ARIMA AND ARCHGARCH MODEL: FORECASTING SHORTTERM VNIBOR AND MANAGING RISK IN VIETNAMESE BANKING SYSTEM45485

VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS ARIMA AND ARCH/GARCH MODEL: FORECASTING SHORT-TERM VNIBOR AND MANAGING RISK IN VIETNAMESE BANKING SYSTEM Tin Khanh Nguyen1*, Mai Thi Nguyen2 University of Economics and Business-Vietnam National University, Vietnam Foreign Trade University, Vietnam ABSTRACT This study applied ARIMA and ARCH/GARCH models in forecasting the volatility of short-term interbank interest rate (VNIBOR) in Vietnam, which is based on data as Vnibor fixing during years from August 1, 2014 to July 14, 2017 The sample included 800 observations extracted from the bank’s Kondor + system The results show that short-term Vnibor could be predicted by previous random errors, expressed in ARIMA models Meanwhile, the ARCH/GARCH models are probably not appropriate A forecasting model with 95% confidence interval, like ARIMA model will help the bank to make business decisions and hedge risks on a quantitative basis Key words: short-term interbank interest rate - VNIBOR, ARIMA model, ARCH/ GARCH model, risk management INTRODUCTION In the field of econometrics, prediction is often based on two main types of models: causalmodel and time series model For the former, regression analysis techniques will be used to understand the relationship between dependent and independent variables For time series data, ARIMA (Autoregressive Intergrated Moving Average) model is used to forecast future value chains This model proposed by Geoger and Gwilym (1976) will predict future values ​​based on its own past value and the weighted sum of the random variables Thus, the ARIMA model could forecast the movement of interest rates in the future based on historical data If the ARIMA model is proven to be appropriate, it will help administrators make decisions to buy or sell assets in order to make a profit; simultaneously, it can also make hedging decisions In addition, this model can also provide a range of fluctuations with different confidence interval, so that administrators can check the accuracy of risk forecasting models, 201 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 such as the VaR model In addition to predicting fluctuations in the price of holding assets, banks need to comprehend and predict the risk of holding such assets One of the important criteria for estimating the risk of an asset is variance Although Engle (2001) mentioned that there are many models researching and estimating the average return (Mean Return), there was no method to estimate the value of variance until the ARCH (Autoregressive Conditional Heteroscedasticity) model was introduced in 1982 Four years after ARCH was− launched, 1986, Bollerslev published studies (𝑌𝑌 = 𝛼𝛼the − 𝜇𝜇) model + 𝛼𝛼, (𝑌𝑌#*, 𝜇𝜇) + ⋯ + 𝛼𝛼in + 𝑢𝑢# # − 𝜇𝜇) ) (𝑌𝑌#*) /𝑌𝑌 #* − 𝜇𝜇0 on the GARCH model (Genaral Autoregressive Conditional Heteroscedasticity) to quantify variance, developed based on Engle’s ARCH model Besides, Bollerslev (1986) assumes that future variance is also affected by the random error Due to the simplicity and effectiveness of ARCH and GARCH models, many studies have used these models to predict variance For example, Tully and Lucey (2007) tested the ability of the GARCH model to forecast Gold market If the ARCH / GARCH model is consistent with historical data, it will help administrators determine the confidence interval of unexpected fluctuations in the future, based on data itself Filling the above gaps in the literature, this project conducts a study to explore how to theory d, q) to + the interest rate of short-term 𝑌𝑌" apply = 𝜇𝜇 +ARIMA 𝛽𝛽' 𝑢𝑢" +model 𝛽𝛽) 𝑢𝑢"*) + 𝛽𝛽+(p, 𝑢𝑢"*+ +⋯ 𝛽𝛽-discount 𝑢𝑢valuable papers From the regression model found, ARCH and GARCH models will be applied to predict variance RELATED LITERATURE 2.1 Basic concepts of time series in econometrics According to Gujarati (2008), although Time series data is used extensively in econometric experiments, researchers still have difficulty in analyzing data: Firstly, previous studies are still based on the assumption of time series being stationary A time series is considered to be stationary if its mean and variance not change over time and the covariance value between two periods depends only on the distance and time lag between the two periods, not depends on the time the covariance is calculated Secondly, regression result is effected by autocorrelation One factor causing autocorrelation is the non-stationary time series Thirdly, when regressing a time series for other time series we usually get high values, despite there is probably no reasonable relationship between them This situation can occur if both two time series under consideration show similar trends (increase or decrease), then high is due to the presence of the trend, not because of real relationship between two time series Finally, some financial time series, such as stock prices, are known as random walk 𝑟𝑟 = 𝜇𝜇" + 𝑢𝑢" that the prediction of tomorrow’s stock price is equal to today’s stock This 𝑟𝑟"" = means 𝜇𝜇" + 𝑢𝑢" price plus a mere random value If this happens, it may be useless to predict the 𝑢𝑢" = of 𝜎𝜎" this × 𝜀𝜀asset " price 𝑢𝑢 " = 𝜎𝜎" × 𝜀𝜀" 202 VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS 2.2 Autoregressive Average Model (ARIMA) 𝜎𝜎" # = 𝛼𝛼& + ∑)/0 𝛼𝛼) Intergrated × 𝑢𝑢"-) # , 𝛼𝛼Moving 𝛼𝛼) ≥ 0 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑖𝑖 >0 & > and According to Ruey (2002) the AR model can be written as follows: Where: μ is the average value of Y; ut is uncorrelated error term, whose average value is zero and the variance is constant (known as pure white noise) Then we say that Yt followed by the autoregression process of order p or AR(p) Here, the Y value in period t depends on its value in p previous periods and a random factor; the values of Y are expressed as deviations from its mean The AR process discussed is not the only mechanism that can produce Y Suppose we construct model Y as follows: Where: μ is a constant; ut-1 is a White noise Here, Y in time t is equal to a constant plus the moving average of current and past errors