The paper provides a framework to model and forecast volatility of EUR/USD exchange rate based on the unbiased AddRS estimator as proposed by Kumar and Maheswaran [1]. The framework is based on the heterogeneous auto-regressive (HAR) model to capture the heterogeneity in a market and to ac-count for long memory in data. The results indicate that the framework based on the unbiased extreme value volatility estimator generates more accurate forecasts of daily volatility in comparison to alternative volatility models.
The volatility of the financial market has implications towards asset pricing, portfolio and fund management and in risk measurement and management. Moreover, more accurate prediction of volatility is important for option valuation, in implementing successful trading strategies and in construction of the optimal hedge using futures. Another important application of volatility is in the estimation and forecasting of Value-at-Risk and expected shortfall.
There are various ways to estimate daily volatility and it depends on the kind of data used. The squared return and the absolute return are the very basic estimates of daily volatility based on close-to-close prices. However, these estimates of daily volatility are highly inefficient in nature [
The third popular way of estimating daily volatility makes use of the daily opening, high, low and closing prices. These include the method of moments estimators [
The main contribution of the paper is to propose the use of heterogeneous autoregressive (HAR) model for the AddRS estimator. The model is named as HAR -AddRS model. The HAR-AddRS model can capture the long memory characteristics and heterogeneity in the markets. We also examine the statistical and distributional properties of the AddRS and Log (AddRS) estimator and obtain similar inference as given in the findings of Kumar and Maheswaran [
The remainder of this paper is organized as follows. Section 2 presents the literature review. Section 3 describes the data, methodology and preliminary analysis. Section 4 reports our empirical findings, discussion and policy implications. Section 5 concludes with a summary of our main findings.
The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) family of models and stochastic volatility models are highly popular in capturing the dynamics of the return based volatility [
Alizadeh, Brandt [
We use the daily opening, high, low and closing values of the EUR/USD exchange rate. The sample period is from 1 January 1999 to 25 August 2014. The choice of the sample period is based on the introduction of Euro currency. We consider last 1000 observations for out-of-sample forecast evaluation. We make use of realized volatility, based on the sum of squares of 5 minutes intra-day high-frequency returns, as measured volatility for out-of-sample forecast evaluation exercises. All the data have been collected from the Bloomberg database. We have used MATLAB to perform the analysis.
Kumar and Maheswaran [
b t = log ( H t O t )
c t = log ( L t O t )
x t = log ( C t O t )
Let u t = 2 b t − x t and v t = 2 c t − x t . Hence, the bias corrected extreme value estimators are given by:
A d d u x = 1 2 ( u t 2 − x t 2 ) + x t 2 ⋅ 1 { b t = 0 or x t = b t } (1)
and
A d d v x = 1 2 ( v t 2 − x t 2 ) + x t 2 ⋅ 1 { c t = 0 or x t = c t } (2)
Therefore, the unbiased AddRS estimator, as proposed by Kumar and Maheswaran [
AddRS = 1 2 [ A d d u x + A d d v x ] (3)
In this section, we examine the properties of the AddRS and the Log (AddRS) estimators of EUR/USD exchange rate.
Null Hypothesis, H 0 : d = 0 , and Alternative Hypothesis, H 1 : d > 0
It can be seen that the fractional integration parameter (d) estimates are significantly greater than zero for both the AddRS estimator and the Log (AddRS) estimator. The d < 0.5 indicate that the AddRS estimator and the Log (AddRS) estimator follow a covariance stationary process.
Corsi [
Mean | Std dev | Skewness | Kurtosis | Q(20) | d | |
---|---|---|---|---|---|---|
AddRS | 0.430 | 0.586 | 8.269# | 120.130# | 6069.820# | 0.237# |
Log(AddRS) | −1.286 | 0.940 | −0.135# | 0.607# | 12953.200# | 0.241# |
# and † mean significant at 1% and 10% level of significance. The estimates of the fractional differencing parameter, d, are based on the Exact Local Whittle estimator.
