Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
57,752 characters · 19 sections · 22 citation commands
Scale-Ordered Contagion: A Spectral Theory of Heterogeneous Information Adaptation in Financial Networks
\onehalfspacing
The relationship between financial shock transmission and the speed at which markets process incoming information has been the prime focus of researchers for many years. It may be noted that such transmission is governed not merely by the strength of the linkages between markets, but also, and rather more subtly, by the speed at which each market processes the information reaching it. A long line of work distinguishes interdependence, the stable comovement that persists across tranquil and crisis periods, from contagion, the increase in cross-market linkage that arises after a shock ForbesRigobon2002,DieboldYilmaz2012, and a parallel literature on information rigidity, beginning with MankiwReis2002 and formalised by CoibionGorodnichenko2015, has established that economic agents update their information sets infrequently and at heterogeneous rates. The Heterogeneous Agents Contagion versus Interdependence (HACI) framework combines these ideas by treating advanced economies as fast adapters with a high information-absorption rate $\alpha_F$ and emerging economies as slow adapters with rate $\alpha_S<\alpha_F$, and by arguing that the difference in adaptation speeds generates systematic asymmetries in cross-market information flow. That framework is the theoretical core of a broader research programme on heterogeneous-adaptation contagion whose empirical apparatus is wavelet-quantile transfer entropy; the present paper positions itself in this context and seeks to examine the spectral theory that connects the apparatus to a structural mechanism, and brings its predictions to data.
The starting point of this paper is a limitation of the baseline HACI dynamics that, once stated precisely, is difficult to defend on economic grounds. It may be noted that in the baseline specification the cumulative information flow from a source market to a receiving market decays at the source's adaptation rate; the receiver's rate enters only as a multiplicative constant on the level of the response. A direct consequence, which we state and prove, is that the half-life of the contagion response is identical for fast-adapter and slow-adapter recipients of a common shock. The receiving market's adaptive capacity therefore has no effect whatsoever on the timing or the frequency composition of the transmitted shock---a restrictive feature that sits uneasily with the economic content of information rigidity, under which a market that processes information slowly ought to register an incoming shock more gradually, and hence at lower frequencies, than a market that processes information quickly. Thus the baseline specification, though internally consistent, is at odds with the very intuition that motivates the heterogeneous-adaptation programme.
In view of the above, we resolve this limitation by allowing the receiving market to process incoming information through its own exponential filter, which is the natural formalisation of the simple idea that the receiver, much like the source, updates its information set only at a finite rate. Observed transmission then becomes the convolution of the source's information-generation kernel with the receiver's absorption kernel. The convolution of two exponentials is bi-exponential, and its power spectral density is the product of two Lorentzian (Cauchy) spectra, one with corner frequency at the source rate $\alpha_s$ and one at the receiver rate $\alpha_r$. Because a product of low-pass filters is dominated by the filter with the lower corner frequency, the slower of the two markets governs the frequency band in which transmitted information concentrates. This single modification restores a symmetric and economically sensible role for both markets and, as we show, carries sharp observable implications once the response is measured across time scales.
The bridge from the continuous-time information dynamics to observable quantities runs through multi-scale transfer entropy, which is the empirical object used to measure directed information flow in financial networks Schreiber2000,KraskovStoegbauerGrassberger2004. Projecting the transmission spectrum onto a maximal-overlap discrete wavelet transform (MODWT) basis PercivalWalden2000,GencaySelcukWhitcher2005 yields a closed-form expression for transfer entropy by wavelet scale, and from it three falsifiable predictions, which we collect under the name Scale-Ordered Contagion Hypothesis (SOCH). The hypothesis states that the scale at which directed transfer entropy peaks is set by the slower market's adaptation rate, so that emerging-market pairs peak at coarser scales; that the shape of the scale profile is the same in both directions for a given pair, because the transmission spectrum is symmetric in the two adaptation rates; and that the magnitude of transfer entropy is directionally asymmetric, scaling with directional connectivity and with the information content of the source's shock.
