A Skewed Studentt ValueatRisk Approach for Long Memory Volatility Processes in Japanese Financial Markets

This paper investigates the relevance of skewed Student‐t distributions in capturing long memory volatility properties in the daily return series of Japanese financial data (Nikkei 225 Index and JPY‐USD exchange rate). For this purpose, we assess the performance of two long memory Value‐at‐Risk (VaR) models (FIGARCH and FIAPARCH VaR model) with three different distribution innovations: the normal, Student‐t, and skewed Student‐t distributions. From our results, we find that the skewed Student‐t distribution model produces more accurate VaR estimations than normal and Student‐t distribution models. Thus, accounting for skewness and excess kurtosis in the asset return distribution can provide suitable criteria for VaR model selection in the context of long memory volatility and enhance the performance of risk management in Japanese financial markets.

ISSN 1598-2769 Journal of International Economic Studies Vol 11, No 1, June 2007 211 A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets Seong‐Min Yoon Professor, Division of Economics, Pukyong National University smyoon@pknu.ac.kr Sang‐Hoon Kang PhD Candidate, School of Commerce, University of South Australia sang.kang@unisa.edu.au This paper investigates the relevance of skewed Student‐t distributions in capturing long memory volatility properties in the daily return series of Japanese financial data (Nikkei 225 Index and JPY‐USD exchange rate) For this purpose, we assess the performance of two long memory Value‐at‐Risk (VaR) models (FIGARCH and FIAPARCH VaR model) with three different distribution innovations: the normal, Student‐t, and skewed Student‐t distributions From our results, we find that the skewed Student‐t distribution model produces more accurate VaR estimations than normal and Student‐t distribution models Thus, accounting for skewness and excess kurtosis in the asset return distribution can provide suitable criteria for VaR model selection in the context of long memory volatility and enhance the performance of risk management in Japanese financial markets Keywords: Value‐at‐Risk, Japanese financial markets, volatility, asymmetry, long memory, skewed Student‐t distribution ISSN 1598-2769 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 비대칭 Student-t 분포를 이용한 일본 금융시장 장기기억 변동성과정의 VaR 분석 尹 盛 民 부경대학교 경제학부 교수 smyoon@pknu.ac.kr 姜 商 勳 School of Commerce, University of South Australia 박사과정 sang.kang@unisa.edu.au 본 연구의 목적은 왜도와 두터운 꼬리를 반영한 비대칭(skewed) Student‐t 분포가 장 기기억 변동성과정에서 어떠한 역할을 하는지를 일본 금융시장의 시계열자료(Nikkei 225 주가지수 및 엔‐달러 환율)를 이용하여 분석하는 것이다 VaR 분석기법을 이용하 여 오차항에 대한 세 가지 다른 분포(정규분포, 대칭적 t 분포 그리고 비대칭 t 분포)를 가정하는 두 가지 대표적 장기기억 모형(FIGARCH 및 FIAPARCH)의 적합성을 비교 분 석하였다 실증분석 결과 비대칭 Student‐t 분포가 정규분포나 대칭적 t 분포에 비해서 장기기억 변동성 모형의 적합도를 높일 수 있음을 발견하였다 이는 일본 금융시장의 경우에는 왜도와 두터운 꼬리 분포를 반영한 VaR 모형이 금융자산 위험관리 측면에서 더 나은 성과를 나타낸다는 것을 의미한다 핵심용어: VaR, 일본금융시장, 변동성, 비대칭성, 장기기억, 비대칭 Student‐t 분포 A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 213 I Introduction Due to ubiquitous risks in financial markets, Value‐at‐Risk (VaR) has become a crucial issue in measuring market risks and assessing the accuracy of asset portfolio volatility risks in financial economics VaR models simply calculate the maximum possible loss of an investment with a specified significance level over a given time period.1) That is, VaR is defined as a quantile of a probability distribution, which is used to quantify market risks and set capital reserves for market risks (Duffe and Pan 1997) The RiskMetrics model of the J P Morgan Group is one of the most popular tools for measuring the market volatility risk of asset portfolios under the assumption of normality.2) Financial asset returns undeniably suffer from excess skewness and kurtosis, implying that the assumption of normal (or Gaussian) distribution is inappropriate for explaining the skewed and fat‐tailed characteristics of return distribution (Fang and Lai 1997; Harvey and Siddique 2000; Smith 2006; Theodossiou 1998) By identifying the appropriate shape of return distributions, investors or risk managers are able to measure investment risk exposure in financial markets (Hogg and Klugman 1983; Premaratne and Bera 2005) To accommodate the characteristics of skewed and fat‐tailed distributions of financial returns, various empirical studies have developed a generalized autoregressive conditional heterosedaticity (GARCH)‐type framework with different distribution innovations For instance, Bollerslev (1987) proposes a Student‐t distribution to capture excess kurtosis of stock returns In addition, the skewed Student‐t distribution of Hansen (1994) allows for asymmetry and tail‐fatness