B.D. McCullough 1315information matrix. The users of these packages will just have to trust that these
standard errors – whatever type they are – are correctly programmed.
28.6.4 Unconditional least squares
We have seen that an important part of ULS is the determination ofQ, the number
of observations to backcast. Many packages claim to use backcasting and offer
only BJ as a reference. Given the atrocious state of software documentation, it
is not surprising that most packages that offer backcasting do not give the rule
used to determine the number of backcasts. What is perhaps surprising is that,
when this author contacted the developers who did not mention the rule in their
documentation, only one of them would reveal its rule.
This is either pathetic or amusing, depending on your point of view, because the
text does not give the rule used to determine the number of backcasts. The back-
casts in an example in BJ (1976, pp. 217–18) “die out quickly and to the accuracy
with which we are working are equal to zero” for backcasts number 4 and greater.
Confusingly, BJ do not declare the accuracy to which they are working! In their
code in the back of the book, BJ (ibid., p. 502) recommend stopping backcasts when
wt−ˆμbecomes negligibly small, but they do not state what constitutes “negligibly
small”: is it 0.1, 0.01, 0.001, or even smaller? For this benchmark we backcast until
wt−ˆμ<0.01. Decreasing this tolerance to 0.001 changes the estimates ofφandθ
in the third decimal.
Analytic derivatives are not used here, as mentioned in the previous section; yet
we can be confident that we are achieving the same level of accuracy as if we were
implementing analytic derivatives. The benchmark is presented in Table 28.9.
Note that this result comports with BJ’s Table 7.13, the BJ results for estimation
of Series A. BJ report (with standard errors in parentheses) a constant of 1.45, aφof
0.92(0.04)and aθof 0.58(0.08). The BJ constant is consistent with our benchmark
constant because 17.066≈1.45× 1 /( 1 −0.9149...). Note that our standard errors
agree with those of BJ. Further, BJ give the residual variance as 0.097 while that
from the benchmark, to four decimals, is 0.0974. So it seems that we are estimating
the same model for which BJ present results.
Of interest is whether any of the three packages that offer backcasting come close
to the benchmark. These results are presented in Table 28.10.
Overall, no package comes close to the benchmark. As mentioned, this is because
they use different, secret methods for determiningQthat, for some reason, they
will not reveal to their users. There is no point in checking the standard errors.
Table 28.9 ULS benchmarkParameter μφ θCoefficient 17.065547663 0.91494836959 0.58268097638
Standard error (MLE) 0.10808561791 0.042209513625 0.083811338527