.4210 0.2040 ρ̂2 0.2160 0.1640 0.2130 0.1780 0.3890 0.3720 ρ̂3 0.1780 0.1860 0.1720 0.2540 0.3920 0.1920 ρ̂4 0.1660 0.1170 0.1550 0.1300 0.2840 0.2880 ρ̂5 0.2130 0.0990 0.2910 0.2420 0.2140 0.3220 Q(12) 1763.4 1069.3 2086.0 2147.1 3960.2 4643.7 21 Table 7: Estimation of the SVM Model using HMM machinery and Importance Sampling HMM MCMC HMM MCMC Parameter Mean 95% interval Mean 95 % interval Parameter Mean 95% interval Mean 95 % interval MERVAL (Argentina) IBOVESPA (Brazil) µ 1.1726 (1.0166,1.3341) 1.1647 (0.9931,1.3291) µ 1.0332 (0.8228,1.2679) 1.2898 (0.9807,1.6022) φ 0.9523 (0.9329,0.9658) 0.9501 (0.9353,0.9629) φ 0.9841 (0.9738,0.9890) 0.9730 (0.9565,0.9863) σ 0.2661 (0.2257,0.3148) 0.2688 (0.2399,0.3044) σ 0.1330 (0.1122,0.1596) 0.1649 (0.1325,0.1968) β0 0.2080 (0.1329,0.2794) 0.2052 (0.1291,0.2815) β0 0.1303 (0.0535,0.2027) 0.2575 (0.1101,0.4091) β1 0.0047 (0.0180,0.0776) 0.0478 (0.0157,0.0783) β1 0.0082 (-0.0213,0.0382) 0.0309 (-0.0161,0.0743) β2 -0.0295 (-0.0501,-0.0091) -0.0287 (-0.0510,-0.0073) β2 -0.0343 (-0.0497,0.0000) -0.0400 (-0.0777,-0.0046) IPSA (Chile) MEXBOL (Mexico) µ -0.3459 (-0.5675,-0.1377) -0.3596 (-0.5706,-0.1555) µ 0.2462 (-0.0427,0.5264) 0.2316 (-0.0634,0.5200) φ 0.9736 (0.9615,0.9812) 0.9697 (0.9599,0.9791) φ 0.9851 (0.9756,0.9895) 0.9803 (0.9725,0.9870) σ 0.1970 (0.1708,0.2325) 0.2111 (0.1880,0.2385) σ 0.1624 (0.1389,0.1937) 0.1859 (0.1655,0.2102) β0 0.0708 (0.0400,0.1041) 0.0710 (0.0385,0.1057) β0 0.1018 (0.0601,0.1414) 0.1039 (0.0655,0.2102) β1 0.1876 (0.1578,0.2167) 0.1885 (0.1578,0.2184) β1 0.0724 (0.0424,0.1009) 0.0742 (0.0438,0.1051) β2 -0.0433 (-0.0843,-0.0035) -0.0442 (-0.0581,-0.0021) β2 -0.0296 (-0.0577,-0.0015) -0.0301 (-0.0581,-0.0020) IGBVL (Peru) S & P 500 (USA) µ -0.0042 (-0.2219,0.1921) -0.0095 (-0.2342,0.2044) µ -0.1355 (-0.3691,0.1058) -0.1332 (-0.4096,0.1375) φ 0.9619 (0.9464,0.9714) 0.9618 (0.9490,0.9732) φ 0.9814 (0.9679,0.9838) 0.9791 (0.9705,0.9864) σ 0.2757 (0.2400,0.3203) 0.2725 (0.2351,0.3102) σ 0.1833 (0.1635,0.2219) 0.1968 (0.1686,0.2311) β0 0.0601 (0.0252,0.0941) 0.0596 (0.0232,0.0960) β0 0.1091 (0.0751,0.1373) 0.1085 (0.0754,0.1404) β1 0.1873 (0.1560,0.2196) 0.1875 (0.1563,0.2199) β1 -0.0600 (-0.0919,-0.0361) -0.0504 (-0.0853,-0.0232) β2 -0.0116 (-0.0398,0.0172) -0.0114 (-0.0414,0.0179) β2 -0.0603 (-0.0889,-0.0259) -0.0595 (-0.0924,-0.0272) 22 0 1000 2000 3000 4000 5000 6000 −6 −4 −2 0 2 4 6 time y t Figure 1: Simulated data set from the SVM model with β = (0.14, 0.03,−0.10)′ , µ = 0.3, φ = 0.98 and σ = 0.2. The vertical dotted lines (red) indicate the sample size T = 1500, 3000 and 6000, respectively. 