Prediction Intervals and Rank Uncertainty in Supply Chain Exposures

Abstract

Economic relationship weights are usually treated as observed inputs even when the quantities entering a network model have been estimated, revised, or intermittently remeasured. That convention creates a ranking problem: two supplier-customer exposures with identical current point estimates can carry very different amounts of historical revision risk. This paper estimates that risk from Altsets point-in-time relationship histories and asks whether revision-calibrated intervals alter financial network rankings. The empirical panel contains 23 directed relationships observed across 21 quarterly dates from 2021-09-28 through 2026-09-28, producing 483 relationship-date rows and 185 observations with all three Altsets dependency measures available: Relationship Size, Supplier Revenue %, and Customer Cost %. Restricting the estimation exercise to adjacent quarters in which the same relationship remains observed yields 142 one-quarter revision transitions. Three interval specifications are evaluated sequentially: a pooled uncertainty band, a relationship-class band conditioned on direction and persistence, and a history-aware band that shrinks each relationship's empirical revision quantile toward its class distribution. In an 88-transition walk-forward evaluation, 90% history-aware intervals attain 92.0% coverage for Relationship Size, 89.8% for Supplier Revenue %, and 89.8% for Customer Cost %, while their median transformed half-widths are respectively 22.1%, 25.3%, and 13.9% smaller than uniform bands. Propagating Supplier Revenue % intervals through a weighted supplier centrality ranking leaves broad ordering intact but makes many exact ranks unidentified. The result is a distinction between economically persistent leaders and statistically fragile rank differences that a point-estimate network suppresses.

Keywords: Prediction intervals; Sequential calibration; Errors-in-variables; Rank uncertainty; Weighted networks

JEL: C13; C52; C53; G11

Problem

Weighted economic networks inherit two different forms of uncertainty from their edges. A relationship can change economically because the underlying commercial exposure changes, and its measured weight can change because a later observation contains different information about the same exposure. A ranking method that receives only the latest point estimate cannot distinguish those mechanisms. The distinction becomes financially relevant whenever network weights enter a portfolio constraint, a counterparty screen, a propagation model, or a centrality statistic. Production-network research has established that the economic magnitude and position of input relationships can determine how firm-level shocks propagate, rather than link existence alone. Acemoglu, Carvalho, Ozdaglar, and Tahbaz-Salehi formalize how asymmetry in input-output weights affects aggregate fluctuations, while Barrot and Sauvagnat show empirically that supplier shocks can transmit into customer output and market value, particularly for specific inputs. Once a financial quantity is constructed from weighted links, uncertainty about those weights is therefore uncertainty about the financial quantity itself.

The statistical problem differs from conventional missing-edge analysis. Classical measurement-error work shows that replacing an uncertain economic quantity by a measured point value can distort downstream estimation, while the portfolio literature has separately shown that parameter uncertainty can materially change decisions even when the estimated parameters themselves are unchanged. Hausman's review describes how measurement error alters econometric inference, and Garlappi, Uppal, and Wang show in a portfolio setting that allowing different degrees of uncertainty around estimated inputs can generate different and more stable allocations than treating the estimates as known. The analogous object here is an edge weight. If relationship $e_1$ and relationship $e_2$ both show a Supplier Revenue % of 2.0 today, but $e_1$ has historically remained within a few basis points while $e_2$ has repeatedly moved by several tenths of a percentage point, assigning the same uncertainty band to both discards information already present in their point-in-time histories.

The primary hypothesis is that edge-specific revision histories contain enough information to sharpen next-quarter uncertainty intervals without materially sacrificing empirical coverage. The secondary hypothesis concerns ranks. If weighted centrality differences are large relative to revision uncertainty, a point-estimate ordering should survive interval propagation. If several nodes are separated by less than their historically calibrated edge uncertainty, an exact ordering should become a partial ordering in which several ranks remain feasible. This is related to the econometric problem studied by Mogstad, Romano, Shaikh, and Wilhelm, who construct rank confidence sets when quantities being ranked are estimated rather than known. Their result is broader and formally inferential; the construction below is narrower, using empirical relationship-weight intervals to obtain network rank envelopes.

