Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 5, 1487-1494 2024 Publisher: Learning Gate DOI: 10.55214/25768484.v8i5.1861 Β© 2024 by the author; licensee Learning Gate Β© 2024 by the author; licensee Learning Gate * Correspondence: Choan@Skuniv.Ac.Kr Research on the volatility forecasting model of KOSPI index returns using AR(M)-GARCH(P,Q) model Chang-Ho An1* 1Department Of Financial Information Engineering, Seokyeong University, Seoul 02713; Choan@Skuniv.Ac.Kr (C.H.A.) Abstract: In this study, we estimated the volatility of the KOSPI index returns and analyzed volatility trends. The data used in the study consisted of monthly observations from January 2005 to December 2022, and the KOSPI index raw data was transformed into log returns. The volatility estimation model used the AR(m)-GARCH(p,q) model, which combines the autoregressive error model and the GARCH model that explains the persistence of volatility at low orders. The goodness of fit of the model was confirmed using the Portmanteau Q-test and LM-test. Applying the autoregressive error model revealed significant autocorrelation in the log returns of the KOSPI index at lags 3 and 6. Residual analysis indicated that the residuals followed white noise, but the squared residuals exhibited heteroscedasticity. Therefore, after fitting the autoregressive error model, we applied the GARCH model and conducted residual analysis, finding both the residuals and squared residuals significant at a 5% significance level. The volatility forecasting results indicated a continuous increase in volatility. The findings of this study are expected to provide important implications for policymakers responsible for risk management in the Korean stock market. Keywords: AR(m)-GARCH(p,q) model, Autoregressive error model, GARCH model, Q-test and LM-test t. Return rate. 1. Introduction Recent uncertainties in the global economy, coupled with the proliferation of advanced financial technologies and algorithmic trading, have led to increased volatility in international financial markets. Key factors contributing to heightened market volatility include US interest rate hikes, political instability, trade wars, regulatory constraints on major financial institutions' risk investments, as well as potential shifts to a tightening monetary policy in China and fiscal crises in Europe. These global issues significantly impact various domestic industries such as financial markets, financial industries, export industries, real estate markets, and more, including the domestic financial market sectors, which involve foreign exchange, stock, bond, and short-term capital markets. Particularly, sustained US dollar strength expands financial market volatility, increasing uncertainty in both domestic and international stock indices, interest rates, exchange rates, and other factors. This acceleration of the safe asset preference by investors can result in significant declines in stock indices and bond prices. It also affects domestic real economies, thereby reducing consumer and investor confidence and hindering economic growth rates. The volatility of financial asset returns, such as derivatives, stocks, futures, exchange rates, etc., is a crucial element in financial trading, encompassing risk management, derivative valuation, and portfolio selection. The stock market connects investors like enterprises, individuals, and institutions, through the securities issued by businesses to raise industrial capital within a country's national economy, enabling efficient distribution of capital. The expansion of stock market volatility can weaken the transmission of information regarding corporate value, resulting in misguided decision-making by enterprises, 1488 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 5: 1487-1494, 2024 DOI: 