Modelling and forecasting long memory in exchange rate volatility vs. stable and integrated GARCH models

Abstract

The purpose of this article is to compare stable, integrated and long-memory generalized autoregressive conditional heteroscedasticity (GARCH) models in forecasting the volatility of returns in the Turkish foreign exchange market for the period 1990–2005 and for the subperiod that covers the floating exchange rate regime 2001–2005. In the first period, we found that long-memory GARCH specifications capture the temporal pattern of volatility for returns in US and Canadian dollars against Turkish lira. For the same period, the temporal pattern of volatility for returns Australian dollar, Japanese yen, Euro and British pound against Turkish lira are best captured by stable GARCH specifications. We found that in the subperiod, only the stable GARCH models are relevant and the return series no longer exhibit the long-memory properties. It was also concluded that all return series except British pound against Turkish Lira have asymmetric effects. Our analysis has shown that when long memory, asymmetry and power terms in the conditional variance are employed, together with the skewed and leptokurtic conditional distribution (of innovations), the most accurate out-of-sample volatility is produced for the first and subperiod. Thus is useful for financial decisions which utilize such forecasts.

Acknowledgements

We would like to thank M. Taylor for his useful comments and suggestions.

Notes

¹For example Fernandez (2006) analysed whether the Asian crisis, Japanese crisis and similar crisis caused permanent volatility shifts in the world stock markets and detected some shifts with two different methods. Similarly, Aggarwal et al. (1999) analysed the emerging stock markets and concluded that local and global events were leading to volatility shifts.

²For example, Felmingham and Mansfield (1997) examined the relationship between depreciation and exchange rate volatility and they found that depreciation creates uncertainty and premium varies continuously until the market settle.

³The volatility models to stock market returns have been extensively analysed in Bollerslev (1987), Engle et al . (1987), Akgiray (1989), Baillie and De Gennaro (1990), Lamureux and Lastrapes (1990), Nelson (1991), Glosten et al . (1993), Rabemananjara and Zakoian (1993), Zakoian (1994), Ding et al . (1993), Hentschel (1995), Bollerslev and Mikkelsen (1996), Loudon et al . (2000), Brooks et al . (2000), Chen and Kuan (2002), Caporin (2003), Sanchez-Fung (2003) and Kılıç (2004), among others.

⁴See De Santis and Imrohorog^ˇlu (1997), Bekaert and Harvey (1997), Aggarwal et al . (1999), Choudhry (2000), Gokcan (2000), Ortiz and Arjona (2001), Al-Loughani and Chappell (2001), Poshakwale and Murinde (2001), Kasch-Haroutounian and Price (2001), Kassimatis (2002), Siourounis (2002), Kilic (2004), Vougas (2004) and Doong et al . (2005), among others.

⁵Under the assumption of normality the entire distribution can be designed by the first two moments. That is, δ is either 1 or 2. When δ = 1, the APARCH model evaluates the SD while the variance is modelled when δ = 2.

⁶In Turkish economy, some other crises occurred, like the financial crisis in April 1994, and the short-lived crisis in November 2000. In the beginning of 1990s, to attract more capital inflows the Central Bank of the Republic of Turkey (CBRT) increased interest rates. With these increase, public deficit was observed. Public sector was financed with domestic borrowing and as a result, domestic debt stock grew. Expectations of devaluations increased at the end of 1993. In January 1994, CBRT abandoned the exchange rate policy and devaluated the exchange rate by 14%. Devaluation of the currency continued until April 1994 and in April 1994, a new programme to overcome the financial crisis was brought into effect. At the end of 1999, the Turkish government signed 17th stand-by agreement with the IMF and started to implement inflation stabilization programme, based on a fixed exchange rate policy. But this programme to be cause of overvalued currency and the overvaluation continued during the programme. These were the leading reasons of the crisis in 2000 and in reality this was the cause of the crisis in 2001. The programme was unsuccessful, besides the real appreciation of the currency, low real interest rates lead to increased imports and a high current account deficit problem with the adverse political developments to be cause of crisis in 23 February 2001. After this, the central bank had to abandon the fixed parity and announced that the country moved into a floating exchange rate regime. Following the crisis, the Turkish government signed 18th stand-by agreement with the IMF and started to implement inflation stabilization programme and this programme is still continuing. For more details see Celasun (1998) and Özatay and Sak (2003).

