Palgrave Handbook of Econometrics: Applied Econometrics

B.D. McCullough 1303

passes that, then give it to Big Crush. If it passes that, then it is fit to use. There are
too many good RNGs to use one that fails any reasonable test of randomness. There
are many bad RNGs, even ones published in journals, so testing is imperative.
It is not enough that the underlying uniform RNG passes tests. Often the need
for random normal variates, or random variables with other distributions, arises.
These are usually obtained from the uniform RNG via transformation. However,
the process of turning seemingly good uniforms into normals, e.g., requires not just
good uniforms, but also a good transformation. The transformations from unifor-
mity to another distribution, e.g., standard normal, can be flawed. The Marsaglia
Multicarry RNG, which passed all the DIEHARD tests and was popular for some
years, does not play well in the tails with the Kinderman–Ramage transform to
normality. So if your econometric software package uses the Multicarry, and it pro-
duces random normals via the Kinderman–Ramage, then the tails of the so-called
random normals actually deviate substantially from true random normals. For a
graphical depiction of this phenomenon, see Figure 28.1. Note the slight gap at
about 3.4 and a pronounced gap at about 3.6. These gaps would wreak havoc upon
any simulation that focused on the tails of the standard normal.^2
Further, the Multicarry fails both Small Crush and Crush batteries in TESUT01
(there was no need to apply Big Crush). Testing random normals, random-t’s, and
random chi-squares needs to be done. As this time, such testing is in its infancy.
A common approach is to back transform the random normals (say) to unifor-
mity, and then apply tests for uniformity to the backtransformed uniforms. This
approach was used successfully by Tirleret al.(2004). Do you think it safe to trust
the random normals in your software package?

0 1000

Z score

2000

Sample

3000 4000

3.0

3.5

4.0

4.5

5.0

5.5

Figure 28.1 The extreme tails of 4,000 samples of random normals from Marsaglia Multicarry
and Kinderman–Ramage