Statistical Analysis of Daily Exchange Rate Data

A A time series {y t } is a discrete time continuous state process where the variable y is identified by the value that it takes at time t denoted y t . Time is taken at equally spaced intervals from –∞ to +∞ and the finite sample size T of data on y is for t = 1,2, . . . ,T. Time series {y t } may emerge from deterministic and/or stochastic influences. For example, a time trend y t = t is a very simple deterministic time series. If {y t } is generated by a deterministic linear process, it has high predictability, and its future values can be forecasted very well from the past values. A basic stochastic time series is “white noise,” yt = εt , where εt is an independent and identically distributed (i.i.d.) variable with mean 0 and variance σ2 for all t, written εt ~ i.i.d.(0, σ2). A special case is “Gaussian white noise,” where the εt are independent and normally distributed variables with mean 0 and variance σ2 for all t, written εt ~ NID(0, σ2). A time series generated by a stochastic process has low predictability, and its past values provide only a statistical characterization of the future values. Predictability of a time series can therefore be considered as the signal strength of the deterministic component of the time series to the whole time series. Usually, a given time series is not simply deterministic or stochastic, but rather some combination of both: (8.1) $$y_t = \alpha + \beta _t + \varepsilon _t$$