Forecasting polish zloty exchange rate volatility

using garch models

 

włodarczyk a., zawada M. (WZ PCz, c. Częstochowa, Poland)

 

As far as investment decisions are concerned, a key role is played by forecasts of exchange rate variability as they reduce uncertainty concerning future changes in prices of individual currencies. Correctly constructed forecasts enable currency market participants to gain proceeds from speculative or arbitrage investments, and are also used in the process of currency risk management.

      Due to the specific properties of financial time series, the approach that assumes constant variability parameter is more and more often replaced in modelling of exchange rate variability by the approach assuming changeable conditional variability. The class of GARCH models, in which the conditional process variance depends not only on previous process values, but also on previous conditional process variances, makes it possible to model the variance cluster effect. 

For these reasons, the main purpose of this paper is to verify the ability of GARCH model to calculate k-period ahead forecasts of the volatility of Polish exchange rates.

1.Introduction

Liberalisation of capital flows, growing internationalisation and changing the structure of turnover on financial markets and dynamic development of financial innovations resulted in an increase in the significance of international exchange, and also in the dependence of economic growth of individual countries on the degree of connections with the world of economy. The outlined changes create favourable conditions for currency crises, which cause a significant increase in variability of market parameters, especially including the level of exchange rates. 

In the activity of business entities a particularly important role is played by financial risk connected to the structure of possible capital flows, which reflects connections of this entity with its environment, and especially with the domestic and international financial market. The importance of volatility in currency market is due to the fact, that volatility estimates are widely used as currency risk measures. Therefore, the volatility of exchange rates determines investment decisions of market participants. For these reasons, the main purpose of this paper is to verify the ability of GARCH model to calculate k-period ahead forecasts of the volatility of Polish exchange rates.

2. Methodology of GARCH models

The theory of financial markets often instead taking into account financial prices focuses on rates of return on these prices,[1] and at the same time logarithmic rates of return are most frequently used:

         for t = 2, ..., n            (1)                                  

where:  r_t – rate of return in t period,

            P_t – price of a financial instrument in t period,

            P_t-1 – price of a financial instrument in t-1 period.

In the literature on the subject aiming at detecting and describing statistical properties of prices of financial instruments a lot of space is taken up by such issues as: nonstationarity and heteroskedasticity of price series, leptokurtosis, asymmetry and “thick tails” of their distributions. Such setting of exchange rates is a result of the nature of the processes taking place in the currency markets. Some of these properties are shown on the following figures:

Fig. 1 Daily USD/PLN exchange rate (top panel) and their logarithmic returns (bottom panel)   in period  01.01.1998 –28.02.2005

Fig. 2 QQ plot (top panel) and empirical distribution (bottom panel) of USD/PLN exchange rate returns.

Such behavior of exchange rates requires the use of special methods that would take into account the dependencies discussed above. There are various approaches in the literature for volatility modelling that try to capture these properties. It is worth to list time series volatility models, such as: random walk, historical average method, exponential smoothing method, autoregressive conditional heteroscedasticity models, stochastic volatility models, regime switching models. The author of this paper examined the forecasting performance of autoregressive conditional heteroscedasticity models  - GARCH model.

Issues related to the modelling of the exchange rate returns series include three areas to search the correct initial model specification:[2]

§  specification of the deterministic part of the model related to the conditional expected process value : μ_t = E(r_t / Φ_t-1);

§  specification of the stochastic part of the model including:

- the conditional process variance equation: h_t = var(r_t / Φ_t-1);

- the selection of the function form of standardised densities, independent of the reminders of the model with a zero average value and unitary variance: ε_t ~ i.i.d. D(0,1). [3]

The following equations may be used to describe the deterministic part of the process:[4]

- if in the exchange rate returns series there is no significant autocorrelation of the order of 1, then:

                                                                                                   (2)

where: ;

- if in the exchange rate returns series there is a significant autocorrelation of the order of 1, then:

                                                                                        (3)                                                                                                                  

- if the relationship between the expected rate of return and risk is taken into account, then:[5]

                                                                                       (4)

where: .