So, in this case, we say that Y follows the moving average process of the order q or MA (q) Thus, for an ARMA (p, q) process, there will be p autoregression terms and q moving 𝑟𝑟" = 𝜇𝜇" +terms 𝑢𝑢" average The time series models are based on the assumption that time 𝑢𝑢" = above 𝜎𝜎" × 𝜀𝜀analyzed " series is weakly1 stationary In other6 words, the mean value and the variance of weakly + + stationary is constant over time 𝜎𝜎" + = 𝛼𝛼- +time series 𝛼𝛼/ × are 𝑢𝑢"0/constant + and 𝛽𝛽5 ×its 𝜎𝜎covariance "05 /23 523 Non-stationary time data is often used by the method of taking differentials to become stationary series The differential of456 (7;9) a function With 𝛼𝛼" > and 𝛼𝛼& , 𝛽𝛽 𝑗𝑗 > and ∑ (𝛼𝛼& + 𝛽𝛽is) ) the < difference of two ) ≥ 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑖𝑖, &:; adjacent values of function Therefore, if we have to calculate differential of a time series d times to make it stationary and then apply the ARMA (p, q) model, we say that the time series is ARIMA (p, d, q), which means that it is an autoregressive intergrated moving average, with p is the number of autoregressive terms, d is the number of differentials until stationary time series, and q is the number of moving average terms 2.3 ARCH/GARCH Model Engle (1982) proposed the ARCH(p) model as following: 𝜎𝜎" # = θ + ' )/0 # 𝛼𝛼) × 𝑢𝑢"-) + ' 1 𝑖𝑖𝑖𝑖 𝑢𝑢"#$ < Where: 𝑆𝑆"#$ = & 0 𝑖𝑖𝑖𝑖 𝑢𝑢"#$ ≥ 2/0 # 𝛽𝛽2 × 𝜎𝜎"-2 +' )/0 # 𝛾𝛾) × 𝑆𝑆"-) × 𝑢𝑢"-) 203 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 ut represents the fluctuations in period t that can affect the value of rt , ut assuming follows a normal distribution ut ~ N(0; ) This means that the fluctuation of interest rates in period t (st) will tend to depend on the interest rate fluctuations of p previous periods However, Tsay (2002) discussed some limitations of the ARCH model as follows: Firstly, the model is based on the assumption that positive and negative shocks have the same effect on volatility, because the model depends on the square of past shocks In fact, prices of financial instruments are often recognized as different responses to positive and negative shocks Second, determining the intercept will become more complicated if the order of the ARCH is greater than Third, the ARCH model does not describe the causes of fluctuations in the time series but only provides a complex method to describe the fluctuations of interest rates Finally, the ARCH model often overestimates the level of volatility because the model responds slowly to large individual shocks In 1986, Bollerslev expanded the ARCH model and named it the general ARCH + + 𝑌𝑌"model 𝑌𝑌= 𝑓𝑓(𝑥𝑥) 𝑓𝑓(𝑥𝑥) ++𝜔𝜔𝜎𝜎 𝜔𝜔𝜎𝜎 +𝑢𝑢"q), 𝑢𝑢" (u (u =σtσ , zt,t,zzt,t z~t ~NN(0,(0,1))1)) t =tfollowing: " = " "+ GARCH (p, as σt2= θ + ∑$%&' βiσt-i2 + ∑(%&' αiut-i2 Thus, the variance depends not only on the shock of the past but also on the variance in the previous period Although the GARCH model is more general than the ARCH model, there are still weak points like the ARCH model The standard GARCH model suggests that negative shocks and positive shocks have the same effect on volatility In other words, the good news and the bad news all have the same effect on the paradigm shift In fact, this assumption is often violated, especially with regard to stock returns, where bad news will cause more volatility than good news To overcome this disadvantage of the GARCH model, the T-GARCH and T-ARCH models were introduced independently by Zakoian (1994), and Glosten et al (1993) The general formula for variance is given as follows: Where: From the above model, based on determining whether the value of smaller than the threshold value 0, will have different effects on 204 is greater or If VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS the effect of on is ; If , the effect of to is This shows that the impact of the bad news will be greater than the good news The GARCH-in mean model is a very important extension of a standard GARCH model and it has broad applications in economics and finance The driving force behind the introduction of an extended GARCH (GARCH-M) model is the interpretation of risk premium in the financial market The theory shows that the yield on asset may depend on its risk, expressed by the variance of the asset (volatility of that security or its risk) However, the traditional GARCH model cannot explain the residual profit because the conditional expectation remains zero throughout the model Therefore, the GARCH-M model was proposed by Engle, Lilien and Robins (1987), by establishing a direct relationship between risk and return where time changes the risk premium, and was shown as a linear function of the magnitude of the current risk The general formula of the model is given as follows: Prospect Model: Variance: STUDY DESIGN Sample selection and baseline survey The overnight interbank funding rate (overnight VNIBOR) used in the model is Vnibor fixing during the period of years from August 1, 2014 to July 14, 2017, including 800 observations, extracted from the bank’s Kondor + system The charts of short-term VNIBOR and its distribution charts are as follows: Figure Overnight VNIBOR 205 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 Figure One-week VNIBOR Figure One-month VNIBOR Figure Three-month VNIBOR The graph shows us the abnormal and sudden fluctuations of short-term VNIBOR Interest rates did not show a clear trend, continuously created new peaks and troughs in a short period of time However, in the first three quarters of 2017, overnight and three-month VNIBOR was in a downward trend (Figure 1) In terms of its distribution chart, the average overnight, one-week, one-month and three-month VNIBOR in the past three years were approximately 3.25%, 3.43%, 206 VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS 4.1% and 4.60%, respectively In which, the lowest interest rates were 0.4%, 0.48%, 1.42%, and 3.39%, and the highest were nearly 5.95%, except for 5.86% (threemonth VNIBOR) Simultaneously, interest rates also not have a clear distribution ESTIMATION STRATEGY AND RESULTS 4.1 Model Estimation and forecast Step Testing stationarity of the data There are two steps in