The HAR model for the Log(AddRS) is given as:
log ( AddRS ) t ( d ) = α 0 + α d log ( AddRS ) t − 1 ( d ) + α w log ( AddRS ) t − 1 ( w ) + α m log ( AddRS ) t − 1 ( m ) + ε t (4)
where log ( AddRS ) t − 1 ( d ) is the lagged daily Log(AddRS) estimator, log ( AddRS ) t − 1 ( w ) = 1 5 ∑ i = 1 5 log ( AddRS ) t − i ( d ) is the lagged weekly volatility component and log ( AddRS ) t − 1 ( m ) = 1 22 ∑ i = 1 22 log ( AddRS ) t − i ( d ) is the lagged monthly volatility component.
To account for the conditional heteroskedasticity in the residuals from the HAR-AddRS model, we implement the GARCH(1, 1) error process. The proposed model is given as:
log ( A d d R S ) t ( d ) = α 0 + α d log ( A d d R S ) t − 1 ( d ) + α w log ( A d d R S ) t − 1 ( w ) + α m log ( A d d R S ) t − 1 ( m ) + ε t
ε t = σ t Z t , Z t ~ N ( 0 , 1 )
σ t 2 = ω + α 1 ε t − 1 2 + β 1 σ t − 1 2 (5)
We implement the HAR model using Log (AddRS) estimator and refer the model as HAR-AddRS model. To capture the heteroskedasticity in the residuals of the HAR-AddRS model, we also implement the heterogeneous autoregressive AddRS generalized autoregressive conditional heteroskedasticity (HAR-AddRS-GARCH) model. We use the Schwarz information criterion (SIC) to select the appropriate orders of the HAR-AddRS-GARCH(p, q) model for Log (AddRS). The orders that minimize the SIC is an HAR-AddRS-GARCH(1, 1) specification.
For both the HAR-AddRS model and the HAR-AddRS-GARCH model, the coefficients of lagged weekly (αw) and monthly (αm) volatility components are significant at 1% level of significance indicating that the lagged weekly and monthly volatility components have the greatest impact on the current volatility of the given exchange rates. For the HAR-AddRS model, the lagged daily (αd) volatility component is significant at 10% level of significance. On the other hand, for HAR-AddRS-GARCH model, the lagged daily (αd) volatility component is significant at 1% level of significance. For both models, the lagged daily volatility has a negative impact on the current volatility of EUR/USD. This indicates that the next day volatility of the given exchange rates is an aggregate effect of short term, medium term and long term volatility components. We also do not find any autocorrelation in the squared residuals of the HAR-AddRS-GARCH model based on the Ljung Box statistic up to 20 lags (Tables 2-3).
This section addresses the forecasting performance of the HAR-AddRS and the HAR-AddRS-GARCH models in predicting realized volatility. The results are compared with the corresponding forecasts of the range based conditional autoregressive range model (CARR) and return-based models which include the GARCH, the GJR-GARCH, the EGARCH, the FIGARCH, the FIEGARCH and the RiskMetrics models. We consider the forecasting horizon of 1 day for all the models. We generate 1000 forecasts of 1-day horizon based on the parameter estimates using rolling windows with fixed window size. We generate the realized volatility measure based on 5 minutes high-frequency return data as a proxy for measured volatility (MVt). Suppose rt,n represents the return for the period from (n-1) to n on day t. The daily realized volatility based on 5 minutes returns is given as:
R V t = ∑ i = 1 j r t , i 2
EUR/USD | |
---|---|
α0 | −0.122# |
(0.027) | |
αd | −0.032† |
(0.019) | |
αw | 0.258# |
(0.040) | |
αm | 0.681# |
(0.039) | |
R2 | 0.372 |
Skewness | −0.084 |
Kurtosis | 3.842 |
JB Stat | 124.629# |
[0.000] | |
Q(20) | 84.775# |
[0.000] | |
ARCH(10) | 1.521 |
[0.125] |
#, * and † mean significant at 1%, 5% and 10% levels of significance, respectively. The terms in parenthesis (.) represent the standard error and the terms in the square braces [.] represent p-value. The JB Stat represents the Jarque Bera statistic, Q(20) represents the Ljung-Box test statistic upto 20 lags and ARCH(10) represents the ARCH Lagrange Multiplier test statistic up to 10 lags.
