Amortized Neural Filtering for Calibrated Posteriors in Statistical Arbitrage

CCS Concepts

• Computing methodologies → Machine learning; Learning in probabilistic graphical models; • Applied computing → Economics.

Keywords

statistical arbitrage, amortized inference, calibration, filtering, simulation-based inference, regime switching

ACM Reference Format:

Minkey Chang. 2026. Amortized Neural Filtering for Calibrated Posteriors in Statistical Arbitrage. In 7th ACM International Conference on AI in Finance (ICAIF ’26), November 14–17, 2026, Milan, Italy. ACM, New York, NY, USA, 9 pages. https://doi.org/10.1145/3858063.3858894

Introduction

In the old tale of the blind men and the elephant, each man touches the animal and describes a different beast. Economics and finance pose the same difficulty. It is nearly impossible to say where the market stands at the moment, and certainty arrives only after the fact. A familiar example is the business cycle, whose recessions are dated by the NBER many months after they begin [21]. The term nowcasting arose from this predicament [8]. Rather than wait for the whole animal to come into view, we estimate the present state from the part within reach. A forecast means little without knowing where one stands. The same problem appears in statistical arbitrage, where knowing what is happening now matters more to the next action than predicting what happens next.

Conventional statistical arbitrage [1, 6] handles this problem with a hand-crafted pipeline. One regresses returns on factors, accumulates the residuals, fits a mean-reversion model on a trailing window, and uses the fitted point estimates to decide when to enter and exit. The procedure treats the assumed process as if it were true, and despite the name statistical arbitrage it discards the uncertainty of the estimates, so that a genuine opportunity and an optimistic misreading of the data become indistinguishable. In practice, moreover, the procedure is applied to a large universe of candidate spreads: the candidates are ranked by these estimates and the top-ranked ones are traded. However, such ranking compounds the problem, because the candidates with the most favorable estimation errors are precisely the ones selected; this is the winner’s curse of post-selection inference [3]. More recent work learns the factors end to end [4, 11]. This improves the construction of the residuals but leaves their use unchanged: the trading decision is still made from a point estimate, with no estimate of the latent state and no measure of its uncertainty.

Despite these differences, the methods above share one structure. Each compresses a panel of returns, one day’s cross-section of many assets, into a few factors, and trades what the factors do not explain. They differ only in how the factors are obtained. Fama–French [5] fix them in advance as characteristic portfolios; Avellaneda–Lee [1] extract them from the data by principal components and trade the residual spread as a mean-reverting process; Epstein et al. [4] learn them end to end from characteristics with a trading objective. Outside finance, tabular foundation models [16, 23] perform the same compression with a network pre-trained on a synthetic prior rather than fitted to each dataset. This paper takes the next step in that sequence. It keeps the residual spread and its mean-reversion state from Avellaneda–Lee, the task-driven factor learning of Epstein et al., and the synthetic-prior pre-training of the tabular models, and it adds what none of them has: a dynamic model of the latent state, with the uncertainty of its filtered estimate. We call the resulting estimator the Neural Filter Estimator (NFE).

Our claim concerns how the regime is defined and estimated. Most statistical-arbitrage strategies already condition on some notion of regime, but in practice that notion is a heuristic, such as a This work is licensed under a Creative Commons Attribution 4.0 International License.

ICAIF ’26, Milan, Italy

© 2026 Copyright held by the owner/author(s).

ACM ISBN 979-8-4007-3087-0/2026/11

https://doi.org/10.1145/3858063.3858894 volatility threshold or a trailing return, or a label assigned ex post, once the outcome is known. We instead define the regime as a latent variable of an explicit generative model of the spread: whether the spread is currently converging to its level or diverging from it, and how fast. We then show that a single amortized filter can estimate this regime for every spread in a cross-section from past data alone, with a calibrated probability attached, and that the trading decision can use that probability in place of a point estimate.

Two choices in this definition need justification: the number of states and the use of a probability. In the regime-switching literature [12], a latent state is called a regime when a change in it changes the distribution of the observed process, for example its mean, its variance or its speed of mean reversion. For a trader, only the changes that alter the trading decision matter, so we define the regime relative to the decision. For a contrarian strategy held over a fixed horizon this leaves two states: one in which enough of the deviation reverts within the horizon for a position taken against it to pay, and one in which it does not, so that the same position earns nothing but its costs. Acting on this regime requires a judgment of how certain its assignment is, because the two states call for different actions and a mistaken assignment is costly. A point estimate assigns the spread to one state outright and gives no measure of that certainty, which makes the assignment difficult to act on; a posterior assigns a probability to each state, which the decision can weigh against the cost of being on the wrong side. Conventional practice lacks this measure not out of neglect but because its plug-in estimates do not provide it.

The rest of the paper supports this claim with three measurements. The first checks the simulator. The filter is trained only on simulated spreads, so the simulated process must resemble the real residuals before anything learned from it can be trusted on real data. We test this directly: a classifier is trained to distinguish simulated windows from real ones, and a candidate process passes only if the classifier largely fails. The Gaussian noise of the classical model is rejected. The only candidate that passes on both cross-sections has both time-varying volatility and asymmetric shocks; fat tails alone are not enough.