order to testing stationarity of the data The first one is using the Correlogram chart to check the data is stationary or not If the data is nonstationary, the Unit Root Test will be used If the data is still non-stationary, taking differentials for the data to be a stationarity series Step Selecting the order of ARIMA model (p, d, q) Correlogram chart is used to find out the lag variables before regressing and eliminating lag variables which its p-value is higher than 5% Besides, indexes AIC and BIC are also used to select the model Step Regressing to find the appropriate ARCH/TARCH model Step Regressing to find the appropriate GARCH/TGARCH model Step Forecating the future interest rate The above model is used to forecast the future interest rates which will be compared to the real interes rates 4.2 Results Regression In terms of overnight VNIBOR, we have appropriate models ARIMA [(1; 2), 1, (1)] (AIC = 0.421969) and ARIMA [(0), 1, (1)] (BIC = 0.445543) Regression based on the remainder of the ARIMA model [(1; 2), 1, (1)], we have an appropriate ARCH (3) model and GARCH(1; 1) Regression based on the remainder of the ARIMA model [(0), 1, (1)], we have appropriate ARCH (3) and TARCH (1) models and GARCH(2,1) In terms of one-week VNIBOR, we have appropriate models ARIMA [(1; 2), 1, (1)] (AIC =  0.078799) and ARIMA [(0), 1, (1)] (BIC =  0.103892) Regression based on the remainder of the ARIMA [(1; 2), 1, (1)], we have an appropriate ARCH(3), TARCH(3), and GARCH(2; 1) Regression based on the remainder of the ARIMA [(0), 1, (1)], we have appropriate ARCH(3), TARCH(3), GARCH(2;1), and TGARCH(2; 1) Regarding to one-month VNIBOR, we have appropriate models ARIMA [(1), 1, (0)] and ARIMA [(0), 1, (1)] ARCH(3) and GARCH(1; 1) could be suitable to the remainder of the ARIMA [(1), 1, (0)] and ARIMA [(0), 1, (1)] In terms of three-month VNIBOR, we have appropriate models ARIMA [(1; 2), 1, (1; 2; 3)] ARCH(1), GARCH(1;1) and TGARCH(1;1) could be suitable to the remainder of the ARIMA [(2), 1, (1;2;3)] 207 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 Forecasting results Table Forecasting results of the model ARIMA Standard Overnight week month VNIBOR VNIBOR VNIBOR months VNIBOR ARIMA [(1; 2), 1, (1)] ARIMA [(0), 1, (1)] ARIMA [(1; 2), 1, (1)] ARIMA [(0), 1, (1)] ARIMA [(1), 1, (0)] ARIMA [(0), 1, (1)] ARIMA [(1; 2), 1, (1; 2; 3)] Mean Error (ME) -0.03237 0.00983 -0.02824 -0.01208 -0.01819 -0.02366 -0.01647 Mean Absolute Error (MAE) 0.07987 0.077289 0.07111 0.07061 0.06348 0.06451 0.04518 Mean Absolute Percentage Error (MAPE) 10.12243 9.71706 7.53642 7.53768 3.48712 3.55572 1.31057 Theil Inequality Coefficient (TIC) 0.08941 0.08481 0.07467 0.07078 0.04011 0.04069 0.01490 Number of range fluctuation violations 0 0 1 (Detailed forecasting results and range fluctuations of models at Appendix 1, 2, and 4) After determining models which have statistics meaning from the above regression step, it is possible to use these models in order to select the most appropriate model with each interest rate by predicting interest rate volatility As can be seen from the Table 1, ARIMA [(1), 1, (0)] could be the most appropriate model for the overnight VNIBOR Furthermore, the suitable models for the one-week, one-month, and threemonth VNIBOR are ARIMA [(0), 1, (1)], ARIMA [(1), 1, (0)], and ARIMA [(1; 2), 1, (1; 2; 3)] respectively Finally, regressing the remainder of the above ARIMA models, this study finds out that the ARCH/GARCH model is not appropriate to forecast the volatility of short-term VNIBOR Particularly, the graph of estimating interest rates, real interest rates and the confidence intervals of each type of VNIBOR of the models as follows: Firstly, overnight VNIBOR is considered below: Figure The graph of estimating interest rates, real interest rates and the confidence intervals of overnight VNIBOR 208 VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS Figure Forecasting range fluctuations of overnight VNIBOR with 95% confidence interval by ARCH/GARCH model (Details in Appendix 5) Figure also supports that ARIMA [(0), 1, (1)] forecasts more accurately than ARIMA [(1; 2), 1, (1)] From the regression results (Figure and Appendix 5) we can see, the TARCH (1) based on the remainder of the ARIMA model [(0), 1, (1)] has the total number of violations confidence interval at least times Although the predicted confidence interval from this model closely follows actual fluctuations, the model has not yet achieved the required reliability of 95% Second, one-week VNIBOR is considered below: Figure The graph of estimating interest rates, real interest rates and the confidence intervals of one-week VNIBOR 209 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 Figure Forecasting range fluctuations of one-week VNIBOR with 95% confidence interval by ARCH/GARCH model (Details in Appendix 6) Figure also supports that ARIMA [(0), 1, (1)] forecasts more accurately ARIMA [(1; 2), 1, (1)] From the regression results (Figure and Appendix 6) we can see, the ARCH/GARCH models are not appropriate to forecast the confidence intervals for one-week VNIBOR because the predicting intervals are violated Third, one-month VNIBOR is considered below: 210 VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS Figure The graph of estimating interest rates, real interest rates and the confidence intervals of one-month VNIBOR Figure 10 Forecasting range fluctuations of one-month VNIBOR with 95% confidence interval by ARCH/GARCH model (Details in Appendix 7) 211 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 Figure shows that forecasting results from models are consistent with one-month interest rate volatility From the regression results (Figure 10 and Appendix 7) we can see, the ARCH/GARCH models are not appropriate to forecast the confidence intervals for one-month VNIBOR because the predicting intervals are violated Finally, three-month VNIBOR is considered below: Figure 11 The graph of estimating interest rates, real interest rates and the confidence intervals of three-month VNIBOR Figure 12 Forecasting range fluctuations of three-month VNIBOR with 95% confidence interval by ARCH/GARCH model (Details in