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Local Gaussian copula inference with structural breaks: testing dependence predictability
\affil[a]{ {\it Erasmus University Rotterdam & Tinbergen Institute, The Netherlands}} \affil[b]{ {\it Keio University, Japan}} \affil[c]{ {\it University of Cologne, Germany}} \thispagestyle{empty}
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Understanding the dynamics of dependence between economic and financial time series is crucial for assessing systemic risk, market connectedness, and the role of common factors (e.g. DieboldYilmaz2014Connectedness, BrownleesEngle2017SRISK). In many applications, researchers are not only interested in the marginal dynamics of time series but also in how their joint distribution evolves over time and whether this evolution is affected by observable state variables (see, e.g., patton:06,cappiello2006asymmetric,adrian:16).
For example, in our empirical application to systemic risk, we ask whether lagged market downturns predict state-dependent quantile dependence between a financial institution's return and the contemporaneous market return, with particular interest in whether market downturns amplify joint downside risk (see also han:2016). A large empirical literature documents that dependence among equity returns strengthens during market stress and is often asymmetric, with different comovements in downturns and in upturns (see, e.g., longin2001extreme,ang2002asymmetric,forbes2002no). Several econometric contributions develop procedures to test whether dependence is constant over time or to detect structural changes in dependence (e.g., wied:2012,buecher:14,stark:22,bomw:24). Our focus is complementary: rather than characterizing dependence through global high/low regimes, we ask whether negative market states predict changes in dependence in particular parts of the joint distribution and during particular time periods.
For illustration, Figure (ref) shows three scatter plots of the empirical ranks of standardized returns (upon filtering out conditional mean and variance dynamics) for major U.S.\ financial institutions, $r_t$, and the (CRSP value-weighted) market return, $r_{m,t}$, based on daily data from January 2000 to December 2019. Similar to han:2016, we consider three financial institutions: JP Morgan ({\sf JPM}), Morgan Stanley ({\sf MS}), and {\sf AIG} that represent Depositories, Broker-Dealers, and Insurers, respectively. The plots are stratified by the sign of the lagged market return $r_{m,t-1}$, distinguishing days following a negative versus a positive market move. Comovement appears stronger following negative market returns \(r_{m,t-1}<0\), suggesting that market stress may increase dependence between \(r_t\) and \(r_{m,t}\). This is confirmed by a higher Spearman’s rho in the down market regime than in the up market regime and also (lower panel) quantile dependence (patton2013copula) appears to be higher following a bearish market. Put differently, Figure (ref) points to two questions that correspond directly to our testing framework: ($i$) whether lagged market states have predictive content for dependence, and ($ii$) whether any such predictive effect is stable over time or instead subject to structural change. Motivated by this pattern, the goal of the present paper is to develop formal tools to test whether such state dependence is statistically significant and to locate where in the joint distribution it occurs (e.g., in lower versus upper quantiles). In addition, we allow the dependence response to vary over time and investigate whether it is stable or exhibits structural change.
The problem studied in this paper is closely related to the Granger-causality literature, which has largely focused on the conditional mean or selected quantiles and therefore does not capture predictability in the full distributional dependence. This has motivated a growing literature on Granger causality in quantiles, including troster:18, songetal:21, and mwt:25, as well as copula-based methods such as Bouezmarni:2012, fuentes:2025, lee:2014, and jang:2024. Moreover, there is a literature on how regressors influence copulas, including, among others, factor copulas mw:2023, vine copulas jobst:2024, and parametric copulas acar:13,gijbels:2017,gijbels:2021. We contribute to this line of work by testing whether a predetermined state vector carries predictive content for the conditional copula, after filtering the marginal dynamics. In this sense, our approach can be interpreted as a test for Granger-type predictability of dependence.
Our test is semiparametric and accommodates flexible AR-GARCH type dynamics in the marginal distributions, thereby allowing for empirically realistic features such as volatility clustering. Building on modern distribution regression (DR) methods (see, e.g. foresiperacchi:1995,cher:13,rw:13,wang:2023,wied:2024,spady:2025), we employ a local Gaussian representation (see, e.g. cherno24), which enables us to remain fully agnostic about the true copula family. This strategy allows the dependence structure to be modelled locally at each individual point $(u,v)\in[0,1]^2$, rather than via global parametric restrictions (as, e.g., acar:13,gijbels:2017,gijbels:2021 do).
