A Primer for Unit Root Testing by K. Patterson

By K. Patterson

This e-book offers an authoritative evaluate of the literature on non-stationarity, integration and unit roots, delivering course and information. It additionally offers precise examples to teach how the options might be utilized in useful occasions and the pitfalls to avoid.

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Example text

These examples illustrate that a random variable is a mapping from the sample space Ω to some function of the sample space; the identity mapping is included to allow for Ω being unchanged. Indeed, the term random variable is a misnomer, with random function being a more accurate description; nevertheless this usage is firmly established. Usually, the new space, that of the random variable, is ᑬ and that of an n-vector of random variables is ᑬn. Consider the coin-tossing experiment, then for each toss separately xi(␻): ⍀1 ‫ ۋ‬ᑬ.

For a single trial and this assignment of x1, say, then E(x1) = 1p + 0q = p, and variance ␴2x1 = (1 – p)2 p + (0 – p)2 q = (1 – p)2 p + p2q = p(1 – p) using q = 1 – p. When the coin is tossed twice in sequence, we can construct a new random variable (which is clearly measurable) that counts the number of heads in the sequence and so maps ⍀2 into N (the set of nonnegative integers) say, S2 = x1 + x2, with sample space ⍀S,2 = {0, 1, 2} and probabilities {q2, 2pq, p2}; if p = q = ½, then these probabilities are {¼, ½, ¼}.

We can now return to the question of what is special about a stochastic process, other than that it is a sequence of random variables. , xn); then it may be of interest to know whether x n converges in a well-defined sense to a random variable or a constant as n → ∞, a problem considered at length in Chapter 4. For example, suppose that xj is distributed as Student’s t with j degrees of freedom; then as n → ∞, xn → x, where x is normally distributed. Such an example occurs when the distribution of a test statistic has a degrees of freedom effect.

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