distributions of asset returns; this innovation was extended by Lambert and Laurent in 2001 1) The Bank for International Settlement (BIS) imposes confidence level 99% and the time horizon at 10 days for the purpose of measuring the adequacy of bank capital 2) The RiskMetrics model is equivalent to a normal integrated GARCH (IGARCH) model where the autoregressive parameter is set at a pre‐specified value λ (0.94), whereas the coefficient of ε t −1 is equal to − λ (0.06) for optimal back-testing VaR forecasts (RiskMetrics Group 1996) 214 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 Subsequent studies dealing with VaR have widely adopted the skewed Student‐t distribution innovation in order to model the appropriate shape of return distributions for financial time series data (Giot and Laurent 2003; Tang and Shieh 2006; Bali and Theodossiou 2007) Another growing issue in financial economics is that of financial asset return volatility, which often exhibits long memory properties where the autocorrelations of the absolute and squared returns are characterized by a very slow decay (Baillie 1996) In order to circumvent this problem, the fractionally integrated GARCH (FIGARCH) model of Baillie, Bollerslev and Mikkelsen (1996) takes into consideration the fractional integration (long memory) of the conditional variance, which dates back to Granger (1980), Granger and Joyeux (1980) and Hosking (1981) Wu and Shieh (2007) compare the VaR performance of the GARCH and FIGARCH models with normal, Student‐t, and skewed Student‐t innovation distributions Their evidence suggests that the FIGARCH model with a skewed Student‐t innovation distribution outperforms the GARCH model with different distribution innovations for US Treasury bond returns However, although the FIGARCH model can capture persistence in conditional variance, it is unrealistic to assume that positive and negative shocks have the same effects on volatility in the FIGARCH specification (Tse 1998; Hwang 2001) It is well known that volatility tends to increase more following a large price fall (i.e bad news) than following a price rise (i.e good news) of the same magnitude (Black 1976; Nelson 1991; Engle and Ng 1993; Hentschel 1995) To incorporate both long memory and asymmetry in volatility, Tse (1998) develops a fractionally integrated asymmetric power ARCH (FIAPARCH) model Degiannakis (2004) finds that the FIAPARCH model with a skewed Student‐t distribution provides more accurate VaR predictions than other variants of GARCH‐class models for the three European stock markets This paper considers the relevance of skewed Student‐t distributions in estimating volatility persistence for daily returns data in Japanese financial markets (stock market and foreign exchange market) using two long memory volatility models, namely the FIGARCH and FIAPARCH To further enhance A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 215 the robustness of the estimation results, we compare the performance of various VaR models with normal, Student‐t, and skewed Student‐t distribution innovations The contribution of this paper is twofold First, we examine the process of long memory volatility with different distribution innovations in Japanese financial markets The empirical results reveal that the FIAPARCH (1, d ,1) model is suitable for the Nikkei 225 Index to capture asymmetric long memory properties, while the FIGARCH (1, d ,1) model outperforms in interpreting symmetric long memory properties for the Japanese Yen–US Dollar (JPY‐USD) exchange rate The second contribution is that VaR analyses offer the relevance of asymmetries and tail‐fatness in the return distribution of Japanese financial data For instance, the models with skewed Student‐t distributions provide more accurate volatility forecasting results than normal and Student‐t distribution VaR models for both long and short positions Consequently, our VaR analyses imply that the assumption of normal distribution is inappropriate for evaluating the accuracy of VaR estimates in Japanese financial markets The rest of this paper is organized as follows Section Ⅱ describes the theoretical properties of symmetric and asymmetric long memory VaR models Section Ⅲ provides the statistical characteristics of sample data and the empirical results The final section contains some concluding remarks II Methodology FIGARCH model Baillie, Bollerslev and Mikkelsen (1996) extend the general GARCH model through the introduction of a fractionally integrated process, I (d ) , otherwise known as the FIGARCH model Unlike the knife‐edge distinction between I (0) and I (1) processes, the fractionally integrated process, I (d ) , distinguishes between short memory and long memory in the time series