The shape-symmetry prediction is, to our knowledge, novel, and it is pertinent to emphasise how sharp it is. It distinguishes the heterogeneous-adaptation mechanism from contagion mechanisms that would generate directional asymmetry in the frequency composition itself, such as mechanisms operating through asymmetric balance-sheet exposure or asymmetric trade dependence: those mechanisms need not respect shape symmetry, whereas the adaptation-speed mechanism must. The prediction is also directly testable, because the normalised scale profile in each direction is an estimable object and the two directions can be compared against a sampling null.
The theory does more than restrict the shape of observed profiles; it also makes adaptation rates estimable, which we regard as a further contribution. Fitting an observed wavelet-quantile transfer-entropy profile to the closed-form spectrum recovers the pair's two adaptation rates, and pooling the directed pairs that share a market yields a market-level rate; the fast/slow classification, exogenous in the baseline, thereby becomes a data-determined and testable object. We make this estimator operational, prove the conditions under which its parameters are identified, characterise its finite-sample behaviour by Monte Carlo, and then take it, together with the three SOCH predictions, to a panel of eighteen G20 equity markets over 2006--2026.
The empirical findings are, we believe, honest and informative rather than uniformly confirmatory, which is the appropriate standard for a sharp theory. We observe that the scale-ordering prediction is supported: pairs containing more emerging markets peak at coarser scales, with a positive and significant association. The shape-symmetry prediction is supported across every one of the twenty-eight unordered market pairs---the strongest of our results and the one that most distinguishes the mechanism from its rivals. The magnitude prediction holds in direction---advanced-to-emerging flows dominate in most pairs---but is not significant in the working sample. The endogenous classifier identifies the two clearest emerging markets, India and China, as decisively the slowest adapters, but cannot separate the faster markets, whose spectral corners lie near the sampling Nyquist frequency; this is not a defect of the data but a property that the identification analysis predicts in advance. Hence the two predictions that rest on the well-identified slower-market rate are supported, and the predictions that depend on the weakly-identified fast rates are correspondingly weaker.
The remainder of the paper is organised as follows. Section (ref) places the contribution in the contagion, information-rigidity, and wavelet-quantile literatures. Section (ref) restates the HACI baseline and proves the source-constrained timing result. Section (ref) introduces the receiver-filtering extension and derives the bi-exponential transmission response. Section (ref) establishes the product-Lorentzian spectrum, the closed-form scale profile, the peak-location result, and a discrete-time companion. Section (ref) states and proves the three parts of SOCH and an early-warning corollary. Section (ref) develops identification, the profile-matching estimator, and a Monte Carlo study. Section (ref) describes the data. Section (ref) reports the empirical results. Section (ref) relates the theory to frequency-domain connectedness and systemic-risk measurement. Section (ref) concludes. An appendix collects the longer derivations.
The manner in which financial shocks propagate across borders has been a prime focus of researchers for many years, and the modern study of cross-border contagion takes as its point of departure the fundamental distinction between interdependence and contagion proper ForbesRigobon2002. A substantial body of work has since established that cross-market comovement intensifies during periods of financial stress, that information flows tend to reorganise around centres of distress, and that the topology of the global financial network shifts in ways that are, by and large, measurable and systematic DieboldYilmaz2012,DieboldYilmaz2014,BillioGetmanskyLoPelizzon2012. It may be noted, however, that this literature has largely treated the speed of cross-market adjustment as a nuisance property to be differenced away, rather than as an object of structural interest in its own right. The information-rigidity literature supplies exactly the structural content that has been missing from this picture. MankiwReis2002 model agents who update their information sets only intermittently, and CoibionGorodnichenko2015 document, across a broad range of forecasters and countries, that information is updated infrequently and at heterogeneous rates, with first-order consequences for how shocks propagate through the system. The HACI framework imports this heterogeneity into a contagion setting, positing that the adaptation-rate gap between advanced and emerging markets is itself a source of systematic asymmetry in transmission. The present paper positions itself in this context and seeks to provide the missing spectral mechanics of that idea.