Appendix 8) From the regression results (Figure 12 and Appendix 8) we can see, the ARCH/ GARCH models are not appropriate to forecast the confidence intervals for threemonth VNIBOR because the predicting intervals are violated 212 VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS After regressing, ARIMA model [(0), 1, (1)] may be the most accurate model to estimate not only fluctuations in points but also the range of fluctuations in the overnight and one-week interbank funding rates Furthermore, ARIMA model [(1), 1, (0)] and ARIMA model [(1; 2), 1, (1; 2; 3)] may be the most accurate model to estimate not only fluctuations in points but also the range of fluctuations in the one-month and threemonth interbank funding rates, respectively [Appendix 1, 2, and 4] CONCLUSIONS From the regression results of interbank lending rate models for overnight, oneweek, one-month and three-month loans with a length of three-year historical data, and 28 observations ** to test for the forecasting ability of the model, the results are shown in the table below Data Model ARIMA ARCH Overnight Vnibor 1-week Vnibor 1-month Vnibor 3-months Vnibor ARIMA models [(0), 1, (1)] forecast close to real interest rate fluctuations There is no day that real interest rates exceed the forecast Therefore, the model is appropriate for simultaneous forecasting of interest rates and interest rate fluctuation range for overnight interbank loans ARIMA models [(0), 1, (1)] forecast close to real interest rate fluctuations There is no day that real interest rates exceed the forecast Therefore, the model is appropriate for simultaneous forecasting of interest rate fluctuation range of interest rate for 1-week interbank loans ARIMA models [(1), 1, (0)] forecast close to real interest rate fluctuations 01/28 days actual interest rate exceeded the forecast Therefore, the model is appropriate for forecasting interest rates and interest rate fluctuation range for 1-month interbank loans ARIMA models [(1; 2), 1, (1; 2; 3)] forecast close to real interest rate fluctuations There is no day that actual interest rates exceed the forecast Therefore, the model is appropriate for simultaneous forecasting of interest rates and interest rate fluctuation range for months interbank loans The regression model shows that overnight interbank rate data has ARCH effect (2) However, the forecast results show that the model is not appropriate for forecasting the interest rate fluctuation range when there are 11 violation days out of 28 test days The regression model shows that -week interbank rate data has ARCH effect (3) However, the forecast results show that the model is not appropriate for forecasting the interest rate fluctuation range when there are up to 15 violation days out of 28 test days The regression model shows that 1-month interbank rate data has ARCH effect (1) However, the forecast results show that the model is not appropriate for forecasting the interest rate fluctuation range when there are 19 violation days out of 28 test days The regression model shows that 3-month interbanl rate data has ARCH effect (1) However, the forecast results show that the model is not appropriate for forecasting the interest rate fluctuation range when there are up to 25 violation days out of 28 test days 213 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 TARCH The regression model shows that the data has a TARCH effect, the forecasting data range is closely follow the interest rate fluctuation However, there are violation days out of 28 test days The regression model shows that 1-week interbank rate data has a TARCH effect (3) However, the forecast results show that the model is not appropriate for forecasting the interest rates fluctuation range when there are 13 violation days out of 28 test days GARCH The regression model shows the data has GARCH effect (2; 1) However, the fluctuation range of the model is continuously violated, up to 16 actual interest rates violate the confidence interval, out of 28 test days The regression model shows that 1-week interbank rate data has a GARCH effect (2; 1) However, the forecast results show that the model is not appropriate for forecasting the interest rate fluctuation range when there are up to 21 days of violation on a total of 28 test days TGARCH 214 The regression model shows that week interbank rate data has TGARCH effect (2; 1) However, the forecast results show that the model is not appropriate for forecasting the interest rate fluctuation range when there are up to 15 violation days out of the 28 test days The regression model shows that 1-month interbank rate data has a GARCH effect (2; 1) However, the forecast results show that the model is not appropriate for forecasting the interest rate fluctuation range when there are 19 violation days out of the 28 test days The regression model shows that 3-month interbank rate data has a GARCH effect (1; 1) However, the forecast results show that the model is not for forecasting the interest rate fluctuation range when there are up to 25 violation days out of the 28 test days The regression model shows that 3-month interbank rate data has TGARCH effect (1; 1) However, the forecast results show that the model is not for forecasting the interest rate fluctuation range when there are 22 violation days out of the 28 test days VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS The results show that VNIBOR for very short terms are overnight and week terms can be predicted by previous random errors, shown by ARIMA model [(0), 1, (1)] For overnight interbank rates, ARIMA models [(0), 1, (1)] tend to forecast lower interest rates than the actual situation, the mean forecast error is 0.077%/day, TIC index of 0.085 is approximately 0, showing that this is a good model for forecasting Similarly for the 1-week term interest rate, ARIMA models [(0), 1, (1)] tend to forecast higher interest rates than reality, the mean forecast error is 0.0706%/days, the TIC is at a low of 0.0708 For longer term interest rates, the model has changed For 1-month term, future Vnibor rates can be reflected from interest rates of the previous period, through ARIMA