Download A Primer for Unit Root Testing by K. Patterson PDF

By K. Patterson

This publication supplies an authoritative review of the literature on non-stationarity, integration and unit roots, delivering course and assistance. It additionally presents distinct examples to teach how the ideas might be utilized in sensible events and the pitfalls to avoid.

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Example text

1 Discrete random variables It is helpful to consider first the case of two random variables, x1 and x2, each of which is discrete. 40) where | indicates that the probability of x 2 is being considered conditional on x1. This notation is shorthand for much more. More explicitly we are concerned with a conditioning event (set) in the range of x1, say X1 ʦ A and an event (set) in the range of x 2, say X 2 ʦ B. Sometimes, in the case of discrete random variables, the sets A and B will comprise single values of x1 and x 2 in their respective outcome spaces, but this is not necessary; for example, in a sequence of two throws of a dice, the conditioning event could be that the outcome on the first throw is an odd number, so that A = (1, 3, 5) and the second event is that the outcome on the second throw is an even number, so that B = (2, 4, 6).

3: Extension of the uniform distribution to two variables In this case we consider two independent random variables x1 and x2, with a uniform joint distribution, implying that each has a uniform marginal distribution. The sample space is a rectangle ʚ ᑬ2, the twodimensional extension of an interval for a single uniformly distributed random variable. Thus, x1 and x2 can take any value at random in the (1) rectangle formed by I(1) 1 = [a1, a 2] on the horizontal axis and I 2 = [b1, b2] on the vertical axis, a1 < a2 and b1 < b2.

The probability density function associated with a discrete random variable is usually referred to as the probability mass function, pmf, because it assigns ‘mass’, rather than density, at a countable number of discrete points. 3. 20) Recall the notational convention that x denotes the random variable, or more precisely random function, and X denotes an outcome; thus x = Xi means that the outcome of x is Xi and P(x = Xi) is the assignment of probability (mass) to that outcome; the latter may more simply be referred to as P(x = Xi) or P(X) when the context is clear.

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