When analysing the properties of the PLN exchange rate returns series, one may suggest the following specifications of the equation of conditional process variance:

  - in the GARCH(1,1) model, when the variance cluster effect is modelled

                                                                                  (5)

- in the EGARCH(1,1) model, when the leverage effect is modelled

                     (6)

where:  , ε_t ~ N(0,1)

-   in the TAGARCH(1,1) model, when the leverage and asymmetry effect is modelled

                  (7)

where: = 1 where u_t-1 < κ₁ and 0 in other situation,

κ₁ – asymmetry parameter responsible for the modelling of skewness of the rates of return distribution,

κ₂ – thresold parameter.

The empirical distribution of standardised reminders of the model is matched in practice, above all, with normal distribution and distributions that enable to obtain thick tails – the t-Student distribution and the general error distribution (General Error Distribution, GED).

3. Volatility forecasting using GARCH models

      Exchange rates volatility forecast are often based on the fact that volatility is time-varying and that periods of high and low volatility tend to cluster in case of high – frequency data.

For the GARCH(1,1) model (see (2) and (5)), the h-period-ahead variance forecast is defined as (cf. Doornik J., Hendry D., Econometric Modelling .., p.8-9):

                        (8)

The quality of the volatility forecasts are investigated using some ex post prediction errors and out-of –sample volatility forecasting statistics. The volatility forecasts should be compared with the realized variance over the forecast period. The most uses measure based on cumulated squared intradaily returns is integrated volatility, which is defined as  (cf. Andersen T., Bollerslev T. 1998, p. 885 – 905):

                                (9)    

In particular  the usual measure for the one-period observed volatility in the literature is the square of the return or the absolute return.

To compare the quality of volatility forecasts set, the following statistical criteria are used (cf. Poon S-H., Granger C. W. 2003):

v    the root mean square error (RMSE)

RMSE =                          (10)

where

- the realized variance over the forecast period T,

- the variance forecast at time T.

v    Mean Error ( ME)

ME =                                   (11)

v    Mean Absolute Error ( MAE)

MAE =                                  (12)

v    Heteroscedasticity Adjusted Mean Absolute Error (HMAE), which is adjusted for ARCH effects

HMAE =                                  (13)

v    Logarithmic Loss Function (LL), which is computed to stress the influence of low volatility periods

LL =                                        (14)

The last discussed in this paper statistical criterion for the quality of volatility forecasts is one based on the standard forecast efficiency regression:

                                       (15)

To compare forecasting performance of different volatility models one can compute the coefficient of determination of the forecast efficiency regression:

                                  (16)

where:  .

The higher value of this coefficient indicates the model, which outperforms other volatility models. 

4. Empirical research

The empirical research were based on the average daily NBP quotations of USD/PLN exchange rate between 01.01.1998 – 28.02.2005. The parameter estimation of GARCH(1,1) models was made in the PC GIVE package (see table 1).

Table 1. Parameters of GARCH (1,1) models – estimation results

Parametr	GARCH(1,1)	GARCH(1,1)	EGARCH(1,1)	EGARCH(1,1)
μ	-0,032265 [0,016]	-0,0331487 [0,013]	-0,0243724 [0,001]	-0,0176051 [0,203]
α0	0,0437925 [0,000]	0,0399029 [0,005]	-0,08914 [0,000]	-0,0831758 [0,000]
α1	0,187454 [0,000]	0,183701 [0,000]	-	-
β1	0,7303 [0,000]	0,743642 [0,000]	0,898192 [0,000]	0,893705 [0,000]
ν(t-Student)	-	8,51076 [0,000]	-	-
υ1	-	-	0,0735715 [0,001]	0,0853723 [0,000]
υ2	-	-	0,323273 [0,000]	0,316875 [0,000]
GED ν*= log(ν/2)	-	-	-0,268214 [0,000]	-
Log-like.	-1796,62297	-1775,49876	-1773,44565	-1790,257

Source: Own calculations; p-value in parentheses.