10.55214/25768484.v8i5.1861 Β© 2024 by the author; licensee Learning Gate individuals, and government officials [1]. Factors influencing stock return volatility include governmental legislation, collective bargaining agreements, and policy decisions impacting the economy, as well as inflation, consumer spending indicators, and GDP indicators potentially influencing market performance. Furthermore, enhanced governmental regulation in specific industries and the performance of individual businesses, such as the launch of innovative products meeting consumer satisfaction, also affect volatility [2]. Volatility serves as an anticipated indicator of financial asset uncertainty and price fluctuations. The increase in stock market volatility serves as a destabilizing factor due to changes in political and economic performance, as well as corporate activities. Managing volatility is crucial for the stability and growth of the national economy, considering it as a key means of corporate financing and a representative savings method for households. Stock returns reflect the intrinsic value of companies and are therefore utilized as a significant indicator in financial markets. Therefore, this study aims to examine the volatility trends of the composite stock index using the AR(m)-GARCH(p,q) model, which combines autoregressive error and GARCH(p,q) models. 2. Prior Research The volatility of financial assets is associated with risk premiums and plays a crucial role in the decision-making of financial participants in financial transactions. Estimating volatility in financial markets has been studied extensively over the years, with a variety of research models developed. Considering the current expansion of financial market volatility due to factors such as intensified inflation, accelerated monetary tightening, and increased risk of economic downturns, volatility estimation remains a significant issue. Various existing research models for volatility estimation include: Engle and Rangel (2008) proposed the spline-GARCH model, which allows for long-term forecasting of volatility using financial market data from 48 countries and provides estimates for expected volatility in newly opened markets [3]. Conrad, Loch, and Rittler (2014) analyzed long-term correlations between oil and US stock returns using the DCC-MIDAS model [4]. Chauvet et al. (2012) analyzed whether the symmetric characteristics of financial market volatility are useful for predicting real economic conditions and argued that stock volatility measures and common factors improve the macroeconomic forecasting of financial indicators in the short term [5]. Conrad and Loch (2014) used the GARCH-MIDAD model to analyze the relationship between stock market risk and the macroeconomic environment and confirmed that the ability to predict long-term volatility of stock prices was improved[6]. Asgharian, Hou, and Javed (2013) demonstrated that extracting common factors from various financial and macroeconomic variables and incorporating them into the GARCH-MIDAS model improves its predictive ability [7]. Seung Hee Lee and Hee Joon Han (2016) analyzed the volatility of KOSPI index returns using a semiparametric single-index volatility model combining GARCH models explaining short-term fluctuations and single-index models explaining long-term volatility fluctuations, demonstrating that the long-term volatility of the Korean stock market is best explained when using the housing price index [8]. Young Im Lee and Jin Lee (2017) showed that the GARCH-MIDAS model effectively explains domestic stock market volatility by reflecting not only domestic macroeconomic variables but also international variables such as overseas prices, oil prices, and production [9]. Do Kyun Chun (2017) compared and analyzed the volatility of exchange rates such as USD-KRW, JPY-KRW, EUR-KRW, and GBP-KRW using stochastic volatility models and GARCH(1,1) models to explain exchange rate volatility [10]. And Conrad and Kleen (2020) argued that the mixed-frequency GARCH model has significant explanatory power for long-memory processes and types of long autocorrelation similar to volatility processes such as squared returns [11]. 