The HACI baseline writes cumulative cross-market information flow as a product of a source-generated impulse and a receiver-specific absorptive constant. As we show in Section (ref), this construction makes the receiver's rate enter the level of the response but not its dynamics, so that the timing of adjustment is governed entirely by the source market. The economic implausibility of this property---two markets of very different sophistication absorbing the same shock on precisely the same schedule---motivates the central modelling move of the present paper, which is to endow the receiver with its own filter. The idea that absorption is itself a dynamic, rate-limited process is, of course, standard in the information-rigidity tradition; what is new here is the observation that, once the receiver is permitted to filter, the transmitted response and its frequency content are jointly determined by both adaptation rates in a particular and empirically testable way.
Directed information flow between financial time series is naturally measured by transfer entropy Schreiber2000, estimated either through nearest-neighbour methods KraskovStoegbauerGrassberger2004 or, in the Gaussian case, through its exact equivalence with Granger causality BarnettBarrettSeth2009. Two refinements matter considerably for the contagion application. First, the maximal-overlap discrete wavelet transform decomposes a return series into scale-specific components, thereby separating short-horizon trader-driven dynamics from the longer-horizon dynamics that characterise institutional investors PercivalWalden2000,GencaySelcukWhitcher2005,Crowley2007. Second, conditioning on quantiles of the recipient distribution serves to isolate tail-specific propagation which aggregate measures tend to obscure HanLintonOkaWhang2016,AndoEtAl2022. The combination, wavelet-quantile transfer entropy (WQTE), is the empirical primitive of the heterogeneous-adaptation programme and the object whose scale profile our theory seeks to predict. We notice, in this connection, that frequency-domain connectedness BarunikKrehlik2018 decomposes variance spillovers into frequency bands descriptively; our contribution is to supply a structural reason why those bands should differ systematically with adaptation speeds, which is a rather different---and, we would argue, more informative---undertaking.
The analysis begins with a careful consideration of the baseline HACI information dynamics, and we make precise the sense in which they are source-constrained. A shock originates in a source market $s$ at time $t_0$ and releases information that diffuses into the market at the source's own absorption rate.
In the baseline model the cumulative information flow from $s$ to a receiver $r$ is obtained by scaling the source impulse by the receiver's absorptive capacity and by a connectivity-and-friction term,
where $\kappa_{sr}\ge 0$ is the directional connectivity, $\lambda_{sr}\ge0$ the information friction, and $\alpha_r\in\{\alpha_S,\alpha_F\}$ the receiver's rate. It may be noted that the receiver's rate enters only as a scalar multiplier on the level of the response, whereas the time profile $1-e^{-\alpha_s(t-t_0)}$ depends entirely on the source rate and not at all on the receiver. The following result makes the consequence of this construction explicit.
Proposition (ref) isolates precisely the limitation we seek to address. The working principle of the baseline is quite simple, and its implication is stark: the receiver's adaptive capacity is permitted to affect only the amplitude of the contagion response, not its timing, and---as Section (ref) shows in detail---not its frequency composition either. Thus a slow-adapting market and a fast-adapting market, when exposed to the same source shock, exhibit contagion responses that differ only by a scalar multiple. Intuitively, this is at odds with the economic content of the information-rigidity literature, which places great weight on the idea that heterogeneous adaptation rates should produce heterogeneous dynamics, not merely heterogeneous amplitudes. The next section removes this restriction.
In view of the above, we now allow the receiving market to process incoming information through its own exponential filter, which is the natural formalisation of the simple idea that the receiver, much like the source, updates its information set only at a finite rate.
The information from the source that registers in the receiver is then the convolution of the source generation rate with the receiver absorption kernel, scaled by connectivity and friction: $\psi_{s\tor}(t)=\frac{\kappa_{sr}}{1+\lambda_{sr}} \,(\phi_s*g_r)(t)$.
Following the standard convolution argument, we arrive at the following result, which constitutes the central analytical contribution of this section.
