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

For time series data, ARIMA Autoregressive Intergrated Moving Average model is used to forecast future value chains.. If the ARCH / GARCH model is consistent with historical data, it wil

ARIMA AND ARCH/GARCH MODEL: FORECASTING SHORT-TERM VNIBOR AND MANAGING RISK IN VIETNAMESE BANKING SYSTEM

1 University of Economics and Business-Vietnam National University, Vietnam

2 Foreign Trade University, Vietnam

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

1 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,

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 the ARCH model was launched, in 1986, Bollerslev published studies

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 apply ARIMA model theory (p, d, q) to the discount interest rate of short-term valuable papers From the regression model found, ARCH and GARCH models will

be applied to predict variance

2 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 do 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 This means that the prediction of tomorrow’s stock price is equal to today’s stock price plus a mere random value If this happens, it may be useless to predict the price of this asset

2.2 Autoregressive Intergrated Moving Average Model (ARIMA)

According to Ruey (2002) the AR model can be written as follows:

Where:

μ is the average value of Y;

utis uncorrelated error term, whose average value is zero and the variance is stant (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

con-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 average terms

The above analyzed time series models are based on the assumption that time series is weakly stationary In other words, the mean value and the variance of weakly stationary time series are constant and its covariance is constant over time

Non-stationary time data is often used by the method of taking differentials to become stationary series The differential of a function is the difference of two 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

6 523With 𝛼𝛼" > 0 and 𝛼𝛼&, 𝛽𝛽) ≥ 0 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑤𝑤, 𝑗𝑗 > 0 and ∑456 (7;9)&:; (𝛼𝛼& + 𝛽𝛽)) < 1

u t represents the fluctuations in period t that can affect the value of r t , u t assuming follows a normal distribution u t ~ 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 re-sponses to positive and negative shocks

Second, determining the intercept will become more complicated if the order of the ARCH is greater than 4

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 model GARCH (p, q), as following:

Thus, the variance depends not only on the shock of the past but also on the ance in the previous period Although the GARCH model is more general than the ARCH model, there are still weak points like the ARCH model

vari-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:

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:

3 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 3 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 1 Overnight VNIBOR

Figure 2 One-week VNIBOR

Figure 3 One-month VNIBOR

Figure 4 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%,

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% (three-month VNIBOR) Simultaneously, interest rates also do not have a clear distribution.

4 ESTIMATION STRATEGY AND RESULTS

4.1 Model Estimation and forecast

Step 1 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 non-stationary, 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 2 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, 2 indexes AIC and BIC are also used to select the model

Step 3 Regressing to find the appropriate ARCH/TARCH model

Step 4 Regressing to find the appropriate GARCH/TGARCH model

Step 5 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)]

Forecasting results

Table 1 Forecasting results of the model ARIMA

Standard

Overnight VNIBOR

1 week VNIBOR

1 month VNIBOR

3 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 0.04518Mean Absolute

range fluctuation

violations

(Detailed forecasting results and range fluctuations of models at Appendix 1, 2, 3 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 three-month 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 5 The graph of estimating interest rates, real interest rates and the confidence

intervals of overnight VNIBOR

Figure 6 Forecasting range fluctuations of overnight VNIBOR with 95% confidence interval

by ARCH/GARCH model (Details in Appendix 5)

Figure 5 also supports that ARIMA [(0), 1, (1)] forecasts more accurately than ARIMA [(1; 2), 1, (1)] From the regression results (Figure 6 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 4 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 7 The graph of estimating interest rates, real interest rates and the confidence

intervals of one-week VNIBOR

Figure 8 Forecasting range fluctuations of one-week VNIBOR with 95% confidence interval

by ARCH/GARCH model (Details in Appendix 6)

Figure 7 also supports that ARIMA [(0), 1, (1)] forecasts more accurately ARIMA [(1; 2), 1, (1)] From the regression results (Figure 8 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:

Figure 9 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)

Figure 9 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 three-month VNIBOR because the predicting intervals are violated