The second measures how well the regime can be identified at all. On simulated spreads the true regime is known, so every method can be compared with the best possible classifier, one that knows the true model, level and volatility. Every practical method, ours included, lands in one narrow band far below that ideal. The shortfall comes not from the estimator but from the window: the level of the spread must be estimated from the same window as its reversion, and that alone accounts for most of the loss. Better inference cannot close this gap; only more information can.

The third asks what the posterior changes in trading. Under fixed trading rules and after costs, posterior sizing earns about the same as the point estimate; the posterior adds no alpha. What changes is the strategy’s own forecast of its return. The point estimate predicts far more than it realizes, a direct consequence of the winner’s curse above; the posterior-based strategies reduce this gap several-fold. The mean-reverting structure in the data is real, but it is smaller than the cost of trading on it, and the value of the posterior lies in saying so.

Related Work

Factor models and statistical arbitrage. The line of work this paper extends begins with static characteristic factors: Fama–French [5] explain the cross-section of returns with a few factor portfolios fixed in advance. Dynamic factor models let the factors evolve as a latent linear state [7, 26]. Statistical arbitrage trades the residuals that the factor models leave unexplained. Gátev et al. [6] trade pairs selected by the distance method, and Avellaneda and Lee [1], in what has become the standard treatment, fit a mean-reversion model to cumulative factor residuals and trade its point estimates. Recent deep-learning work trades factor portfolios instead of selected pairs and learns the trading policy from residual histories to end [11], and Epstein et al. [4] learn conditional factors by attention over characteristics, jointly with an after-cost trading objective. In all of these methods the trading decision rests on a point estimate, and the uncertainty of the latent state plays no part in it.

Tabular foundation models. Outside finance, a parallel line of work arrived at the same computation. Prior-data fitted networks [20] train a transformer on draws from a synthetic prior, so that one forward pass approximates a posterior predictive. Hollmann et al. [13] make the approach practical for small tables with TabPFN [14], Qu et al. [23] scale it with a column-then-row encoding, and Kong and Das [16] combine these in TabFM, which alternates attention across columns and rows, compresses each row to one vector, and predicts from those vectors in context. A factor model also compresses each cross-section into a few numbers; in this sense the conditional factor of Epstein et al. [4] is the finance special case of the learned row embedding. These models are not built for financial panels: their in-context stage attends over rows as an exchangeable set, and a time-ordered panel is not one. This paper closes part of that gap by placing a latent-state dynamic model under the row embedding and inferring the state from the past alone.

Filtering and amortized inference. In state-space models, filtering refers to the estimation of a latent state from past observations alone [15]. Hamilton [12] introduced its regime-switching form to economics, and modern deep variants learn the state-space model itself [17] or amortize the filter with a variational inference network [19]. When the latent state can be simulated together with the observations, a different approach becomes possible. Neural posterior estimation trains a conditional density estimator on simulator draws and then evaluates the posterior in one forward pass [22]; it is the core of simulation-based inference [2], and its calibration can be checked by simulation [27]. The NFE applies neural posterior estimation to the spreads of a time-ordered panel, restricts the prior to a dynamic process whose innovation law is tested against real data, and amortizes the filtering posterior over the state.

Method

Notation

We consider $N$ assets observed on trading days $t = 1, \ldots, T$, with daily returns $\mathbf{R}_t \in\mathbb{R}^{N}$ and $M$ characteristics $\mathbf{X}_t \in\mathbb{R}^{N \times M}$, following the notation of Epstein et al. [4]. A characteristic is an attribute of a firm known at the start of day $t$, such as its market capitalization, book-to-price ratio or recent return and volatility. From returns and characteristics a residual return $\varepsilon_{i,t}$ is constructed for each asset, and its cumulative sum $S_{i,t} = \sum_{s \le t} \varepsilon_{i,s}$ is the spread. We drop the asset index $i$ where it is immaterial. Two horizons are used throughout: the filter reads a look-back window of $L$ trading days, and a position is held for $H$ trading days ($L = 60$ and $H = 20$ in § 4).

Data-Generating Process

The filter is trained by neural posterior estimation (§ 3.5): on simulated draws in which the latent state is known, it learns the map from an observed window to the posterior over the state, and it is then applied to real windows without further fitting. This amortization requires a process to simulate from, and the posterior is worth only as much as that process resembles the data, so the data-generating process is the object on which the method rests and must be stated explicitly. Conventional strategies assume one implicitly when they expect the spread to revert to a stationary level, and the end-to-end approaches leave theirs in the training distribution.

The process describes the increment of the spread: since the spread is a cumulative sum of residual returns, its daily increment is the residual return itself, and mean reversion means that this increment tends to move the spread back toward a level $m_{i,t}$. We write the law of the increment conditional on where the spread stands, indexed by a latent variable $z_{i,t} \in\mathcal{Z}$ and by parameters $\theta$,

Given the latent variables, the spread is Markov: the next increment depends on the past only through $S_{i,t}$, so a filter can carry its state forward one day at a time, and $z_{i,t}$ selects which conditional law is in force on day $t$. The following definition fixes which values of the index count as regimes.