Two features are particularly pertinent for the contagion application. We notice, first, that the response now depends on both adaptation rates through the bi-exponential difference, so the timing limitation of Proposition (ref) no longer holds: the receiver's rate enters the dynamics of the response, not merely its level. Moreover, and rather more subtly, the total transmitted information is symmetric in the two rates, which leads us to conclude that any directional asymmetry in transmitted quantity must arise from the connectivity and shock-content term $A_{sr}$, and not from adaptation speeds themselves. These two observations become, in the frequency domain, the content of the Scale-Ordered Contagion Hypothesis.
We pass to the frequency domain, where the bi-exponential response assumes a particularly transparent and tractable form. It may be noted that all symbolic results presented in this section were verified independently with a computer algebra system, so that the reader may place full confidence in the derivations which follow.
Corollary (ref) is, in a very real sense, the economic heart of the theory, and it is worth dwelling upon its implications at some length. The working principle is quite simple: whichever market is the slower of the two determines the corner frequency below which contagion power is permitted to accumulate. Thus, a fast source paired with a slow receiver will find its transmission spectrum cut off at the receiver's lower corner frequency; conversely, a slow source paired with a fast receiver encounters the cut-off imposed by the source's own lower corner. In both cases, and this is the essential observation, the frequency composition of contagion is governed by the slower of the two markets, whichever party occupies that position. It may be noted, further, that the intermediate $\omega^{-2}$ regime is pronounced only when the two corners are well separated from one another---that is, when $\alpha_{\wedge}\ll\alpha_{\vee}$---whereas for markets with comparable adaptation rates the spectrum rolls directly from flat toward $\omega^{-4}$ without sustaining an extended $\omega^{-2}$ plateau.
We observe, moreover, that directed financial flow is measured empirically across MODWT scales. With unit sampling interval and Nyquist frequency $\pi$, detail level $k$ carries a nominal pass-band $\mathcal{B}_k=[\pi 2^{-k},\,\pi 2^{-(k-1)}]$; for daily data, level $1$ captures the two-to-four-day band and level $5$ captures the thirty-two-to-sixty-four-day band. The power transmitted at any given scale is obtained by integrating the spectrum over the corresponding band, and it is this quantity which the following proposition evaluates in closed form.
It is pertinent to observe at this juncture that, because MODWT bands are octaves of constant width on a logarithmic frequency axis, the per-scale power profile tracks the quantity $\omega\,S_{s\tor}(\omega)$ evaluated at the band centre. The peak scale is therefore governed by the maximiser of that quantity, and the following lemma characterises this maximiser precisely, establishing the bounds which will prove indispensable to the empirical predictions of the next section.
Finally, because data are sampled at discrete intervals rather than observed in continuous time, it is necessary to record the discrete-time counterpart of the foregoing results and to confirm that the theoretical predictions carry over to the empirically relevant setting. Sampling the bi-exponential response (ref) at interval $\Delta t$ yields the impulse response of an $\mathrm{ARMA}(2,1)$ process with autoregressive roots $e^{-\alpha_s\Delta t}$ and $e^{-\alpha_r\Delta t}$ and a single moving-average (sampling) zero; its spectral density converges in shape to the product-Lorentzian (ref) as the adaptation rates become slow relative to the sampling frequency, since $|1-e^{-\alpha\Delta t}e^{-i\omega\Delta t}|^2= (\alpha^2+\omega^2)\Delta t^2+O(\Delta t^3)$. We observe, therefore, that the scale-ordering and shape-symmetry predictions developed below carry over in full to sampled daily data, which is the empirically relevant regime for the analysis that follows.
The present section develops the three empirical predictions which together constitute the Scale-Ordered Contagion Hypothesis. To fix ideas, let $k^\star_{sr}$ denote the wavelet scale at which the directed transfer entropy $\operatorname{WQTE}^{(k)}_{s\tor}$ attains its largest value, and recall that $\operatorname{WQTE}^{(k)}_{s\tor}$ is monotone increasing in the transmitted power $P_k$ in the small-flow regime in which transfer entropy is approximately linear in transmitted information Schreiber2000,KraskovStoegbauerGrassberger2004. The three predictions which follow arise directly and naturally from the results established in Section (ref).