A particular emphasis of our framework is on quantile dependence (see, e.g., patton:06,patton:12,patton2013copula,patton:13,oh2017modeling). By examining the conditional copula along the main diagonal, we can study how co-movements, especially in the tails, vary with the state of the market. This perspective allows us, for example, to address whether the dependence between two return series changes with market conditions and whether such changes differ across quantiles and/or across time periods. Related ideas appear in, for instance, patton:06, who examines time-varying quantile dependence in copula models. He shows that allowing copula parameters to vary over time is essential for capturing the asymmetric and state-dependent dependence observed in exchange rate returns, particularly the stronger lower-tail dependence during turbulent periods. His empirical results demonstrate that conditional copulas can reveal substantial time variation in co-movement that would remain hidden under static dependence models, motivating the need for flexible, state-dependent modelling frameworks like the one developed in this paper.
Our procedure is designed to be robust to structural breaks in the joint dependence structure in the spirit of sowell:96 and rossi:2005. Using residuals from univariate AR-GARCH models removes marginal conditional mean/variance dynamics. A major contribution of this paper is thus a theory for time-series DR with estimated marginal processes. The cornerstone is a functional central limit theorem for a marked sequential empirical copula process with estimated pseudo-observations based on bracketing arguments from andrews1994introduction and, in particular, the extensions to sequential processes by mohr2020weak and scholze2024weak. This result connects recent distributional regression ideas to the empirical copula process literature (e.g., buecher:14, bucher2016dependent, nasri2022change) and provides the basis for our break-robust testing framework. Finally, we suggest a moving block bootstrap suitable for multi-stage estimators gonccalves2023.
The remainder of this paper is organized as follows: Section (ref) introduces the model and the testing problem. The test statistics and the bootstrap are described in Section (ref), the asymptotic properties of which are derived in Section (ref). Finite sample evidence in the form of a Monte Carlo experiment and an estimation exercise using financial data are presented in Sections (ref) and (ref), respectively, before Section (ref) concludes. All proofs are delegated to the appendix.
Notation. For any random variable $X$ in $L^p$, let $\|X\|_p \coloneqq \textnormal{\textsf{E}}[|X|^p]^{1/p}$; for any vector $x = (x_1,\dots,x_m)^\top$, let $|x|=\sqrt{x^\top x}$ and $|x|_\infty = \operatorname*{\textnormal{\textsf{max}}}\{|x_1|,\dots,|x_m|\}$; for $u = (u_1,u_2) \in [0,1]^2$, let $u^{(1)} \coloneqq (u_1,1)$ and $u^{(2)} \coloneqq (1,u_2)$.
Consider the following location-scale specification
where $\mu_{j,t}$ and $\sigma_{j,t}$ are known parametric functions up to $\lambda = \lambda_0$ that are ${\cal F}_{t-1}$-measurable. The innovations $\varepsilon_{j,t}$ are {\it individually} independent of ${\cal F}_{t-1}$ for each $j \in \{1,2\}$, while $\varepsilon_t = (\varepsilon_{1,t},\varepsilon_{2,t})^\top$ might depend {\it jointly} on ${\cal F}_{t-1}$. This framework rules out time-varying parameters beyond the chosen parametric marginal model, but incorporates the important class of ARMA-GARCH models, similarly to what e.g. chen2006,chen2006estimation or patton:13, oh2017modeling consider.
We assume that the marginal cdfs $Y_{j,t} \mid {\cal F}_{t-1} \sim {\sf H}_{j,t}$ are continuous. Following patton:06, we can then decompose the conditional joint distribution $Y_t \mid {\cal F}_{t-1} \sim {\sf H}_t$, $Y_t \coloneqq (Y_{1,t},Y_{2,t})^\top$, in terms of a unique copula ${\sf C}_t$, i.e. \[ {\sf H}_t(x_1,x_2) = {\sf C}_t({\sf H}_{1,t}(x_1),{\sf H}_{2,t}(x_2)). \] Due to the structure of Eq. (ref), it is known that we can express the preceding copula in terms of the innovation ranks:
with
Moreover, upon filtering out the effect of ${\cal F}_{t-1}$ on $Y_t$, assume furthermore that for a $k \times 1$ ${\cal F}_{t-1}$-measurable vector $Z_t \coloneqq (Z_{1,t},\dots,Z_{k,t})^\top$, the remaining dependence-relevant information in ${\cal F}_{t-1}$ is summarized by $Z_t$:
Assumption (ref) is a maintained assumption that defines the information set with respect to which we assess predictability of the conditional copula. Although restrictive, Assumption (ref) is not uncommon in the literature and less restrictive than, for example, assuming that the copula is invariant to the conditioning information, i.e., ${\sf C}_t(\cdot\mid{\cal F}_{t-1})={\sf C}(\cdot)$ (see, e.g., patton:13, oh2017modeling). In contrast, following mw:2023 we allow dependence dynamics through a predetermined state vector $Z_t$. Extending our limit theory beyond such a state reduction would be technically non-trivial as discussed below. Therefore, dependence non-predictability in copulas corresponds to whether the copula ${\sf C}_t(u \mid Z_t)$ varies with $Z_t$; i.e. we test whether $Z_t$ has predictive content for the conditional dependence between $Y_{1,t}$ and $Y_{2,t}$. In doing so, we also want to be robust to temporal instabilities.
