Research article

Conditional Beta and Residual Volatility with Supply Chain Data

Abstract

Market beta and residual volatility are usually estimated from return histories after the firm's underlying economic exposures have already changed. This paper examines whether changes in economically weighted supply-chain position provide an earlier observable state variable for changes in those financial risk characteristics. Rather than testing whether highly central firms earn different returns, the analysis focuses on centrality migration: changes in a firm's upstream or downstream position caused by changes in the economic importance of persistent commercial relationships. Historical Altsets relationship snapshots are used to construct directional centrality contributions from Relationship Size, Supplier Revenue %, and Customer Cost %, then changes in those measures are matched to subsequent changes in market beta and market-adjusted residual volatility. A six-relationship case panel spanning Intel-SK Hynix, Caterpillar-Teijin, Boeing-Korean Air, Nike-MAP Aktif, Broadcom-Samsung Electronics, and PepsiCo-Seven & i produces a directional result. Across the usable upstream transitions, absolute economic-centrality migration has a 0.553 correlation with subsequent absolute beta change, compared with 0.294 for subsequent residual-volatility change. Across downstream transitions, the beta relation is weak, while migration has a 0.583 correlation with subsequent residual-volatility change. Economic weights carry information that binary links cannot represent because several of the largest migrations occur while the relationship itself remains present. Static centrality also performs substantially worse than centrality change in the downstream residual-risk test. The effect is concentrated in the largest migrations and weakens sharply when the largest observations are removed, which favors a threshold or state-transition interpretation over a stable linear centrality coefficient. The evidence therefore supports a narrower hypothesis than conventional network-centrality asset pricing: sufficiently large changes in directional economic position may identify periods when a firm's covariance structure is changing.

Keywords: conditional beta; time-varying risk; production networks; covariance estimation; residual volatility; state transitions; economic centrality

Introduction

A firm's observed market beta is an estimate of a covariance structure that may already be stale when the economic exposures generating that covariance have changed. Rolling regressions reduce the influence of distant observations, and conditional-beta models allow factor loadings to vary through time, but both approaches infer the transition primarily from financial variables. They provide limited information about why the loading should change at a particular date. For firms whose revenues and production costs depend materially on other companies, one candidate state variable exists outside the return process itself: the firm's changing economic position among suppliers and customers. The relevant object is not whether the firm is central in a static network. It is whether the economic weight carried by its existing relationships is being materially reallocated.

Production-network research provides an economic reason for such reallocations to affect financial risk. Ahern studies network centrality as a determinant of aggregate risk exposure, while Herskovic links production-network structure to systematic risk in asset prices. Barrot and Sauvagnat show that idiosyncratic shocks can propagate through supplier-customer relationships, with the magnitude of propagation depending on the economic characteristics of the input relationship. These results establish that commercial links can transmit economically relevant shocks. They leave a different empirical question open: whether a change in the intensity and direction of those links precedes a change in the firm's own factor loading or residual variance. A firm may occupy a highly central position for years while its beta remains comparatively stable. Conversely, a firm may begin from an ordinary network position and experience a substantial change in financial exposure when one or several counterparties become much more important to its cost structure or revenue base.

That distinction makes centrality change economically different from centrality level. Let $C^d_{i,t}$ denote the economically weighted centrality of firm $i$ at time $t$ in direction $d$ , where $d$ distinguishes upstream supplier exposure from downstream customer exposure. The central object in this paper is

M^d_{i,t} = C^d_{i,t} - C^d_{i,t-1}.

$M^d_{i,t}$ is centrality migration. A large positive or negative value represents a material movement in the firm's economically weighted network position even when its set of counterparties changes little. The financial quantity of interest is correspondingly a change in risk loading rather than a return premium. The primary hypothesis is that large absolute upstream migrations should be followed by larger absolute changes in market beta when the altered supplier exposures load differently on aggregate production conditions. A secondary hypothesis is that downstream migration may operate through a different channel: changing dependence on major customers can alter firm-specific cash-flow variance even when the customer's shock has relatively little covariance with the market. Theory therefore motivates separate tests for $\Delta\beta$ and $\Delta\sigma_{\epsilon}$ , rather than assuming that all forms of network migration should move the same financial-risk statistic.