EUR/USD | |
---|---|
α0 | −0.146# |
(0.031) | |
αd | −0.068# |
(0.023) | |
αw | 0.340# |
(0.048) | |
αm | 0.607# |
(0.046) | |
ω | 0.164† |
(0.091) | |
α | 0.035* |
(0.018) | |
β | 0.667# |
(0.171) | |
LL | −3423.049 |
AIC | 2.245 |
BIC | 2.259 |
JB Stat | 97.293# |
[0.000] | |
Q(20) | 51.979# |
[0.000] | |
Qs(20) | 21.641 |
[0.248] | |
ARCH(10) | 0.842 |
[0.588] |
#, * and † mean significant at 1%, 5% and 10% levels of significance, respectively. The terms in parenthesis (.) represent the standard error and the terms in the square braces [.] represent p-value. The JB Stat represents the Jarque Bera statistic, Q(20) represents the Ljung-Box test statistic upto 20 lags and ARCH(10) represents the ARCH Lagrange Multiplier test statistic up to 10 lags.
where j depends on the number of 5 minutes returns in a day. We undertake statistical forecast evaluation exercises based on the error statistics test and superior predictive ability test to test the forecasting performance of the HAR-AddRS and the HAR-AddRS-GARCH models.
We use the following five loss functions for evaluating the forecasting performance of the models.
1) Mean squared errors (MSE)
M S E ( m , h ) = 1 T ∑ t = 1 T ( F V t + h ( m ) − M V t + h ) 2
2) Mean absolute errors (MAE)
M A E ( m , h ) = 1 T ∑ t = 1 T | F V t + h ( m ) − M V t + h |
3) Heteroskedasticity adjusted mean square error (HMSE)
H M S E ( m , h ) = 1 T ∑ t = 1 T ( 1 − M V t + h F V t + h ( m ) ) 2
4) Heteroskedasticity adjusted mean absolute error (HMAE)
H M A E ( m , h ) = 1 T ∑ t = 1 T | 1 − M V t + h F V t + h ( m ) |
5) Logarithmic loss function (LL)
L L ( m , h ) = 1 T ∑ t = 1 T ( ln ( M V t + h F V t + h ( m ) ) ) 2
where m represents the models under consideration, h can be related to the h-step(s) ahead forecasts, MVt represents the measured volatility at time t (realized volatility at time t), FVt(m) represents the volatility forecast made by the model m and T represents the number of out-of-sample volatility forecasts. Here, h is taken to be 1 and T is taken to be 1000.
We also make use of Hansen [
1) Hansen’s [
HAR | HAR-G | CARR | GARCH | GJR-GARCH | EGARCH | FIGARCH | FIEGARCH | Risk-Metrics | |
---|---|---|---|---|---|---|---|---|---|
MSE | 0.031 | 0.032 | 0.312 | 0.033 | 0.033 | 0.033 | 0.037 | 0.034 | 0.033 |
MAE | 0.113 | 0.115 | 0.532 | 0.118 | 0.118 | 0.117 | 0.123 | 0.120 | 0.114 |
HMSE | 0.345 | 0.361 | 14.611 | 0.650 | 0.661 | 0.640 | 0.672 | 0.705 | 0.525 |
HMAE | 0.361 | 0.370 | 2.800 | 0.462 | 0.462 | 0.461 | 0.461 | 0.473 | 0.408 |
LL | 0.245 | 0.266 | 1.678 | 0.247 | 0.245 | 0.246 | 0.265 | 0.257 | 0.245 |
Where HAR and HAR-G represent the HAR-AddRS model and the HAR-AddRS-GARCH model.