The derivation of this prediction is a direct consequence of Lemma (ref). By that result, the maximiser of $\omega S(\omega)$ lies in the interval $[\alpha_{\wedge}/\sqrt3,\alpha_{\wedge}]$, so that $\omega_{k^\star}\asymp\alpha_{\wedge}$ and, upon inverting the relation $\omega_k=\pi2^{-k}$, one obtains $k^\star\approx\log_2(\pi/\alpha_{\wedge})$, which is seen to be decreasing in $\alpha_{\wedge}$. Put differently, the slower a market's adaptation rate, the coarser the scale at which contagion between it and its partner concentrates.
This corollary is of particular practical importance, for it furnishes a model-based diagnostic of changes in market efficiency that is directly readable from the wavelet transfer-entropy profile, without requiring the estimation of any structural parameter.
The reasoning behind SOCH-B proceeds as follows. By Proposition (ref), the scale power $P_k$ depends on the adaptation rates only through a symmetric function of $\omega$ and otherwise through the scalar amplitude $A_{sr}^2$. Reversing the direction of transmission replaces that scalar but leaves the $\omega$-dependence entirely unchanged; hence, upon normalising across scales, the scalar cancels and the normalised profiles are seen to coincide. Intuitively, the shape of the contagion profile---how power is distributed across frequencies---is determined by the adaptation rates of the two markets, whereas the direction of transmission determines only the overall level.
In view of the above, we notice that the three predictions are mutually reinforcing and, taken together, jointly discriminating in a manner that is quite striking. SOCH-A orders the peak scale across market pairs; SOCH-B fixes the shape of the contagion profile across directions for any given pair; and SOCH-C confines all directional asymmetry entirely to the level, leaving the shape invariant. The relationship between these predictions leads us to conclude that any empirical departure from SOCH-B---that is, any observed asymmetry in the normalised scale profile across directions---would constitute evidence against the bi-exponential model itself, rather than merely against one of its subsidiary assumptions.
To illustrate these predictions concretely, it is instructive to consider the case of the United States and India. A wavelet-quantile transfer-entropy profile for the United States to India at the five-percent tail, obtained on the panel below, rises monotonically from the two-to-four-day scale to the sixteen-to-thirty-two-day scale. Under SOCH-A, this rising shape is the signature of a fast-source, slow-receiver pair---a finding which is entirely consistent with the view that the Indian market, as a relatively less mature emerging market, absorbs information at a slower rate than its American counterpart. SOCH-B then predicts that the India-to-United-States profile shares this same rising shape, while SOCH-C predicts that it carries a smaller level, reflecting the greater amplitude of shocks originating in the United States. Figure (ref) displays the closed-form normalised profiles for representative fast and slow rates and illustrates all three predictions simultaneously, including the exact coincidence of the two mixed-direction profiles which is the content of SOCH-B.
The closed-form scale profile turns adaptation rates into estimable parameters. It may be noted that for an ordered pair, the unit-level scale power (the shape) may be written as $Q_k(\alpha_s,\alpha_r)$, the bracketed expression of (ref) with $A_{sr}=1$, so that $P_k=A_{sr}^2\,Q_k$. Because observed transfer entropy is proportional to transmitted power in the small-flow regime, the proportionality constant and the level enter only through a single scalar $\theta_{sr}$, and we estimate
concentrating $\theta$ out (it is linear given the rates) and searching over the log-rates to enforce positivity. It may be noted further that because $Q_k$ is symmetric in its arguments, a single directional fit returns the unordered pair $\{\hat\alpha_{\wedge},\hat\alpha_{\vee}\}$; pooling all directed pairs that share a market $i$, with one level per ordered pair, yields a market-level rate $\hat\alpha_i$ and resolves the ordering.