Our approach is based on the following (conditional) Gaussian representation (see kolev2006copulas and cherno24 as well as the references therein): Let $\Phi_2(y_1,y_2)$ be the bivariate normal cdf. Then, for each $(u,z)$, there exists a function $\rho_t(u;z) \in [-1,1]$ such that
Note that the local Gaussian representation is a pointwise reparametrization and thus not a parametric assumption. However, in order to make (ref) operational, we impose a parametric structure on the local dependence parameter $\rho_t(u;z)$ using the Fisher transform:
For a subset of interest ${\cal U} \subset (0,1)^2$, the null hypothesis $H_{0} = H_{0,1} \cap H_{0,2}$ can thus be rephrased as
Under $H_0$, the restricted population parameter obtains as $\theta_0 = (\alpha_0(u),0)^\top$ so that, in view of Eq. (ref), we get $$\varrho(u,\alpha_0) \coloneqq \varrho(u,\theta_0;z) ,\; C(u, \alpha) \coloneqq C(u; \varrho(u,\alpha)), \,\text{ and }\; C(u, \alpha_0) = {\sf C}_t(u \mid z) = {\sf C}(u).$$ The testing problem is thus similar in nature to sowell:96 and rossi:2005 (see also mwt:25).
Since the population ranks $U_t = (U_{1,t},U_{2,t})^\top$ are usually unobserved, we will replace them with sequentially computed sample counterparts. In particular, define as $$\hat U_{j,t,s} \coloneqq \hat F_{j,s}(\hat \varepsilon_{j,t}), \qquad j \in \{1,2\}, \quad s \in [0,1],$$ the ranks of the residuals $\hat\varepsilon_{j,t} \coloneqq \varepsilon_{j,t}(\hat\lambda)$, $\varepsilon_{j,t}(\lambda) = (Y_{j,t}-\mu_{j,t}(\lambda))/\sigma_{j,t}(\lambda)$, calculated using the sequential empirical distribution function $\hat F_{j,s}(x) \coloneqq \hat F_{j,s}(x;\hat\lambda)$, with
using observations up to $\floor{s n}$, with $n$ denoting the length of the sample; the convention $\hat F_{j,s}(x;\lambda) = 0$ is used at $s = 0$. These pseudo observations $\hat U_{t,s} = (\hat U_{1,t,s},\hat U_{2,t,s})^\top$ involve two sources of first-step sampling uncertainty due to (1) estimation of $\lambda$ and (2) estimation of marginals ${\sf F}_j(\cdot)$ that will be taken into account when deriving the analytical properties of our test.
Now, at $u = (u_1,u_2)^\top$, the log-likelihood is given by $$\hat\ell(u,s,\theta) \coloneqq \frac1{n}\sum_{t=1}^{\floor{sn}} \hat\ell_{t}(u,s,\theta;Z_t),$$ where
with $C_t(u,\theta;z) \coloneqq C(u; \varrho_t(u,\theta; z))$. Note that, for a given $z$, $\hat\ell_{t}(u,s;\theta,z)$ is strictly concave in $C$, and, for a given $(u,z)$, $\theta \mapsto C(u,\theta;z)$ is monotone in $\theta$. It is convenient to represent the likelihood more succinctly via \[ \hat\ell_{t}(u,s,\theta;z)= \hat d_{t}(u,s)^\top \operatorname*{\textnormal{\textsf{log}}} p(u;\varrho_t(u,\theta;z)) = \sum_{j=1}^4 \hat d_{j,t}(u,s) \operatorname*{\textnormal{\textsf{log}}} p_j(u;\varrho_t(u,\theta;z)), \] with
denoting the $4\times 1$ vector of empirical events and
represent its population counterparts.