23 MERVAL 0 5 10 15 20 01/08/1998 03/22/2001 07/14/2004 09/25/2007 01/03/2011 04/24/2014 IBOVESPA 0 5 10 15 20 25 30 01/07/1998 04/02/2001 06/22/2004 09/12/2007 10/12/2010 03/07/2014 IPSA 0 5 10 15 01/07/1998 03/19/2001 06/04/2004 08/16/2007 11/03/2010 01/20/2014 MEXBOL 0 5 10 15 01/07/1998 03/14/2001 05/26/2004 07/23/2007 09/30/2010 12/11/2013 IGBVL 0 5 10 15 01/07/1998 04/18/2001 08/18/2004 12/09/2007 03/25/2011 07/07/2014 S & P 500 0 5 10 15 01/07/1998 03/14/2001 05/26/2004 07/23/2007 09/30/2010 12/11/2013 Figure 2: MERVAL, IBOVESPA, IPSA, MEXBOL, IGBVL and S&P 500 returns data sets: absolute returns (full gray line), e ht 2 estimator usign HMM machinery (dotted red line) and posterior smoothed mean of e ht 2 of the SVM model using MCMC (black line). 24 ÚLTIMAS PUBLICACIONES DE LOS PROFESORES DEL DEPARTAMENTO DE ECONOMÍA  Libros Alfredo Dammert Lira 2021 Economía minera. Lima, Fondo Editorial PUCP. Adolfo Figueroa 2021 The Quality of Society, Volume II – Essays on the Unified Theory of Capitalism. New York, Palgrave Macmillan. Carlos Contreras Carranza (Editor) 2021 La Economía como Ciencia Social en el Perú. Cincuenta años de estudios económicos en la Pontificia Universidad Católica del Perú. Lima, Departamento de Economía PUCP. José Carlos Orihuela y César Contreras 2021 Amazonía en cifras: Recursos naturales, cambio climático y desigualdades. Lima, OXFAM. Alan Fairlie 2021 Hacia una estrategia de desarrollo sostenible para el Perú del Bicentenario. Arequipa, Editorial UNSA. Waldo Mendoza e Yuliño Anastacio 2021 La historia fiscal del Perú: 1980-2020. Colapso, estabilización, consolidación y el golpe de la COVID-19. Lima, Fondo Editorial PUCP. Cecilia Garavito 2020 Microeconomía: Consumidores, productores y estructuras de mercado. Segunda edición. Lima, Fondo Editorial de la Pontificia Universidad Católica del Perú. Adolfo Figueroa 2019 The Quality of Society Essays on the Unified Theory of Capitalism. New York. Palgrave MacMillan. Carlos Contreras y Stephan Gruber (Eds.) 2019 Historia del Pensamiento Económico en el Perú. Antología y selección de textos. Lima, Facultad de Ciencias Sociales PUCP. Barreix, Alberto Daniel; Corrales, Luis Fernando; Benitez, Juan Carlos; Garcimartín, Carlos; Ardanaz, Martín; Díaz, Santiago; Cerda, Rodrigo; Larraín B., Felipe; Revilla, Ernesto; Acevedo, Carlos; Peña, Santiago; Agüero, Emmanuel; Mendoza Bellido, Waldo; Escobar Arango y Andrés. 2019 Reglas fiscales resilientes en América Latina. Washington, BID. José D. Gallardo Ku https://departamento.pucp.edu.pe/economia/libro/10743/ https://departamento.pucp.edu.pe/economia/libro/la-economia-ciencia-social-peru-cincuenta-anos-estudios-economicas-la-pontificia-universidad-catolica-del-peru/ https://departamento.pucp.edu.pe/economia/libro/la-economia-ciencia-social-peru-cincuenta-anos-estudios-economicas-la-pontificia-universidad-catolica-del-peru/ https://departamento.pucp.edu.pe/economia/libro/la-historia-fiscal-del-peru-1980-2020-colapso-estabilizacion-consolidacion-golpe-la-covid-19/ https://departamento.pucp.edu.pe/economia/libro/la-historia-fiscal-del-peru-1980-2020-colapso-estabilizacion-consolidacion-golpe-la-covid-19/ http://departamento.pucp.edu.pe/economia/libro/the-quality-of-society-essays-on-the-unified-theory-of-capitalism/ 2019 Notas de teoría para para la incertidumbre. Lima, Fondo Editorial de la Pontificia Universidad Católica del Perú. Úrsula Aldana, Jhonatan Clausen, Angelo Cozzubo, Carolina Trivelli, Carlos Urrutia y Johanna Yancari 2018 Desigualdad y pobreza en un contexto de crecimiento económico. Lima, Instituto de Estudios Peruanos. Séverine Deneulin, Jhonatan Clausen y Arelí Valencia (Eds.) 2018 Introducción al enfoque de las capacidades: Aportes para el Desarrollo Humano en América Latina. Flacso Argentina y Editorial Manantial. Fondo Editorial de la Pontificia Universidad Católica del Perú. Mario Dammil, Oscar Dancourt y Roberto Frenkel (Eds.) 2018 Dilemas de las políticas cambiarias y monetarias en América Latina. Lima, Fondo Editorial de la Pontificia Universidad Católica del Perú. http://departamento.pucp.edu.pe/economia/libro/las-alianzas-publico-privadas-app-en-el-peru-beneficios-y-riesgos/ http://departamento.pucp.edu.pe/economia/libro/las-alianzas-publico-privadas-app-en-el-peru-beneficios-y-riesgos/ http://departamento.pucp.edu.pe/economia/libro/las-alianzas-publico-privadas-app-en-el-peru-beneficios-y-riesgos/  Documentos de trabajo No. 501 “El impacto de políticas diferenciadas de cuarentena sobre la mortalidad por COVID-19: el caso de Brasil y Perú”. Angelo Cozzubo, Javier Herrera, Mireille Razafindrakoto y François Roubaud. Octubre, 2021. No. 500 “Determinantes del gasto de bolsillo en salud en el Perú”. Luis García y Crissy Rojas. Julio, 2021. No. 499 “Cadenas Globales de Valor de Exportación de los Países de la Comunidad Andina 2000-2015”. Mario Tello. Junio, 2021. No. 498 “¿Cómo afecta el desempleo regional a los salarios en el área urbana? Una curva de salarios para Perú (2012-2019)”. Sergio Quispe. Mayo, 2021. No. 497 “¿Qué tan rígidos son los precios en línea? Evidencia para Perú usando Big Data”. Hilary Coronado, Erick Lahura y Marco Vega. Mayo, 2021. No. 496 “Reformando el sistema de pensiones en Perú: costo fiscal, nivel de pensiones, brecha de género y desigualdad”. Javier Olivera. Diciembre, 2020. No. 495 “Crónica de la economía peruana en tiempos de pandemia”. Jorge Vega Castro. Diciembre, 2020. No. 494 “Epidemia y nivel de actividad económica: un modelo”. Waldo Mendoza e Isaías Chalco. Setiembre, 2020. No. 493 “Competencia, alcance social y sostenibilidad financiera en las microfinanzas reguladas peruanas”. Giovanna Aguilar Andía y Jhonatan Portilla Goicochea. Setiembre, 2020. No. 492 “Empoderamiento de la mujer y demanda por servicios de salud preventivos y de salud reproductiva en el Perú 2015-2018”. Pedro Francke y Diego Quispe O. Julio, 2020. No. 491 “Inversión en infraestructura y demanda turística: una aplicación del