By K. Patterson
This booklet supplies an authoritative assessment of the literature on non-stationarity, integration and unit roots, supplying path and assistance. It additionally offers certain examples to teach how the concepts will be utilized in useful events and the pitfalls to avoid.
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Additional info for A Primer for Unit Root Testing
A convex function requires that the second derivative of g(x) with respect to x is positive; for example, for positive x, the slope of g(x) increases with x. An example will illustrate the application of this inequality. Consider x to be the random variable with outcomes equal to the number of spots on the face of a rolled dice and z = g(x) = x 2; x is positive and the second derivative is 2x, which is positive for x positive, hence the function g(x) is convex. The expected value of z is E[g(x)] = E(x2) = 91/6, whereas E(x)2 = (7/2)2 = 49/4 < 91/6.
The measurable spaces are (⍀, –F) and (ᑬ, B), with corresponding probability space (Ω, –F, P) and derived probability space (ᑬ, B, PX), respectively. The requirement is that of measurability defined as follows. The function x: ⍀ ۋᑬ, is said to be measurable, relative to –F and B, if x(–1) (B) ʚ –F. (The operator indicated by the superscript (–1), to distinguish it from the inverse operator, is the pre-image) That is the pre-image of X is in the –field of the original random experiment. Intuitively, we must be able to map the event(s) of interest in x back to the field of the original sample space.
41) would not be defined. To outline the approach, but to avoid this difficulty in the first instance, we consider the conditioning event to have a non-zero probability. 49) These conditions are not problematical as assuming that the density functions exist then all component functions are well defined. In seeking a conditional expectation, we could approach the task by first defining a conditional distribution function, by analogy with the discrete case, as the ratio of the joint distribution function to the (conditioning) marginal distribution function, or in terms of density functions as the ratio of the joint density function to the (conditioning) marginal density function.
A Primer for Unit Root Testing by K. Patterson