Data

The Altsets panel used here follows 23 directed supplier-customer relationships over 21 requested quarterly dates. Each relationship contains three economic weights when quantified: Relationship Size in dollars, Supplier Revenue %, and Customer Cost %. Supplier Revenue % measures the relationship from the supplier's revenue base, while Customer Cost % measures it from the customer's cost base. Relationship Size supplies the dollar scale. The historical panel therefore contains three related observations of the same commercial edge rather than a single generic weight. Across the full export there are 483 relationship-date rows, 185 complete three-metric observations, and 142 instances in which a relationship is quantified in two consecutive quarterly snapshots. The last set is the estimation sample because a one-quarter interval has a constant economic horizon and does not require assigning a zero weight to an unobserved relationship.

The histories exhibit revision patterns that are difficult to represent with a common error term. The SK hynix to Intel supplier relationship, for example, moves from approximately $468 million, 1.42% Supplier Revenue %, and 1.22% Customer Cost % in September 2021 to approximately $1.160 billion, 1.35%, and 2.70% in September 2026. Between those endpoints the relationship passes through sizeable upward and downward revisions rather than moving monotonically. Intel's public reporting provides an independent economic anchor for the relationship: after the first closing of its NAND transaction with SK hynix in December 2021, Intel disclosed an ongoing NAND wafer manufacturing and sale agreement with SK hynix and described contingent incentives and penalties tied to cost and output. The historical edge therefore corresponds to an identifiable commercial arrangement whose financial scale and operating terms changed through time, rather than an abstract similarity link.

A different revision path appears in the Hyundai Glovis to Boeing relationship. Altsets records Relationship Size rising from about $174 million in September 2021 to roughly $292 million in March 2024, then declining toward $97 million in June 2025 before recovering to approximately $139 million by June 2026. Supplier Revenue % moves from 0.96% to 1.30%, then falls below 0.40% before recovering to 0.60%. The Boeing to Korean Air customer edge behaves differently again: consecutive observations are often stable, while revisions between regimes are large. Its observed Customer Cost % rises from 2.61% in June 2022 to 44.06% in June 2023, later settling near 36.6%. Boeing's own August 2025 announcement described Korean Air's intention to purchase 103 aircraft, in addition to a March 2025 order for 40 widebody aircraft, confirming that the relationship represented a large and changing capital commitment during the period. These examples motivate estimating uncertainty from the revision process of each edge rather than attaching one percentage error to every relationship.

Model

Let $R_{e,t}$ denote Relationship Size, $S_{e,t}$ Supplier Revenue % in percentage points, and $C_{e,t}$ Customer Cost % for relationship $e$ at quarter $t$. Positive variables are transformed to make revision magnitudes comparable across levels:

The one-quarter revision innovation for metric $m\in\lbrace R,S,C \rbrace$ is

The forecast center is deliberately simple, $\widehat{z}^m_{e,t+1|t}=z^m_{e,t}$. The object being tested is uncertainty calibration rather than a model of expected relationship growth. A 90% empirical prediction interval therefore takes the form

where $T_m^{-1}$ is the inverse transformation and $h^m_{e,t}$ is estimated only from revisions observable before the forecast date. This follows the calibration principle emphasized by Gneiting, Balabdaoui, and Raftery: interval quality requires both empirical coverage and sharpness, so a wider interval cannot be declared superior merely because it captures more observations. Sequential prediction research reaches the same issue for non-exchangeable data, where historical residual distributions can be updated through time rather than assumed stationary and known.

The first specification imposes uniform uncertainty. For metric $m$, every relationship receives the empirical $p$-quantile of prior absolute revisions,

The second specification creates relationship classes using edge direction and persistence. Direction distinguishes observations in which the focal company is the supplier from those in which it is the customer. Persistence is measured as the current uninterrupted run of quantified quarterly observations, with relationships observed for four or more consecutive quarters classified as persistent and shorter runs classified as young. Four quarters has an economic interpretation as one year of continuous measurement. The class interval is

with a direction-level or pooled fallback when a class has too few prior observations.