models [(1), 1, (0)] The prediction of the model tends to be higher than the actual fluctuation, the forecast error is approximately 0.0635% / day, the TIC index is as low as 0.0401 In contrast, Vnibor interest rates with 3-month terms can be predicted by the higher latent variables, based on ARIMA model [(1; 2), 1, (1; 2; 3)] with errors The forecast is approximately 0.0452%/day and the TIC is approximately 0.0149 REFERENCES [1] Box, G.E.P and Jenkins, G (1970) Time Series Analysis, Forecasting and Control, HoldenDay, San Francisco [2] Edel Tully, Brian M Lucay (2007) A power GARCH examination of the gold market, Research in International Business and Finance volume 21, Isue 2, pp 316 - 325 [3] Engle R, Lilien D, Robins R (1987) Estimating Time Varying Risk Premia in the Term Structure: The Arch-M Model Econometrica 55: 391-407 [4] Gujarati D N.và Porter D C (2008) Basic Econometric, Mc Graw-Hill, Fifth Edition, chapter 21-22, pp.737 - 799 [5] Robert Engle (2001) GARCH 101: The use of ARCH/GARCH models in applied econometrics The Journal of Economic Perspectives Vol 15, No.4 (Autumn, 2001) pp 157 - 168 [6] Robert F Engle (1982) Autoregressive Conditonal Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica, Volume 50, Issue 4, pp 9871008 [7] Ruey S Tsay (2002) Analysis of Financial Time Series, John Wiley & Son, Inc [8] Tim Bollerslev (1986) Generalized Autogregressive Conditional Heteroskedasticity, North-Holland, Journal of Econometrics 31 (1986) , pp.307 - 327, North-Holland [9] Zakoian J (1994) Threshold Heteroscedastic Models Journal of Economic Dynamics and Control 18, pp.931-944 215 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 APPENDIX FORECASTING OVERNIGHT VNIBOR Date Real overnight VNIBOR (%) ARIMA [(1; 2), 1, (1)] (%) Forecast Upper bound Lower bound ARIMA [(0), 1, (1)] (%) Forecast Upper bound Lower bound 17/07/2017 1.54173 1.32399 1.908051 0.739939 1.29885 1.88513 0.712561 18/07/2017 1.49098 1.62094 2.204661 1.037228 1.61678 2.202984 1.030577 19/07/2017 1.20721 1.49934 2.082721 0.91595 1.45218 2.037949 0.866405 20/07/2017 0.97471 1.14752 1.731144 0.563896 1.13144 1.71704 0.545834 21/07/2017 0.75280 0.94236 1.525745 0.358969 0.92616 1.51146 0.340867 24/07/2017 0.74274 0.73187 1.315151 0.148585 0.69905 1.284123 0.113975 25/07/2017 0.70245 0.79138 1.374275 0.208486 0.75628 1.340968 0.171585 26/07/2017 0.73266 0.73740 1.319971 0.15483 0.68577 1.270105 0.10143 27/07/2017 0.60180 0.78577 1.367971 0.203573 0.74719 1.331173 0.163213 28/07/2017 0.58167 0.60113 1.183209 0.019051 0.55677 1.140168 -0.02662 31/07/2017 0.54146 0.61654 1.198217 0.034863 0.58938 1.172372 0.006383 01/08/2017 0.67224 0.56715 1.148515 -0.01421 0.52663 1.109347 -0.05609 02/08/2017 0.65211 0.74679 1.327789 0.165789 0.71729 1.299719 0.13487 03/08/2017 0.63198 0.67156 1.252262 0.090851 0.63194 1.21398 0.049909 04/08/2017 0.54145 0.64979 1.230108 0.069467 0.63199 1.213678 0.050309 07/08/2017 0.56157 0.53688 1.116907 -0.04315 0.51344 1.095142 -0.06826 08/08/2017 0.58168 0.59207 1.171633 0.012512 0.57645 1.157798 -0.0049 09/08/2017 0.63198 0.60857 1.187754 0.029391 0.58330 1.164274 0.00232 10/08/2017 0.57162 0.66459 1.243379 0.085802 0.64704 1.227676 0.066405 11/08/2017 0.57161 0.56701 1.145508 -0.01148 0.54830 1.128615 -0.03201 14/08/2017 0.64205 0.58657 1.164609 0.008523 0.57882 1.158771 -0.00113 15/08/2017 0.66218 0.67626 1.253909 0.098614 0.66160 1.241208 0.081989 16/08/2017 0.62192 0.67649 1.253779 0.099206 0.66236 1.241607 0.083104 17/08/2017 0.63198 0.61647 1.193397 0.039535 0.60941 1.188321 0.030507 18/08/2017 0.67223 0.64280 1.2193 0.066296 0.63896 1.217518 0.060406 21/08/2017 0.70245 0.69061 1.266731 0.114497 0.68252 1.260728 0.104313 22/08/2017 0.77296 0.71511 1.290855 0.13936 0.70861 1.28647 0.130756 23/08/2017 0.90405 0.79555 1.370932 0.220169 0.79287 1.370395 0.215336 Mean Error (ME) -0.03237 0.00983 0.07987 0.077289 Absolute Error 10.12243 9.71706 Mean Absolute (MAE) Mean Percentage (MAPE) Error Theil Inequality Coefficient (TIC) Number of range fluctuation violations 216 0.08941 0.08481 0 0 VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS APPENDIX FORECASTING WEEK VNIBOR ARIMA [(1; 2), 1, (1)] (%) ARIMA [(0), 1, (1)] (%) Date Real week VNIBOR (%) Forecast Upper bound Lower bound Forecast Upper bound Lower bound 17/07/2017 1.71425 1.55682 2.053637 1.060001 1.53832 2.032372 1.044269 18/07/2017 1.69391 1.77857 2.275138 1.282011 1.77443 2.268352 1.280499 19/07/2017 1.36911 1.70784 2.204086 1.211594 1.66641 2.159976 1.172836 20/07/2017 1.13628 1.27674 1.773591 0.779887 1.26711 1.760644 0.773579 21/07/2017 0.93425 1.08845 1.585042 0.591855 1.09136 1.584581 0.598139 24/07/2017 0.89389 0.90384 1.400344 0.407333 0.88030 1.373251 0.387343 25/07/2017 0.85355 0.92396 1.420155 0.427755 0.89856 1.391183 0.40594 26/07/2017 0.85355 0.87984 1.375768 0.383914 0.83810 1.330416 0.345781 27/07/2017 0.75276 0.89054 1.386161 0.394914 0.85886 1.351169 0.366547 28/07/2017 0.75276 0.74996 1.245447 0.254468 0.71634 1.208431 0.224253 31/07/2017 0.70241 0.78544 1.28058 0.29031 0.76526 1.257035 0.273491 01/08/2017 0.78299 0.71331 1.2082 0.218414 0.68084 1.172331 0.189349 02/08/2017 0.73262 0.83800 1.332546 0.343462 0.81802 1.309282 0.326764 03/08/2017 0.75276 0.73487 1.229206 0.24053 0.70335 1.194339 0.21236 04/08/2017 0.68227 0.77924 1.273175 0.285305 0.76970 1.260377 0.279016 07/08/2017 0.68227 0.67376 1.167453 0.180068 0.65233 1.142754 0.1619 08/08/2017 0.70241 0.69939 1.192661 0.206111 0.69253 1.182636 0.202417 09/08/2017 0.73262 0.72424 1.217164 0.231314 0.70579 1.195592 0.215996 10/08/2017 0.68227 0.75504 1.247618 0.262455 0.74181 1.231309 0.252304 11/08/2017 0.68227 0.67377 1.166075 0.181474 0.66188 1.151109 0.172652 14/08/2017 0.77291 0.69239 1.184295 0.200486 0.68926 1.178179 0.200331 15/08/2017 0.76284 0.81306 1.30464 0.321471 0.80157 1.290234 0.312904 16/08/2017 0.74269 0.76272 1.254028 0.271419 0.74957 1.237945 0.261201 17/08/2017 0.75276 0.73912 1.230075 0.248172 0.74033 1.228399 0.252267 18/08/2017 0.82331 0.76014 1.25074 0.269549 0.75702 1.24479 0.269252 21/08/2017 0.85355 0.85057 1.340855 0.360288 0.84601 1.333511 0.358506 22/08/2017 0.88381 0.86319 1.353167 0.373207 0.85614 1.343335 