Finally, the GARCH(1,1) model with the t-Student distribution and the EGARCH(1,1) model with the GED distribution were selected as prognostic models of conditional variance of the USD/PLN exchange rate returns.[6]

Variance forecasts at 1, 10, 21 days horizons are constructed for estimated GARCH models, in relation to formulas (8). Then variance forecasts are compared with the realized variance over the forecast period, where realized variance is given by the formula (9). Summary statistics are shown in table 2.

Table 2. Verification of conditional variance forecasts for GARCH models

	Out-of-sample volatility forecasting statistics
Model	RMSE	ME	MAE	LL	HMAE	R2
one-day-ahead forecasts
GARCH-t(1,1)	0,867719	0,023363	0,648457	1,17738	54,76239	0,012751
EGARCH-GED(1,1)	0,796429	0,083739	0,541921	0,940238	45,84107	0,050661
ten-day-ahead forecasts
GARCH-t(1,1)	4,148559	0,41881	2,983632	0,057469	0,700954	0,001462
EGARCH-GED(1,1)	4,110228	1,27938	3,040334	-0,11098	0,432136	0,008231
twenty one-day-ahead forecasts
GARCH-t(1,1)	5,57499	1,518658	4,398723	-0,0531	0,369342	2,93E-06
EGARCH-GED(1,1)	6,483768	3,440364	4,633188	-0,25594	0,336313	0,0007388

Source: Own calculations

Taking into account the values of ME errors one may notice that in each horizon discussed forecasts of exchange rate variability constructed based on models that belong to the GARCH family, in contrast to forecasts constructed based on switching models, are more often underestimated than overestimated. In a 1-day forecast horizon, according to all criteria taken into account, the EGARCH-GED(1,1) model is characterised by better forecast properties. For a 10-day forecast horizon, more precise forecasts of variability according to most ex post measures (except MAE) are constructed based on the EGARCH-GED(1,1) model, whereas only the HMAE criterion indicates significant differences in the accuracy of forecasts. In the case of the longest forecast horizon, more precise forecasts according to the RMSE and MAE criteria are constructed based on the GARCH-t(1,1) model.

5. Summary

When comparing the values of all determined measures that evaluate the accuracy of forecasts concerning the USD/PLN the exchange rate variability one could state that the evaluation of forecast capabilities of the model depends to a large extend on the adopted comparison criterion and the forecast horizon.

It is worth to emphasis that the information about the exchange rate volatility is one of the most important pieces of information in financial markets. It is due to the fact that volatility becomes a key variable in the pricing of derivative securities, in establishing monetary policy, in risk management. Also volatility forecast enters option pricing formulas derived from the Black – Scholes model and its various extensions. For hedging against risk, portfolio management, volatility estimates are crucial too. Nowadays, banks and trading houses have to set aside reserve capital of at least three times that of value – at – risk. (VaR is defined as the minimum expected loss with a 1- percent confidence level for a given time horizon (usually one or ten days). Volatility forecast is needed to obtain such VaR estimates.

 

Referneces:

 

1.     Andersen T., Bollerslev T., Answering the Sceptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts, “International Economic Review” 39/1998.

2.     Beine M., Laurent S. , Lecourt Ch., Official central bank interventions and exchange rate volatility: Evidence from a regime-switching analysis, “European Economic Review” 47/2003.

3.     Doornik J., Hendry D., Econometric Modelling Using PcGive, Timberlake Consultants LTD., London 2001.

4.     Poon S-H., Granger C. W., Forecasting Volatility in Financial Markets: A Review, “Journal of Economic Literature” 41/2003.

5.     Tsay R., Analysis of Financial Time Series, Wiley & Sons, Chicago 2002.