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...addressed examine the euro/dollar exchange rate and the related financial returns in the context of detecting exact and stochastic unit roots, and in the consequence, modelling them using time varying parameters model. The estimated STUR models are compared with standard ARMA-GARCH representations. We also examine causal relationships in the Granger sense. Upon the results of causality testing, some ADL-GARCH models are built, which are further used to examine their forecasting performance. (JEL C10, G15)
Introduction
Financial prices and returns have been the subject of empirical and theoretical analysis for many years, and their dynamic properties and other characteristics are still of interest. Many different tools have been applied to describe financial markets. Some hypotheses, such as efficient market hypothesis, refer to the stochastic nature of the financial processes. On the other hand, some researchers studied chaotic properties of the financial phenomena. As there is only little evidence in favour of chaotic dynamics of the empirical financial time series, we have adopted the assumption of the stochastic nature of the processes. In many papers, the following properties of the financial time series were examined: Stationarity, randomness, autocorrelation, ARCH effect, normality, long-run relationships, calendar effects, and many others. The aim of the presented paper is to examine the euro/dollar exchange rate and the related financial prices in the context of detecting exact and stochastic unit roots, and in the consequence, modelling them in the proper way. The methodology by Leybourne et al. [1996a, 1996b] was used. We also detected some causal relationships in the Granger sense. Upon the results of the tests some models are built, which are further used to examine their forecasting performance.
Testing for Exact and Stochastic Unit Root
Non-stationarity in variance is typical for financial prices, while rates of return are usually variance-stationary, which is based on the definition of the standard as well as logarithmic rate of return. This gives rise to use a random walk to describe the prices of financial assets. Empirical evidence indicates that such a hypothesis is too simple to characterize financial prices and returns which are strongly volatile. Moreover, standard unit root tests, such as Dickey-Fuller (DF) [Dickey and Fuller, 1979], Augmented Dickey-Fuller (ADF) [Dickey and Fuller, 1981], Philips-Perron (PP) [Phillips and Perron, 1998], and even Kwiatkowski-Phillips-Schmidt-Shin (KPSS) [Kwiatkowski et al., 1992] tests are not able to distinguish between the exact and stochastic unit roots. The former results in the difference stationary process, while in the latter case, the time series cannot become stationary after taking differences of any order. It can be shown that the process that has an exact unit root also has the stochastic one. The stochastic unit root (STUR) concept can be considered as an extension of the exact unit roots processes in the context of time varying parameter models [Granger and Swanson, 1997]. It draws our attention to the nonlinear models' specification. This type of models is able to describe the conditional mean and the conditional variance of the time series and can compete with standard ARIMA-GARCH approach.
The article concerning the STUR identification [Leybourne et al., 1996a] suggests the following simple random coefficient autoregressive model describing a stochastic unit root:
[y.sub.t] = [[alpha].sub.t][y.sub.t-1] + [[epsilon].sub.t] (1)
where [[alpha].sub.t] = [[alpha].sub.0] + [[delta].sub.t], [[alpha].sub.0] = 1.
[[delta].sub.t] = [rho][[delta].sub.t-1] + [[eta].sub.t] (2)
[[delta].sub.0] = and |[rho]| [less than or equal to] 1.
Stochastic processes [[epsilon].sub.t] ~ N (0, [[sigma].sup.2]) and [[eta].sub.t] ~ N (0, [[omega].sup.2]) are assumed to be independent. If |[rho]| 0, a process with a unit root in mean, called a stochastic unit root process, is observed.
Leybourne et al. [1996b] have proposed a testing procedure (LMT test), where under the null the exact unit root is tested, while under the alternative the stochastic unit root is assumed [Leybourne et al., 1996a]. Hypotheses in the LMT test concern the variance [[omega].sup.2] in the model (2). The null is [H.sub.0] : [[omega].sup.2] > 0, which means the random walk process or ARIMA(p,1,0), while the alternative is [H.sub.1]: [[omega].sup.2] > 0. The interpretation of the alternative depends on the value of the [rho] parameter in (2). When |[rho]| < 1, [[delta].sub.t] in (2) is a stationary process with a zero mean. For [rho] = 1, it follows a random walk process.
To avoid the influence of deterministic trend and autocorrelation, the model may include the linear or quadratic time trend and the autoregressive lags of the dependent variable as well, so it takes the following form:
[y*.sub.t] = [[alpha].sub.t][y*.sub.t-1] + [[epsilon].sub.t] (3)
where:
[y*.sub.t] = [y.sub.t] - [P.sub.t] - [p.summation over (i=1)][[phi].sub.i][y.sub.t-i] (4)
[P.sub.t] means a deterministic component, for example, trend;
[P.sub.1t] = [beta] + [gamma]t or [P.sub.2t] = [beta] + [gamma]t + [theta]t(t + 1)/2.
The autoregressive part in (4) is stationary, and its role is similar to the augmentation in the ADF test. If in [H.sub.1] |[rho]| < 1, then the Z statistic is computed in the following way:
1. estimate ordinary last square (OLS) the equation (5)
[DELTA][y.sub.t] = [DELTA][P.sub.t] + [p.summation over (i=1)][[phi].sub.i][DELTA][y.sub.t-i] + [[epsilon].sub.t] (5)
2. compute the statistics
Z = [T.sup.-[3/2]][[sigma].sup.-2][[kappa].sup.-1][T.summation over (t=2)]([t-1.summation over (j=1)][[epsilon].sub.j])[.sup.2] ([[epsilon].sub.t.sup.2] - [[sigma].sup.2]) (6)
where [[sigma].sup.2]...
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