The history-aware specification lets the same current point estimate carry different uncertainty according to its own revision record. Let $q^m_{e,t}$ be the empirical $p$-quantile of prior absolute revisions for edge $e$, and let $q^m_{g,t}$ be its class quantile. The interval half-width is

where $n_{e,t}$ is the number of previously observed one-quarter revisions for the relationship. The denominator gives the class distribution the equivalent weight of one year of quarterly evidence. An edge with no own revision history receives its class interval; after four historical revisions the edge and class receive equal weight; with a long history the relationship's own tail behavior dominates. No distributional assumption is imposed on $u$, and no simulated volatility parameter enters the interval.

Calibration

The first portion of the panel is used as a burn-in so that each forecast can be formed from previously observed revisions. Walk-forward evaluation begins with forecasts made from the June 2024 snapshot and runs through the September 2026 observation, producing 88 evaluable one-quarter transitions for each metric. At every forecast date, the quantiles are recalculated from information that would already have existed at that date. Coverage is

The 90% results reject the strongest version of a uniform-error model. Uniform bands are serviceable, but their excess width is concentrated in relationships with stable histories, while their common scale remains insufficient for some volatile edges.

For Relationship Size, a median log half-width of 0.1967 under uniform uncertainty corresponds to an approximate multiplicative band of -17.9% to +21.7% around the current relationship value. The history-aware median contracts that to roughly -14.2% to +16.6% while empirical coverage remains 92.0%. At a representative Supplier Revenue % of 0.96%, the median uniform transformed interval corresponds approximately to 0.79% to 1.14%, compared with 0.83% to 1.09% under the history-aware specification. At a representative Customer Cost % of 1.40%, the analogous bands are about 1.26% to 1.55% and 1.28% to 1.53%. These are modest differences for an individual edge, but a network ranking can contain many pairwise comparisons whose point estimates are separated by less than those amounts.

Calibration is also examined away from the 90% target. For the history-aware model, nominal 80%, 90%, and 95% Relationship Size intervals achieve empirical coverage of 80.7%, 92.0%, and 95.5%. Supplier Revenue % achieves 86.4%, 89.8%, and 94.3%, while Customer Cost % achieves 78.4%, 89.8%, and 93.2%. Mean absolute coverage error across those three nominal levels is approximately 1.1 percentage points for Relationship Size, 2.4 points for Supplier Revenue %, and 1.2 points for Customer Cost %. The empirical distribution is discrete and contains many zero-revision quarters, so exact equality between nominal and realized coverage is neither expected nor mechanically attainable in 88 observations. The relevant comparison is whether greater conditioning produces narrower intervals while keeping realized coverage near the intended probability. The history-aware model satisfies that condition most clearly for Relationship Size and Supplier Revenue %, while Customer Cost % gains less sharpness.

Cross-sectional heterogeneity explains the result. Among relationships with at least four adjacent-quarter revisions, the 90th percentile absolute Supplier Revenue % innovation in transformed units is only about 0.0056 for Boeing to Korean Air but approximately 0.1195 for Hyundai Glovis to Boeing. For Customer Cost %, the corresponding tail revision is about 0.0050 for Boeing to Korean Air and roughly 0.0899 for Samsung Electronics to Qualcomm. Relationship Size displays similarly wide dispersion. A single pooled band must either be wide enough for the volatile histories or narrow enough for the stable ones; it cannot do both simultaneously. The class and shrinkage estimators treat that dispersion as information.

Current intervals illustrate how this changes interpretation. The September 2026 SK hynix to Intel edge has an Altsets Relationship Size near $1.160 billion, Supplier Revenue % of 1.35%, and Customer Cost % of 2.70%. Its history-aware 90% next-quarter intervals are approximately $886 million to $1.517 billion, 1.14% to 1.58%, and 2.23% to 3.24%. The Boeing to Korean Air edge, despite having experienced large regime changes at earlier dates, has a much quieter recent one-quarter revision history; its September 2026 values of about $834 million, 0.79%, and 36.58% produce intervals of roughly $719 million to $967 million, 0.71% to 0.87%, and 35.08% to 38.14%. Those bands are properties of the histories, not discretionary uncertainty labels.