Our test will be based on the sequential score evaluated at the restricted ($\beta = 0$), full-sample estimator (defined below) $\hat\theta \coloneqq (\hat\alpha,0)^\top$:
where $(e_1,\dots,e_4)^\top \coloneqq (1,-1,-1,1)^\top$ and $$\tau(u,\alpha) \coloneqq (1-\varrho^2(u,\alpha))C_\varrho(u,\alpha), \quad C_\varrho(u,\alpha) \coloneqq \frac{\partial}{\partial \varrho} C(u,\varrho) \Big\vert_{\varrho = \varrho(u,\alpha)}.$$ In order to ascertain the validity of the joint null in Eq. (ref), we consider a test statistic that explicitly takes potential deviations from the two sub-hypotheses into account:
where $\omega$ is a suitable {\it aggregation} function (e.g. $\operatorname*{\textnormal{\textsf{sup}}}_u \Delta(u)$ or $\operatorname*{\textnormal{\textsf{log}}} \int \exp(\Delta(u)/2) \mathrm{d} u$, andpol:1994), and $$ \Delta_1(u) \coloneqq \operatorname*{\textnormal{\textsf{sup}}}\limits_{s \in [0,1]} \sqrt{n}|\nabla_\beta\hat\ell(u,s;\hat\theta)-s \nabla_\beta\hat\ell(u,1;\hat\theta)|_\infty, \quad \Delta_2(u) \coloneqq \sqrt{n}|\nabla_\beta\hat\ell(u,1;\hat\theta)|_\infty, $$ are the CUSUM detector and the LM statistic, respectively, tailored to detect deviations from $H_{0,1}$ (parameter stability) and $H_{0,2}$ (constant non-Granger causality). Note that the CUSUM detector $\Delta_1$ does not require any trimming. Intuitively, $\Delta_1$ has no power against constant deviations from the null, and $\Delta_2$ lacks power if Granger causality is unstable; hence, combining both detectors ensures non-trivial power in both cases, an idea that goes back to sowell:96 and rossi:2005 (see also mwt:25).
As it turns out, the test statistic $T$ has an asymptotic distribution which depends on nuisance parameters (see Section (ref)), so we propose to use a moving block bootstrap approximation for the critical values. For a given bootstrap draw $b \in \{1,\dots,B\}$, one proceeds as follows:
Because, as discussed below, the first-step estimation error enters the $\Delta_2$ component through an additional drift term that does not cancel in $\Delta_2(u)= \sqrt{n}|\nabla_\beta \hat\ell(u,1;\hat\theta)|_\infty$, we center the bootstrap analogue by subtracting the original sample score, i.e., $\Delta_2^{\,b}(u)=\sqrt{n}|\nabla_\beta \ell_n^{\,b}(u,1;\hat\theta^{\,b})-\nabla_\beta \ell_n(u,1;\hat\theta)|_\infty$, so that the bootstrap replicates the correct score fluctuations. On the other hand, if one is only interested in break testing, one can use an {\sf IID} bootstrap similar to nasri2022change, because, as explained below, the limit of $\Delta_1$ is nuisance parameter free.
In order to derive the properties of $T$, we impose the following assumptions:
The smoothness condition is due to segers2012asymptotics and is needed to apply the extended continuous mapping theorem. Assumptions (ref) and (ref) are needed in order to apply mohr2020weak, which is an extension of the empirical process CLT of andrews1994introduction to the sequential case with unbounded classes of functions. Note that this involves a trade-off between moments of $Z$ and dependence allowed. In our empirical application $Z_t$ is an indicator and thus surely bounded. If one were interested only in the full-sample statistic ($s=1$) and $\beta$-mixing can be assumed, then $\textnormal{\textsf{E}}[|Z_1|^{4+2\delta}]$, $\delta > 0$ suffices. This can be shown using doukhan1995invariance as in neumeyer2019copula. Assumption (ref) is similar to conditions imposed by neumeyer2019copula (see also mw:2023) and rules out marginal distributions with bounded support.
Because our test statistic will be based on the score (with respect to $\beta$) of the {\it un}restricted DR problem evaluated at the {\it restricted} estimator, we first need to understand the properties of the latter. Under the restriction $\beta(u) = 0$, we have $\theta(u) = (\alpha(u),0)^\top$ so that the full-sample ($s=1$) likelihood contribution $$\hat\ell_{t}(u,1,\alpha,0;z) \eqqcolon \hat\ell_{t}(u,\alpha)$$ is independent of $z$. Define the restricted estimator that imposes $\beta(u) = 0$:
Note that, by strict concavity of $\hat\ell(u,\alpha)$ in $C(u,\alpha) \coloneqq C(u, \varrho(u,\alpha))$ and strict monotonicity of $\alpha \mapsto C(u,\alpha)$, the maximizer $\hat\alpha(u)$ is, for each $u \in {\cal U}$, unique.
