A Primer for Unit Root Testing by K. Patterson

By K. Patterson

This booklet provides an authoritative review of the literature on non-stationarity, integration and unit roots, delivering path and suggestions. It additionally presents targeted examples to teach how the options should be utilized in functional occasions and the pitfalls to avoid.

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2) 12 A Primer for Unit Root Testing An equivalent way of writing this for a random variable x is F(X) = P(x(␻) ≤ X). The properties of a distribution function are: D1. F(a) is bounded, F(–∞) = 0 and F(∞) = 1; D2. it is non-decreasing, F(b) – F(a) ≥ 0 for a < b; D3. it is continuous on the right. For example, consider a continuous random variable that can take any value on the real line, R, and A = [a, b] where –∞ < a ≤ b < ∞, then what is the probability measure associated with A? 3) An identifying feature of a discrete random variable is that it gives rise to a distribution function that is a step function; this is because there are gaps between adjacent outcomes, which are one-dimensional in ᑬ, so the distribution function stays constant between outcomes and then jumps up at the next possible outcome.

If the n tosses of the coin are consecutive, then the sample space, of dimension 2n, is denoted ⍀n, where the generic element of ⍀n, ␻i, refers to an n-dimensional ordered sequence. In the usual case that the coin tosses are independent, then the sample space Introduction to Probability and Random Variables 19 ⍀n is the product space, ⍀n = ⍀1 ϫ ⍀1 ϫ ... ϫ ⍀1 = ⍀n1 (where the ϫ symbol indicates the Cartesian product, see glossary). We now understand by fixing ␻ that we fix a whole path, not just a single element at time j (or t); thus as ␻ is varied, the whole sample path is varied, at least potentially.

57) for all B. 51), seem straightforward extensions of the discrete random variable case, they raise a problem. From the start, the situation in which ͐X1ʦA f(X1)dX1 = 0 was ruled out, so that the set A could not be a single point in the range of X1 or a set with measure zero. To see the problem, note that in the case of a discrete random variable, a conditional probability mass function is obtained by taking a Introduction to Probability and Random Variables 35 value for, say, X1, as given; this fixes a row of the joint event table.

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