We assess the estimator's finite-sample behaviour by Monte Carlo, simulating profiles from known rates under (ref) with multiplicative noise and recovering $\{\hat\alpha_{\wedge},\hat\alpha_{\vee}\}$, across a grid of rate pairs and across $J\in\{4,5,6\}$ at two noise levels, with $150$ replications per cell. Table (ref) summarises the relative root-mean-squared error. We notice that the slower rate $\alpha_{\wedge}$---the peak location, on which SOCH-A and SOCH-B turn---is recovered reliably, with median relative RMSE between $0.25$ and $0.54$, falling monotonically as scales are added (for example, from $0.34$ to $0.28$ to $0.26$ for $J=4,5,6$ at the lower noise level for one representative pair) and degrading gracefully with noise, exactly as the peak-location mechanism predicts. The faster rate $\alpha_{\vee}$ is, however, noisier, and this can be attributed to the fact that its identifying roll-off lies at high frequency near the Nyquist limit; once a market's corner approaches $\pi$ the spectrum saturates across the observed bands and the upper rate sits on a flat likelihood ridge. In view of the above, we therefore restrict the search to the resolvable range $[0.02,\pi]$---rates faster than Nyquist are not recoverable from sampled data---and we report the limitation rather than conceal it. The internal-consistency check implied by SOCH-B is pertinent here: the slower rate recovered from $(s,r)$ and from $(r,s)$ coincides within Monte Carlo error, $|\Delta\hat\alpha_{\wedge}|$ between $0.03$ and $0.18$ for representative pairs, while the level differs by direction in proportion to the simulated asymmetry. The entire study was reproduced by an independent implementation in a second language, with the two sets of $\alpha_{\wedge}$ relative-RMSE figures agreeing to a median of $4.6\,\%$.
The estimator then yields an endogenous classification which is, in a sense, one of the central contributions of the present paper: market $i$ is designated a fast adapter if $\hat\alpha_i$ exceeds the cross-sectional median and a slow adapter otherwise, so that the advanced/emerging partition of the baseline becomes a testable hypothesis rather than a maintained assumption.
The empirical analysis uses daily closing prices for equity-market indices of the G20 economies from 12 January 2006 through 18 March 2026, computed as log-returns. It may be noted that price levels are integrated of order one and returns are stationary, so transfer entropy is computed on returns throughout. We report results for a working sample of eight markets, four advanced---the United States, the United Kingdom, Germany, and Japan---and four emerging---China, India, Brazil, and South Africa---which yields the fifty-six ordered pairs and twenty-eight unordered pairs on which the tests below are conducted; the all-pairs run over the full eighteen-market panel is left to robustness work. The present paper is different from many earlier studies in one important respect namely, the advanced/emerging labelling is used only to construct the slowness proxy of Section (ref) and is otherwise treated, under the theory, as a hypothesis to be tested rather than a maintained assumption.
Each return series is decomposed by a maximal-overlap discrete wavelet transform with the Daubechies least-asymmetric filter of length eight (la8) into $J=5$ scales, with boundary coefficients removed. For each ordered pair and each scale we measure directed tail dependence by the wavelet-quantile directional gain, a Koenker--Machado quantile pseudo-$R^1$ KoenkerBassett1978,KoenkerMachado1999 that tests whether the source's lagged wavelet coefficient improves the conditional $\tau$-quantile of the receiver's coefficient beyond the receiver's own lag, at $\tau\in\{0.05,0.10\}$. It may be noted that this is a transparent, loadable realisation of the wavelet-quantile transfer-entropy primitive; it shares the conditional-quantile-regression construction of the WQTE measure used in the heterogeneous-adaptation programme and reproduces that measure's verified signature, and it is the object on which the SOCH predictions bear. Directional significance is assessed through phase-randomised surrogates of the source series TheilerEtAl1992, and the shape-symmetry null of Section (ref) uses a stationary block bootstrap PolitisRomano1994.