0.368939 23/08/2017 1.03522 0.89212 1.381776 0.402458 0.89329 1.380194 0.406379 0 Mean Error (ME) -0.02824 -0.01208 0.07111 0.07061 Mean Absolute Percentage Error (MAPE) 7.53642 7.53768 Theil Inequality cient (TIC) 0.07467 0.07078 Mean (MAE) Absolute Error Coeffi- Number of range fluctuation violations 0 217 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 APPENDIX FORECASTING MONTH VNIBOR Date Real month VNIBOR (%) 17/07/2017 ARIMA [(1), 1, (0)] (%) ARIMA [(0), 1, (1)] (%) Forecast Upper bound Lower bound Forecast Upper bound Lower bound 2.54815 2.55305 2.79968 2.30642 2.56338 2.813221 2.313544 18/07/2017 2.52648 2.53024 2.776832 2.283654 2.54299 2.792592 2.293386 19/07/2017 2.23657 2.51801 2.76422 2.271803 2.52089 2.770314 2.271475 20/07/2017 2.15233 2.12303 2.370375 1.875677 2.14016 2.390904 1.889426 21/07/2017 2.19134 2.11940 2.366214 1.872596 2.15644 2.406502 1.906376 24/07/2017 1.99703 2.20657 2.453225 1.959909 2.20316 2.452969 1.953358 25/07/2017 2.01561 1.92128 2.168442 1.674126 1.92713 2.177551 1.676704 26/07/2017 1.86184 2.02283 2.269724 1.775927 2.04543 2.295624 1.795231 27/07/2017 1.84798 1.80216 2.049286 1.555032 1.80024 2.050869 1.549614 28/07/2017 1.83304 1.84261 2.08946 1.59575 1.86393 2.114096 1.613756 31/07/2017 1.74335 1.82725 2.073918 1.580574 1.82270 2.072611 1.572782 01/08/2017 1.80009 1.70858 1.955184 1.46197 1.71674 1.966519 1.466954 02/08/2017 1.68690 1.82206 2.068554 1.575558 1.82803 2.077758 1.578296 03/08/2017 1.68191 1.64316 1.889731 1.396588 1.63976 1.889684 1.389835 04/08/2017 1.56537 1.67998 1.926326 1.433635 1.69595 1.945553 1.44634 07/08/2017 1.58921 1.52037 1.766752 1.273994 1.52192 1.771665 1.272183 08/08/2017 1.55220 1.59841 1.844596 1.352217 1.61153 1.861048 1.362012 09/08/2017 1.52556 1.53794 1.783974 1.291901 1.53254 1.781875 1.2832 10/08/2017 1.51055 1.51529 1.761146 1.26944 1.52325 1.772331 1.274161 11/08/2017 1.54864 1.50476 1.750434 1.259094 1.50634 1.755239 1.257435 14/08/2017 1.57427 1.56332 1.808835 1.3178 1.56267 1.811425 1.313917 15/08/2017 1.57486 1.58414 1.829478 1.338809 1.57811 1.826668 1.329559 16/08/2017 1.50948 1.57509 1.82024 1.329935 1.57378 1.822152 1.325407 17/08/2017 1.52288 1.48429 1.72933 1.239245 1.48815 1.736418 1.239884 18/08/2017 1.57724 1.52804 1.772901 1.283179 1.53439 1.782486 1.286301 21/08/2017 1.59693 1.59818 1.842899 1.353453 1.59145 1.839361 1.343534 22/08/2017 1.62678 1.60452 1.849054 1.359988 1.59875 1.846482 1.351024 23/08/2017 1.66690 1.63828 1.882643 1.393917 1.63608 1.883643 1.388516 Mean Error (ME) -0.01819 -0.02366 0.06348 0.06451 Mean Absolute Percentage Error (MAPE) 3.48712 3.55572 Theil Inequality Coefficient (TIC) 0.04011 0.04069 Mean Absolute (MAE) Error Number of range fluctuation violations 218 VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS APPENDIX FORECASTING MONTH VNIBOR ARIMA [(1; 2), 1, (1; 2; 3)] (%) Date Real month VNIBOR (%) Forecast Upper bound Lower bound 17/07/2017 3.90536 3.88778 4.043914 3.731652 18/07/2017 3.86627 3.91803 4.073977 3.76208 19/07/2017 3.87485 3.86284 4.018861 3.706818 20/07/2017 3.87245 3.86595 4.021806 3.710096 21/07/2017 3.72925 3.87004 4.025824 3.714265 24/07/2017 3.63134 3.72248 3.878932 3.56603 25/07/2017 3.63157 3.62264 3.779075 3.4662 26/07/2017 3.63226 3.61268 3.768906 3.456454 27/07/2017 3.61001 3.63336 3.789479 3.477249 28/07/2017 3.48121 3.62797 3.784177 3.471762 31/07/2017 3.48894 3.45361 3.609683 3.297527 01/08/2017 3.40889 3.48042 3.63642 3.324426 02/08/2017 3.42952 3.40391 3.560028 3.24779 03/08/2017 3.35908 3.45406 3.61045 3.297678 04/08/2017 3.28311 3.33107 3.487061 3.175086 07/08/2017 3.30697 3.26401 3.420026 3.107997 08/08/2017 3.27873 3.32689 3.482698 3.171078 09/08/2017 3.30949 3.27288 3.428482 3.117273 10/08/2017 3.28966 3.30020 3.455992 3.144407 11/08/2017 3.24214 3.28613 3.441691 3.130575 14/08/2017 3.23774 3.25274 3.408479 3.096998 15/08/2017 3.26849 3.23204 3.387416 3.076663 16/08/2017 3.26488 3.25588 3.411509 3.100248 17/08/2017 3.22589 3.27476 3.429979 3.119549 18/08/2017 3.22911 3.22847 3.383537 3.073407 21/08/2017 3.27447 3.21907 3.374074 3.064075 22/08/2017 3.24630 3.27451 3.429435 3.119593 23/08/2017 3.34916 3.25375 3.408701 3.098794 0 Mean Error (ME) -0.01647 Mean Absolute Error (MAE) 0.04518 Mean Absolute Percentage Error (MAPE) 1.31057 Theil Inequality Coefficient (TIC) 0.01490 Number of range fluctuation violations 219 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 0.720316 0.598817 0.646103 0.661936 0.700469 0.619044 0.820643 0.703966 0.740011 0.749431 0.783596 0.703755 0.626659 0.696138 0.595761 0.68033 0.485208 0.70226 0.607781 0.74576 0.550981 0.72634 0.465314 0.76397 0.526813 0.618316 0.650815 0.678918 0.71207 0.628724 0.812264 0.706655 0.730446 0.753775 0.783345 0.702221 0.490798 0.593894 0.735172 0.641671 0.777876 0.588241 0.79867 0.564857 0.667294 0.803291 0.614566 0.82477 0.591626 0.594872 0.756914 0.78157 0.630464 0.657179 0.721792 0.626697 0.626631 0.651861 0.75276 0.70241 0.78299 0.73262 0.75276 0.68227 0.68227 0.70241 0.73262 0.68227 0.68227 0.77291 28/07/2017 31/07/2017 01/08/2017 02/08/2017 03/08/2017 04/08/2017 07/08/2017 08/08/2017 09/08/2017 10/08/2017 11/08/2017 14/08/2017 0.784029 0.75276 27/07/2017 0.759181 0.735417 0.85355 26/07/2017 0.733891 0.595748 0.838873 0.70762 0.85355 25/07/2017 0.842034 0.798872 0.655419 0.89389 0.797526 0.899314 0.770288 0.753585 24/07/2017 0.90466 0.777101 0.636253 0.82321 0.674494 0.93425 0.776446 0.847984 0.713853 0.740708 0.64562 21/07/2017 0.850656 0.81994 0.668285 0.715645 1.13628 0.814633 0.951289 0.824663 0.674039 0.600868 20/07/2017 0.953584 0.949201 0.807698 0.90789 0.672467 1.36911 0.948569 0.997338 0.851693 0.520913 0.573973 19/07/2017 0.997866 1.015562 0.789668 1.195366 0.971055 1.430161 1.129803 11 1.795286 0.698406 1.616841 0.890347 1.868671 0.744588 1.692955 0.544441 0.650322 0.720299 0.73032 0.626939 0.513068 0.624682 0.748671 0.704026 0.614738 0.626602 0.565452 0.701757 0.533997 0.750605 0.545372 0.508032 0.603338 0.78017 0.597313 0.730255 0.56523 0.537596 0.603403 0.758786 0.521586 0.729621 0.622761 0.700824 0.690141 0.541813 0.617278 0.464379 0.810016 0.77828 0.601152 0.577723 0.690717 0.564954 0.689427 0.812652 0.760396 0.630662 