Evidence of Long Memory and Asymmetry in the EUR/PLN Exchange Rate Volatility

Abstract

The EUR/PLN exchange rate is of great importance to Polish exporting industries as well as to investors in the Polish region. As a consequence, there is a particular interest in managing foreign exchange (FX) risks due to short and long FX exposures. By implementing a variety of GARCH models under different return distributions, we forecast the exchange rate volatility of daily returns of EUR/PLN exchange rates. Confirming findings of recent literature, we ap-ply Fractionally Integrated GARCH (FIGARCH) and Asymmetric Power ARCH (APARCH) which reveal a statistically significant long memory effect and an asymmetry in volatility in both time series. These characteristics implicate some challenges in volatility fore-casting and Value-at-Risk (VaR) predictions in short and long trading positions. Therefore, we combine these two effects in the Fractionally Integrated Asymmetric Power ARCH (FIAPARCH) modeling framework which yields the best goodness-of-fit of all aforementioned models. The FIAPARCH outperforms all other models in regard to the applied loss functions and is found to provide the best VaR prediction results. Our findings contribute to research on volatility of financial instruments in Poland and expand the findings of existing literature, raise awareness of combined effects to practitioners, and may give insights to Polish financial regulators, as well.

Full text

Faculty of Business and Economics Finance and Financial Services
EMPIRICAL EVIDENCE OF LONG
MEMORY AND ASYMMETRY IN
EUR/PLN EXCHANGE RATE
VOLATILITY
Tony Klein, Hien Pham Thu, and Thomas Walther
WROFIN 2015, Wrocław, September 2015

Contents
1Introduction
2Methods and Models
3Data
4Results and Discussion
5Conclusion
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 2 / 23

01 Introduction
EUR/PLN FX rate volatility? What for?
•EURO area most important trading partner to Poland
•51.5% of exports, 54.5% of imports (2013)
•EUR/PLN exchange rate of interest
•EUR/PLN free float regime
•Exporting/Importing companies & investors have short and long
exposures
=⇒In need for forecasting, quantifying, and hedging FX risks
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 3 / 23

01 Introduction
Long Memory and Asymmetry
•popular volatility modeling with GARCH (volatility clusters)
•GARCH drawbacks:
(1) short memory (exponential decay of shocks)
(2) no asymmetric effects
0 20 40 60 80 100
0.0 0.4 0.8
Lag
ACF
y^2 GARCH(1,1) Simulation
0 20 40 60 80 100
0.0 0.4 0.8
Lag
ACF
y^2 EUR/PLN
•Long memory models (e.g. FIGARCH) with hyperbolic persistence
(e.g. Baillie et al., 1996, Bollerslev & Mikkelsen, 1996)
•Asymmetric models (APARCH, EGARCH)
(e.g. Ding et al., 1993, Nelson, 1991)
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 4 / 23

01 Introduction
Literature Review
•EUR/PLN FX series features a wide range of ”stylized facts”
•addressed in
– Koˇ
cenda & Valachy (2006): volatility clustering, asymmetry
– Bedowska-S ´
ojka & Kliber (2010): infinite persistence of shocks
•research questions:
– long memory and asymmetry still present in a more recent time
series (up to 05/31/15)
– combining both effects in a unified model beneficial to modeling
and forecasting
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 5 / 23

02 Methods and Models
•modeling of conditional variance in order to:
– evaluate the goodness-of-fit (loglikelihood, BIC)
– forecast future variances and evaluation (loss functions)
– Value-at-Risk (VaR) prediction and evaluation (coverage)
•performance comparison of GARCH, APARCH, FIGARCH, and FIAPARCH
•we set
yt=µ+εt,
εt=ztphtwith zt∼Dist(0,1) i.i.d.,
ht=Var (yt|Ft−1),
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 6 / 23

02 Methods and Models
Conditional variance models
GARCH(1,1)
Var (yt|Ft−1) = ht=ω+αε2
t−1+βht−1,
•standard model
•see further Bollerslev (1986)
APARCH(1,1)
hδ/2
t=ω+α(|εt−1| − γεt−1)δ+βhδ/2
t−1,
•variable power of volatility (δ), δ= 2 variance default
•incorporation of asymmetric effects of upward/downward movements (if
γ6= 0)
•see further Ding et al. (1993)
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 7 / 23

02 Methods and Models
Conditional variance models
FIGARCH(1,d,1)
ht=ω+1−βL−(1 −αL) (1 −L)dε2
t+βht−1
=ω
1−β+
∞
X
i=1
λiε2
t−i
•long memory model, introduced by fractional differencing parameter d
•see further Baillie et al. (1996)
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 8 / 23