The sign of a beta change is not determined by the magnitude of centrality migration alone. Increasing dependence on a counterparty with strongly procyclical shocks could raise the firm's beta, while reallocating exposure toward a counterparty with lower aggregate covariance could reduce it. The primary empirical prediction is consequently expressed in absolute changes:

|M^d_{i,t}| \longrightarrow |\Delta \beta_{i,t+1}|

for systematic-risk migration, and

|M^d_{i,t}| \longrightarrow |\Delta \sigma_{\epsilon,i,t+1}|

for residual-risk migration. The hypotheses concern whether the magnitude of financial risk changes with the magnitude of network movement. A signed specification would require additional information about the systematic loading carried by each entering, expanding, contracting, or disappearing counterparty. This paper therefore treats a large migration followed by little change in either beta or residual volatility as evidence against the proposed mechanism rather than forcing a directional prediction unsupported by the economic weights alone.

The analysis is deliberately separated from expected-return prediction. No centrality factor is constructed, firms are not ranked by future performance, and the commercial network is not used to generate a long-short portfolio. The problem is conditional risk estimation. If changes in network position coincide with subsequent changes in beta or residual variance, then relationship migration may help identify periods during which a backward-looking covariance estimate deserves less persistence. This interpretation also creates direct baselines. Centrality migration can be compared with centrality level to determine whether movement contains information beyond location. Economically weighted centrality can be compared with binary centrality to determine whether changing relationship intensity matters when topology is unchanged. Upstream and downstream migration can then be compared to test whether relationship direction corresponds to different components of financial risk.

Data

The empirical exercise uses historical Altsets relationship observations rather than the full relationship universe. Six relationships are selected because they contain repeated historical measurements and economically meaningful variation in the three available dependency measures: Relationship Size, Supplier Revenue %, and Customer Cost %. The panel contains Intel-SK Hynix and Caterpillar-Teijin as upstream relationships, together with Boeing-Korean Air, Nike-MAP Aktif, Broadcom-Samsung Electronics, and PepsiCo-Seven & i as downstream relationships. The purpose of the sample is not population-wide inference. It is a historical case panel in which actual changes in relationship intensity can be measured and matched to independently observed changes in financial risk.

The three relationship variables describe different economic margins. Relationship Size measures the absolute economic scale assigned to an observed supplier-customer link. Supplier Revenue % measures the relationship relative to the supplier's revenue base. Customer Cost % measures the relationship relative to the customer's cost base. They are therefore not interchangeable measures of "importance." In an upstream test centered on the customer, Customer Cost % has the most direct interpretation as exposure to the supplier because it measures the supplier relationship against the customer's input economics. In a downstream test centered on the supplier, Supplier Revenue % more directly represents revenue dependence on the customer. Relationship Size can remain informative in both directions because a large absolute commercial exposure can produce economically relevant cash-flow consequences even when the corresponding percentage is moderate.

For relationship $e$ at time $t$ , the three economic measures are first standardized within their respective historical series:

z^{RS}_{e,t} = \frac{RS_{e,t}-\mu_{RS,e}}{s_{RS,e}},

z^{SR}_{e,t} = \frac{SR_{e,t}-\mu_{SR,e}}{s_{SR,e}},

and

z^{CC}_{e,t} = \frac{CC_{e,t}-\mu_{CC,e}}{s_{CC,e}},

where $RS$ is Relationship Size, $SR$ is Supplier Revenue %, and $CC$ is Customer Cost %. Standardization is required because the variables are expressed in different units and differ considerably in scale. The relationship-level economic contribution is then constructed as an equal-weight composite,

w_{e,t} = \frac{ z^{RS}_{e,t} + z^{SR}_{e,t} + z^{CC}_{e,t} }{3}.

The equal-weight construction is intentionally transparent. The small panel cannot justify estimating optimal weights without introducing severe overfitting, while choosing a single measure in advance would impose an economic interpretation that may differ between upstream and downstream relationships. The individual components are therefore retained for a later decomposition, allowing the composite result to be challenged by examining which underlying measure actually carries the association.

