By K. Patterson
This e-book provides an authoritative evaluation of the literature on non-stationarity, integration and unit roots, supplying course and assistance. It additionally offers targeted examples to teach how the concepts might be utilized in useful occasions and the pitfalls to avoid.
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Additional info for A Primer for Unit Root Testing
38) Consider z = x2 where x = (X1 = –1, X2 = 0, X3 = 1) with probabilities then z = (Z1 = 0, Z2 = 1) with probabilities (–13, –32), so that working with z directly E(z) = 0 ϫ –13 + 1 ϫ –32 = –32; in terms of x, E(z) = (–1)2 ϫ –16 + 0 ϫ –13 + (1)2 ϫ –12 = –32. The answers are, of course, the same. In this case it is simple enough to obtain the pmf of z from the pmf of x, however, this is not always the case and it is in any case unnecessary. 39) −∞ where G(Z) is the distribution function of z and, as usual, F(X) is the distribution function of x, see Rao (1973, p.
Following the notational convention in this chapter the components of the stochastic process will be denoted y t() for a discrete-time process and y(t, ) for a continuous-time process; the reference to ʦ Ω is often suppressed. ). In the continuous-time case, T is an interval, for example T = ᑬ, or the positive half line T = ᑬ+ or an interval on R, for example T = [0, 1]. A stochastic process is a collection of random variables, denoted Y, on a probability space (see, for example, Billingsley, 1995), indexed by time t ʦ T and elements, , in a sample space Ω.
46) this will be equal to E(x2). 2 for a development of this concept. Of course the equality follows trivially for independent random variables, but remember this is just a special case of the general result: E[E(x2 | x1)] = E(x2). Other moments, such as the variance, can be conditioned on events in the space of the random variables. 47) It is also of note that whilst the independence of x1 and x2 implies that their covariance is zero, the reverse implication does not hold unless x1 and x2 are normally distributed.