Different regimes thus have different conditional laws, as in Hamilton [12], and a change of index is a change of regime only if it changes what a trade can expect. We instantiate the kernel as Ornstein–Uhlenbeck (OU) mean reversion with regime-indexed speed,

where the innovation scale $\sigma_{i,t}$, the level $m_{i,t}$ and the law of innovation $\zeta_{i,t}$ are themselves latent processes (Appendix § 8). For $\kappa$ constant over the horizon, $\rho_H = 1-e^{-\kappa H}$ (exact in continuous time), so a threshold on $\rho_H$ is equivalent to a threshold $\tau^*$ on the half-life $\tau= \ln2/\kappa$, and the two regimes are $\mathrm{C} = \{\tau\leq\tau^*\}$ and $\mathrm{D} = \{\tau> \tau^*\}$, converging and diverging. We set $\tau^* = L = 60$ days, because a slower reversion cannot be told from none within the window and the decision cannot act on it; with $H = 20$ this is $\rho^* = 1 - 2^{-H/\tau^*} \approx0.21$, so in C at least a fifth of the deviation reverts within the holding period and in D less, down to the boundary $\kappa= 0$ at which the pull vanishes. The two speeds $\theta= (\kappa(\mathrm{C}),\kappa(\mathrm{D}))$ lie on either side of $\tau^*$ (half-lives of 10 and 120 days in the simulations of Section 4); the hyper-parameters of the innovation processes are fixed by moment matching (Appendix § 8). The regime path follows a first-order chain with transition matrix $A$, so the prior factorizes as $p(\theta,z_{1:T}) = p(\theta)\,p(z_1)\prod_{t=2}^{T} A_{z_{t-1}z_t}$.

The mean-reversion kernel is the standard model of the literature with extensions [1], and adopting it entails little loss of generality: the filter can be trained on any kernel of (1) that can be simulated, and the direct variant of Section 3.3 drops the kernel’s link between the state and the forecast. The innovation law, by contrast, is chosen by a two-sample test [18]: a candidate law is retained only if a classifier cannot separate its simulated windows from real residual windows beyond a pre-specified threshold, and only stochastic volatility with asymmetric innovations passes on both cross-sections (Appendix § 8).

The regime is a latent variable of the process; the data provide only an estimate of it. With the whole path in hand the best estimate is the smoothed assignment $\hat{z}_{i,t} = \arg\max_{z} p(z_{i,t}=z \mid S_{i,1:t},\hat{\theta})$, available only ex post. What a trader needs at time $t$ is the filtering posterior, which conditions only on the past,

Its likelihood is marginalized over the earlier states and the latent scale and level paths, so the posterior is intractable; computing it is the filtering problem of state-space models. The NFE is a neural network trained to return the marginal of this posterior that the decision uses, the posterior over the current speed, in a single forward pass for every window and every asset: an amortized filter.

Neural Filter Estimator

Once the process is fixed, the question is what information to give the filter at time $t$. A single spread’s window identifies its state only partly, since the level must be estimated from the same window; the rest of the panel, the same day’s returns and characteristics of the other assets, enters through the cross-sectional step. As shown in Figure 1, the panel is a table with days as rows and assets as columns, and the NFE processes it along both axes: a cross-sectional step across the assets of one day, which removes what the assets share and leaves each asset’s own spread, and a time-series step along each asset’s window, which reads the spread’s dynamics. The tabular foundation models [16, 23] organize in-context learning on tables the same way, and the cross-sectional step corresponds to their row embedding (§ 2), with one deliberate difference: those models treat rows as exchangeable, which a time-ordered panel is not, so the NFE’s time-series step is a recursion that keeps the order of days, while its cross-sectional step, a factor model, is indifferent to the order of assets.

Cross-sectional step. The classical construction uses returns alone: the factors are the leading principal components of the recent return covariance, and each asset’s residual is what they leave. The Attention Factor Model (AFM) [4] instead computes the factor weights each day by attention over the characteristics, so that assets with similar characteristics are pooled into the same factor portfolio and the factors adapt as the characteristics change. In either construction a factor model maps returns and characteristics to one residual return per asset,

where $\Omega_{t-1}$ holds the factor weights and $\beta_{t-1}$ the loadings, both estimated with information up to $t-1$; the cumulative sum of $\varepsilon_t$ is the spread the time-series step reads. The filter takes this residual from whichever factor model supplies it rather than learning its own: the factor construction is where the earlier generations of the lineage in § 2 differ, and holding it fixed isolates the contribution of the posterior. § 4 evaluates both constructions; the NFE of that section uses AFM’s factors, refit annually as in the AFM strategy.

Time-series step. A filter is a recursion: the estimate of the state is updated from the previous estimate and each new observation without revisiting the past, which is what makes it causal. A recurrent network has this form, so we encode each spread’s look-back window with a gated recurrent unit (GRU), $h_j = \operatorname{GRU}_{\psi}(h_{j-1}, w_{i,j})$ for $j = t-L+1,\ldots,t$, where $w_{i,j}$ is the day-$j$ entry of the window $w_{i,t}$, the spread and its increment standardized by the window’s own mean and standard deviation, and the final state $h_t$ summarizes the window. The standardization makes the input invariant to location and scale, so the filter reads the shape of the window and not its units, and the simulator has to match the data in shape only. A panel form, both steps as alternating attention in one network, is tested only on simulated draws in which the latent truth is known (truth-known draws hereafter) and is left for future work (§ 4.2).