3. Research Model 3.1. Data Transformation The data consists of monthly data of the composite stock index from January 2005 to December 2022. 1489 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 5: 1487-1494, 2024 DOI: 10.55214/25768484.v8i5.1861 Β© 2024 by the author; licensee Learning Gate In this study, to estimate the volatility of the composite stock index, we transformed the original data by taking the natural logarithm to use relative changes. Specifically, at time 𝑑 , if the value of the composite stock index is 𝑃𝑑, the formula for calculating returns is given by Equation (1). 𝑍𝑑 = π‘™π‘œπ‘” ( 𝑃𝑑 π‘ƒπ‘‘βˆ’1 ) (1) where, the logarithmic return of the composite stock index at time [𝑑 βˆ’ 1, 𝑑] derived from Equation (1) is referred to as log-return. 3.2. AR(m)-GARCH(p,q) Model The AR(m)-GARCH(p,q) model combines an autoregressive model of order m for the errors and a model where the variance of the errors follows a GARCH(p,q) process, and is expressed as follows. 𝑍𝑑 = π‘₯𝑑 β€² 𝛽 + πœ€π‘‘ πœ€π‘‘ = βˆ’πœ™1 πœ€π‘‘βˆ’1 βˆ’ β‹― βˆ’ πœ™π‘šπœ€π‘‘βˆ’π‘š + πœˆπ‘‘ πœˆπ‘‘ = πœŽπ‘‘ 𝑒𝑑 , 𝑒𝑑~𝑖. 𝑖. 𝑑 𝑁(0,1) πœŽπ‘‘ 2 = 𝛼0 + βˆ‘ π›Όπ‘–πœˆπ‘‘βˆ’π‘– π‘ž 𝑖=1 + β‹― + βˆ‘ 𝛽𝑗 πœŽπ‘‘βˆ’π‘— 2𝑝 𝑗=1 (2) where, (Equation 2) must satisfy 𝛼0 > 0, 𝛼𝑖 β‰₯ 0, 𝛽𝑗 β‰₯ 0, βˆ‘ 𝛼𝑖 π‘ž 𝑖=1 + βˆ‘ 𝛽𝑗 𝑝 𝑗=1 < 1. An autoregressive error model utilizes local information when there is autocorrelation among error terms, allowing for more accurate estimation and prediction. The GARCH(p,q) model, on the other hand, uses the squares of all past residuals to explain variance, enabling estimation of volatility over long lags with fewer parameters. 3.3. Heteroscedasticity Test Statistical tests such as the Portmanteau Q-test and the Lagrange multiplier test (LM-test) are employed to test for the presence of autocorrelation among squared errors. The statistic for the Portmanteau Q-test is as follows [12], 𝑄 = 𝑇(𝑇 + 2) βˆ‘ πΆπ‘œπ‘Ÿπ‘Ÿ(πœˆπ‘‘ 2Μ‚,πœˆπ‘‘βˆ’π‘– 2Μ‚ ) (𝑇 βˆ’ 𝑖) π‘ž 𝑖=1 (3) and the statistics for Lagrange multiplier test is as follows [13]. 𝐿𝑀 = 𝑇 π‘Šβ€² 𝑍(𝑍′𝑍)βˆ’1 𝑍′ π‘Š π‘Šβ€² π‘Š (4) 4. Research Results 4.1. Volatility of Log Returns and Heteroskedasticity Test It was observed that the squared data of the KOSPI index's log returns exhibit a clustering and persistence phenomenon of volatility, where once volatility increases, it tends to remain high for a period, and similarly, once it decreases, it stays low for a certain duration Figure 1. 1490 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 5: 1487-1494, 2024 DOI: 10.55214/25768484.v8i5.1861 Β© 2024 by the author; licensee Learning Gate Figure 1. Volatility of squared log returns. The Portmanteau Q-test results for the squared log returns of the KOSPI index show that the p - values of the chi-square statistics are smaller than the significance level at all lags (lag 1 to lag 30). This indicates that the squared log returns of the KOSPI index exhibit significant autocorrelation in variance, suggesting that they are not independent Table 1. Table 1. Portmanteau Q-test of squared log returns. To lags Chi-square Pr > ChiSq Autocorrelation coefficient 6 14.11 0.0285 0.010 0.057 0.054 0.104 0.142 0.062 12 37.90 0.0002 -0.019 0.073 0.308 -0.053 0.044 -0.008 18 45.09 0.0004 -0.018 0.072 0.077 -0.079 0.101 0.053 24 46.16 0.0042 0.004 0.023 0.020 0.027 0.043 -0.030 30 50.82 0.0102 0.014 0.088 0.032 0.031 0.084 0.044 Furthermore, heteroskedasticity testing of