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3 3.5 4 4,5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Evaluation points

Figure 17.2. Quadrature bias with 3, 6, and 9 evaluation points.

mined percentage, then increase the number of draws some more to make sure the stability was not a random fluke.

A body ofliterature is now emerging on simulation with non-random draws; that is, taking systematic draws from the density of interest. Halton draws are one example, and a few applications with environmental implications are available.13 These methods have the potential to significantly reduce simulation noise and the required number of draws. Train, 2003 devotes much of his Chapter 9 to a discussion of simulating with systematic draws. After estimating the same model twice, first with 1,000 random draws and then with 100 Halton draws, Train summarizes:

These results show the value of Halton draws. Computer time can be reduced by a factor of ten by using Halton draws instead of random draws, without reducing, and in fact increasing, accuracy. These results need to be viewed with caution, however. The use of Halton draws and other quasi-random numbers in simulation-based estimation is fairly new and not completely understood.

Future research might compare quadrature with simulation using systematic methods of drawing parameter vectors.

In closing, consider a direct mathematical comparison of the simulation formula and quadrature formula for Pj,. Consider the case of one random param-

3 3.5 4 4,5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Evaluation points

Figure 17.2. Quadrature bias with 3, 6, and 9 evaluation points.

13See Goett et al., 2000b, Hess et al., 2004, and Sandor and Train, 2004.

eter. With simulation and for quadrature where h(n) = H $ j=i

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