0.656724 0.620123 0.75496 0.42468 0.630611 0.459419 0.707525 0.69521 0.653403 0.481347 0.769581 0.678895 0.635144 0.532682 0.71373 0.67753 0.664738 0.445704 0.824194 0.646433 0.621245 0.755148 0.551221 0.674812 0.455117 0.781224 0.69309 0.876572 0.476179 0.652643 0.680627 0.750215 0.531967 0.724812 0.385325 0.824806 0.64769 0.678483 0.409312 0.795865 0.612656 0.826741 0.472152 0.936857 0.511936 0.809413 0.75014 0.933931 0.53348 0.576073 0.752977 0.726143 0.594208 0.582078 0.732436 0.509287 0.682783 0.636085 0.770605 0.80686 0.696298 0.471614 0.724549 0.776272 0.657353 0.707711 0.776368 0.679512 0.804978 0.679132 0.590329 0.523562 1.69391 1.004101 0.646454 0.490814 18/07/2017 1.18724 0.644756 0.585277 1.656746 1.388761 0.675762 1.71425 1.79488 0.619316 17/07/2017 1.879054 0.66988 0.510545 1.455533 0.809832 0.675668 Date 0.829464 0.762162 0.610053 Real week VNIBOR (%) 1.636582 0.706941 0.741471 0.76019 0.806203 0.885433 0.640955 0.518163 0.850174 1.03522 0.74919 0.826081 0.836206 0.770183 0.739903 0.986205 23/08/2017 0.625453 0.8082 1.023757 0.639435 0.600714 0.851488 0.801428 0.726617 1.031338 0.770793 0.589824 0.812545 0.88381 0.601182 0.6544 0.83469 0.762549 0.610204 0.718689 0.83265 0.843801 0.675988 0.770821 0.405853 0.894946 0.65154 0.747044 0.425233 0.837324 0.624343 0.782984 0.501069 0.977698 22/08/2017 0.784734 0.778206 0.680762 0.731889 0.467271 0.99353 0.85355 0.656072 0.627724 0.70715 1.266933 1.383612 0.75942 0.482234 1.54968 1.484519 0.628001 0.700544 1.165278 0.550339 0.971101 0.729848 0.520229 0.714594 0.60773 0.546925 1.70432 Lower bound 1.190595 0.694693 0.410474 1.150246 0.82331 0.582755 1.449028 0.578781 0.71314 1.3771 1.100642 0.700137 0.764085 0.717565 Upper bound 0.722289 0.44087 1.642684 0.75276 1.615012 0.585582 0.630201 GARCH(2; 1) 0.576922 0.664277 1.762039 0.488978 0.613837 0.728714 0.518545 1.709824 0.464493 0.707005 0.812611 0.62464 1.846916 1.696276 0.573326 0.655402 0.702607 0.86555 0.839769 0.647783 0.703011 0.850332 0.65023 0.646598 0.757831 0.718961 0.774516 0.563736 0.625665 0.81252 0.88883 0.582142 0.751494 0.785205 0.749094 0.660943 0.607434 0.632878 0.834147 0.87455 0.699604 0.644067 0.787634 0.872988 0.640199 0.707825 1.477776 1.443892 0.515395 0.655994 0.535203 1.618426 0.644664 0.747838 0.87176 0.924362 0.767616 0.595833 0.627925 0.56869 0.79468 0.76284 0.704533 0.771719 0.822544 0.926365 0.677069 0.732636 0.867441 0.621524 0.908369 0.77291 0.811978 0.751719 0.71936 0.813162 0.699208 0.768314 0.763487 0.654142 0.638425 0.634243 0.660767 0.768841 0.640735 1.09055 0.907148 0.68227 0.43062 0.695203 0.725214 0.8081 0.910645 0.794822 0.783392 0.974705 0.753525 0.547697 0.695765 0.609438 0.779961 0.894476 0.765307 1.064026 0.728872 0.762548 0.644092 0.786398 0.8759 0.790529 0.927895 0.621203 0.759565 0.915462 0.738807 0.812143 0.986029 0.836711 0.962071 0.824819 0.825821 0.82555 0.68227 Lower bound 0.918925 0.976883 0.696734 Lower bound 0.704639 0.771394 0.986997 0.79966 0.95735 0.769783 0.73262 Upper bound 0.866202 1.35553 0.985812 09/08/2017 Lower bound 0.837002 0.987182 0.773864 0.493124 Upper bound 0.906813 0.975791 0.687053 Upper bound 1.377321 0.70241 Upper bound 08/08/2017 GARCH(2; 1) 0.625616 1.749478 0.650893 0.68227 TARCH(3) 0.827338 0.985378 1.013422 0.468897 07/08/2017 ARCH(3) 0.880083 1.267925 1.193576 0.503898 0.741839 ARIMA [(1; 2), 1, (1)] 1.090046 1.011254 0.990908 1.621464 0.481047 0.68227 220 1.303113 1.188446 0.562959 Number of range fluctuation violations 1.317605 1.407196 0.769987 21/08/2017 1.60030 1.139643 0.75276 18/08/2017 1.306855 1.753623 0.853217 1.880475 0.73262 Lower bound 02/08/2017 ARCH(3) 0.658238 17/08/2017 1.586534 1.570172 0.691752 0.74269 1.393653 1.866764 0.885058 0.78299 16/08/2017 1.478033 1.687873 0.75276 01/08/2017 15/08/2017 1.73515 1.639116 0.639995 0.726115 14/08/2017 1.447989 1.438343 0.834765 Upper bound 0.85355 0.70241 11/08/2017 1.802032 1.168152 0.684662 31/07/2017 10/08/2017 1.502419 0.987552 0.598665 0.705354 04/08/2017 1.056416 1.158787 Lower bound 0.905435 0.778768 0.75276 03/08/2017 1.56040 1.37022 0.85355 0.979679 1.612231 Lower bound 1.143164 1.119683 25/07/2017 ARIMA [(0), 1, (1)] 1.196653 1.59883 1.732211 0.848269 1.361553 1.753121 Upper bound 1.439218 1.607781 0.89389 1.457749 Lower bound ARIMA [(0), 1, (1)] 1.209486 1.722345 24/07/2017 28/07/2017 Upper bound 1.574234 1.085539 27/07/2017 1.132386 1.698384 1.853403 0.93425 26/07/2017 1.458751 Lower bound 1.706951 1.284198 21/07/2017 1.163398 Upper bound 1.84647 1.13628 Upper bound 1.630421 20/07/2017 1.398761 1.448189 1.625662 1.80546 1.36911 TARCH(3) 19/07/2017 1.452195 Lower bound TARCH(1) ARCH(2) 1.689992 1.69391 Upper bound GARCH(1; 1) 1.465487 18/07/2017 Lower bound 1.71425 GARCH(2; 1) 17/07/2017 ARIMA [(1; 2), 1, (1)] ARCH(2) 1.606674 Date Upper bound Real overnight VNIBOR (%) 1.430549 Lower bound TGARCH(2; 1) APPENDIX FORECASTING RANGE FLUCTUATIONS OF WEEK VNIBOR WITH 95% CONFIDENCE INTERVAL BY ARCH/GARCH MODEL APPENDIX FORECASTING RANGE FLUCTUATIONS OF OVERNIGHT VNIBOR WITH 95% CONFIDENCE INTERVAL BY ARCH/GARCH MODEL 0.948569 0.85355 0.88381 1.03522 1.71425 1.69391 1.36911 1.13628 0.93425 0.89389 0.85355 0.85355 21/08/2017 22/08/2017 23/08/2017 17/07/2017 18/07/2017 19/07/2017 20/07/2017 21/07/2017 24/07/2017 25/07/2017 26/07/2017 Number of range fluctuation violations 0.997866 0.82331 18/08/2017 1.004101 1.18724 1.388761 1.79488 1.879054 1.636582 0.950295 0.921386 0.912437 0.817892 Upper bound Date 0.807698 0.851693 0.789668 0.971055 1.129803 1.616841 1.692955 1.455533 0.830158 0.80077 0.789328 0.697287 Lower bound ARCH(3) Real week VNIBOR (%) 0.949201 0.997338 1.015562 1.195366 1.430161 1.795286 1.868671 1.656746 0.94826 0.919273 0.908571 0.816012 Upper bound 11 0.806203 0.851488 0.784734 0.971101 1.100642 1.615012 1.696276 1.443892 0.830125 0.800668 0.788997 0.697622 Lower bound TARCH(3) ARIMA [(1; 2), 1, (1)] 0.933931 0.986205 0.977698 1.165278 1.3771 1.762039 1.846916 1.618426 0.920001 0.891309 0.881553 0.788122 Upper bound 14 0.809413 0.850174 0.812545 0.99353 1.150246 1.642684 1.709824 1.477776 0.86064 0.83012 0.815466 0.724833 Lower bound GARCH(2; 1) 0.907148 0.976883 0.975791 1.188446 1.377321 1.749478 1.880475 