the KOSPI index's log return data using the Portmanteau Q-test and the Lagrange multiplier test indicates that conditional heteroskedasticity is present from lag 5 onwards in the Portmanteau Q-test and from lag 6 onwards in the Lagrange multiplier test Table 2. Table 2. Heteroscedasticity test of log returns. Order Q Pr > Q LM Pr > LM 1 0.3601 0.5484 1.3854 0.5796 2 1.6216 0.4445 3.0364 0.4969 3 2.7409 0.4333 4.2241 0.5218 4 5.4209 0.2468 5.1208 0.3625 1491 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 5: 1487-1494, 2024 DOI: 10.55214/25768484.v8i5.1861 Β© 2024 by the author; licensee Learning Gate 5 11.8788 0.0635 8.2247 0.0937 6 20.0082 0.0028 10.2041 0.0193 7 20.0326 0.0055 16.2311 0.0290 8 22.2291 0.0045 18.5124 0.0408 9 45.1552 <.0001 21.3645 0.0001 10 45.9518 <.0001 22.0142 <.0001 11 46.5010 <.0001 22.5716 <.0001 12 46.5207 <.0001 23.8715 0.0001 4.2. Autoregressive Error Model The AR order of the autoregressive error model was determined to be 𝑇1/3 , and significant parameters estimated by backward elimination showed πœ™3, πœ™6 to be significant at the 5% significance level (Table 3). Table 3. Estimation of parameters for the AR (3,6) model. AR (3,6) model parameter estimates Variable Estimate S. E t -value Pr >|t| Intercept 0.0004024 0.000136 2.01 0.0224 AR3 -0.1423 0.0680 -2.21 0.0366 AR6 0.1484 0.0695 2.13 0.0339 Therefore, the autoregressive error model estimated using Table 3 can be represented as follows. 𝑙𝑛 𝑍𝑑 = 0.0004024 + πœ€π‘‘ πœ€π‘‘ = 0.1423 πœ€π‘‘βˆ’3 βˆ’ 0.1484πœ€π‘‘βˆ’6 (5) 4.3. Conditional Heteroskedasticity Model After fitting the autoregressive error model in Section 3.2, the Portmanteau Q-test results for the residuals and squared residuals at all lags indicate that while the residuals show no significant autocorrelation at the 5% significance level, the squared residuals exhibit significant autocorrelation (Table 4). Table 4. Portmanteau Q-test for residuals and squared residuals. AR (3,6) model residual analysis Residual To lags Chi-square Pr > ChiSq Autocorrelation coefficient 6 1.38 0.9668 -0.031 0.025 0.001 0.060 -0.030 -0.010 12 5.28 0.9478 0.099 -0.043 0.006 -0.045 0.017 -0.057 18 13.82 0.7407 -0.054 -0.129 0.042 -0.082 -0.072 -0.056 24 17.31 0.8352 0.032 0.045 -0.027 0.072 -0.040 0.061 30 26.44 0.6523 -0.120 -0.137 0.026 0.018 0.014 -0.051 Residual squared To lags Chi-square Pr > ChiSq Autocorrelation coefficient 6 13.34 0.0379 0.013 0.051 0.110 0.106 0.123 0.137 12 36.27 0.0003 -0.006 0.079 0.298 -0.065 0.042 0.002 18 41.92 0.0011 -0.014 0.067 0.037 0.089 0.085 0.053 1492 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 5: 1487-1494, 2024 DOI: 10.55214/25768484.v8i5.1861 Β© 2024 by the author; licensee Learning Gate 24 42.59 0.0111 0.004 0.006 -0.006 0.016 0.025 0.042 30 45.61 0.0338 0.010 0.078 0.025 0.014 0.066 0.027 In the AR (3,6) model, the fact that the residual square does not follow white noise means that heteroscedasticity exists. When heteroscedasticity exists, the volatility estimation model mainly used in reality is the low-order GARCH (1,1). Therefore, in this study, the AR (3,6)-GARCH (1,1) model was applied to estimate the model, and as a result of the estimation, all parameters were found to be significant at the significance level of 5% (Table 5). Table 5. Parameter estimation of AR (3,6)-GARCH (1,1) model. AR (3,6)-GARCH (1,1) Model Parameter Estimates Variable Estimate S. E t -Value Pr >|t| Intercept 0.000513 0.000197 2.19 0.0201 AR3 -0.1123 0.0324 -5.09 <.0001 AR6 -0.0632 0.0212 -2.13 0.0216 ARCH0 0.0000496 0.0000333 1.49 0.1368 ARCH1 0.0812 0.0483 2.30 0.0216 GARCH1 0.8523 0.0477 18.49 <.0001 4.4. Diagnosis of AR (3,6)-GARCH (1,1) Model After fitting the AR (3,6)-GARCH (1,1) model, the Portmanteau Q-tests for residuals and squared residuals showed no significant autocorrelation at a 