02 Methods and Models
Conditional variance models
FIAPARCH(1,d,1)
hδ/2
t=ω+1−βL−(1 −αL) (1 −L)d(|εt−1| − γ εt−1)δ+βhδ/2
t−1
=ω
1−β+
∞
X
i=1
λi(|εt−i| − γεt−i)δ
•combines long memory and asymmetry by applying fractional
differencing on APARCH innovations
•see further Tse (1998)
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 9 / 23

02 Methods and Models
Distributions
•Normal (Gaussian)
•Standardized Student-t(Bollerslev, 1987)
•Skewed Student-t(Hansen, 1994)
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 10 / 23

03 Data
1999 2001 2003 2005 2007 2009 2011 2013 2015
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
Time
Log Returns
Figure: EUR/PLN daily log returns, 01/01/1999 – 05/31/2015, T= 4279, out-of-sample window is plotted
in red.
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 11 / 23

03 Data
EUR/PLN
Descriptive Statistics
T4279
Mean 0.0000
Standard deviation 0.0068
Minimum −0.0466
Maximum 0.0553
Skewness 0.3599
Kurtosis 8.3450
Preliminary Tests
Jarque-Bera 5186.0∗∗∗
Ljung-Box Q02(65) 3430.9∗∗∗
AG-LME y2
t0.2512∗∗
ADF −69.48∗∗∗
KPSS 0.0267
Log-returns feature:
•Assumption of non-normality
– kurtosis, skewness, rejected
Jarque-Bera test
•heteroskedastic properties
– Ljung-Box test up to lag 65
•long memory
– significant Andrews &
Guggenberger LME (modified
GPH)
•stationarity confirmed by
– rejected ADF test
– non-rejection of KPSS
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 12 / 23

04 Results and Discussion
Parameter estimates and goodness-of-fit
•FIAPARCH with skewed-tinnovations yields the best LL and BIC
– FIAPARCH-sk-t: LL = 16 087, BIC =−32 122
– GARCH-n: LL = 15 941, BIC =−31 862
•long memory parameter dfulfills stationarity conditions, statistically
significant
=⇒shocks level out at a slower rate, no infinite persistence detected
•sign. asymmetry parameter γ, negative (γ≈ −0.20)
=⇒upward movements higher impact on cond. variance as downward
moves,
so-called ”conditional skewness”
•skewness parameter of skewed-tdistribution, ξ, stat. sign.
so-called ”unconditional skewness”
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 13 / 23

04 Results and Discussion
Forecasting Performance
Variance Forecast
•1-, 5-, and 20-day ahead variance forecasts, out-of-sample: 01/01/13 -
05/31/15
•forecasts tested against realized volatility with different loss functions
•FIAPARCH significantly outperforms all other models (with n-, t-, sk-t dist.)
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 14 / 23

04 Results and Discussion
Forecasting Performance
Value-at-Risk Predictions
•short pos. (selling PLN): FIAPARCH features best coverage and
test-statistics
•long pos. (buying PLN): all long memory and asymmetric models show
good performance
=⇒superiority of FIAPARCH over standard models
=⇒accounting for asymmetry and long memory improves forecasting quality
and VaR predictions
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 15 / 23

04 Results and Discussion
Forecasting Performance
100 200 300 400 500 600
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
EUR/PLN Returns
GARCH(1,1)-n
FIAPARCH(1,d,1)-Sk-t
Figure: Value-at-Risk 1-day ahead forecast (α= 0.05) of EUR/PLN FX-rate 2013-2015.
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 16 / 23

05 Conclusion
•evidence of long memory and asymmetry in conditional variance
•effects of major importance to risk management
•biased forecasts and exposure evaluation if not introduced to modeling
•superiority of FIAPARCH over all tested models reveals necessity
=⇒more accurate forecasts and VaR predictions
”Essentially, all models are wrong, but some are useful.”
— George E. P. Box (1919 – 2013)
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 17 / 23

07 Bibliography
Baillie, R. T., Bollerslev, T., & Mikkelsen, H. O. (1996). Fractionally integrated
generalized autoregressive conditional heteroskedasticity. Journal of
Econometrics,74, 3–30. URL:
http://linkinghub.elsevier.com/retrieve/pii/S0304407695017496.
doi:10.1016/S0304-4076(95)01749- 6.
Bedowska-S ´
ojka, B., & Kliber, A. (2010). Realized volatility versus GARCH and
stochastic volatility models. The evidence from the WIG20 index and the
EUR/PLN foreign exchange market. Przeglad Statystyczny (Statistical
Review),57, 105–127.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity.
Journal of Econometrics,31, 307–327. URL:
http://linkinghub.elsevier.com/retrieve/pii/0304407686900631.
doi:10.1016/0304-4076(86)90063- 1.
Bollerslev, T. (1987). A Conditionally Heteroskedastic time series model for
speculative prices and rates of return. The Review of Economics and
Statistics,69, 542–547. doi:10.2307/1925546.
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 18 / 23

