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Starting the estimate phase by selecting the issue that earlier phases suggest is the most important is a simple and effective rule of thumb. This rule of thumb generalizes to successive selections based on an overall ordering of the issues for treatment in terms of a perceived order of importance, usually within activities or other subdivisions of the project, which in turn may be selected because of the relative importance of issues as currently perceived. No matter how crude the basis of this judgement, it is preferable to using an alphabetic ordering or some other totally arbitrary ordering. Any estimation process involves getting up to speed , even if everyone involved is experienced in this kind of analysis. If inexperienced people are involved, going down a learning curve is part of the process. It makes sense to spend extra time as part of this process on the most important issues. It also makes sense to grip people s interest and involvement by starting with particularly important issues. When in doubt, err on the side of issues that have aspects that intrigue or touch on common experience, provided this is a cost-effective use of training time. Having identi ed a suitable candidate in these terms, a key question follows: is this a useful issue to quantify

Starting with a guess x0, the next point, xn+1, can be found from the previous point, xn, by x n+1 = x n - H -1 f (x n ) (1.17)

6.13 Two sources A and B produce the following electromagnetic fields on an aperture through a periodic conducting scatterer: Ea = V1+ + V1 e1 x y +

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In this section we have analyzed measurement error effects in detail and demonstrated that outcome variable measurement error is benign, and explanatory variable measurement error is dangerous Perturbations are essentially researcherimposed measurement error Having discussed the deleterious modeling problems with measurement error, we advise against intentionally including data perturbations in published estimates, although some researchers have explicitly modeled known data measurement error using an approach similar to perturbation analysis (Beaton et al 1976) Instead, the perturbations act like unintended, but completely known measurement error that provide a means of testing the behavior of estimators and algorithms That is, models that react dramatically to modest levels of measurement error are models that one should certainly be cautious about (particularly in the social sciences) In summary, perturbation may introduce bias, but if the problem is wellconditioned and the algorithm and implementation accurate, the bias should be small.

Moreover, any bias introduced by perturbations should be the same when the same model and perturbed data are used in different implementations So if two implementations of the same model show marked differences in pseudostability with respect to similar perturbation analyses, the root cause is asserted to be computational and not statistical1 4323 Some Rami cations of Perturbed Models Using the core idea of random perturbation, we can assess whether results are reliable, whether they are consistent with respect to small perturbations of the data This methodology complements standard diagnostic plots in two ways First, one can use the strictly numerical results as an unambiguous check: Simply evaluate whether the range of results across input perturbations still ts the original substantive conclusions about the results Second, this methodology may sometimes reveal numerical problems that may be missed in standard diagnostic plots.

With regard to input perturbation, what is considered small for any particular case is a matter of subjective judgment There is an obvious lower limit: Perturbations of the data that are at the level below the precision of the machine should not be expected to cause meaningful changes in output The upper limit on perturbations is less clear, but should be bounded by the accuracy of data measurement In many of the social sciences, measurement error certainly dominates machine precision as a source of input inaccuracy Many of the signi cant digits of macro data are reported as rounded, for example, to thousands Introducing perturbations to the rounding error of these data is a tractable problem to solve Sometimes data are bounded, which introduces complications to perturbations.

The simplest way of avoiding the bounding problem is to truncate any illegal value generated by perturbations to the constraint, but this introduces mass at the boundary points..

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