For firm $i$ , directional economic centrality is the sum of the relevant relationship contributions at each snapshot:

C^{up}_{i,t} = \sum_{e \in \mathcal{E}^{up}_{i,t}} w_{e,t},

C^{down}_{i,t} = \sum_{e \in \mathcal{E}^{down}_{i,t}} w_{e,t}.

In the six-relationship case panel, most firms contribute one focal relationship, so these quantities should be interpreted as directional centrality contributions rather than estimates of total global network centrality. That restriction is useful for identification because the observed change can be attributed to a known relationship rather than to an opaque aggregate over hundreds of edges. The broader research object remains general: with full historical snapshots, the same construction can sum every economically weighted incoming or outgoing edge and convert the resulting distribution into a cross-sectional percentile or standardized network position.

Centrality migration is measured as the first difference of that economic position:

M^d_{i,t} = C^d_{i,t} - C^d_{i,t-1}.

The binary comparison replaces $w_{e,t}$ with an indicator $b_{e,t}$ equal to one when a relationship exists and zero otherwise. Binary centrality migration is therefore

M^{d,B}_{i,t} = \sum_e b_{e,t} - \sum_e b_{e,t-1}.

This comparison isolates a specific measurement question. If a Boeing-Korean Air or Nike-MAP Aktif relationship remains present while its economic weight changes sharply, binary centrality records no migration. The economically weighted measure does. Relationship births and disappearances remain visible to both constructions, but only the weighted version records substantial changes inside continuing relationships.

The historical observations contain precisely that type of variation. Boeing's downstream Korean Air relationship produces the largest standardized migration in the downstream panel, increasing by 1.897 composite units between 2022 and 2023. Nike's MAP Aktif relationship produces a 1.264-unit increase over the same interval. Intel's SK Hynix history contains a later large upstream migration, while Caterpillar-Teijin provides a second upstream series with a different path through time. Broadcom-Samsung Electronics and PepsiCo-Seven & i provide additional downstream histories that prevent the downstream result from reducing entirely to the Boeing and Nike cases. These values are used as state changes, not as standalone evidence that any particular counterparty caused a subsequent stock-market outcome.

Risk measure

The financial side of the test asks whether those economic migrations are followed by changes in systematic or residual risk. Historical market beta is used as the systematic-risk measure because it directly captures the sensitivity of the firm's return to aggregate market returns. Total historical volatility alone cannot separate a change in market exposure from a change in firm-specific uncertainty. A company can become more volatile because the market itself becomes more volatile, because its beta rises, because its residual variance rises, or because several of these quantities move simultaneously.

The case-panel construction therefore separates the market component from the remaining variance. Let the one-factor return process be

r_{i,t} = \alpha_i + \beta_{i,t}r_{M,t} + \epsilon_{i,t},

with

\operatorname{Cov}(r_{M,t},\epsilon_{i,t})=0.

Under this decomposition,

\sigma^2_{i,t} = \beta^2_{i,t}\sigma^2_{M,t} + \sigma^2_{\epsilon,i,t}.

For each firm-year, the required inputs are annualized firm volatility $\sigma_{i,t}$ , market volatility $\sigma_{M,t}$ , and beta $\beta_{i,t}$ . Under a one-factor variance decomposition, the residual-volatility proxy is

\widehat{\sigma}_{\epsilon,i,t} = \sqrt{ \max \left[ \sigma_{i,t}^{2} - \left( \beta_{i,t}\sigma_{M,t} \right)^{2}, 0 \right] }.

The calculation removes the portion of total variance attributable to the contemporaneous market loading. It is deliberately narrower than an estimated multifactor residual because its role in the case panel is to distinguish changes that can be represented as market exposure from changes that remain after that exposure is removed. The external series show enough movement for the distinction to be economically relevant. Intel's one-year volatility rises from 31.38% in 2021 to 77.49% by September 2026, while its beta moves from 1.32 to 2.95. Caterpillar's volatility moves from 25.59% to 39.87% over the same endpoints, and Nike's volatility passes through 42.29% in 2022, 26.94% in 2023, 42.05% in 2025, and 36.21% in 2026. Broadcom, Boeing, and PepsiCo display different combinations of beta and total-volatility movement, which prevents a mechanical equivalence between changing beta and changing total risk.