07 Bibliography
Bollerslev, T., & Mikkelsen, H. O. (1996). Modeling and pricing long memory in
stock market volatility. Journal of Econometrics,73, 151–184. URL:
http://linkinghub.elsevier.com/retrieve/pii/0304407695017364.
doi:10.1016/0304-4076(95)01736- 4.
Ding, Z., Granger, C. W., & Engle, R. F. (1993). A long memory property of stock
market returns and a new model. Journal of Empirical Finance,1, 83–106.
URL:
http://linkinghub.elsevier.com/retrieve/pii/092753989390006D.
doi:10.1016/0927-5398(93)90006- D.
Hansen, B. E. (1994). Autoregressive Conditional Density Estimation.
International Economic Review,35, 705–730.
Koˇ
cenda, E., & Valachy, J. (2006). Exchange rate volatility and regime change: A
Visegrad comparison. Journal of Comparative Economics,34, 727–753.
doi:10.1016/j.jce.2006.07.003.
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 19 / 23

07 Bibliography
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new
approach. Econometrica: Journal of the Econometric Society,59, 347–370.
URL: http://www.jstor.org/stable/2938260?origin=crossrefhttp:
//www.jstor.org/stable/2938260. doi:10.2307/2938260.
Tse, Y. K. (1998). The conditional heteroscedasticity of the yen-dollar exchange
rate. Journal of Applied Econometrics,13, 49–55. URL:
http://doi.wiley.com/10.1002/(SICI)1099-1255(199801/02)13:
1<49::AID- JAE459>3.0.CO;2- O.
doi:10.1002/(SICI)1099-1255(199801/02)13:1< 49::
AID-JAE459> 3.0.CO;2-O.
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 20 / 23

08 Backup
GARCH(1,1) APARCH(1,1) FIGARCH(1,d,1) FIAPARCH(1,d,1)
Normal Skewed-t Normal Skewed-t Normal Skewed-t Normal Skewed-t
ω0.0000 0.0000 0.0 000 0.0000 0.000 0 0.0000 0.0000 0.0 000
(0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
α0.0899 0.0864 0.0865 0.0869 0.1878 0.2222 0.1586 0.1888
(0.0146) (0.0112) (0.0147) (0.0136) (0.0550) (0.0733) (0.0711) (0.1524)
β0.9083 0.9100 0.9135 0.9131 0.5354 0.5728 0.4624 0.5024
(0.0144) (0.0105) (0.0136) (0.0130) (0.0735) (0.1368) (0.1157) (0.3535)
γ-0.1846 -0.1679 -0.2130 -0.1819
(0.0636) (0.0546) (0.0597) (0.0793)
δ1.7821 1.7903 1.9544 1.9535
(0.1471) (0.1370) (0.0758) (0.6304)
d0.4250 0.4432 0.3702 0.4000
(0.0566) (0.1009) (0.0703) (0.2383)
ν6.7518 6.7420 7.0430 6.9448
(0.6769) (0.6957) (0.7578) (0.9103)
ξ0.0908 0.0919 0.0933 0.0950
(0.0200) (0.0200) (0.0213) (0.0232)
LL 15 941 16068 15 948 16 072 15964 16 079 15979 16 087
BIC −31 862 −32 104 −31 864 −32 098 −31 903 −32 119 −31 919 −32 122
Parameter estimates for EUR/PLN log returns, 01/01/1999 – 05/31/2015, n= 4279. Robust standard
errors are in parenthesis. Bold numbers indicate the best model regarding the best goodness-of-fit (LL)
and information criterion (BIC).

08 Backup
An Example – December 2012
•suppose polish airline LOT buys aircrafts for EUR 1bn from Airbus
•the EUR/PLN Spot FX rate is 4.08
•aircrafts will be delivered and payment is due in December 2014
•How should LOT save up the money?
•in theory it does not matter due to FX interest rate parity
•say iPLN = 4.25%, iEUR = 0.75%
It=EUR 1bn
(1 + 0.0075)2·4.08 PLN / EUR =PLN 4.02bn (1)
FXEUR/PLN
t+2 = 4.08 ·(1 −0.0075 + 0.0425)2= 4.37 (2)
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 21 / 23

08 Backup
An Example – December 2014
•LOT saved up PLN 4.37bn
•say the yearly volatility of FX rate is σ= 0.06
•in 99 of 100 cases the FX rate will be between 4.15 and 4.59
•due to volatility (uncertainty) LOT can either win EUR 53mn or lose EUR
48mn
4.1 4.2 4.3 4.4 4.5 4.6 4.7
PLN / EUR
pd
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 22 / 23

08 Backup
Further Research
•extend the study to other currencies (Cz Korona, etc.)
•out-of-sample forecast analysis of a more volatile time period (2008-2010)
•moving time window, to examine, weather parameters are robust over
time
•including intra-day data for comparison and as benchmark
Walther, Thomas Long Memory and Asymmetry in EUR/PLN FX Volatility 23 / 23