The FIGARCH ( p, d , q ) 216 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 model is defined by: yt = µ + ε t , ε t = zt σ t , zt ~ N (0,1) , φ (L )(1 − L )d ε t2 = ω + [1 − β (L )]ν t , (1) 2 where ν t ≡ ε t − σ t All roots of φ (L ) and [1 − β (L )] lie outside the unit root circle Equation (1) can be re‐written as { } σ t2 = ω + − [1 − β (L )]−1φ (L )(1 − L )d ε t2 ≡ ω + λ (L )ε t2 , (2) where λ (L ) = λ1 L + λ2 L + ⋅ ⋅ ⋅ and ≤ d ≤ For the FIGARCH (p, d, q) process to be well‐defined and the conditional variance to be positive for all t , all coefficients in the infinite ARCH representation must be non‐negative, i.e λ j ≥ for j = 1,2,⋅ ⋅ ⋅ We consider the conditions imposed by Bollerslev and Mikkelsen (1996) as necessary and sufficient to ensure the non‐negativity of λ j : β1 − d ≤ φ1 ≤ 1− d ⎤ ⎡ 2−d d ⎢φ1 − ≤ β1 (φ1 − β1 + d ) and ⎥⎦ ⎣ (3) For the FIGARCH model in Equation (3), the persistence of shocks to the conditional variance or the degree of long memory is measured by the fractional differencing parameter d Thus, the attractiveness of the FIGARCH model is that for < d < , it is sufficiently flexible to allow for an intermediate persistence range FIAPARCH model The FIAPARCH model extends the FIGARCH model with the APARCH A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 217 model by Ding, Granger and Engle (1993) to capture asymmetry in the conditional variance The FIAPARCH ( p, d , q ) model is specified as follows: [ σ tδ = ω + − (1 − β (L ))−1 (1 − φ (L ))(1 − L )d ] (ε − γε t ) , δ t (4) where δ > , − < γ < , and < d < When γ > , negative shocks give rise to higher volatility than positive shocks, and vice versa The FIAPARCH model also nests the FIGARCH model, when δ = and γ = Thus, we can argue the FIAPARCH model is superior to the FIGARCH model since the former model can capture asymmetric long memory features in the conditional variance (Tse 1998) Model densities The parameters of the volatility models can be estimated by using non‐linear optimization procedures to maximize the logarithm of the Gaussian likelihood function Under the assumption that the random variable is zt ~ N (0,1) , the log‐ likelihood of Gaussian or normal distribution (LNorm ) can be expressed as LNorm = − [ ( ) ] T ∑ ln(2π ) + ln σ t2 + zt2 , t =1 (5) where T is the number of observations However, it is widely recognized that residuals suffer from excess kurtosis In order to capture fat‐tails, the Student‐t distribution is incorporated into this study If the random variable is zt ~ ST (0,1,υ ) , the log‐likelihood function of the Student‐t distribution (LStud ) will be defined as follows: ⎧ ⎛υ +1⎞ ⎫ ⎛υ ⎞ LStud = T ⎨ln Γ⎜ ⎟ − ln Γ⎜ ⎟ − ln[π (υ − 2)]⎬ ⎝2⎠ ⎩ ⎝ ⎠ ⎭ 218 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 − ⎡ ⎛ ⎞⎤ ⎤ z2 T ⎡ ⎟⎥ ⎥ ⎢ln σ t + (1 + υ )⎢ln⎜⎜1 + t ∑ ⎟ t =1 ⎣⎢ ⎢⎣ ⎝ σ t (υ − 2) ⎠⎥⎦ ⎦⎥ , ( ) (6) where < υ ≤ ∞ and Γ(⋅) is the gamma function In contrast to the normal distribution, the Student‐t distribution is estimated with an additional parameter υ , which stands for the number of degrees of freedom measuring the degree of fat‐tails in the density Despite accounting for tail‐thickness, a Student‐t distribution alone cannot capture the asymmetric feature of density To incorporate excess skewness and kurtosis, we consider a skewed Student‐t distribution proposed by Lambert and Laurent (2001) If zt ~ SKST (0,1, k ,υ ) , the log‐likelihood of the skewed Student‐t distribution (LSkSt ) is as follows: LSkSt ⎧ ⎫ ⎛ ⎞ ⎜ ⎟ ⎪⎪ ⎛ υ + ⎞ ⎪ ⎛υ ⎞ ⎟ + ln (s )⎪⎬ = T ⎨ln Γ⎜ ⎟ − ln Γ⎜ ⎟ − ln[π (υ − )] + ln⎜ ⎜k + ⎟ ⎝2⎠ ⎪ ⎝ ⎠ ⎪ ⎜ ⎟ ⎪⎩ ⎪⎭ k⎠ ⎝ − ⎡ (sz t + m )2 − I t ⎤ ⎤ T ⎡ + + ( ) k ⎥⎥ ln σ υ ln ⎢ ⎢1 + ∑ t t =1 ⎣⎢ υ −2 ⎢⎣ ⎥⎦ ⎦⎥ , ( ) (7) where I t = if z t ≥ − m s or I t = −1 , if z t < − m s , k is an asymmetry parameter The constants m = m(k ,υ ) and s = s (k ,υ ) are the mean and standard deviations of the skewed Student‐t distribution: ⎛υ −1⎞ Γ⎜ ⎟ υ −2 1⎞ ⎠ ⎛ m(k ,υ ) = ⎝ ⎜k − ⎟ k⎠ ⎛υ ⎞ π Γ⎜ ⎟ ⎝ , ⎝2⎠ ⎛ ⎞ s (k ,υ ) = ⎜ k + − 1⎟ − m k ⎝ ⎠ (8) The value of ln(k ) can also represent the degree of asymmetry in the residual distribution For example, if ln(k ) > (ln(k ) < 0) , the density is right (left) skewed A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 219 When k = , the skewed Student‐t distribution equals the general Student‐t distribution, i.e zt ~ ST (0,1,υ ) in Equation (6) VaR models and tests 1) VaR models Today, traders or portfolio managers are increasingly finding that their portfolios change dramatically from one day to the next, and must concern themselves with not only long trading positions, but also short trading positions So the performance of each VaR model should be compared on the basis of both long trading positions (the left tail of