New PDF release: A Primer for Unit Root Testing

By K. Patterson

ISBN-10: 0230248454

ISBN-13: 9780230248458

ISBN-10: 1403902054

ISBN-13: 9781403902054

This booklet supplies an authoritative evaluate of the literature on non-stationarity, integration and unit roots, delivering course and assistance. It additionally offers precise examples to teach how the innovations may be utilized in functional events and the pitfalls to avoid.

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Extra info for A Primer for Unit Root Testing

Example text

3. 6: The coin-tossing experiment with n = 2 Consider the coin-tossing experiment with n = 2, with random variables {xj}2j=1, where the outcome on the j-th toss is mapped into (+1, –1). Then there are four sets of outcomes: ⍀x,2 = {(1, 1), (1, –1), (–1, –1), (–1, 1)}. Under independence the joint event table has the following entries. 25. The probabilities in the final row and final column are just the probabilities of the events comprising x1 and x2, respectively; these are referred to as the marginal probabilities, and their whole as the marginal distribution(s), to distinguish them from the conditional probabilities and conditional distributions in the body of the table.

It turns out that this is an application of a theorem that greatly simplifies the evaluation of the expectation for nonlinear functions; for a formal statement of the theorem and proof see, for example, Ross (2003). 37) This implies that to obtain E(z), we appear to need the pmf of z, that is, in effect the distribution of z. 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.

Thus, Ω comprising the set of all positive integers, Ω = (1, 2, ... ,) is countably infinite; outcomes of the form ␻i = ␣ + hi, so that h is the step size, are countably infinite. Condition F3 limits the subsets in the field to a finite union of events, which means that many subsets of interest, including Ω itself, cannot be generated. What is required is an extension of conditions F3 and P3, to allow infinite unions of events. When this is done the field F is known as a ␴–field or ␴–algebra (of Ω).

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A Primer for Unit Root Testing by K. Patterson


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