A worked example with `mtram`: oil into the Spanish CPI¶

The pass-through case. Output IPC_ES, input WTI, monthly 2002-01…2019-12, 216 observations. Every figure is from the suite's own regression test (drtran-python/tests/test_end_to_end_passthrough.py) and from the runs made while fixing BUG-8; the oracle column is the independent TASTE implementation.

The nodes are the ones drtran/docs/DECISION_NODES.md defines. What this document adds is what each one looked like on a real pair, including the two places where the honest answer was "the program cannot tell you that".

Before any node · the gate¶

Two .pre files arrive from art — one per series, each the optimum of its own univariate specification. load_pre checks the crossing before anything else, and the check is complete rather than indicative:

With a diagonal structure the exact likelihood factorises. So the joint diagonal fit must reproduce the SUM of the two univariate log-likelihoods. If the transformation, the differencing, the deterministics or the seeds did not arrive intact, the identity fails.

Measured here: −1.50e−07. And it earns its place — it caught a real error this year that nothing else could:

	joint diagonal	sum of univariate	gap
WTI over 2005-2019 (180 obs)	−629.168775	−629.168775	0.000000
WTI over 2005-2019 vs a 2002-2019 output	−681.034215	−689.960006	8.93

Both series ended 12/2019, so the alignment check passed — correctly, the dates really did line up. But one carried 36 months more history, the cast trims to the common window, and the longer series' .pre was therefore the optimum of a sample that was not being fitted. check_operators now reports the common window and who has spare history, so this is caught before rather than after.

The lesson for an analyst: a gate that only ever says "fine" is decoration. This one says "fine" on every legitimate case and fired on the first illegitimate one it met.

N0 · Which series is the output¶

Not a computation. The transfer model is directional: WTI drives the CPI and not the reverse, and that is economics, not a correlogram. load_pre takes the first path as the output and asks for confirmation, because getting it backwards produces a perfectly estimable model of nothing.

N1 · The link order `(b, r, s)`¶

Evidence: identify_link prewhitens — filters the output through the INPUT's ARMA — and reads the cross-correlation, with the band that corresponds to the real overlap.

Decision here: b = 0, r = 0, s = 1. A contemporaneous response with one extra lag; no rational denominator. Oil reaches a consumer price index within the month it moves, through fuel, and the second weight picks up the rest.

Two things worth an analyst's attention at this node:

The band is 2/√n of the OVERLAP, not of either series. When that band and the real overlap disagree, something upstream is wrong — that mismatch was the only visible symptom of the misalignment bug that produced a b = 18 proposal in earnest.
If the output carries stochastic seasonality and the input does not, this node fails silently. The prewhitening filter carries nothing seasonal, so the seasonal component survives it and the CCF comes out structured everywhere; the heuristic reads an order off it anyway. That is why the univariate work for a transfer model should choose deterministic seasonality from the start (WORKED_EXAMPLE_ART.md, Part II).

N1b · Do the two operators agree?¶

A node that did not exist until August 2026, and the one that would have corrupted the answer in silence.

The transfer relates the levels, with the differencing carried by the noise. So the input must enter differenced by the OUTPUT's operator. check_operators answers before anything is estimated:

IPC_ES <- WTI     Delta(1) = 1    operadores IGUALES, nada que hacer

Here both series are d = 1, D = 0: Δ = 1 and there is nothing to do. Had the output been the airline variant (∇∇₁₂), Δ would be 1 − B¹² and Δ(1) = 0 — the gain annihilated. Measured on this very series: the five airline variants reported a pass-through of ≈ 0.006 while the three harmonic ones reported ≈ 0.027, and the factor of five was arithmetic, not economics.

And do NOT divide the gain by Δ(1) to repair it. The law ν̂(1) = ν(1)·Δ(1) only holds once the fitted (b, r, s) has the reach to see where Δ puts its weight; the real error is partial and is not knowable from the output.

N5 · The cast¶

estimate uses the embedded cast by default: the transfer becomes off-diagonal VARMA coefficients, nothing is subtracted, and the exact likelihood does its own pre-sample initialisation — so the reported likelihood is the likelihood of the data and LR tests and information criteria mean what they say.

When the operators differ the fit is dispatched to the subtracting cast, because the input then needs a second, re-differenced vector and the embedded cast has one column per series. estimate says so when it happens, and warns that the two casts' likelihoods are not comparable with each other.

Here Δ = 1, so the embedded cast runs and nothing is announced.

N6 · Estimation, and what the transfer bought¶

ω₀ = 0.016402   ω₁ = −0.010748      φ_N = 0.295216   μ = 0.140321
gain ν(1) = 0.027149

Against the diagonal rung — the same model with the transfer restricted to zero, which is the right null because it is nested:

diagonal  ℓ = −1744.135582
joint     ℓ = −1704.423918
LR = 79.42 on 2 df

The transfer is not decoration. And the gain has an economic reading: a permanent 1 % move in WTI passes through to about 0.027 % of the Spanish CPI level.

Corroborated independently. TASTE shares no code with this family and estimates by unconditional sum of squares with backforecasting rather than exact ML, so agreement to the low single digits is the most that can be asked:

	TASTE	drtran	difference
ω₀	0.016410	0.016402	8.0e−06
ω₁	−0.010790	−0.010748	4.2e−05

N6b · Diagnosis, and one number that has to be right¶

diagnose reports residual whiteness, the cross-correlations between residual and input, normality and extreme residuals.

One detail worth carrying: the degrees-of-freedom correction for a per-series statistic uses that series' parameters, not the joint vector's. Using all seventeen where three belong turned a p-value of 0.34 into 0.0017 — the difference between "clean" and "reject", from an arithmetic slip in a correction nobody looks at.

N7 · Does it earn its keep?¶

In-sample the transfer model can never fit worse: it has more free parameters, so a significant ω and a higher likelihood are necessary and not sufficient. The rule in drtran/docs/FORECAST_DIAGNOSIS.md is utilitarian:

A transfer model is kept over its univariate iff it forecasts better out of sample — recursive origins, parameters fixed, Diebold-Mariano with the HAC variance and the HLN correction.

The univariate model is the measuring stick. If the transfer cannot beat it, the transfer is what needs rethinking, not the stick.

And what the model leaves behind¶

write_inp writes each series back with the JOINTLY estimated values — as .inp, deliberately not as .pre. A .pre asserts "these values are the optimum of this specification", and a univariate block that was optimised with the transfer in the model is not a fixed point of the univariate operator. Measured: re-running fue on a genuine .pre moves the numbers by 0.000000; on drtran's jointly-fitted block, by 13.109261.

That is the whole convention in one measurement, and it is what lets the ladder be climbed by trusting the files.

A worked example with mtram: oil into the Spanish CPI¶