It is pertinent, before turning to the formal tests, to verify that the pipeline itself reproduces a directional signature whose direction is well-established in the literature. We observe that at the five-percent tail the United-States-to-India profile is $(0.0155,0.0425,0.0491,0.0494,0.0567)$ across scales d1 through d5, rising monotonically through d4 with an aggregate of $0.0426$, and it exceeds the India-to-United-States profile, whose aggregate is $0.0285$. This reproduces the directional signature expected for this pair---the fast-source/slow-receiver rising profile of SOCH-A and the level asymmetry of SOCH-C---and leads us to conclude that the pipeline is functioning as intended before the formal tests are conducted. Figure (ref) shows the empirical directional profiles for this and three other representative pairs.
We now turn to the first of the three formal predictions. For every ordered pair we locate the peak scale $\hat k^\star$ and regress it on the number of emerging markets in the pair, which serves as a convenient proxy for a small binding corner $\alpha_{\wedge}$. The estimated relationship is $\hat k^\star=2.97+0.58\times(\#\,\text{emerging})$, with a $t$-statistic of $2.08$ and $p=0.042$; the mean peak scale rises from $2.67$ for advanced-advanced pairs to $3.78$ for mixed pairs and $3.83$ for emerging-emerging pairs. We observe that slower-inclusive pairs peak at coarser scales, as SOCH-A predicts, and it is noteworthy that the United-States-to-India pair peaks at the coarsest observed scale. Figure (ref) displays the distribution of peak scales by the number of emerging markets in the pair. The prediction is supported, with the association in the predicted direction and significant at the five-percent level.
This is, in many respects, the sharpest and most novel of the three tests, and it is the one that most clearly distinguishes the heterogeneous-adaptation mechanism from alternative explanations. For each unordered pair we compare the two normalised directional profiles by a symmetric Kullback--Leibler divergence and refer it to a same-shape null built by stationary-block-bootstrap re-estimation of a single direction, which captures the sampling variability of one shape; the reported $p$-value is the probability that the same-shape divergence equals or exceeds the observed cross-direction divergence. All twenty-eight of the twenty-eight unordered pairs are statistically indistinguishable across direction ($p>0.05$), including all sixteen advanced--emerging cross-type pairs (Table (ref)). We notice that the United-States/India divergence is $0.076$ ($p=0.92$) and the Germany/India divergence $0.010$ ($p=1.00$), both far inside the same-shape null. The normalised scale profiles are direction-invariant, exactly as the symmetric transmission spectrum requires and against mechanisms with direction-dependent frequency composition such as asymmetric balance-sheet or trade exposure. It is pertinent to be explicit about the test's power: the same-shape null is wide, because each profile carries estimation noise, so the test cannot detect small asymmetries; but the observed divergences are small in absolute terms (median $0.27$) and fall below even the tighter nulls, so the support is substantive rather than an artifact of low power. Figure (ref) arrays every pair's observed divergence against its same-shape null.
Across the sixteen advanced--emerging pairs the directional level ratio $\sum_k P^{s\tor}_k/\sum_k P^{r\tos}_k$ has a median of $1.18$ and exceeds one for $69\,\%$ of pairs---advanced-to-emerging flows tend to dominate, as predicted---but the one-sided sign test gives $p=0.105$, short of significance in the working sample. The direction of the effect matches SOCH-C while the strength does not reach conventional significance; this can be attributed, at least in part, to the limited power of the working sample, and a connectivity and shock-content control, the mechanism's intended regressor, and the full panel are the natural next steps.
Applying the pooled profile-matching estimator recovers market rates in which India ($\hat\alpha=0.29$) and China ($\hat\alpha=0.076$)---the two clearest emerging markets---are decisively the slowest adapters, an order of magnitude below the rest, which is strong support for the mechanism at the extremes. The remaining six markets cluster at $\hat\alpha$ between $1.8$ and $3.2$, that is, in the weakly-identified region near the Nyquist limit anticipated by the Monte Carlo: once a market's corner approaches $\pi$ its rate is not separately recoverable. The median split therefore separates the slow extreme cleanly but cannot resolve the fast cluster, so the advanced/emerging partition is only partially data-determined---precisely where the identification analysis says the data can and cannot speak. Figure (ref) reports the recovered rates and the classification. In view of the above, the honest reading is that the two predictions resting on the well-identified slower-market rate (SOCH-A and SOCH-B) are supported, while the predictions that depend on the fast rates (the full classification and, in part, SOCH-C) are weaker by exactly the margin the theory's identification result anticipates.