Posterior head. The state the decision needs is the half-life of the current regime, $\tau_t = \ln2/\kappa(z_t)$. From the final state $h_t$ a mixture-density head outputs a posterior $q_{\psi}(\tau\mid w_{i,t})$, a five-component Gaussian mixture in $\log_2 \tau$ whose weights, locations and widths are functions of $h_t$. The half-life spans two orders of magnitude, hence the logarithmic scale, and a mixture rather than a single Gaussian lets an ambiguous window spread its mass over both regimes instead of committing to one. The regime probability of Definition 3.1 is the tail mass of this posterior beyond $\tau^*$,

so the regime probability and the half-life posterior are consistent by construction. The decision layer uses the half-life posterior through its moments; the regime probability is reported but is not an input to the sizing rule. A direct variant places the same head on the standardized $H$-day increment of the spread, $q_{\psi}(\Delta S \mid w_{i,t})$. It is a control for the functional form and not a deployable filter, since the decision layer needs a posterior over the half-life: the OU-form posterior ties the predictive mean to the half-life through the kernel (3), and the comparison of § 4.2 tells whether that form limits the forecast.

Decision layer

The decision layer is pre-specified: the rules below were fixed before any run and apply with the same conventions to every strategy, and the filter is never fitted to realized profit, so the posterior is the only interface between the filter and the portfolio. If the rule were learned, as in AFM’s end-to-end trading network, a difference in performance could come from the rule as much as from the state estimate; with the rule fixed, the strategies differ only in the residual they trade and in whether the position is sized from a point estimate or from the posterior. Three components are selected on the training split: the threshold rule’s entry level and AFM’s characteristic set, by their training-split Sharpe ratios, and the per-asset prior below, from the training split’s half-lives.

The signal is the standardized deviation $d$, the last entry of the window’s spread channel: the spread’s distance from its window mean in units of the window’s standard deviation, clipped at $\pm3$. This is the signal of the classical pipeline [1]. The base rule sets the position proportional to $-d$, a contrarian position against the deviation with no tuned parameter; the threshold rule of Avellaneda and Lee [1] opens a unit position once $\lvert d\rvert$ exceeds an entry threshold tuned on the training split. Both rules act on the deviation as if it would revert; the NFE instead sizes positions on the same signal from the posterior. Under the kernel (3) with reversion rate $\kappa$, the expected change of the spread over the holding period is $-d(1-e^{-\kappa H})$ in standardized units, and with the scale and level at their window estimates, its variance is $1-e^{-2\kappa H}$; when $\kappa$ is uncertain, the law of total variance adds the variance of that expected change across the posterior, and the mean–variance optimal position is the expected change divided by the total variance,

with all moments taken under $q_\psi(\cdot\mid w_{i,t})$ through $\kappa=\ln2/\tau$. The term $d^2\operatorname{Var}(e^{-\kappa H})$ is identically zero for any point estimate, so (7) with a point estimate is the plug-in rule of the classical pipeline; Section 4 compares the posterior with the base rule on the same residuals and, at matched signal and window, with this plug-in rule. The term shrinks the position when the posterior is wide and leaves it unchanged when the posterior is narrow. Positions are then demeaned across assets and scaled to unit gross exposure, the dollar-neutral convention under which every strategy is evaluated, so within a fixed gross the penalty moves exposure from spreads whose posterior is wide to those whose posterior is narrow. It accounts for uncertainty in the reversion speed only; the level appears in $d$ as a window estimate and is not penalized, the omission Section 4.2 finds to be the largest source of identification loss.

The training prior over $\tau$ is deliberately wide, flat in $\log_2\tau$ over [3, 400] days, so that one network serves every spread. At deployment it is replaced, for each stock, by a narrower Gaussian prior in $\log_2\tau$ centred on that stock’s own half-life, estimated on $L$-day blocks of the training split and shrunk toward the panel mean by empirical Bayes, so that a stock with few or noisy blocks is pulled further toward the mean. Because the head is trained under a flat prior, its output is likelihood-shaped, and the new prior is applied by reweighting the posterior draws used in (7); the network is not retrained.

Training

Real data never reveal the latent state, but simulated draws include it by construction, so the simulator supplies the labels. The filter is trained as in neural posterior estimation [22], by maximizing the likelihood of the known half-life under the head, $\mathcal{L}(\psi)=-\mathbb{E}_{(\tau,w)\sim p_{\mathrm{train}}}[\log q_\psi(\tau\mid w)]$, on windows drawn from the process of Section 3.2 with one simplification: in training the half-life is held fixed over the window and drawn uniformly in $\log_2\tau$ on [3, 400] days, with the innovation hyper-parameters at their moment-matched values (Table 5). Each window then carries one half-life, so $\theta$ reduces to $\tau$, and the flat prior gives the likelihood-shaped head on which the per-asset prior relies. The two-state chain of Section 3.2 is the process of the two-sample test and of the identification experiments of Section 4.2; the calibration check of Figure 2(b) uses the training draws themselves. No real returns enter the training of the network. This keeps the posterior checkable: its calibration is measured on simulated draws, where the truth is known (Section 4.2), and it cannot have been fitted to the outcomes it is later scored on. Real data enter the filter only through the choice of innovation law, by the two-sample test on full-sample windows (Appendix A), and the moment matching of its hyper-parameters on the training split; at deployment they enter through the per-asset prior, the split-conformal layer [24] on every learned predictive distribution of Section 4.2, and the direct variant’s target scale, all fitted on the training split (Appendix B).

