Eleven out of 10 for effort, minus infinity
for believability, I'm afraid, Stace.
The gaussian distribution ranges from minus infinity
to plus infinity, so as long as the SD is not zero, it is a mathematical certainty that the probability of results larger than 15.7% is greater than zero.
Do these zeros converge when c tends to infinity or minus infinity
? If so, what is the rate of convergence?
One implication of these results is that, if divergence is towards minus infinity, the null hypothesis of no cointegration will be spuriously rejected, and this size distortion will increase with the sample size, approaching one asymptotically.
Given that the Engle-Granger DF-based test for cointegration is a left tail test, result a) in theorem 1 is not enough to establish the presence of spurious cointegration; for this the t-statistic has to diverge to minus infinity, since divergence in the opposite direction would imply nonrejection asymptotically.
As shown in theorem 2, the EG test will diverge to minus infinity, thus rejecting the null hypothesis [H.sub.0] : [gamma] = 0, and (correctly) indicating the presence of cointegration.
Should the t-statistic diverge to minus infinity, the null hypotheses [H.sub.0] : [[gamma].sub.i] = 0, i == 1,2 in regression model (7) will be correctly rejected in large samples (see below).
Finally, as one might expect, if in the cointegrating equations (5) or (6) we include a trend break dummy variable (assuming of course we know the timing of the break), then the Engle-Granger t-statistic for testing cointegration will diverge to minus infinity, thereby correctly rejecting the null of no cointegration.
where M(V) is the velocity-dependent "mass" of the said quasi-particle, and inertial motion requires the vanishing of the right-hand side, here written symbolically as the difference ("jump") between the values of a driving force between plus and minus infinity
, again defined in terms of the value of the "Eshelby stress" at infinities.
Individual chapters discuss cosmic acceleration, dark matter, the cosmic ray paradox, renormalization (necessary because infinity minus infinity
does not equal zero in physics), the theorized Higgs particle, quantum gravity, string theory, the origin of the universe, and other mysteries.
For high values of k, Xnext becomes completely unstable, swinging back and forth between plus and minus infinity
. For intermediate values of k, Xnext oscillates, but in a bounded, repeating fashion.