SPY's one-year historical volatility is used for $\sigma_{M,t}$ . It changes from 13.10% in 2021 to 24.27% in 2022, falls to 13.18% in 2023 and 12.58% in 2024, rises to 19.48% in 2025, and is 13.00% in September 2026. These shifts are large enough that simply comparing company volatility before and after a network migration would confound the firm transition with the market volatility regime. Applying the decomposition gives, for example, an Intel residual-volatility estimate of 25.40% in 2022, 33.94% in 2023, 43.50% in 2024, 56.44% in 2025, and 67.33% in 2026. Boeing's corresponding residual estimate falls from 36.41% in 2022 to 23.93% in 2023, then rises to 33.17% in 2024. The object being tested is therefore a change in risk after market exposure has been accounted for, rather than raw volatility alone.

The timing convention is conservative. A migration observed from $t-1$ to $t$ is paired with the financial-risk change from $t$ to $t+1$ :

Y^{\beta}_{i,t+1} = \left| \beta_{i,t+1}-\beta_{i,t} \right|,

Y^{\epsilon}_{i,t+1} = \left| \widehat{\sigma}_{\epsilon,i,t+1} - \widehat{\sigma}_{\epsilon,i,t} \right|.

The primary explanatory quantity is $|M^d_{i,t}|$ . This timing prevents the same annual financial observation from defining both the network transition and the response. It also makes the hypothesis more demanding than a contemporaneous association. A firm can experience a large network reallocation without a subsequent loading change, in which case the observation works directly against the hypothesis. Because the theory does not identify whether the gaining exposure carries higher or lower aggregate covariance than the displaced exposure, the primary test concerns the magnitude of loading migration. A signed-beta specification becomes appropriate only when the factor loading of the entering or expanding counterparty can also be measured.

Mechanism

The relation between network migration and beta can be derived without assuming that centrality itself is a priced factor. Let the unexpected component of firm $i$ 's return be

u_{i,t} = a_{i,t}m_t + \sum_j \theta_{ij,t}\eta_{j,t} + \nu_{i,t},

where $m_t$ is the aggregate market shock, $\eta_{j,t}$ is an economically transmitted shock associated with counterparty $j$ , and $\theta_{ij,t}$ is the exposure created by the commercial relationship. The market beta implied by this representation is

\beta_{i,t} = a_{i,t} + \sum_j \theta_{ij,t} \frac{ \operatorname{Cov}(\eta_{j,t},m_t) }{ \operatorname{Var}(m_t) }.

A change in relationship weight therefore changes beta whenever the counterparty shock has nonzero covariance with the market. To a first-order approximation,

\Delta\beta_{i,t} \approx \sum_j \Delta\theta_{ij,t}\lambda_{j,t} + \sum_j \theta_{ij,t-1}\Delta\lambda_{j,t},

where

\lambda_{j,t} = \frac{ \operatorname{Cov}(\eta_{j,t},m_t) }{ \operatorname{Var}(m_t) }.

The first term is the migration channel examined here. The second represents changes in the systematic character of existing counterparties. A static-centrality regression mixes these channels because a high $\theta$ can persist while $\Delta\theta$ is close to zero. Economic migration isolates periods in which the firm's exposure map is actually changing.

The same expression also explains why direction can alter the observable financial outcome. An important supplier enters the customer's production function before revenue is realized. If dependence on that supplier expands, input shortages, capacity shocks, freight conditions, semiconductor cycles, commodity costs, or other production shocks can alter the customer's covariance with broad economic activity. Barrot and Sauvagnat's evidence that supplier shocks propagate more strongly when inputs are difficult to substitute gives an economic basis for this channel. Herskovic's production-network model likewise makes changes in network structure a source of systematic asset-pricing risk. A downstream customer shock enters through a different margin. When a supplier becomes increasingly dependent on one customer, customer-specific demand, procurement, inventory, or contract decisions can raise firm-specific cash-flow variance even when the customer's shock has modest covariance with the aggregate market.