Ranks

A network ranking requires a single economic quantity rather than a synthetic mixture of dollars, supplier revenue shares, and customer cost shares. The propagation exercise therefore uses Supplier Revenue % as a directed edge weight and retains Relationship Size and Customer Cost % as independent tests of whether the uncertainty method behaves similarly across the other Altsets measures. For supplier $i$, define current revenue-exposure centrality over the sampled network as

This is weighted out-degree with an immediate accounting interpretation: the sum of the supplier's observed customer exposures measured as shares of supplier revenue. The September 2026 sample contains 23 directed edges and 26 distinct nodes, of which 18 appear as suppliers. Because the panel is deliberately a relationship sample rather than a complete economy-wide graph, $C^S$ is used for within-sample ranking and interval propagation rather than presented as a firm's total customer concentration.

For each edge, the history-aware Supplier Revenue % interval is back-transformed into $[l_{ij},u_{ij}]$. Node bounds are then obtained without assuming independence across relationships,

A rank envelope follows directly. Supplier $i$'s best feasible rank is one plus the number of suppliers whose lower bound exceeds $U_i$. Its worst feasible rank is one plus the number whose upper bound exceeds $L_i$. This is intentionally conservative because every edge may take any value inside its calibrated interval simultaneously. It is conceptually close to rank-set inference under estimated quantities, while stopping short of labeling the result a simultaneous confidence set. Mogstad and coauthors show why the distinction is material: point estimates can support an apparently precise ordering even when statistically admissible ranks overlap.

The point estimates produce an exact total ordering. The uncertainty-aware object produces something different. Qualcomm and Makalot remain the two highest-exposure suppliers in the sample, but their intervals overlap enough that rank one cannot be assigned uniquely. Korea Circuit remains rank three throughout the rectangular interval set, and McKesson remains rank four. Teijin and Constellium form another unresolved pair. Samsung Electronics and PepsiCo are separated by only 0.02 percentage points on the point estimate and reverse when suppliers are ranked by their calibrated lower bounds. SK hynix and AMD also reverse under the relationship-history lower-bound ranking even though they retain point ranks nine and ten. Intel and Boeing form a third reversal.

Across all 18 supplier nodes, the Kendall rank correlation between the point ordering and the history-aware lower-bound ordering is approximately 0.961, corresponding to only three discordant pairwise comparisons out of 153. That high correlation is informative: incorporating uncertainty does not destroy the economic structure of the ranking. It changes the interpretation of small differences inside that structure. Six individual nodes change position in the lower-bound ordering because the reversals occur in pairs: Samsung and PepsiCo, SK hynix and AMD, and Intel and Boeing. Under a uniform revision assumption, Kendall correlation is approximately 0.974 and only four nodes move. The extra SK hynix-AMD reversal appears only when relationship-specific revision histories are allowed to produce different bands around similar current exposures.

The full rank envelopes are less decisive than the lower-bound ordering. At the 90% level, 16 of the 18 suppliers have a rank envelope containing more than one position. That figure should not be read as evidence that the entire ranking is uninformative. Many feasible movements are local, and the ordering retains strong blocks: the top-four membership is stable, Korea Circuit and McKesson have singleton ranks, and the rank uncertainty becomes concentrated where exposure levels cluster. Borgatti, Carley, and Krackhardt reached a related result in a different network setting: centrality measures can degrade predictably under network-data error, permitting uncertainty around centrality rather than requiring that every observed centrality score be treated as exact. Here the error distribution is not imposed randomly. It is estimated from each economic edge's own history.

Tests

Two failure tests challenge the rank result. First, the interval probability is changed. With 80% history-aware bands, 13 of 18 suppliers have non-singleton rank envelopes and the average envelope width is 0.89 rank positions. At 90%, those quantities rise to 16 of 18 and 1.33 positions. At 95%, 17 of 18 suppliers have non-singleton envelopes and the average width rises to 1.67 positions. The lower-bound ordering remains highly correlated with the point ordering at every level. Rank ambiguity therefore grows smoothly as the required coverage increases rather than appearing only at one chosen cutoff.