distribution) and short trading positions (the right tail of distribution) We also compare the performance of the FIGARCH and FIAPARCH models estimated on the assumption of three innovation distributions including the normal, Student‐t, and skewed Student‐t distributions, as discussed above In addition, a one‐step‐ahead VaR is computed along with the results of the estimated volatility models and the given distribution The VaR of α quantiles for long and short trading positions are computed as follows Under the assumption of the normal distribution, VaRlong = µt − zασ t , and VaRshort = µt + zα σ t , (9) where zα is the left or right quantile at α % of the normal distribution in Equation (5) Under the assumption of the Student‐t distribution, VaRlong = µt − stα ,υσ t , and VaRshort = µ t + stα ,υ σ t , (10) where stα ,υ is left or right quantile at α % of the Student‐t distribution in 220 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 Equation (6) Under the assumption of the skewed Student‐t distribution, VaRlong = µ t − skstα ,υ ,k σ t , and VaRshort = µ t + skstα ,υ ,k σ t (11) where stα ,υ ,k is the left or right quantile at α % of the skewed Student‐t distribution in Equation (7) If ln(k ) < , the VaR for long trading positions will be larger for the same conditional variance than the VaR for short trading positions When ln(k ) > , the opposite holds true 2) Tests of accuracy for VaR estimates We calculate the VaR at the pre‐specified significance level of α , ranging from 5% to 0.25%, and then evaluate their performance by calculating the failure rate for both the left and right tails of the distribution in the sample return series The failure rate is widely applied in evaluating the effectiveness of VaR models (Giot and Laurent 2003; Tang and Shieh 2006) The failure rate is defined as the ratio of the number of times (x) in which returns exceed the forecasted VaR to the sample size (T ) Following Giot and Laurent (2003), testing the accuracy of the model is equivalent to testing the hypothesis H : f = α versus H1 : f ≠ α , where f is the failure rate: if the VaR model is correctly specified, the failure rate should be equal to the pre‐specified significance level of α This test is also called the Kupiec LR test, which examines the null hypothesis using a likelihood ratio test (Kupiec 1995) The LR statistic is: [ LR = −2 ln (1 − α ) ∧ T −x ] ∧ ⎡ α x + ln ⎢⎛⎜1 − f ⎞⎟ ⎣⎢⎝ ⎠ T −x x ⎛∧⎞ ⎤ ⎜ f ⎟ ⎥ ~ χ (1) ⎝ ⎠ ⎦⎥ (12) where f is the estimated failure rate Under the null hypothesis, the Kupeic 228 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 long memory volatility process of the Nikkei 225 Index returns The coefficients of asymmetric responses to volatility news (γ ) are positive and highly significant at the 1% level That is, unexpected negative returns result in more volatility than unexpected positive returns of the same magnitude, which is in favor of the negative relationship between current returns and future volatility observed by Black (1976) Besides, the power term (δ ) takes a value of 1.490, which is significantly different from Thus, the skewed Student‐t FIAPARCH model is preferred over the Student‐t FIGARCH model for Nikkei 225 Index returns In the case of the JPY‐USD exchange rate, asymmetry volatility seems to be inconsistent and disappears with the use of different distribution innovations The normal distribution innovation exhibits a significant coefficient value of (γ ) , supporting the evidence of volatility asymmetry in the JPY‐USD exchange rate.7) However, for this case, the non‐negativity conditions of Equation (3) in the conditional variance are violated, implying that the FIAPARCH model under the normal distribution is not well defined for the JPY‐USD exchange rate On the other hand, other non‐normality innovations eliminate volatility asymmetry and satisfy sufficient conditions to ensure the non‐negativity of the conditional variance in the FIAPARCH process As a result, there is no evidence of volatility asymmetry in the case of the JPY‐USD exchange rate Additionally, the power term (δ ) ranges in value from 1.803 to 1.992, implying that a squared error term suits the conditional variance specification in favor of the FIGARCH specification for the JPY‐USD exchange rate Consequently, in the case of JPY‐USD exchange rate returns, the skewed Student‐t FIGARCH (1, d ,1) model outperforms the skewed Student‐t FIAPARCH model Overall, the evidence of asymmetric long memory volatility is in favor of the skewed Student‐t FIAPARCH (1, d ,1) model for the Nikkei 225 Index returns according to the lowest value of AIC and P(70) But for JPY‐USD exchange 7) This finding is inconsistent with that of Tse (1998) who n little evidence of volatility