B.D. McCullough 1309In whatever way it is determined, the choice ofQ, the number of observations
to backcast, will affect the final estimation results. What is surprising is that many
packages that offer the ULS method do not mention the stopping rule employed.
Even more surprising, when this researcher contacted many such developers, they
refused to reveal their stopping rules. Not only is this tantamount to refusing to
reveal the method of computation, but it also makes provision of a benchmark
impossible for these packages. Thus, users of such packages are in the unenviable
position of relying on unproven and unverifiable code.
The variation of the Marquardt algorithm proposed by BJ for estimating CLS and
ULS models is the essence of simplicity. Given initial estimates of the coefficients,
μ 0 ,φ 0 andθ 0 , compute the vector of residualsa, which will have lengthn−pin
the case of CLS and lengthn+Q+1 in the ULS case. Compute the derivative ofa
with respect to the parameters, denoteda(μ),a(φ)anda(θ), and run the regression:
a=bμa(μ)+bφa(φ)+bθa(θ), (28.9)to obtain estimatesbˆμ,bˆφandbˆθ. Compute the coefficient estimates at the end of
the first iteration asμ 1 =μ 0 +bˆμ,φ 1 =φ 0 +bˆφandθ 1 =θ 0 +bˆθ. To commence the
second iteration, based onμ 1 ,φ 1 andθ 1 , computea(note that the value ofQon
this iteration may well not be equal to the value ofQon the previous iteration) and
repeat the process until a termination criterion is achieved (e.g., successive sum of
squared residuals is less than , etc.). Suppose the termination test is successful at
the end of thecth iteration. Then the procedure is said to have terminated afterc
iterations. Note, though, that the gradients currently in the computer’s memory
are computed based onμc− 1 ,φc− 1 andθc− 1.
Computation of the standard errors is effected in the usual fashion. First, the
gradients must be recomputed usingμc,φcandθc, and each gradient will have
lengthn−p(CLS) orQ+ 1 +n(ULS). In the latter case, drop the firstQ+1 elements
of each vector. Form the matrix with three columns and either(n−p)(CLS) orn
(ULS) rows:g=[a(μ)a(φ)a(θ)]. The covariance matrix is given by(g′g)−^1 which,
when multiplied by
∑
a^2 i/n, has as its trace the variances of the coefficients.28.6.2 Calculation of derivatives
Given the general superiority of analytical derivatives over numerical derivatives,
no benchmark for a nonlinear procedure should be attempted on the basis of
numerical derivatives alone, except in exceptional circumstances (e.g., when cal-
culation of the derivatives is nearly impossible). Comparing the performance of
numerical and analytic derivatives in a benchmark setting can determine whether
it is safe for a user to rely on numerical derivatives or whether, as Fiorentini, Calzo-
lari and Panattoni (1996) found in the case of GARCH models, analytic derivatives
are necessary to achieve decent accuracy.
Computation of numerical derivatives is easy. Consider computinga(φ)on the
ith iteration. Compute the residuals based onμi,φiandθi, and call this vectorai.
Now for some differencing intervalh, computeahi based onμi,φi+handθi. Then
the numerical estimate ofa(φ)is given byai−ahi. The choice of differencing interval