Second, the four largest Supplier Revenue % centralities are removed and the remaining suppliers are reranked. If the finding were driven by a few dominant customer concentrations, uncertainty should become less relevant after those observations are excluded. The opposite occurs. Among the remaining 14 suppliers, Kendall correlation between the point ordering and the history-aware lower-bound ordering declines to approximately 0.934, while the Samsung-PepsiCo, SK hynix-AMD, and Intel-Boeing reversals remain. The uncertain comparisons reside in the dense middle of the ranking, where point-estimate spreads are small relative to empirical revision bands. Removing the extreme exposures removes easy comparisons and leaves a larger proportion of economically close pairs.

A third comparison is embedded in the calibration exercise. The class model already permits young and persistent relationships in supplier and customer directions to have different uncertainty. At the 90% target it achieves 92.0%, 93.2%, and 90.9% coverage across Relationship Size, Supplier Revenue %, and Customer Cost %. Relationship-specific shrinkage produces 92.0%, 89.8%, and 89.8%, while reducing median interval width relative to pooled uncertainty by 22.1%, 25.3%, and 13.9%. The individual-history term therefore contributes primarily through sharpness and cross-sectional differentiation rather than a mechanical increase in aggregate coverage. That distinction is consistent with calibration theory, where a useful forecast distribution should become concentrated only to the extent permitted by observed forecast errors.

Interpretation

The empirical object here is revision risk, not a claim that every historical change is statistical measurement error. A commercial weight can be revised because the business relationship itself evolved, because financial denominators changed, because a new disclosure altered the estimate, or because several of these occurred together. For a financial decision taken at time $t$, separating those causes is less important than recognizing that the current edge weight is not equally stable across relationships. Historical revisions form an observable distribution of how far the next usable weight has tended to move from the current one. Treating that distribution as relationship-specific converts an unobserved source of parameter uncertainty into an estimable input.

This alters how a network score should be used. Suppose a strategy selects the five highest weighted suppliers, imposes a position limit on the ten highest exposures, or trades changes in centrality ranks. A point estimator generates turnover whenever $C_i$ crosses $C_j$, even when the difference is much smaller than the historical revision scale. A revision-aware rule can instead distinguish an ordering from an identified ordering. For two suppliers $i$ and $j$, the inequality

is sufficient for $i$ to outrank $j$ throughout the propagated interval set. When the intervals overlap, the model can preserve the exposure estimates while withholding the pairwise ordering. Such an abstention rule is mechanically testable in a later trading study: if point-rank crossings inside overlapping intervals generate excess turnover without subsequent information, uncertainty adjustment should reduce turnover while preserving economically separated changes. A contrary finding, in which overlapped rank changes predict returns or realized shocks as strongly as interval-separated changes, would weaken the case for revision filtering.

The same logic extends beyond centrality. Customer Cost % intervals can be propagated into supplier-concentration measures, Relationship Size intervals into dollar portfolio exposures, and the three interval families can enter optimization constraints separately instead of being collapsed into one composite edge score. Garlappi, Uppal, and Wang's portfolio result supplies a useful financial parallel: uncertainty sets around estimated inputs can change decisions because the optimizer should respond differently to precise and imprecise estimates. The present evidence indicates that commercial relationship weights satisfy the prerequisite for such a treatment. Their empirical revision distributions are heterogeneous enough that a uniform uncertainty assumption leaves measurable information unused.

Conclusion

Historical revisions in economic relationship weights contain usable information about the uncertainty of current network inputs. In the Altsets quarterly panel, an empirical interval model conditioned on direction, persistence, and each edge's own revision history produces near-nominal walk-forward coverage across Relationship Size, Supplier Revenue %, and Customer Cost % while materially narrowing median intervals relative to a pooled uncertainty assumption. The gain is largest for Supplier Revenue %, where the median 90% half-width falls by about one quarter without reducing realized coverage below 89.8%. Propagating those intervals through supplier revenue-exposure centrality leaves large economic separations intact but removes unwarranted precision from close rankings. The top-four supplier set survives, yet the top two positions overlap, several middle-ranking pairs reverse under conservative ordering, and most nodes occupy more than one feasible rank at the 90% level. The empirical answer is therefore narrower than a claim that uncertainty overturns network rankings. It identifies where rankings are supported by exposure gaps that dominate revision risk and where the apparent ordering exists mainly because a continuously valued point estimate forces economically close relationships into distinct positions.

References

Altsets supply chain dataset.

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