This formulation differs from the centrality-level hypothesis studied in earlier work. Ahern's centrality argument links central industries with greater exposure to sectoral shocks that aggregate into market risk. The migration hypothesis instead concerns the derivative of that exposure with respect to network position. A company can remain economically central while its risk loading is stable, and a company starting from a peripheral position can undergo a large risk transition if one relationship rapidly becomes economically important. The empirical question is consequently whether $\Delta C$ , rather than $C$ , contains the stronger relation with $\Delta\beta$ or $\Delta\sigma_{\epsilon}$ .

Results

The upstream panel contains eight usable migration-to-outcome transitions from the Intel-SK Hynix and Caterpillar-Teijin histories. Across those observations, the correlation between absolute economic centrality migration and the absolute subsequent change in market beta is

\rho \left( |M^{up}_{i,t}|, |\Delta\beta_{i,t+1}| \right) = 0.553.

A univariate regression,

|\Delta\beta_{i,t+1}| = \alpha + \gamma |M^{up}_{i,t}| + e_{i,t+1},

produces $\gamma=2.788$ and $R^2=0.306$ . Since $M$ is expressed in standardized economic-centrality units, the coefficient should not be read as a structural elasticity. Its useful interpretation is comparative: within this small upstream panel, periods with greater reallocation in the relationship's economic importance are followed by larger changes in the firm's measured market loading. The corresponding correlation between upstream migration and subsequent residual-volatility change is only 0.294, with $R^2=0.087$ . The upstream evidence therefore aligns more closely with a changing systematic-loading channel than with a generic increase in unexplained volatility. The historical beta observations underlying these changes include Intel's move from 1.39 in 2023 to 2.15 in 2024, its decline to 1.54 in 2025, and its increase to 2.95 in 2026, together with Caterpillar's progression from 0.78 in 2022 to 1.22 in 2024, 1.13 in 2025, and 1.72 in 2026.

The downstream panel contains eleven usable transitions across Boeing-Korean Air, Nike-MAP Aktif, Broadcom-Samsung Electronics, and PepsiCo-Seven & i. Pooling those transitions against beta gives almost the opposite result. The correlation between absolute downstream migration and subsequent absolute beta change is $-0.238$ , corresponding to an $R^2$ of only 0.057. Large changes in customer-side economic position therefore do not appear to map consistently into subsequent market-beta movement in this panel. The residual-volatility result is materially different:

\rho \left( |M^{down}_{i,t}|, |\Delta\widehat{\sigma}_{\epsilon,i,t+1}| \right) = 0.583,

and

|\Delta\widehat{\sigma}_{\epsilon,i,t+1}| = 2.499 + 2.262 |M^{down}_{i,t}| + e_{i,t+1}

has $R^2=0.340$ . The magnitude is measured in annualized volatility percentage points. Boeing provides the largest downstream transition: its Korean Air centrality contribution rises by 1.897 standardized units between 2022 and 2023, followed by a 9.25 percentage-point increase in the market-adjusted residual-volatility estimate from 2023 to 2024. Nike's 1.264-unit increase from 2022 to 2023 precedes a 7.15-point increase in residual volatility from 2023 to 2024. These observations do not establish a population parameter, but they materially affect the form of the hypothesis: downstream migration appears more compatible with a change in non-market cash-flow risk than with a common shift in beta.

The level comparison separates migration from the familiar centrality result. For downstream relationships, absolute centrality level has a correlation of only $-0.142$ with the subsequent absolute residual-volatility change, giving $R^2=0.020$ , compared with $R^2=0.340$ for centrality migration. The information resides in movement rather than location in this part of the panel. The distinction is weaker upstream. Absolute upstream centrality level produces an $R^2$ of 0.278 against subsequent beta change, compared with 0.306 for migration. The upstream sample therefore does not support a strong claim that changes dominate levels. It supports the narrower proposition that migration contains information about beta instability that is at least comparable to, and slightly stronger than, the static economic position in these observations.

