ARCH/GARCH Modelling of Exchange Rates in Turkey Prof. Dr. Mustafa ÖZER Anadolu University Department of Economics muozer@gmail.com 00(90) 533 372 4704

DATA: MONTHLY DOLLAR-TRY EXCHANGE RATE PERIOD (1999:01-2008:05)

Figure 1: Nominal Exchange rate series

Figure 2: Logarithmic values of nominal exchange rates

Figure 3:Logarithmic first differences of series

DF, ADF, PP, DF-GLS de-trended test UNIT ROOT TESTS DF, ADF, PP, DF-GLS de-trended test Null hypothesis state that a series does contain a unit root (i.e. İt is non-stationary). KPSS Null hypothesis state that a series is stationary. Unit-root tests with structural breaks Perron test Zivot-Andrew test.

UNIT ROOT TESTS ADF (0)

PP(1)

UNIT ROOT TEST FOR LEXC ADF(0)

PP(1)

UNIT ROOT TEST FOR DLEXC ADF(10)

PP(2)

ARCH/GARCH MODELS-I Financial time series are often available at a higher frequency than macroeconomic time series and many high-frequency financial time series have been shown to exhibit the property of 'long-memory' (the presence of statistically significant correlations between observations that are a large distance apart). Another distinguishing feature of many financial time series is the time-varying volatility or 'heteroscedasticity' of the data. It is typically the case that time series data on the returns from investing in a financial asset contain periods of high volatility followed by periods of lower volatility (visually, there are clusters of extreme values in the returns series followed by periods in which such extreme values are not present). For example, stock markets are typically characterized by periods of high voltility and more ‘relaxed’ periods of low volatility (volatility clustering-shocks tend to be followed by big shocks in either direction, and small shocks tend to follow small shocks).

ARCH/GARCH MODELS-II When discussing the volatility of time series, econometricians refer to the 'conditional variance' of the data, and the time-varying volatility typical of asset returns is otherwise known as 'conditional heteroscedasticity'. The concept of conditional heteroscedasticity was introduced to economists by Engle (1982), who proposed a model in which the conditional variance of a time series is a function of past shocks; the autoregressive conditional heteroscedastic (ARCH) model. Because of the presence of ‘asymmetry effect’ (In the context of financial time series analysis the asymmetry effect refers to the characteristic of time series on asset prices that an unexpected drop tends to increase volatility more than an unexpected increase of the same magnitude (or, that 'bad news' tends to increase volatility more than 'good news'); also known as the ‘leverage effect’, extensions of ARCH model is developed, such as GARCH, EGARCH, and TGARCH.

The Threshold GARCH (T-GARCH) Model  

DETERMINING THE MEAN EQUATION ARMA(1,1)

AR(1)

MA(1)

Chosen model should have the significant coefficients and also yield lowest information criterion values of AIC, SC, HQ and largest value of LogL. Based on these criterions, the selected model is MA(1). For MA(1) model, the results of ARCH-LM Test are as follow: ARCH-LM(1) Since TR² =17,26104 > =3,8415, we reject the null hypothesis and conclude that there is statistically significant ARCH effects in the errors of MA(1) model. Therefore, we have to model the exchange rate series by using ARCH/GARCH type models.

THE RESULTS OF ARCH/GARCH MODELS

ARCH(1) MODELİ

GARCH(1,1)

GARCH(1,1)-M (Garch in Mean) (According to standart deviation)

GARCH(1,1)-M (Garch in Mean) (According to variance)

In the models of ARCH(1), GARCH(1,1) and GARCH(1,1)-M, it is assumed that effects of negative and positive shocks on conditional variance are symmetric. If it is believed that they are asymmetric, then, EGARCH and TARCH models should be used. Following slights are presenting the results of these models.

%10 anlam düzeyinde anlamlı kabul edilebilir. TARCH(1,1) MODEL %10 anlam düzeyinde anlamlı kabul edilebilir.

EGARCH(1,1) MODEL

Among all these models, best model chosen is the TARCH(1,1) model, since it has lowest AIC, SC, HQ criterion values and largest LogL value as well as it has the significant coefficients. Moreover, since TARCH model is taking into account of asymmetry. Therefore, we decide to continue with use of TARCH(1,1) model.

Since the chosen model’s error shouldn’t have the significant ARCH effects, we carried out another ARCH-LM test. ARCH-LM(1) Since TR²=2.224655 < =3,8415, we fail to reject the null hypothesis that there is a statistically significant ARCH effects in the errors of TARCH(1,1) model.

Conditional Variances of TARCH(1,1) Model