asymmetry in the JPY‐USD exchange rate (January 1987 ‐ June 1994) using the FIAPARCH model under the normal distribution innovation A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 229 Table Estimation results from the FIGARCH (1, d ,1) model Series Nikkei 225 Index JPY‐USD rate Distributions Normal Student Sk‐ST Normal Student Sk‐ST µ 0.075 0.069 0.060 0.0002 0.013 ‐0.0001 (0.013)*** (0.012)*** (0.013)*** (0.009) (0.008) (0.009) ω α1 d β1 0.038 0.026 0.025 0.036 0.025 0.024 (0.007)*** (0.008)*** (0.008)*** (0.012)*** (0.008)*** (0.008)*** 0.284 0.146 0.148 0.440 0.385 0.387 (0.038)*** (0.039)*** (0.039)*** (0.079)*** (0.050)*** (0.048)*** 0.582 (0.062)*** 0.558 (0.074)*** 0.560 (0.076)*** 0.244 (0.035)*** 0.314 (0.053)*** 0.316 (0.052)*** 0.672 0.627 0.629 0.602 0.653 0.657 (0.054)*** (0.074)*** (0.075)*** (0.083)*** (0.058)*** (0.056)*** 6.725 6.802 5.116 5.250 (0.517)*** (0.530)*** (0.400)*** (0.422)*** υ ‐0.043 (0.018)** ln(k ) ‐0.070 (0.020)*** ln(L ) ‐9068.50 ‐8875.29 ‐8872.54 ‐4258.88 ‐4102.99 ‐4097.20 AIC 3.20562 3.13771 3.13709 1.98597 1.91383 1.91160 Q2 (20) 16.14 19.06 19.01 11.49 15.99 16.38 ARCH(10) 0.254 0.893 0.872 0.615 0.983 0.998 P(70) 185.37*** 111.53*** 78.83 233.01*** 90.34** 65.23 Notes: Sk‐ST represents the skewed Student‐t distribution Standard errors in parentheses correspond to parameter estimates ln (L ) is the value of the maximized Gaussian log likelihood, and AIC is the Akaike information criteria ARCH(10) represents the F‐statistic value of ARCH test statistics with a lag of 10 The ARCH test is based on standardized residuals P(70) is the value of the Pearson goodness‐of‐fit statistic with 70 cells ** and *** indicate rejection of the null hypothesis at the 5% and 1% significance levels, respectively See Table rate returns, the skewed Student‐t FIGARCH (1, d ,1) model is preferred for modeling a symmetric long memory volatility process based on the same criteria Hence, we designate the FIAPARCH (1, d ,1) model for the Nikkei 225 Index and the FIGARCH (1, d ,1) model for the JPY‐USD exchange rate to calculate the in‐ sample and out‐of‐sample VaR estimations 230 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 Table Estimation results from the FIAPARCH (1, d ,1) model Series Nikkei 225 Index µ ω α1 d β1 δ γ JPY‐USD rate Student Sk‐ST Normal Student Sk‐ST 0.040 0.048 0.040 ‐0.002 0.012 ‐0.001 (0.013)*** (0.012)*** (0.013)*** (0.009) (0.008) (0.009) 0.078 0.080 0.083 0.034 0.027 0.025 (0.017)*** (0.020)*** (0.020)*** (0.014)** (0.013)** (0.012)** 0.264 0.141 0.140 0.576 0.398 0.401 (0.047)*** (0.053)*** (0.053)*** (0.103)*** (0.062)*** (0.061)*** Distributions Normal 0.438 0.405 0.409 0.235 0.307 0.301 (0.050)*** (0.054)*** (0.055)*** (0.054)*** (0.088)*** (0.086)*** 0.542 0.466 0.469 0.701 0.654 0.653 (0.070)*** (0.085)*** (0.086)*** (0.075)*** (0.068)*** (0.066)*** 1.527 (0.091)*** 1.488 (0.101)*** 1.490 (0.102)*** 1.803 (0.225)*** 1.944 (0.301)*** 1.992 (0.293)*** 0.489 0.523 0.516 0.241 0.066 0.056 (0.052)*** (0.072)*** (0.071)*** (0.088)*** (0.076) (0.072) 7.279 7.350 5.144 5.271 (0.613)*** (0.623)*** (0.406)*** (0.428)*** υ ‐0.043 (0.018)** ln(k ) ‐0.070 (0.020)*** ln(L ) ‐8982.47 ‐8820.67 ‐8818.03 ‐4253.50 ‐4102.56 ‐4096.86 AIC 3.17593 3.11912 3.11854 1.98440 1.91456 1.91237 Q2 (20 ) 18.73 21.52 21.45 14.16 15.12 15.37 ARCH (10) 0.230 0.860 0.854 1.009 0.945 0.947 P(70) 152.33*** 87.07 86.80 233.17*** 78.99 62.59 Note: See Table 3 Empirical results of the VaR analyses In this section, we move to compute not only the in‐sample VaR values for examination of the estimated goodness‐of‐fit ability, but also the out‐of‐sample VaR values to evaluate the forecasting performance of the estimated models Under the assumption of different distribution innovations, the VaR values A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 231 using the long memory FIGARCH and FIAPARCH models are tested at a significance level of α , ranging from 5% to 0.25%, and their performance is assessed by computing the failure rate If the VaR models are correctly specified, the failure rate would be equal to the pre‐specified significance level of α This information provides us with a more accurate assessment of possible trading losses 1) In‐sample VaR analysis The empirical results for the in‐sample VaR analysis of the Nikkei 225 Index and JPY‐USD exchange rate returns are summarized in Table and Table 6, respectively These tables contain the failure rates, Kupiec LR test statistics, and P‐values According to these tables, we observe that the normal and Student ‐t distribution innovations have poor performance for long and short trading positions In particular, as α ranges from 0.025 to 0.0025, the failure rates significantly exceed the prescribed