universe	top-200 KRX by median traded value; 120 names with complete history, 2010–2026 (4,072 d)
characteristics	ROE, D/E, net margin, revenue growth, B/P, E/P (DART, point-in-time, from 2016)
split	2018–04; the first 30% of each period trains
window, hold	$L = 60$, $H = 20$ (non-overlapping); $\tau= 60$ d
residuals	PCA, $K = 3$ on $52$ d: first 1 [1] size, $B/P$
sizing	64 posterior draws; $d \in[-3, 3]$; dollar-neutral at unit gross
costs	5 bp per unit of turnover; 1 bp per day on the short notional
benchmark, nulls	random signal held $H$ d; vol-matched random walk; 10d/120d switching
encoder, training	GRU-64; five-component MDN; Adam, 4,000 steps, batch 512, rate $3 \times10^{-3}$ cosine; 60,000 draws
half-life prior $(\phi,\gamma)$, training split	$\log U(3, 400)$ d; per-stock centrs at test (0.09, 0.08) in 2010–18; (0.28, 0.12) in 2018–26
two-state chain	half-lives 10 d (C) and 120 d (D)
AFM	$K = 5$; annual refits; 40 epochs
panel form	2 blocks, width 64, 4 heads; 30k panels, 3k steps, batch 16, rate $3 \times10^{-4}$

Experiment

We report one empirical study: walk-forward statistical-arbitrage portfolios on 120 Korea Exchange (KRX) stocks over 2010–2026, with annual refits (Table 5), net of transaction and shorting costs; the package, the experiment scripts and the data pipeline are available at https://github.com/minkeymouse/nfe. One strategy represents each generation of the line of work in Section 2. Fama–French three-factor characteristic residuals (FF3) [5], rolling principal-component residuals (PCA) and Attention Factor Model residuals (AFM) [4] are each traded by the same base rule; the NFE, posterior sizing on the AFM factors, is the fourth generation. In Table 1, AFM denotes AFM’s residuals under the base rule in our own implementation; its learned trading network is reported among the variants. We compare orderings, not levels, so absolute Sharpe ratios are not compared with the published U.S. figures. The conventions, fixed in advance and identical across strategies, are collected in Table 5 and Appendix B, and every label-free diagnostic is reported with its null model and positive control. Results come from one training seed and are exploratory unless stated; a result is called confirmed when it was reproduced on a disjoint period or on disjoint seeds against criteria fixed beforehand.

strategy	SR	$\mu$ (%)	$\sigma$ (%)	MDD (%)	turnover	$\Delta$SR
FF3	−0.12	−1.1	8.8	23.0	0.052	−0.01
PCA	0.57	3.8	6.7	7.1	0.053	+0.59
AFM	1.45	10.4	7.2	4.2	0.055	+1.35
NFE	1.49	10.6	7.1	4.1	0.055	+1.38
market-adjusted	−0.12	−1.2	9.2	21.8	0.052	−0.01
random benchmark	−0.23	−1.0	4.4	16.0	0.071

Results

Table 1 reports the later period, 2018 to 2026, on the days after its training split, one row per generation, with a market-adjusted residual as a negative control; $\Delta$SR is the annualized Sharpe ratio of the daily return difference to the random benchmark (Appendix B).

The four generations. FF3’s fixed characteristics earn no excess return over the random benchmark, and neither does the market-adjusted control. Statistical factors rank above the benchmark (PCA, Sharpe 0.57) and learned factors higher still (AFM, 1.45). In the earlier period no strategy outperforms the benchmark. The ordering therefore depends on the period, and because the panel is selected at the end of the sample, the levels are an upper bound.

The fourth generation. The NFE matches AFM’s Sharpe, a difference in level of +0.04, within the period’s noise, at slightly lower volatility and drawdown and with equal turnover. The standard error of either strategy’s $\Delta$SR is 0.41, so a difference of this size is not resolvable here. Posterior sizing also raises the PCA-residual strategy’s Sharpe ratio from 0.57 to 0.68, a difference subject to the same noise. The per-asset prior of Section 3.4 was confirmed on three disjoint training seeds. With the training prior the NFE’s mean Sharpe ratio is 1.424 in this period and −0.259 in the earlier one; with the per-asset prior it is 1.470 and $-0.193$; with the per-asset centres permuted across stocks it is 1.458 and $-0.215$. The gains are consistent in sign across seeds but small relative to the period’s noise, and most of them come from calibrating the prior’s centre and width to the panel rather than from the per-stock centres. Figure 2(a) shows the cumulative returns.

Window length. A posterior over the reversion speed conditioned on a 250-day window, applied to the same 60-day signal, has a larger $\Delta\mathrm{SR}$ than AFM on two training seeds. The gain over AFM, however, is a window effect: a pre-specified control that trades the 250-day deviation directly reaches a Sharpe ratio of 2.77 in this period, above every strategy, and $-0.91$ in the other, so we do not claim it and retain the 60-day window. What survives at matched signal and window is that this posterior outperforms the plug-in rule with a 250-day least-squares point estimate of the speed.