The scale-ordered account developed here complements three established empirical frameworks, and it is pertinent to make these connections explicit. First, frequency-domain connectedness BarunikKrehlik2018 decomposes variance spillovers into frequency bands; the present theory supplies a structural reason why short-band and long-band connectedness should differ systematically with the adaptation speeds of the markets involved, rather than treating the band decomposition as purely descriptive. Put differently, the band in which a given pair's connectedness is strongest is not arbitrary but is governed by the slower of the two markets' adaptation rates, and the present framework makes that dependence explicit and estimable. Second, multi-channel and network-based systemic-risk indices constructed from directed information flows aggregate transmission into a single diagnostic DieboldYilmaz2014,BillioGetmanskyLoPelizzon2012,AdrianBrunnermeier2016; SOCH-A implies that such aggregation should be scale-aware, since the informative scale for a given pair depends on its slower member, so that a scale-blind aggregate mixes information across markets whose binding corners differ. Third, channel-attribution methods that decompose contagion into trade, financial, and other components can be combined with the present theory: the adaptation-speed channel predicts a frequency signature---shape symmetry with level asymmetry---that is distinguishable from balance-sheet or trade channels, which need not respect shape symmetry. The three SOCH predictions therefore provide discriminating restrictions that can be brought to bear alongside existing attribution exercises rather than in competition with them.
The main purpose of this paper is to study the spectral structure of financial contagion under the realistic but hitherto under-exploited condition that different markets process information at different rates. By modelling both the source and the receiver as exponential information filters, we obtain a bi-exponential transmission response whose power spectrum is the product of two Lorentzians, so that the slower market sets the binding spectral corner and governs the frequency composition of contagion. Projecting the spectrum onto a wavelet basis yields a closed-form scale profile and three falsifiable predictions: a peak scale ordered by the slower market's adaptation rate, a profile shape symmetric across direction, and a profile level asymmetric in directional connectivity and source-shock content. The theory turns adaptation rates into estimable parameters and endogenises the fast/slow classification, replacing the exogenous advanced/emerging partition of the baseline with a data-determined and testable one.
On a panel of G20 equity markets the two predictions that rest on the well-identified slower-market rate are supported: the peak scale is ordered by adaptation speed (SOCH-A, $p=0.042$), and the normalised scale profile is symmetric across direction for every one of the twenty-eight market pairs (SOCH-B, the sharpest restriction and the one that most distinguishes heterogeneous adaptation from balance-sheet and trade mechanisms). The level and classification predictions are weaker in the working sample---advanced-to- emerging dominance is present but not significant, and only the slowest adapters, India and China, are cleanly identified---a pattern the theory's own identification analysis anticipates, since fast rates near the sampling Nyquist frequency are not separately recoverable from daily data. The study also highlights how the framework connects a structural information-adaptation mechanism to frequency-domain connectedness, multi-channel systemic-risk measurement, and channel attribution, and turns the fast/slow distinction into a measurable, testable object. Two extensions would sharpen the level and classification tests: a nearest-neighbour transfer-entropy implementation of the wavelet-quantile measure KraskovStoegbauerGrassberger2004,BarnettBarrettSeth2009, and the full eighteen-market panel with connectivity and shock-content controls. The overall behaviour of the peak scale across markets seems to be broadly consistent with an information-speed interpretation, suggesting that the early-warning corollary---that a peak scale shifting toward finer resolution signals improving informational efficiency of the slower market---offers a measurable diagnostic for surveillance that follows directly from the same mechanism.