Binary centrality is less ambiguous. Every relationship used in the transition tests remains present across the adjacent observations that generate the large economic migrations. For such an edge,

\Delta b_{e,t}=0,

even when $\Delta w_{e,t}$ is large. The Boeing-Korean Air jump, Nike-MAP Aktif expansion and reversal, Intel-SK Hynix increase, and Caterpillar-Teijin contraction therefore produce no binary migration signal. The comparison establishes a measurement result rather than a forecasting victory: binary topology discards within-relationship state changes that are required to formulate the risk-migration hypothesis at all. A binary network could capture births and disappearances, but it cannot distinguish a persistent marginal relationship from one whose share of costs, revenue, or transaction value has changed sharply.

Components

Decomposing the composite weight identifies which economic margin carries the directional association. For the upstream observations, migration in Customer Cost % has the strongest association with subsequent beta change. The correlation between the absolute standardized change in Customer Cost % and subsequent absolute beta change is 0.707, with $R^2=0.499$ . Relationship Size follows at 0.652 and $R^2=0.425$ , while Supplier Revenue % is much weaker at 0.281 and $R^2=0.079$ . This ordering is economically coherent for a customer receiving inputs. Customer Cost % measures how much of the customer's cost structure is represented by the supplier relationship, so movement in that variable changes the fraction of production economics directly exposed to the upstream counterparty. Relationship Size captures scale, while Supplier Revenue % primarily describes how important the focal customer is to the supplier rather than how exposed the customer is to the supplier.

The downstream decomposition reverses part of that ordering. Relationship Size has the strongest association with subsequent residual-volatility movement, with a correlation of 0.629 and $R^2=0.395$ . Customer Cost % follows with a correlation of 0.522 and $R^2=0.272$ , while Supplier Revenue % produces 0.424 and $R^2=0.180$ . Because the focal company is the supplier in a downstream relationship, Supplier Revenue % is conceptually the direct customer-concentration measure. Its weaker result is therefore informative. The small panel does not behave as though downstream risk migration is merely a monotonic customer-revenue-share effect. Absolute economic scale contains at least as much information. One possible interpretation is that the cash-flow consequence of a changing customer relationship depends jointly on concentration and the absolute size of the activity exposed to customer-specific decisions. The observation argues against reducing downstream centrality to a single concentration percentage.

Failure

The strongest challenge to the migration result is concentration in a few large state changes. Removing the largest upstream migration, Intel's 2025 increase of 0.275 standardized units, reduces the upstream beta $R^2$ from 0.306 to 0.044. That observation is followed by Intel's beta change from 1.54 at the end of 2025 to 2.95 in September 2026. Removing Boeing's 1.897-unit 2023 migration from the downstream sample reduces the residual-volatility $R^2$ from 0.340 to 0.063. The linear relationship is consequently unstable under a leave-largest-transition test. A smooth specification in which every tenth of a standard unit of migration produces the same expected change in financial risk is difficult to defend from this panel.

The instability suggests a different model rather than a simple rejection of the economic mechanism. Production relationships often adjust incrementally for several periods and then undergo discrete reallocations when contracts, capacity, sourcing, customer demand, or strategic dependence changes materially. If financial exposures respond only after the commercial reallocation crosses an economically meaningful threshold, a linear regression will concentrate its apparent explanatory power in the largest transitions. The panel is consistent with such a threshold mechanism. Splitting observations by migration magnitude illustrates the pattern: the four larger upstream migrations are followed by an average absolute beta change of 0.63, compared with approximately 0.40 for the four smaller migrations. For downstream observations, the larger-migration group is followed by an average absolute residual-volatility change of 4.68 percentage points, compared with 2.53 points among the smaller migrations. Those differences are descriptive, and the sample is too small for a stable threshold estimate, but they specify the next falsifiable form of the hypothesis more precisely than the original linear relation.

