0134 — ARIMA, SARIMAX and state-space models are not written; VAR is the one that could be

Status: accepted · Date: 2026-09-15 · Applies: 0074, 0075, 0095, 0105 (as amended by 0129), 0130

Context

Decision 0105 delegated forecasting to Microsoft.ML.TimeSeries, wrote the diagnostics, and left ARIMA, SARIMAX, VAR and state-space estimation to a later lot. The lot named for it, #617, closed on the diagnostics, and docs/migration/statsmodels.md kept pointing at it. #772 exists to read the subject before anything is claimed, and allowed the verdict to be not written. 0130 had just found statsmodels' MixedLM unpinnable at the corpus tolerance, and a time-series model is a likelihood optimisation too, so that finding was the first thing to check.

The searches

nuget.org, 2026-09-15:

query	hits	anything that estimates the model
arima	4	Cortex.TimeSeries 1.1.0; Dew.Stats, Dew.Stats.Core, Dew.Stats.Linux 6.3.10 — read below
sarima, vector autoregression	0	—
kalman	18	none — signal filters (MathNet.Filtering.Kalman), GPS smoothing, trading strategies
state space	22	none — PDF, planning and UI libraries
time series forecasting	9	Cortex.TimeSeries, Dew.Stats.Core again; NW.UnivariateForecasting, Tsfm.Forecasting forecast without an ARIMA estimator

The reading

Surfaces with tools/survey.cs (decision 0110), licences from the package (decision 0075), pattern (Arima|ARIMA|Sarima|VectorAutoregress|VarModel|StateSpace|Kalman|UnobservedComponent|ExponentialSmoothing|Garch):

package	licence, from the package	surface	what the pattern returns
Cortex.TimeSeries 1.1.0	MIT, <license type="expression">	27 types, 130 members	ARIMA(p, d, q), SARIMA, AutoARIMA: AR and MA coefficients, intercept, AIC, BIC, forecasts; no standard error, no likelihood, no VAR, no state space
Numerics.NET 10.7.0	commercial (0129)	13,839 members	ArimaModel, ExponentialSmoothingModel, GarchModel; no VAR, no SARIMAX, no state space
Dew.Stats.Core 6.3.10	commercial (0130)	727 members	ARIMASimulate — a simulator, no estimator
Microsoft.ML.TimeSeries 5.0.0	MIT	read for 0105	singular spectrum analysis; no ARIMA

The commercial libraries are read and not run: nothing is timed or fitted under a trial licence. Julia's StateSpaceModels.jl was not measured: no Julia toolchain is installed here, and statsmodels is the reference every corpus in this repository is frozen from, so a second reference would not change what parity means.

What Cortex.TimeSeries' ARIMA estimates

One series of 300 points simulated from ARMA(2,1) with φ = (0.6, −0.2), θ = 0.3 and a fixed numpy seed, fitted as ARIMA(2,0,1) with a constant on both sides:

estimator	φ₁	φ₂	θ₁
statsmodels ARIMA(...).fit() — state-space MLE	0.5312	−0.1028	0.4706
statsmodels, method="innovations_mle"	0.5311	−0.1028	0.4706
conditional sum of squares, fitted with scipy	0.5347	−0.1048	0.4701
Hannan–Rissanen, statsmodels	0.6076	−0.1563	0.3862
AR(2) by ordinary least squares, the MA term ignored	0.9365	−0.3837	—
Cortex.TimeSeries ARIMA(2, 0, 1).Fit	0.9337	−0.3787	0.0466

Every ARMA estimator lands near (0.53, −0.10, 0.47); Cortex lands on the pure autoregression's coefficients, and its AIC reads 47.8 where the likelihood's is 895.8. It does not estimate an ARIMA model in the sense a statsmodels caller means, so it cannot be delegated to as one — and, reported with no standard errors, it would not carry the inference table this project's thesis is about even if it did.

What parity would mean

The same series, fitted by statsmodels 0.15.0 with its own optimisers:

comparison	largest relative gap in the parameters
ARIMA.fit() (state space) against method="innovations_mle"	8.8e-5
SARIMAX.fit() L-BFGS against Nelder–Mead	2.6e-4 (log-likelihoods −442.89174442 and −442.89174456)
SARIMAX.fit(method="bfgs")	1.27 — reports non-convergence

The reference does not pin its own answer to the tolerance every corpus here is held to. A corpus frozen from one optimiser is a record of that optimiser's stopping point, and a C# fit that agreed with it at 1e-9 would be agreeing with a path, not with the model: 0130's finding for MixedLM, measured again. State-space models (UnobservedComponents, DynamicFactor) are the same likelihood optimised the same way.

VAR is not. statsmodels.tsa.api.VAR(...).fit(p) is equation-by-equation least squares on the stacked lags, and on a two-variable series it matches numpy.linalg.lstsq on that design to a relative gap of 0.0. Its standard errors, t statistics and information criteria are the least-squares table Lodestar.Stats.Regression already builds.

Decision

1. ARIMA, SARIMAX and state-space estimation are not written. There is no parity target at the corpus tolerance, and no free .NET incumbent to delegate to: Cortex.TimeSeries' ARIMA is a pure autoregression's least squares under an ARIMA name. Forecasting stays delegated to Microsoft.ML.TimeSeries (0105).
2. The lot waits for a caller, under 0095's rule. The caller who reopens it inherits two questions this record cannot answer: what tolerance a likelihood fit is held to (a log-likelihood match, not parameters, is the candidate), and which of the reference's optimisers is the reference.
3. VAR is the one model of the four that could be written at parity, as a least-squares system with the table beside it, and no .NET package read here estimates one. It is not written by this record: whether it earns a lot is a question for an issue, proposed rather than opened.
4. docs/migration/statsmodels.md points at this record instead of the closed #617, and names Cortex.TimeSeries' ARIMA for what it computes.

Rejected

• Writing ARIMA against a looser tolerance now. A tolerance chosen to make the corpus pass is a tolerance chosen by the implementation; with no caller to say what agreement they need, there is nothing to choose it by.
• Recommending Cortex.TimeSeries for ARIMA. Its coefficients are a different model's.
• Timing Numerics.NET's ArimaModel. Commercial, trial licence; read only.

Consequences

• The migration row for ARIMA/SARIMAX/VAR/state space reads not written, with VAR named as the writable one.
• #772 closes on this record. A VAR lot, if Cyril wants one, is a new issue that starts from the least-squares identity measured above.