⁷ r_t  = 100 * [ln(p_t ) − ln(p_{t −1})] where p_t is the price on day t.

⁸For some emerging stock markets similar results are found by Bekaert and Harvey (1997), Aggarwal et al . (1999) and Kapopoulos and Siokis (2005).

⁹Bollerslev (1987) was suggested the use of student-t distribution and followed by the others. See Baillie and Bollerslev (1989), Baillie and De Gennaro (1990), Pesaran and Robinson (1993), Tse and Tsui (1997), Choudhry (2000), Brooks et al . (2000), Solibakke (2001), Beine et al . (2002), Morana (2002), McKenzie and Mitchell (2002), Jacobsen and Dannenburg (2003) and Sanchez-Fung (2003).

¹⁰See Peters (2001), Laurent and Peters (2004) and Lambert and Laurent (2002) extended the skewed student-t density proposed by Fernandez and Steel (1998) to the GARCH framework. According to them, the models which takes into consideration both the conditional heteroscedasticity in the conditional variance as well as the skewed and leptokurtic conditional distribution of innovations, produces the most volatility forecasts.

¹¹Models which parameters significant and satisfy the set of nonnegativity conditions are given in the table. Asymmetric GARCH models (GJR, EGARCH, APARCH, FIEGARCH, FIAPARCH) were also estimated.

¹²As a rule, model yielding the minimum AIC, SIC or maximum LL is deemed to be most appropriate model.

¹³Similar results with our findings have been found for in other studies. e.g. Engle and Bollerslev (1986), Bollerslev (1987), Baillie and Bollerslev (1989), Lastrapes (1989), McCurdy and Morgan (1988), Hsieh (1989), Phylaktis and Kassimatis (1997), Tse (1998), Morana (2002) and Baillie et al . (2000), among others.

¹⁴These results are especially similar with the findings of Tse and Tsui (1997) and Tsui and Ho (2004), but contrary to the findings of Tse (1998), Morana (2002) and Baillie et al . (2000) that their analysis were made for developed countries, sum of α and β was found approximately unity but this hypothesis was rejected and found that conditional heteroscedasticity was not integrated. Besides, the results of Tsui and Ho (2004) is very important for the relationship between persistence in volatility and structural break, because they were excluded observations to avoid the possible distortions caused by the outbreak of the 2-year Asian financial crisis and hence they found that sum of α and β were still equivalently unity.

¹⁵This is taken as the sign of the Central Bank interventions and it is well known that Central Banks in emerging market countries intervene in the foreign exchange market frequently. In Turkey, especially after the 2001 financial crisis, the CBRT intervenes in the market in the case of excessive volatility. For the results of these interventions on the exchange rate volatility, see Akıncı et al . (2005a b) and Guimarães and Karacadag (2004).

¹⁶IGARCH model implies persistence in variance that is any shocks to the conditional variance affect the entire future path of h_t and persistence in variance is equivalent to a unit root in traditional volatility series.

¹⁷See Nelson (1991), Ding et al . (1993), Rabemananjara and Zakoian (1993), Glosten et al . (1993), Hentschel (1995), Hu et al . (1997), Loudon et al . (2000) and Brooks et al . (2000), among others.

¹⁸While some of these findings are consistent with the previous ones of the literature, some are not. In the study of Tse and Tsui (1997), the value was found negative and significant for Malaysian ringgit; negative but insignificant for Singapore dollar against US dollar (shown as MYR/USD and SGD/USD, respectively) exchange rates returns (under managed-floating regime). Besides, in the study of Tsui and Ho (2004), the value was found negative and significant for Malaysian ringgit and for Singapore dollar against Japanese yen, but found negative and insignificant for MYR/USD, positive and insignificant for SGD/USD. On the other hand, in the study of McKenzie and Mitchell (2002), the value was found negative and significant for Deutschemark against Francis frank; positive but insignificant for Canadian dollar against US dollar exchange rates returns; as a result among 17 heavily traded bilateral exchange rate, only four of them are found statistically significant.