enfoque de control sintético para el caso Kuéalp, Perú”. Erick Lahura y Rosario Sabrera. Julio, 2020. No. 490 “La dinámica de inversión privada. El modelo del acelerador flexible en una economía abierta”. Waldo Mendoza Bellido. Mayo, 2020. No. 489 “Time-Varying Impact of Fiscal Shocks over GDP Growth in Peru: An Empirical Application using Hybrid TVP-VAR-SV Models”. Álvaro Jiménez y Gabriel Rodríguez. Abril, 2020. No. 488 “Experimentos clásicos de economía. Evidencia de laboratorio de Perú”. Kristian López Vargas y Alejandro Lugon. Marzo, 2020. No. 487 “Investigación y desarrollo, tecnologías de información y comunicación e impactos sobre el proceso de innovación y la productividad”. Mario D. Tello. Marzo, 2020. No. 486 “The Political Economy Approach of Trade Barriers: The Case of Peruvian’s Trade Liberalization”. Mario D. Tello. Marzo, 2020. No. 485 “Evolution of Monetary Policy in Peru. An Empirical Application Using a Mixture Innovation TVP-VAR-SV Model”. Jhonatan Portilla Goicochea y Gabriel Rodríguez. Febrero, 2020. No. 484 “Modeling the Volatility of Returns on Commodities: An Application and Empirical Comparison of GARCH and SV Models”. Jean Pierre Fernández Prada Saucedo y Gabriel Rodríguez. Febrero, 2020. No. 483 “Macroeconomic Effects of Loan Supply Shocks: Empirical Evidence”. Jefferson Martínez y Gabriel Rodríguez. Febrero, 2020. No. 482 “Acerca de la relación entre el gasto público por alumno y los retornos a la educación en el Perú: un análisis por cohortes”. Luis García y Sara Sánchez. Febrero, 2020. No. 481 “Stochastic Volatility in Mean. Empirical Evidence from Stock Latin American Markets”. Carlos A. Abanto-Valle, Gabriel Rodríguez y Hernán B. Garrafa- Aragón. Febrero, 2020. No. 480 “Presidential Approval in Peru: An Empirical Analysis Using a Fractionally Cointegrated VAR2”. Alexander Boca Saravia y Gabriel Rodríguez. Diciembre, 2019. No. 479 “La Ley de Okun en el Perú: Lima Metropolitana 1971 – 2016.” Cecilia Garavito. Agosto, 2019. No. 478 “Peru´s Regional Growth and Convergence in 1979-2017: An Empirical Spatial Panel Data Analysis”. Juan Palomino y Gabriel Rodríguez. Marzo, 2019.  Materiales de Enseñanza No. 5 “Matemáticas para Economistas 1”. Tessy Váquez Baos. Abril, 2019. No. 4 “Teoría de la Regulación”. Roxana Barrantes. Marzo, 2019. No. 3 “Economía Pública”. Roxana Barrantes, Silvana Manrique y Carla Glave. Marzo, 2018. No. 2 “Macroeconomía: Enfoques y modelos. Ejercicios resueltos”. Felix Jiménez. Marzo, 2016. No. 1 “Introducción a la teoría del Equilibrio General”. Alejandro Lugon. Octubre, 2015. Departamento de Economía - Pontificia Universidad Católica del Perú Av. Universitaria 1801, San Miguel, 15008 – Perú. Telf. 626-2000 anexos 4950 - 4951 http://departamento.pucp.edu.pe/economia/ DDD502 caratula DDD502-Segunda hoja DDD502-Contratapa DDD502 texto DDD502-ultimas publicaciones