Other variants. A cross-sectional step learned on the panel itself, asset attention trained within the NFE in place of AFM’s factors, reaches a Sharpe ratio of $-0.03$, within the random benchmark’s noise, so the NFE keeps AFM’s factor extraction. Among the rule variants, the threshold rule reaches 0.75 on PCA residuals, above both the base rule and posterior sizing there, and AFM’s own end-to-end trading network, evaluated under this paper’s 60-day window and 20-day holds instead of its native daily rebalancing, has the lowest Sharpe ratio of any strategy on its factors (0.25); under a convention that is not its own, that row is not a replication of Epstein et al. [4], and the NFE adopts AFM’s factors and not its trading network.

Overshoot. One secondary finding contradicts the pre-stated prediction that the contrarian rule earns more after a window labeled converging than after one labeled diverging: with the label computed from the deviation about a level $g(X)$ learned from characteristics (Appendix § 8; this pre-registered label is not the posterior of (6)), the base rule’s realized P&L per holding period is +118 bp after a diverging window and +27 bp after a converging one, a converging-minus-diverging difference of $-92 \pm25$ bp. The difference survives its controls: the injected positive control recovers the predicted sign ($+81 \pm20$), the random-walk null shows no difference ($+8 \pm21$), the ordering persists when P&L is normalized by position weight, which rules out a sizing artifact, and it reproduces on a disjoint period ($-90 \pm37$, with the fundamentals at their neutral fill before 2016). On this panel a 60-day divergence behaves as an overshoot: the label orders expected return within a profitable rule and does not separate a profitable state from an unprofitable one, so its use is as a sizing input and not as a signal to stand aside. The sizing rule (7) does not use it, and the shrinkage it applies to slow-reversion windows runs against this finding, which is left open.

Analysis

Identification. On truth-known draws of the special case of the generative model with Gaussian innovations of constant scale, the only case in which the exact likelihood-ratio detector can be computed, every practical estimator and the reference detectors fall within one band of 0.61–0.66 AUC. The Bayes bound, i.e. the exact likelihood-ratio detector given the true level and scale, computed on draws whose state is constant over the window, reaches 0.93; any detector that sees only the window is bounded by it. In accuracy the amortized posterior is indistinguishable from the plug-in estimator of Avellaneda and Lee [1]: the window limits accuracy whatever the estimator, and the largest loss comes from the unknown level, since with the level known separation rises from 0.62–0.64 to 0.79–0.81. Over 60 days the slow state (D, half-life 120 days) completes 29% of its reversion and the fast state (C, 10 days) 98%, so a level estimated from the window sits near the current value and makes the slow state resemble the fast one. In a separate input-channel experiment, a channel carrying the un-demeaned spread closes almost none of that gap: every three seeds it adds $+0.0002 \pm0.0005$ AUC alone and $+0.0010 \pm0.0004$ together with the demeaned channel, against $+0.139 \pm0.002$ when the encoder is given the true level. On the half-life itself, an unamortized particle filter over the same model (25 × 256 particles) is the more accurate, with rank correlation 0.74 against 0.59 for the NFE on truth-known draws, at 7.9 s against 2.5 ms per cross-section on a desktop CPU (training the NFE one takes 104 s); the NFE trades that fidelity for amortization.

Calibration. The paper’s central claim is that the posterior is calibrated. Figure 2(b) tests the half-life posterior on truth-known draws under the training prior, before the per-asset reweighting, which is not separately checked: its coverage tracks the nominal level throughout, so the regime probability (6), the mass of the same posterior beyond $\tau^*$, is calibrated in the same marginal sense. The same posterior at half its width under-covers at every level: the maximum deviation from nominal is 0.022 on the seed shown and at most 0.010 on two further seeds, against 0.08–0.12 at half width. A posterior that ignored the window would also pass this check, so it is read together with the identification band above. On real data, the predictive distribution of the $H$-day change is scored on the held-out rolling-PCA residual series, which need no learned factors, with identical features and the same conformal layer for every method, against an unconditional baseline $\mathcal{N}(0,\hat{\sigma}\sqrt{H})$, $\hat{\sigma}$ the asset’s trailing 60-day volatility. Table 2 reports the continuous ranked probability skill score against the baseline (CRPSS), the dispersion of the predictive mean relative to the variance of the outcome, and the coverage and width of the nominal 90% interval, crediting sharpness only subject to calibration [9]. Its last column is the forecast–realized gap: with positions formed for every row by the same mean–variance rule from that row’s predictive mean and variance, the strategy’s own forecast of its return over the holding period minus the return it realizes, in basis points per holding period at unit gross. This is the postdecision surprise of Smith and Winkler [25], the optimizer’s curse, and the decision-level form of the winner’s curse of § 1: a forecast that is too confident about the positions it selects overstates what they earn. The gap is smallest for a forecast of zero, so it is read together with the forecast’s dispersion.

predictive distribution	CRPSS	disp.	cov.	width	gap (bp)
unconditional baseline	0.00	0.00	0.89	2.30	0
OU point estimate	−0.97	0.75	—	—	1084
mixture density net	−0.50	0.71	0.87	3.99	798
amortized OU-form (ours)	−0.11	0.22	0.89	2.40	588
amortized direct (ours)	0.00	0.02	0.89	2.17	137
boosted quantiles	−0.03	0.08	0.88	2.28	277