HAR | HAR-G | CARR | GARCH | GJR-GARCH | EGARCH | FIGARCH | FIEGARCH | Risk-Metrics | |
---|---|---|---|---|---|---|---|---|---|
EUR/USD | |||||||||
MSE | 1.000 | 0.000 | 0.000 | 0.207 | 0.599 | 0.525 | 0.006 | 0.028 | 0.458 |
MAE | 1.000 | 0.000 | 0.000 | 0.274 | 0.328 | 0.387 | 0.027 | 0.024 | 0.682 |
HMSE | 1.000 | 0.004 | 0.000 | 0.013 | 0.028 | 0.007 | 0.037 | 0.017 | 0.026 |
HMAE | 1.000 | 0.000 | 0.000 | 0.002 | 0.002 | 0.000 | 0.002 | 0.000 | 0.022 |
LL | 0.935 | 0.000 | 0.000 | 0.463 | 0.857 | 0.787 | 0.007 | 0.009 | 1.000 |
Where HAR and HAR-G represent the HAR-AddRS model and the HAR-AddRS-GARCH model.
For the HAR-AddRS-GARCH, CARR, FIGARCH and FIEGARCH models acting as a benchmark model, the null hypothesis is rejected for all the loss functions. The GARCH, GJR-GARCH, EGARCH and RiskMetrics models exhibit an equal number of successes (3 out of 5). Overall, it can be seen that the HAR-AddRS model outperforms other models in generating superior forecasts of realized volatility.
Predicting volatility in exchange rate more accurately has implications for regulators, traders, market makers, fund managers and financial institutions. The continuous presence of high volatility in a market with a decline in levels can signal for the forthcoming crisis and recession. This can help the regulators in implementing appropriate strategies to stabilize the economy. The trader can implement profitable trading strategies by generating more accurate forecasts of volatility to take out better short-term gains. For financial institutions and fund managers, the precise forecasts of exchange rate volatility can help in pricing derivatives securities and in portfolio rebalancing at the appropriate time. Hedgers can mitigate risk by estimating appropriate hedge ratio based on more accurate forecasts of volatility. The portfolios of institutional investors usually contain a variety of derivatives. Moreover, options are traded in terms of the volatility of the underlying asset. Furthermore, the more accurate forecast of the exchange rate volatility has important implications for investors, portfolio managers and risk managers having off-shore exposure. Volatility plays an important role in portfolio theory and helps in optimal allocation of investors’ money in different asset classes or securities. The choice of the optimal portfolio is usually based on minimizing the risk (measured by volatility). Another important implication of the findings is the estimation and forecasting of value-at-risk (VaR) and Expected Shortfall. The Basle Committee on Banking Supervision has recommended the compulsory implementation of risk management practices in financial institutions around the world. Financial institutions, regulators, business practitioners and portfolio managers use VaR as a measure of market risk. It helps the financial institutions to determine the minimum capital to deal with catastrophic event in the market.
In this paper, we propose to use a simple HAR model-based framework to model and to generate more accurate volatility forecasts based on the AddRS estimator. We also propose to use GARCH specification with HAR-AddRS model to capture heteroscedasticity in the AddRS volatility series using HAR-AddRS-GARCH model. To evaluate the out-of-sample forecasting performance of the proposed framework, we make use of realized volatility, based on high-frequency data, as a measure of actual volatility. The statistical and distributional properties of the AddRS and Log (AddRS) estimators also support the use of the linear Gaussian model to model them. The findings based on the in-sample analysis support the evidence of better fit by HAR-AddRS model for Log (AddRS). We evaluate the out-of-sample forecasting performance of the HAR-AddRS and HAR-AddRS-GARCH models using error statistic approach and the SPA test. We make use of five loss functions (MSE, MAE, HMSE, HMAE and LL) for the error statistic approach. The findings indicate that the HAR-AddRS model outperforms the alternative models in generating more accurate forecasts of realized volatility by providing lowest value for all error statistics. The results based on the SPA tests also confirm the similar findings that the HAR based models outperform return based and range based volatility models in generating superior forecasts of volatility. The study does not incorporate the impact of structural breaks, leverage effect and jumps in volatility while modelling and generating forecasts of the same. Further research can be undertaken by incorporating the impact of structural breaks, leverage effect and jumps in volatility.
Kumar, D. (2018) Modelling and Forecasting Unbiased Extreme Value Volatility Estimator: A Study Based on EUR/USD Exchange Rate. Theoretical Economics Letters, 8, 1599-1613. https://doi.org/10.4236/tel.2018.89102