quantiles and the null hypothesis ( f = α ) of Kupiec LR test is often rejected in the normal and Student‐t distribution innovations This finding indicates that the normal and Student‐t distribution models tend to underestimate the in‐sample VaR values On the other hands, the skewed Student‐t distribution FIGARCH and FIAPARCH models significantly improve on the in‐sample VaR performance for long and short trading positions This finding seems to support the view that the skewed Student‐t VaR models predict crucial loss more accurately than models with normal and Student‐t distribution innovations in the in‐sample VaR analysis In addition, the FIGARCH and FIAPARCH models with the skewed Student‐t distribution innovation outperform those with the normal and Student‐t innovations to capture asymmetric and fat‐tailed distributions of the Japanese financial data In addition, the models based on the Student‐t distribution innovation yield lower Kupiec LR test values in contrast to the models based on the normal distribution Nevertheless, their P‐values are still statistically significant, rejecting 232 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 Table In‐sample VaR estimation for the Nikkei 225 Index α Short positions Failure Kupiec rate LR Long positions P‐value α Failure Kupiec rate LR 0.0517 0.364 P‐value Normal distribution FIAPARCH 0.950 0.9602 13.43*** 0.000 0.05 0.975 0.9784 0.99 0.9881 0.995 0.9975 0.546 2.895* 0.088 0.025 0.0293 4.116** 0.042 1.819 0.177 0.01 0.0153 14.15*** 0.000 0.9918 9.341*** 0.002 0.005 0.0090 14.75*** 0.000 0.9938 21.76*** 0.000 0.0025 0.0061 21.76*** 0.000 Student‐t distribution FIAPARCH 0.950 0.9567 5.629** 0.017 0.05 0.0572 5.970** 0.014 0.975 0.9798 5.875** 0.015 0.025 0.0287 3.188* 0.074 0.99 0.9913 1.082 0.298 0.01 0.0111 0.702 0.401 0.995 0.9952 0.061 0.804 0.005 0.0058 0.742 0.388 0.9975 0.9975 0.001 0.967 0.0025 0.0024 0.001 0.967 Skewed Student‐t distribution FIAPARCH 0.950 0.9531 1.236 0.266 0.05 0.0531 1.175 0.278 0.975 0.9773 1.368 0.242 0.025 0.0257 0.143 0.704 0.99 0.9904 0.123 0.725 0.01 0.0093 0.237 0.625 0.995 0.9941 0.742 0.388 0.005 0.0047 0.061 0.804 0.9975 0.9971 0.231 0.630 0.0025 0.0024 0.001 0.967 Note: *, ** and *** indicate the rejection of null hypothesis at the 10%, 5% and 1% significance levels, respectively the null hypothesis of f = α in both cases In Table 6, the Student‐t FIGARCH (1, d ,1) model performs more unfavorably for in‐sample VaR calculations Thus, symmetric Student‐t models may mislead the selection of the appropriate model and provide a spurious assessment of in‐sample VaR estimates 2) Out‐of‐sample VaR analysis We further assess the performance of the model with the normal, Student‐t, and skewed Student‐t innovations by computing out‐of‐sample VaR forecasts A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 233 Table In‐sample VaR estimation for the JPY‐USD exchange rate α Short positions Failure Kupiec rate LR Long positions P‐value α Failure Kupiec rate LR 0.0500 0.000 P‐value Normal distribution FIGARCH 0.950 0.9583 6.607*** 0.010 0.05 0.983 0.975 0.9771 0.859 0.353 0.025 0.0351 16.19*** 0.000 0.99 0.9883 1.113 0.291 0.01 0.0207 38.11*** 0.000 0.995 0.9939 0.899 0.342 0.005 0.0118 29.39*** 0.000 0.9975 0.9962 2.247 0.133 0.0025 0.0090 44.28*** 0.000 Student‐t distribution FIGARCH 0.950 0.9548 2.168 0.140 0.05 0.0561 3.267* 0.070 0.975 0.9795 3.807* 0.051 0.025 0.0337 12.22*** 0.000 0.99 0.9946 11.25*** 0.000 0.01 0.0123 2.215 0.136 0.995 0.9983 13.29*** 0.000 0.005 0.0069 3.029* 0.081 0.9975 0.9988 3.837** 0.050 0.0025 0.0051 9.071*** 0.002 Skewed Student‐t distribution FIGARCH 0.950 0.9483 0.258 0.611 0.05 0.0496 0.014 0.905 0.975 0.9762 0.278 0.598 0.025 0.0288 2.524 0.112 0.99 0.9923 2.525 0.112 0.01 0.0100 0.000 0.992 0.995 0.9969 3.912** 0.047 0.005 0.0060 0.899 0.342 0.9975 0.9986 2.494 0.114 0.0025 0.0039 3.109 0.077 Note: See Table Following the analysis procedure of Tang and Shieh (2006), we adopt an iterative procedure in which the estimated model for the whole sample is predicted and then compared with a one‐day‐ahead VaR for both the long and short positions.8) The empirical results of the out‐of‐sample VaR analyses for the Nikkei 225 8) Tang and Shieh (2006) compute 500 out‐of‐sample VaR estimates for three stock indices Following their analysis, however, we encountered many cases of zero failure rates in the result To overcome this problem in our out-of-sample forecasting, we limited the use of samples to only the last four years (1000 observations) and the models are re-estimated every 50 observations 234 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 Table Out‐of‐sample VaR estimation for the Nikkei 225 Index α Short positions Failure Kupiec rate LR Long