On real data, as reported in Table 2, no method has positive skill against the baseline, and the point estimate and the mixture density net score well below it, with returns alone or with fundamentals added (the latter not tabulated). Where the methods differ is in how accurately they assess their own error. The OU point estimate, the unshrunk window plug-in that mean–variance sizing requires, has the largest forecast dispersion in the table and gives no interval; its forecast exceeds its realized return by roughly a thousand basis points per holding period. The gap of the amortized methods is roughly two to eight times smaller, and the ordering holds on three training seeds: 572–593 bp for the OU-form posterior and 120–150 for the direct variant, against 1084 for the point estimate and 658–838 for the mixture density net. The reduction is a uniform shrinkage rather than a conditional one: the gap falls with forecast dispersion in every row of Table 2, and a point forecast scaled by a single constant, chosen so that its mean absolute forecast on the training split equals the posterior’s, reaches the posterior’s gap within sampling error. The OU-form posterior’s gap minus the rescaled point forecast’s is −1 to −5 bp over three seeds on truth-known draws of the admitted process, and +1 ± 17 and −13 ± 12 bp in the two periods of a KRX panel rebuilt under the same inverse rule from a second price source; on the no-reversion null it is −38 ± 10 and −42 ± 10 bp, so there the posterior is the more cautious (Appendix B). The gap that remains for every method we attribute to the level estimated from the window, the loss identified by the level-known comparison above. The conformal layer brings every learned method to nominal coverage, but coverage says nothing about the conditional mean. The amortized OU-form posterior is constrained by its functional form: its forecast dispersion on the volatility-matched random-walk null (Table 5) is as large as on real data. The amortized direct variant matches the baseline’s skill with a narrower interval, has the smallest gap of any learned method, and shows no dispersion on that null. Figure 3 shows the mechanism on truth-known draws: the encoder state is organized by the

latent half-life, which it never receives as input, and an ambiguous window yields a wide posterior and, through (7), a smaller position; whether the sizing results of Section 4.1 come from the width varying across windows or from the dependence of the penalty on $d^{2}$, which reweights large deviations even at a fixed width, has not been separated on real data.

Panel form. On truth-known panel draws, the panel form of Section 3.3, which is not the filter used in the portfolio, outperforms the recurrent time-series step, and by as much on panels whose innovation hyper-parameters are independent across assets, so that margin reflects capacity and not pooling. Pooling is measured separately, as the contrast between panels whose levels share a common factor and an independent-factor control. On shared-factor panels the held-out negative log-likelihood of the half-life falls from 2.088 to 1.762, of which 0.179 ± 0.003 is the contrast with the independent-factor control; the panel form recovers most of that factor and stays calibrated, and the true innovation hyper-parameters, the most that pooling a shared volatility state could supply, bring no further improvement, so the gain comes from the shared factor. The contrast reproduces on disjoint seeds (0.170 ± 0.002) in a confirmation run that also replaces the exploration’s rolling-PCA cross-sectional step by asset attention.

Conclusion

This paper nowcasts the state of a residual spread instead of forecasting its next move. It builds on the classical and the end-to-end approaches to statistical arbitrage, trains its filter on a simulated process whose innovation law was chosen by testing candidates against real residuals, and uses the uncertainty of the estimate in the trading decision. Three measured results follow. Within them, four findings have passed a confirmation run: the overshoot finding, the panel-form pooling contrast, the per-asset prior and the ordering of the forecast–realized gaps.

First, the regime is identifiable, but only marginally: every practical estimator and the strong reference detectors fall in one narrow band 0.61–0.66 AUC, against 0.93 for a fully informed detector, and the largest loss comes from estimating the level from the same window as the reversion. Second, a calibrated posterior gives an accurate account of its own error. A point estimate’s forecast exceeds its realized return by roughly a thousand basis points per hold and gives no interval; the amortized methods reduce that gap by a factor of roughly two to eight, and since a point forecast rescaled by one constant reaches the same gap, the reduction is a uniform shrinkage, and the interval, which the point estimate lacks, is what the posterior adds. No method shows skill against an unconditional baseline, with returns alone or with fundamentals added. Third, in the portfolio evaluation the residual construction matters more than the trading policy. Posterior sizing matches AFM’s Sharpe, within the period’s noise, at no more risk. No pre-specified decision-layer configuration adds return of the size the residual construction adds; the gain from a longer window is unstable across periods and not claimed; and the fully learned cross-sectional step does no better than the random benchmark.

One methodological lesson runs through all three results: a label-free diagnostic needs a null model and a positive control before its output can be interpreted. Without them several of ours lacked power or were biased upward, and we withdrew the conclusions drawn from them, the 250-day gain among them. A simulator can pass a two-sample test that has almost no power against the kernel it appears to validate: changing the kernel moves the AUC by at most 0.014 (Appendix § 8).

Since the largest identification loss comes from the level estimated from the same window, the natural next step is the panel form of the NFE: one causal network over the cross-section and time, pretrained on a panel-level generative model, that pools the cross-section inside the network and returns its posteriors jointly, with a selective state-space encoder [10] as its time-series step. On truth-known draws with a shared factor in the levels, the panel form already improves the half-life posterior over the per-asset filter, in part through capacity, and stays calibrated (§ 4.2); on real data the two cross-sectional routes tried here, a learned cross-sectional step and a level learned from characteristics, did not help, so its value there is untested. Whether on real data the information is worth more than the cost of acting on it is the question this work leaves open.

