fue

ARMAX by exact unconditional maximum likelihood, with rational transfer functions on the inputs and seasonality resolved one frequency at a time.

Three things, and no other program in this space has all three:

1. The exact unconditional likelihood — not a conditional sum of squares and not an approximation — of a model that carries deterministic inputs.

2. A rational transfer function on each input, ω_s(B)/δ_r(B), so an intervention has a dynamic response: an effect that builds up, decays or settles at a long-run gain g = ω(1)/δ(1). statsmodels' exog gives a static coefficient — the effect is the same in the period of the event and ten years later. That is a different model, not a simplified one.

3. Seasonality frequency by frequency. SARIMAX gives you the whole seasonal operator or none of it; fue lets a series be stochastic at f=1 and deterministic at f=2, which is what real series usually are.

What that versatility is for

Two published applications, both of which need the transfer function to be a model of a process rather than a dummy variable:

• García-Hiernaux & Guerrero (2021), "Price convergence: representation and testing", Economic Modelling 104, 105641. A general notion of price convergence — steady-state and catching-up, weak and strong — in which the transition phase is an exogenous deterministic input passed through a transfer function: ω is how far the process travels, δ its speed, and the starting date is the input's date. The shape and starting point are then estimated from the data rather than assumed, and the definitions imply parameter restrictions that are tested. Applied to the aggregate price levels of Germany, France and Italy after the euro, and to transatlantic 19th-century wheat prices.

• García-Hiernaux, González-Pérez & Guerrero (2023), "Eurozone prices: a tale of convergence and divergence", Economic Modelling 126, 106418. The same framework on relative prices across the EMU, 2001-2020, identifying the date, shape and velocity of each convergent or divergent process: convergence for over 80% of the EMU between 2001 and 2011, and price-level divergence from 2012 onwards.

Neither study is a regression with a break dummy. The object being estimated is a transition path — a dynamic response with a shape, a speed and a long-run gain — inside a model whose stochastic part is estimated jointly by exact ML. That is the class of question fue's casting opens.

The engine is José Alberto Mauricio's C, embedded and unmodified: Algorithm AS 311 (1997) for the likelihood, AS 197 (Melard 1984) for the scalar case, and his own quasi-Newton optimiser from JASA 90, 282-291. See PROVENANCE.md for who wrote what and what checks it.

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What it fits

  zₜ = ξₜ + Nₜ                                     (Box-Cox transformed level)

  ξₜ = Σ  ω_s(B)/δ_r(B) · xᵢₜ                      inputs with a DYNAMIC response
                                                   (step, impulse, ramp, Easter,
                                                    harmonics, your own column)

  φ_p(B) Φ_P(B^s) (wₜ − μ) = θ_q(B) Θ_Q(B^s) aₜ ,  wₜ = ∇^d ∇ₛ^D Nₜ

with ∇ₛ replaceable, factor by factor, by the frequencies you actually want to difference. MODEL.md writes it out without ambiguity.

Ten lines that run

import fue
from fue.datasets import ripc

ts = ripc()                       # Spanish CPI, monthly, 2002-2007
m = fue.Model(ts, d=1, boxlam=0.0, refactor=100.0,
              ar=[[0.3]], ar_free=[[True]],
              mu=0.1, estimate_mu=True)
m.fit()

print(m.ar[0][0], m.loglik)                    # 0.430784  -48.995195
print(m._result.termination, m._result.gnorm)  # gradient criterion, 3.9e-06

Five graded examples are in examples/, from this to a mixed MEG; they run in CI, so they still work.

Why the numbers can be trusted

Because that is checkable rather than asserted, and the checks are of four different kinds:

	what it proves
the paper's own FORTRAN	Melard's AS 197 listing, transcribed and compiled: agreement to 5e-08 in log-likelihood on nine Box-Jenkins specifications
the paper's own identities	AS 311's equations (2)-(4) on the engine's outputs; its quadratic form matches Melard's to 1e-14
an implementation with no shared ancestry	statsmodels agrees on all nine to ~1e-7 in the parameters
the original program's preserved runs	28 archived .out files reproduced exactly — termination code, iteration count and gradient norm

And the limits are documented too, because a document that lists only what works is an advertisement: PROVENANCE.md §6, and bugs/, which is public.

The documentation

GETTING_STARTED.md	install, first model, reading the .out
MODEL.md	the class of models, formally
FILE_CONTRACT.md	.inp / .out / .pre, field by field
FORMAL_TESTS.md	non-stationarity, non-invertibility, MEG; with the critical values
CONVERGENCE.md	what the optimiser reports and what to do about it
PROVENANCE.md	which algorithm, from which paper, verified how
PORT.md	how the port was done, and what it found
PERFORMANCE.md	why the wheel: the two engines measured, in speed and in answers
API.md	every public symbol, generated from the docstrings
MIGRATION.md	for users of the FUE in C: what changes and what does not

fue is the engine. The Box-Jenkins-Treadway orchestration — identification, diagnosis, the decision rules — is art; transfer functions are drtran, and multivariate is drvarma. The separation is deliberate: the engine supplies the fits and stays out of the decisions.

Licence and lineage

GPL. The C is Mauricio's, the file format and the user manual are Arthur B. Treadway's, and both authorised the licence. The port is 2026.