Palgrave Handbook of Econometrics: Applied Econometrics

B.D. McCullough 1311

More sophisticated BJ models are much more difficult. Were it the case that the
most estimated ARIMA model was an ARIMA(1,0,1) then it would be worthwhile
for a developer to implement analytic derivatives. For example, the FCP GARCH
benchmark (Fiorentini, Calzaroli and Panattoni 1996; McCullough and Renfro,
1999; Brooks, Burke and Persand, 2001) is based on a GARCH(1,1) model, with
analytic first and second derivatives because most applications of GARCH involve
the GARCH(1,1). The same is not true of ARIMA models.
Nonetheless, it is of interest to know what price is paid for the use of numerical
derivatives instead of analytic derivatives. Moreover, the use of analytic derivatives
can shed light on the appropriate choice ofh, the differencing interval. Since
analytic derivatives are easily implemented in CLS, this will be done in the next
section.

28.6.3 Conditional least squares

We are now in a position to create a CLS benchmark. We use the 197 observa-
tions from Box–Jenkins Series A. In the CLS case, the analytic derivatives given by
equations (28.17)–(28.22) reduce to:

a(φ)t =(wt− 1 −μ)+θa(φ)t− 1 (28.23)

a(θ)t =at− 1 +θa(θ)t− 1 (28.24)

a(μ)t =− 1 −φ+θa(μ)t− 1 (28.25)

which can be calculated recursively after settinga(φ) 0 =a(θ) 0 =a(μ) 0 =0.
Carrying 50 digits through all calculations,^3 using the maximum of the relative
change in the coefficients as the convergence criterion, setting the convergence tol-
erance to 1E-13, and rounding to 11 digits produced the benchmark. To determine
what we can expect from ordinary double-precision calculation, we also re-ran the
program using ordinary double precision. As can be seen in Table 28.4, double
precision delivers the benchmark answer for the first seven digits of the constant,
five digits ofφ, and four digits forθ. The standard errors are computed on division
byn, notn−k.
Recall that the benchmark was produced with analytic derivatives. We can
use some capabilities of Mathematica to obtain this level of accuracy without

Table 28.4 CLS benchmark with analytic first derivatives

Parameter μφ θ

Double precision 17.093753099 0.90658876305 0.56881361427

Benchmark 17.093752390 0.90658703600 0.56880910281

Standard error (MLE) 0.10520938686 0.045388753586 0.086811221485

Note:MLE = maximum likelihood estimation.