Acknowledgments

We thank Jae Young Kim (Seoul National University) and Young Jae Cho (ATZ Trading) for their guidance and support, and the three anonymous ICAIF reviewers for their careful reading and constructive comments.

Ethics and Privacy Statement

This work uses only public daily market data and simulated data; no vendor prices are redistributed and no personal data is used.

References

The Generative Model

The scale, level, and innovation law of (3) evolve as

with $\nu,\ u,\ \xi\sim\mathcal{N}(0,1)$. Table 4 lists the parameters of each component and how each is fixed. Only the innovation law and the scale are fitted to data, by moment matching on the training split’s real windows and never on the two-sample test; the kernel carries the two speeds, $\eta$ is fixed so that $\log\sigma$ has stationary standard deviation $0.5$, and the level step $\delta$ is held at a fixed default. The two-sample test constrains the innovation law and the scale but not the kernel, so the number of regimes is a modeling choice, two being the smallest set that separates a reversion the trade can use from one it cannot. The level is untested, every candidate being simulated with $m \equiv0$; a level $g(X)$ learned from characteristics by gradient boosting on slow window statistics and the six fundamentals, fitted causally, does not outperform a 250-day trailing mean of the spread on held-out data, and is retained only as the pre-registered anchor of the overshoot test in § 4.1. The injected positive controls of § 4 add a trend away from the level.

component	form	parameters	fixed by	tested
kernel	OU, regime-indexed speed	$\kappa(\mathrm{C}), \kappa(\mathrm{D})$	prior	no
innovations	centered lognormal	$\gamma$	moment matching	yes
scale	AR(1) log-volatility	$\phi, \eta$	$\phi$ matched; $\eta$ fixed	yes
level	random walk, step $\delta$	$\delta$	fixed	no

The two-sample test. For each candidate innovation law a classifier is trained to distinguish simulated residual windows from real ones and its AUC recorded (Table 3); a candidate passes when its AUC plus two standard errors lies below 0.75.

innovation law	crypto	KRX	verdict
Gaussian, constant $\sigma$	0.903 ± .002	0.904 ± .001	fail
Student-$t$	0.809 ± .002	0.788 ± .002	fail
lognormal	0.890 ± .001	0.885 ± .001	fail
stochastic vol (SV)	0.744 ± .004	0.712 ± .002	fail/pass
SV + lognormal	0.686 ± .003	0.678 ± .002	pass
random walk, Gaussian	0.895 ± .001	0.898 ± .001	fail

Windows real: one fixed set of cross-sectionally demeaned cumulative residuals over the full sample (the PCA and AFM series are not re-tested); simulated: five seeds, which give the standard error.

Classifier gradient boosting on scale-invariant window statistics: tail weight, skewness, autocorrelations of the increments and of their absolute values, zero crossings, range, half-window volatility ratio.

Controls same-process windows score 0.50 and a wrong process 0.90; the kernel moves the AUC by at most 0.014, so the statistics constrain the innovation law, not the kernel.

Candidates fifteen: the laws of Table 3 and their combinations, plus random walks under the same laws as controls without mean reversion; the table reports one representative per feature.

Student-$t$ innovations score 0.809/0.788 and worsen the fit when combined with stochastic volatility, so the data are better matched by a day-to-day varying scale with asymmetric innovations than by a fixed fat-tailed law. The bar of 0.75, halfway between chance and perfect separation, was registered before any run as a convention, not derived; at 0.678 the selected process remains distinguishable from the data, and the posterior claims on the KRX portfolio should be interpreted in light of that residual discrepancy.

Experiment Details

Data. Both universe filters use end-of-sample information (liquidity ranking at the sample end; delisted stocks dropped). The six characteristics are point-in-time: each becomes available on the first trading day after disclosure, flows are trailing-twelve-month, capitalization is split-consistent, and a missing stock-day receives a neutral rank.

Definitions.

Weights and costs defined on the residual portfolios of (5) [4]; the implied factor-hedge legs are not separately traded or costed.

Turnover the mean daily absolute change in these weights as a fraction of gross.

ΔSR the annualized Sharpe ratio of the daily return difference between a strategy and the random benchmark (bootstrap SE 0.41–0.42); the benchmark redraws its random positions every $H$ days at the same gross, hence its higher turnover.

Not modeled the Korean transaction tax and short-sale restrictions in force during part of the window; levels are therefore not implementable returns.

Estimation notes.

FF3 needs size and B/P on the same stock-day, with joint coverage only from 2017, so part of its weakness, in the earlier period especially, reflects limited coverage.

Conformal layer widths are multiplicative per quantile; the outcome-fitted methods fit on the first 80% of the training split and calibrate on the remaining 20% after an $L$-day gap, simulator methods on the full training split.

Shrinkage control keeps the point estimate’s positions and rescales only its forecast by one constant fixed on the training split; its KRX run uses a panel rebuilt under the same rule from a second vendor’s adjusted prices, on which Table 2’s ordering also holds.