5% significance level across all lags Table 6. Table 6. Portmanteau Q-tests for residuals and squared residuals. AR (3,6)-GARCH (1,1) Model Residual Analysis Residual To lags Chi- square Pr > chisq Autocorrelation coefficient 6 0.38 0.9990 -0.017 0.017 -0.008 0.003 0.033 -0.003 12 4.35 0.9763 0.088 -0.022 0.010 0.026 -0.006 -0.091 18 12.65 0.8122 -0.075 -0.105 0.044 -0.068 -0.081 -0.075 24 17.50 0.8265 -0.043 0.076 -0.051 0.020 -0.037 0.090 30 24.15 0.7651 -0.091 -0.133 -0.016 0.011 0.002 -0.027 Residual squared To Lags Chi- Square Pr > ChiSq Autocorrelation Coefficient 6 1.96 0.9230 -0.072 -0.042 0.032 -0.016 0.027 0.003 12 13.29 0.3484 -0.040 0.087 0.166 -0.112 -0.023 -0.018 18 24.50 0.1394 -0.069 0.013 -0.019 -0.080 0.189 0.020 24 26.57 0.3247 -0.009 -0.034 -0.032 0.017 -0.036 -0.069 30 28.25 0.5573 0.051 0.053 -0.036 -0.007 0.003 -0.007 Using Table 5, the final estimated form of the AR (3,6)-GARCH (1,1) model is represented as follows. 𝑙𝑛𝑍𝑑 = 0.000513 + πœ€π‘‘ πœ€π‘‘ = 0.1123πœ€π‘‘βˆ’3 + 0.0632πœ€π‘‘βˆ’6 (6) 1493 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 5: 1487-1494, 2024 DOI: 10.55214/25768484.v8i5.1861 Β© 2024 by the author; licensee Learning Gate πœŽπ‘‘ 2 = 0.0000496 + 0.0812πœˆπ‘‘βˆ’1 + 0.8523πœŽπ‘‘βˆ’1 2 4.5. Forecasting Volatility Using AR (3,6)-GARCH (1,1) Model The one-step ahead forecast of volatility and multi-step ahead forecasts of volatility for the forecasting period based on the AR (3,6)-GARCH (1,1) model are shown in (Fig. 2). It is observed that volatility increases gradually from January 2023 to December 2023. Figure 2. Volatility Forecasting by AR (3,6)-GARCH (1,1) Model. 5. Conclusion There are various models for predicting the volatility of financial assets' returns. In this study, we used the AR(m)-GARCH(p,q) model to forecast monthly volatility of KOSPI index returns in order to understand the trend in volatility. Analysis using data from 2005 to 2022 revealed the following: The squared log returns of the KOSPI index exhibited clustering and persistence in volatility, confirmed by Portmanteau Q-tests showing significant autocorrelation from lag 5 onwards, and Lagrange multiplier tests indicating heteroscedasticity from lag 6 onwards. To estimate KOSPI index volatility, we applied the autoregressive conditional heteroskedasticity model after employing backward elimination, revealing significant parameters at the 5% significance level for lags 3 and 6. Testing the independence of residuals using the AR (3,6) model showed that residuals followed white noise, although squared residuals exhibited autocorrelation. Adding the GARCH (1,1) model to the AR (3,6) model resulted in all parameters being significant, and both residuals and squared residuals following white noise. Therefore, the final model for KOSPI index return volatility was determined as AR(3,6)-GARCH (1,1), allowing for forecasts of volatility during the fitting period and the forecasting period. Given the current uncertainty in the global economy leading to increased stock market volatility, this study emphasizes the importance of risk management practices, strategic diversification in investments, and adjustments in macroeconomic and financial regulatory policies for policymakers and financial market participants. For more refined predictions of KOSPI index return volatility, future research could explore 1494 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 5: 1487-1494, 2024 DOI: 10.55214/25768484.v8i5.1861 Β© 2024 by the author; licensee Learning Gate models such as EGARCH, QGARCH, TGARCH, PGARCH, as well as mixed-frequency GARCH models, SVM models, and AI-GARCH(1,1) models. These remain as future research topics. Copyright: Β© 2024 by the authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). 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