1.621464 0.957206 0.918414 0.912336 0.819642 Upper bound 0.763487 0.82555 0.773864 0.990908 1.139643 1.570172 1.687873 1.438343 0.831597 0.792765 0.783954 0.694806 Lower bound ARCH(3) 0.908369 0.974705 0.985812 1.193576 1.407196 1.753623 1.866764 1.639116 0.955324 0.917801 0.909137 0.818511 Upper bound 0.765307 0.824819 0.769783 0.987552 1.119683 1.574234 1.689992 1.430549 0.831949 0.794142 0.784325 0.695694 Lower bound TARCH(3) 0.894476 0.962071 0.95735 1.168152 1.37022 1.722345 1.84647 1.606674 0.921279 0.8838 0.877095 0.785959 Upper bound 13 0.779961 0.836711 0.79966 1.013422 1.158787 1.607781 1.706951 1.465487 0.865745 0.828288 0.815736 0.728175 Lower bound GARCH(2; 1) ARIMA [(0), 1, (1)] 0.915462 0.986029 0.986997 1.196653 1.398761 1.732211 1.853403 1.625662 0.925629 0.888832 0.881651 0.791649 Upper bound 11 0.759565 0.812143 0.771394 0.985378 1.132386 1.59883 1.698384 1.448189 0.861359 0.823256 0.811173 0.722485 Lower bound TGARCH(2; 1) APPENDIX FORECASTING RANGE FLUCTUATIONS OF WEEK VNIBOR WITH 95% CONFIDENCE INTERVAL BY ARCH/GARCH MODEL VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS 221 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 222 VIETNAM NATIONAL UNIVERSITY - UNIVERSITY OF ECONOMICS AND BUSINESS APPENDIX 7: FORECASTING RANGE FLUCTUATIONS OF MONTH VNIBOR WITH 95% CONFIDENCE INTERVAL BY ARCH/GARCH MODEL Date Real month VNIBOR (%) 17/07/2017 ARIMA [(1), 1, (0)] ARCH(1) ARIMA [(0), 1, (1)] GARCH(1;1) ARCH(1) GARCH(1;1) Upper bound Lower bound Upper bound Lower bound Upper bound Lower bound Upper bound Lower bound 2.54815 2.581653 2.529235 2.575037 2.536676 2.596242 2.540699 2.587875 2.547169 18/07/2017 2.52648 2.553783 2.508761 2.549066 2.51384 2.565456 2.51957 2.561073 2.524039 19/07/2017 2.23657 2.540935 2.496056 2.534999 2.502159 2.544833 2.499235 2.538939 2.504738 20/07/2017 2.15233 2.195966 2.062939 2.169795 2.091519 2.22803 2.08608 2.19511 2.112761 21/07/2017 2.19134 2.144309 2.098185 2.154108 2.089106 2.174044 2.128043 2.185729 2.118044 24/07/2017 1.99703 2.231092 2.180274 2.23453 2.176551 2.226469 2.178698 2.231785 2.173776 25/07/2017 2.01561 1.972189 1.879271 1.964056 1.888864 1.987332 1.892239 1.975224 1.899666 26/07/2017 1.86184 2.049566 1.995222 2.055945 1.988701 2.06293 2.010226 2.071361 2.00477 27/07/2017 1.84798 1.842449 1.768892 1.842194 1.770412 1.854478 1.773204 1.849019 1.773981 28/07/2017 1.83304 1.866762 1.81909 1.873586 1.812382 1.88104 1.833508 1.889981 1.826651 31/07/2017 1.74335 1.850484 1.804708 1.853903 1.801416 1.849818 1.803028 1.853005 1.798723 01/08/2017 1.80009 1.737283 1.684038 1.736084 1.686036 1.747581 1.693595 1.745493 1.694213 02/08/2017 1.68690 1.847881 1.793668 1.844793 1.796104 1.848458 1.795183 1.847264 1.798463 03/08/2017 1.68191 1.678195 1.613387 1.673099 1.619735 1.683716 1.616823 1.675626 1.62147 04/08/2017 1.56537 1.703608 1.656594 1.703796 1.656458 1.714023 1.666865 1.715117 1.667374 07/08/2017 1.58921 1.553087 1.493429 1.54863 1.498935 1.563601 1.499717 1.55601 1.504233 08/08/2017 1.55220 1.622899 1.572712 1.620858 1.574525 1.629517 1.579752 1.629317 1.581943 09/08/2017 1.52556 1.562743 1.515025 1.560355 1.51777 1.562721 1.513657 1.559258 1.515538 10/08/2017 1.51055 1.538773 1.493173 1.535466 1.496756 1.545197 1.49916 1.541961 1.502575 11/08/2017 1.54864 1.527814 1.482477 1.523085 1.487373 1.530391 1.484484 1.52537 1.489247 14/08/2017 1.57427 1.585971 1.538725 1.579387 1.544865 1.583462 1.535854 1.577356 1.542749 15/08/2017 1.57486 1.60613 1.560856 1.599637 1.567037 1.601082 1.555266 1.594459 1.561926 16/08/2017 1.50948 1.597621 1.552524 1.590607 1.559531 1.596713 1.551232 1.589351 1.558531 17/08/2017 1.52288 1.510742 1.461142 1.502574 1.470169 1.517332 1.467135 1.507761 1.475695 18/08/2017 1.57724 1.550875 1.504522 1.543407 1.511803 1.554268 1.507859 1.546926 1.515934 21/08/2017 1.59693 1.620521 1.573062 1.612229 1.580583 1.613435 1.565726 1.60526 1.574403 22/08/2017 1.62678 1.626376 1.581665 1.618961 1.588788 1.621509 1.576289 1.613604 1.584169 23/08/2017 1.66690 1.660113 1.614939 1.651961 1.62263 1.658224 1.610798 1.648826 1.620058 11 10 10 Number of range fluctuation violations 11 223 IN TERNATIONAL CONFERENCE ON - CIFBA 2020 APPENDIX 8: FORECASTING RANGE FLUCTUATIONS OF MONTH VNIBOR WITH 95% CONFIDENCE INTERVAL BY ARCH/GARCH MODEL Date Real month VNIBOR (%) ARIMA [(1; 2), 1, (1; 2; 3)] ARCH(1) Upper bound Lower bound 17/07/2017 3.90536 3.89158519 3.872744 18/07/2017 3.86627 3.90979753 3.891584 19/07/2017 3.87485 3.85552478 3.836319 20/07/2017 3.87245 3.88233195 3.863686 21/07/2017 3.72925 3.8723795 3.855101 24/07/2017 3.63134 3.69738747 3.651252 25/07/2017 3.63157 3.61220902 3.591911 26/07/2017 3.63226 3.63209499 3.613347 27/07/2017 3.61001 3.62993211 3.612438 28/07/2017 3.48121 3.60027372 3.582767 31/07/2017 3.48894 3.44831703 3.412048 01/08/2017 3.40889 3.50013361 3.47737 02/08/2017 3.42952 3.37866334 3.351457 03/08/2017 3.35908 3.44408767 3.420242 04/08/2017 3.28311 3.32962426 3.304037 07/08/2017 3.30697 3.2632321 3.244078 08/08/2017 3.27873 3.31325575 3.291544 09/08/2017 3.30949 3.26432538 3.246087 10/08/2017 3.28966 3.32060939 3.298777 11/08/2017 3.24214 3.27951357 3.261544 14/08/2017 3.23774 3.22784352 3.209306 15/08/2017 3.26849 3.23884131 3.220986 16/08/2017 3.26488 3.27773638 3.258202 17/08/2017 3.22589 3.26298449 3.245742 18/08/2017 3.22911 3.21542803 3.19699 21/08/2017 3.27447 3.23540645 3.21742 22/08/2017 3.24630 3.29143449 3.270722 23/08/2017 3.34916 3.23863869 3.219616 13 12 Number of range fluctuation violations * Corresponding author Email address: nguyenthinhung.1684@gmail.com 224 ... Standard Overnight week month VNIBOR VNIBOR VNIBOR months VNIBOR ARIMA [(1; 2), 1, (1)] ARIMA [(0), 1, (1)] ARIMA [(1; 2), 1, (1)] ARIMA [(0), 1, (1)] ARIMA [(1), 1, (0)] ARIMA [(0), 1, (1)] ARIMA. .. one-month, and threemonth VNIBOR are ARIMA [(0), 1, (1)], ARIMA [(1), 1, (0)], and ARIMA [(1; 2), 1, (1; 2; 3)] respectively Finally, regressing the remainder of the above ARIMA models, this study finds... examination of the gold market, Research in International Business and Finance volume 21, Isue 2, pp 316 - 325 [3] Engle R, Lilien D, Robins R (1987) Estimating Time Varying Risk Premia in the