positions P‐value α Failure Kupiec rate LR 0.048 0.085 P‐value Normal distribution FIAPARCH 0.950 0.951 0.021 0.884 0.05 0.770 0.975 0.977 0.168 0.681 0.025 0.031 1.373 0.241 0.99 0.992 0.433 0.510 0.01 0.019 6.472*** 0.010 0.995 0.997 0.939 0.332 0.005 0.010 3.888** 0.048 0.9975 0.998 0.107 0.742 0.0025 0.008 7.640*** 0.005 Student‐t distribution FIAPARCH 0.950 0.946 0.328 0.566 0.05 0.055 0.510 0.474 0.975 0.980 1.099 0.294 0.025 0.029 0.624 0.429 0.99 0.996 4.706** 0.030 0.01 0.012 0.379 0.537 0.995 0.999 4.797** 0.028 0.005 0.008 1.529 0.216 0.9975 1.000 NaN 1.000 0.0025 0.003 0.094 0.758 Skewed Student‐t distribution FIAPARCH 0.950 0.941 1.616 0.203 0.05 0.051 0.020 0.884 0.975 0.974 0.040 0.840 0.025 0.027 0.160 0.689 0.99 0.994 1.886 0.169 0.01 0.011 0.097 0.754 0.995 0.998 2.343 0.125 0.005 0.008 1.529 0.216 0.9975 1.000 NaN 1.000 0.0025 0.003 0.094 0.758 Note: NaN represents the statistics is not available See Table Index and JPY‐USD exchange rate returns are reported in Table and Table 8, respectively From these VaR Tables, we see that our results from the out‐of‐ sample VaR analysis are slightly different from those of the in‐sample VaR analysis In particular, the models based on symmetric distributions (normal and Student‐t distributions) provide a relatively good performance for the short and long positions in both cases For example, when compiling short positions for the Nikkei 225 Index, the normal distribution innovation is better at estimating VaR values at the 5% level, whereas in the case of long positions for the JPY‐ USD exchange rate, the Student‐t distribution innovation performs better than A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 235 Table Out‐of‐sample VaR estimation for the JPY‐USD exchange rate α Short positions Failure Kupiec rate LR Long positions P‐value α Failure Kupiec rate LR P‐value Normal distribution FIGARCH 0.950 0.966 6.042** 0.013 0.05 0.047 0.193 0.660 0.975 0.980 1.099 0.294 0.025 0.030 0.964 0.325 0.99 0.988 0.379 0.537 0.01 0.013 0.830 0.362 0.995 0.994 0.188 0.663 0.005 0.007 0.714 0.397 0.9975 0.995 1.937 0.163 0.0025 0.006 3.517* 0.060 Student‐t distribution FIGARCH 0.950 0.962 3.293* 0.069 0.05 0.05 0.000 1.000 0.975 0.982 2.224 0.135 0.025 0.028 0.355 0.550 0.99 0.995 3.093* 0.078 0.01 0.007 1.015 0.313 0.995 0.997 0.939 0.332 0.005 0.003 0.939 0.332 0.9975 1.000 NaN 1.000 0.0025 0.002 0.107 0.742 Skewed Student‐t distribution FIGARCH 0.950 0.950 0.000 1.000 0.05 0.046 0.345 0.556 0.975 0.979 0.693 0.404 0.025 0.020 1.099 0.294 0.99 0.990 0.000 1.000 0.01 0.006 1.886 0.169 0.995 0.996 0.215 0.642 0.005 0.003 0.939 0.332 0.9975 0.998 0.107 0.742 0.0025 0.002 0.107 0.742 Note: NaN represents the statistics is not available See Table other innovations (at the 5% level) More specifically, for the long trading position, the VaR performance of the Student‐t distribution innovation is equal or better to that of the skewed Student‐t distribution Nevertheless, accounting for the overall shape of return distributions, VaR models based on the skewed Student‐t distribution generally provide more accurate forecasting results than the normal and Student‐t distribution VaR models for both long and short trading positions in the Nikkei 225 Index and JPY‐USD exchange rate The values of the Kupiec LR test are insignificant and so the null hypothesis of f = α at most α levels for both long and short trading positions can be rejected 236 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 In the case of the Nikkei 225 Index, the null hypothesis in the normal distribution FIGARCH model for long positions is rejected when α is between 0.0025 to 0.01, while the null hypothesis is rejected in the Student‐t distribution FIGACH model for short positions when α lies in between 0.995 to 0.99 In the case of the JPY‐USD exchange rate, the normal distribution FIGARCH model rejects the null hypothesis when α is 0.0025 for long positions and when α is 0.95 for short positions, whereas the Student‐t distribution FIGARCH model rejects the null hypothesis for short positions when α is 0.95 and 0.99 But the skewed Student‐t distribution FIGARCH model accepts the null hypothesis for all cases Thus, the out‐of‐sample analysis reveals that the skewed Student‐t distribution FIGARCH model serves as a flexible and accurate tool for estimating asymmetry and tail‐fatness in return distributions Several empirical studies have reached the same conclusion that various GARCH class models based on the skewed Student‐t distribution innovation produce accurate VaR values (Bali and Theodossiou 2007; Degiannakis 2004; Giot and Laurent 2003; Wu and Shieh 2007) Among these studies, Degiannakis (2004) concludes that the FIAPARCH model based on the skewed Student‐t