¹⁹Davidian and Carroll (1987) argued that the SD specification is more robust than variance based estimates. According to these approach, conditional SD can be defined as σ _t  = α₀ + Σα _i ϵ _t − i  + Σβ _j σ _t − j or as Taylor/Schwert ST-GARCH model σ _t  = α₀ + Σα _i |ϵ _t − i | + Σβ _j σ _t − j or as Zakoian (1994)'s TGARCH model, .

²⁰The conditional volatility of GJR model is given as h _t  = α₀ + α₁ϵ _t − 1² + γ₁ϵ _t − 1²I _t − 1 + β₁ h _t − 1 and while indicator variable I_t is defined in original paper as for ϵ _t − 1 > 0 I _t − 1 = 1 and for ϵ _t − 1 < 0 I _t − 1 = 0 and γ₁ < 0 indicate the asymmetry (or leverage) effect, in G@RCH package I_t is defined as for ϵ _t − 1 < 0 I _t − 1 = 1, for ϵ _t − 1 > 0 I _t − 1 = 0 and γ₁ > 0 indicate the asymmetry effect; it means that an unexpected negative returns (or bad news) increases conditional variance of the next period exchange return, while an unexpected positive return (or good news) decreases conditional volatility.

²¹ Q statistics for standardized residuals for AUD/TL and JPY/TL models are significant at lag 5 but insignificant at lag 10 for the first period.

²²In the literature, the evidence is somewhat mixed as to whether volatility increases or decreases after market reforms. Some countries argue that the opening of financial markets to foreign investors results in increased volatility because of ‘hot money’ following in and out easily. As noted by Aggarwal et al . (1999) hot money which flows more easily into and out of emerging markets after market liberalization, induces greater volatility. Besides, like the other emerging countries, foreign exchange rate markets in Turkey are relatively small suggests that these markets may be susceptible to speculative trading. Moreover, persistence in volatility may be the result of heterogeneous expectations. Traders in the foreign exchange market may have heterogeneous expectations about the market movements. As noted in Hogan and Melvin (1994) and following them in Tse and Tsui (1997), the persistence in the conditional volatility in the foreign exchange market is positively related to the degrees of heterogeneous expectations in responds to market news. In this content, the consistency of policy interventions and the efficiency of the Central Bank play an important role in influencing the market expectations.

²³Similar criteria are used in the study of Swanson and White (1997), Walsh and Tsou (1998), Brooks and Burke (1998), Franses and Ghijsels (1999), Gokcan (2000), Peters (2001), Yu (2002) and Sarno and Valente (2005), among others.

²⁴According to Morana (2002), while the GARCH model may not reflect the true DGP of the data, it may allow for an approximation useful for forecasting purposes. As the forecasting horizon increases, integrated models become less and less useful. Then medium/long-term forecasts should be preferably generated by long-memory models. For short-term forecasting, GARCH models appear to be useful approximations to a data generating process. Besides Morana and Beltratti (2004) found that modelling the break process was not important for very short term forecasting once the model considers a long-memory component. Moreover, the conclusion of Diebold and Inoue (2001) that a pure long-memory model may be useful for forecasting purposes. At long forecasting horizons, (5–10 steps ahead), accounting for both long memory and structural change leads to superior forecasting model. Besides, according to Baillie et al . (1996), when lengthening the forecast horizon beyond 1 day, the proper modelling of long-term volatility dependencies become especially important. To see whether these conclusions are relevant for the best models in this article and which type of changes occurs in terms of criteria and forecasting values, we made out-of-180-days and out-of-5-days forecast for the full period and we compared the results. After comparisons were made, we can conclude the followings: as expected MSE, MAE and TIC criteria become quite smaller for 5-days forecast, that is 5-steps ahead forecasts of volatility outperform the forecasts using MSE, MAE and TIC and as a result for all exchange rate returns volatility is increasing slowly and no convergence to the steady state, but for AUD/TL volatility is increasing very slowly and similar to constant level.