A second failure condition concerns direction. If economic centrality migration were merely proxying for periods in which firms are generally unstable, upstream and downstream movements should have similar associations with beta and residual volatility. They do not. Upstream migration has the stronger beta relation and weak residual relation; downstream migration has the stronger residual relation and weak beta relation. The pooled centrality statistic would therefore discard information that appears economically relevant. This directional asymmetry is also difficult to reconcile with a generic "network activity" explanation, since such an explanation provides no reason for the financial outcome to switch from systematic loading upstream to residual volatility downstream.

Interpretation

The six relationships support a narrower claim than a conventional centrality premium. Economically weighted network movement can coincide with movement in the composition of firm risk, and the relevant financial quantity depends on the direction of the relationship. Upstream changes are more closely associated with market-beta migration in this panel. Downstream changes are more closely associated with market-adjusted residual-volatility migration. Static centrality performs poorly for the downstream residual-risk problem and only approximately as well as centrality change for the upstream beta problem. Binary centrality cannot represent the largest observed transitions because the commercial links persist while their economic weights change.

This distinction has consequences for conditional-risk estimation. A rolling beta model treats changing covariance as an object to be estimated from returns. A network-conditioned model can instead treat large economic migrations as candidate state transitions in the parameters of that return process. One specification is

\beta_{i,t} = \beta_{i,t-1} + \kappa_{up} M^{up}_{i,t-1} + \omega_{i,t},

with a separate variance equation

\log \sigma^{2}_{\epsilon,i,t} = \mu_i + \phi \log \sigma^{2}_{\epsilon,i,t-1} + \kappa_{down} |M^{down}_{i,t-1}| + \xi_{i,t}.

The case results suggest that imposing a common $\kappa$ on both directions would be poorly motivated. A larger panel could estimate these equations in a state-space system and allow the centrality variables to modify the transition equations for market beta, industry beta, and idiosyncratic variance. The economic state would then help identify when a factor loading is allowed to move rather than serving as another additive return predictor.

The result also changes the interpretation of network risk. A high centrality level describes where a firm sits in an economic network. Migration describes whether the firm's exposure structure is being rewritten. Those are different financial states. A persistently central company can have stable covariance parameters, while a company undergoing rapid relationship reweighting can experience a large change in beta or residual variance even if its final centrality percentile is unexceptional. For risk models, the second state is the one most likely to make a historical covariance estimate stale.

Conclusion

The evidence from the six-relationship case panel rejects the simplest version of the hypothesis that increasing economic centrality mechanically produces increasing market beta. The sign of beta movement depends on the systematic exposure carried by the relationships being gained or lost, and downstream centrality migration shows little relation with subsequent beta changes. The data instead support a directional risk-migration hypothesis. Changes in upstream economic position are associated with larger subsequent movements in market beta, while changes in downstream economic position are associated more strongly with movements in market-adjusted residual volatility.

Economic weighting is necessary for observing these transitions. Relationship Size, Supplier Revenue %, and Customer Cost % change substantially while the underlying links remain alive, leaving binary centrality unchanged. The metric decomposition further indicates that the financially relevant weight is direction dependent: Customer Cost % is the strongest upstream beta component in this panel, while Relationship Size is the strongest downstream residual-volatility component. Static centrality also fails to reproduce the downstream result, where migration explains substantially more variation in subsequent residual-risk movement than the level of economic centrality.

The leave-largest-transition tests prevent a stronger conclusion. Much of the linear explanatory power disappears when the largest upstream or downstream migration is removed. The empirical pattern is therefore better represented as a candidate state-transition effect concentrated around large reallocations than as a smooth centrality coefficient. That finding gives the research question a sharper econometric form. The relevant variable for conditional risk may be neither network centrality nor a generic network factor, but the occurrence and direction of sufficiently large changes in economically weighted network position. A larger historical panel can test whether such migrations identify breaks in market beta, industry beta, and residual covariance more efficiently than price histories alone.

References

Altsets supply chain dataset.

Methodology

Read the methodology for this research.

Cite this research

Altsets Research. "Conditional Beta and Residual Volatility with Supply Chain Data." Published September 30, 2026. https://www.altsets.com/research/conditional-beta-residual-volatility-supply-chain-data

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