distribution innovation generates superior out‐of‐sample volatility forecasting of stock indices Similarly, Wu and Shieh (2007) support the relevance of skewed Student‐t distribution innovation in modeling long memory characteristics in the volatility of T‐bond futures As a result, we conclude that irrespective of in‐sample and out‐of‐sample VaR calculations for both long and short positions, the skewed Student‐t distribution innovation is more suitable for modeling long memory properties of volatility in Japanese financial returns Ⅳ Conclusions Recent econometric literature has focused on the distributional properties of financial assets, which exhibit fatter tails and skewer means than the normal A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 237 distribution The correct assumption of return distribution might improve the estimated performance of Value‐at‐Risk models in financial markets In addition, it is well known that the volatility of financial time series exhibits such stylized factors as volatility clustering, volatility asymmetry, and volatility persistence Accurate volatility modeling is an important component of market risk management, investment portfolio and the pricing of derivative securities (Poon and Granger 2003) To this extent, we have investigated two Japanese financial data sources, the Nikkei 225 Index and JPY‐USD exchange rate, using the symmetric FIGARCH and asymmetric FIAPARCH models with normal, Student‐t, and Skewed Student‐t distribution innovations From the results of our analysis, we find that the skewed Student‐t FIGARCH and FIAPARCH VaR models for long and short positions predict critical loss more accurately than models with normal and Student‐t innovations For example, the FIAPARCH model with a skewed Student‐t distribution is preferred for the Nikkei 225 Index, while the FIGARCH model with a skewed Student‐t distribution is regarded as optimal for the JPY‐ USD exchange rate Skewed Student‐t distribution models improve VaR forecasting and provide a more rigorous criteria of model selection regarding long memory volatility processes In all, our findings provide important implications for investors and portfolio managers For example, risk‐adverse investors or portfolio managers may prefer to evaluate VaR measurements for Japanese financial markets using the skewed 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holds a PhD in Economics from Korea University He has been a visiting scholar to the University of Washington in Seattle and the University of Colorado in Denver His research fields include financial market dynamics, corporate finance, small and medium size firms, regional economics, and applied microeconomics He has published over 70 journal papers on those fields and various books including Intelligent Finance (University of Ballarat, 2004), Regional Innovation and Industrial Networks in Busan (Kuyngsung University Press, 2004), Business Democracy and Corporate Governance (Baeksan Pub Co., 2002), and Economic Policy (Pukyong National University Press, 1996) Sang‐ Hoon Kang Sang-Hoon Kang is a PhD candidate at the University of South Australia He is interested in the long memory properties of financial time series He has recently published several journal papers, which include “Long‐Term Dependence in the Foreign Exchange Markets: International Evidence,” (The Journal of the Korean Economy, 2006) and “Asymmetric Long Memory Feature in the Volatility of Asian Stock Markets,” (Asia‐Pacific Journal of Financial Studies, 2006) 242 󰡔對外經濟硏究󰡕 제11권 제1호 2007년 6월 尹盛民 부경대학교 경제학부 교수로 재직 중이다 고려대학교 경제학과에서 경제학 학사, 석사 및 박사 학위를 취득하였고, University of Washington at Seattle, University of Colorado at Denver에서 방문교수를 하였다 주요 연구분야는 금융시장 동학, 기업금융, 중소기업, 지역경제, 소비자이론 응용 등이다 Intelligent Finance (University of Ballarat, 2004), 󰡔지역혁신과 부산지역의 산업네트워크󰡕(경성대학교 출판부, 2004), 󰡔기업민주주의 와 기업지배구조󰡕(백산서당, 2002), 󰡔경제정책󰡕(부경대학교 출판부, 1996) 등의 저서 외에 70여 편의 학술논문을 발표하였다 姜商勳 현재 University of South Australia 에서 PhD 과정에 있다 주 관심 분야는 금융시 계열의 장기기억 분석이다 주요 논문으로는 “Long‐Term Dependence in the Foreign Exchange Markets: International Evidence,” (The Journal of the Korean Economy, 2006), “Asymmetric Long Memory Feature in the Volatility of Asian Stock Markets,” (Asia‐Pacific Journal of Financial Studies, 2006) 등이 있다 ... long memory volatility models, namely the FIGARCH and FIAPARCH To further enhance A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 215... is clearly observable in the graphs A Skewed Student-t Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 225 Table Unit root tests and Lo’s R/S analysis... Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets 233 Table In? ??sample VaR estimation for the JPY‐USD exchange rate α Short positions Failure Kupiec rate