On the other hand, the population value $\alpha_0(u)$ can be viewed as the unique maximizer of the population objective function
where, in contrast to $\hat\ell_t(u,\alpha)$ in Eq. (ref), the log-likelihood contribution $\ell_t(u,\alpha)$ involves the true $d_{j,t}(u)$ that are based on the population ranks $U_{j,t} \coloneqq {\sf F}_j(\varepsilon_{j,t})$. If $H_0$ is true, then ${\mathbb P}\{d_{j,t}(u)=1\}=p_j(u,\alpha_0(u)) \eqqcolon p_j(u)$ so that the Kullback–Leibler divergence satisfies
with equality if, and only if, $p_j(u,\alpha) = p_j(u)$. Since $\alpha(u) \mapsto C(u, \alpha)$ is strictly increasing $$ \alpha_0(u) \coloneqq \operatorname*{\textnormal{\textsf{arg\,max}}}\limits_{\alpha \in {\cal A}} \ell(u,\alpha) $$ is the unique maximizer for each $u \in {\cal U}$. Because $(u,\alpha) \mapsto \ell(u,\alpha)$ is continuous on a compact set ${\cal U} \times {\cal A}$ we obtain uniform separation:
Put differently, if $H_0$ is true, then $\alpha_0(\cdot)$ is uniformly identified on ${\cal U}$ and there is hope that it can be estimated using a sample of estimated ranks $(\hat U_1,\dots,\hat U_n)$. In order to show that this is indeed the case, we assume that the parameter space is compact:
The limiting distribution is governed by the same Gaussian process ${\mathbb C}$ that appears in fermanian:2004 based on the Kiefer-Müller process ${\mathbb B}_C(u,1)$. Importantly, the well known (e.g. gijbels2015estimation, remillard2017goodness) invariance of the empirical copula process based on pseudo observations from location-scale models with respect to the first-step estimation error carries over to the DR regression framework considered here.
To analyse the test statistic, it turns out convenient to define for any bivariate function $h: [0,1]^2 \rightarrow \mathbb{R}$ the real-valued functional
where we recall that $p_j(u) = p_j(u,\alpha_0(u))$. Now, the crucial observation is that the $(k+1) \times 1$ gradient vector when evaluated at the true parameter vector $\theta_0 = (\alpha_0,0)^\top$ obeys: \[ \sqrt{n}\nabla_\theta \hat\ell(u,s,\theta_0) = \sqrt{n}
= \tau(u) \varphi_u(\mathbb{H}_n(\cdot,s;\hat\lambda)), \] where $\varphi_u(\cdot)$, as defined in Eq. (ref), is applied to each of the $k+1$ elements of the $(k+1) \times 1$ empirical process ${\mathbb H}_n$ given by
where, by convention, $H_n(u,0;\lambda) \coloneqq 0$, and $X_t \coloneqq (1, Z_t^\top)^\top$, $m \coloneqq (1,\mu^\top)^\top$, with $\mu \coloneqq \textnormal{\textsf{E}}[Z_1]$.
Hence, it is clear that, in a first step, the properties of $\mathbb{H}_n$ need to be established to obtain the weak limit of the score. In a second step, the estimation error of $\hat\theta = (\hat\alpha,0)^\top$ from the previous section can be accounted for.
The main idea is to relate the properties of the process ${\mathbb H}_n$ in Eq. (ref), based on the empirical cdf's, to its counterpart $\tilde {\mathbb H}_n$ that is based on the true cdf's:
where we keep the convention $\tilde H_n(u,0;\lambda) \coloneqq 0$. In particular, upon decomposing $\tilde H_n = (\tilde C_n, \tilde G_n^\top)^\top$ according to the elements of $X_t = (1,\ Z_t^\top)^\top$, the following well-known identity (see, e.g., segers2012asymptotics) summarizes this idea:
where $\tilde C_{j,n}(u_j,s;\lambda) \coloneqq \tilde C_n(u^{(j)},s;\lambda)$, $j \in \{1,2\}$, are the marginal ecdf's corresponding to the empirical copula $\tilde C_n$. Thus, as argued elsewhere for the special case $Z_t = 1$ (see e.g. bucher2013empirical or neumeyer2019copula), we can think of Eq. (ref) as a (two-argument) mapping that operates on a suitable function space and relates $H_n$ to $\tilde H_n$. The difference to the previous literature is that we also consider a marked version of the residual copula process, where the marks ($X_t$) depend on the regression residuals. For that reason, since the processes in Eq. (ref) are not necessarily mean-zero even as $n$ diverges, we also define centred counterpart
Note that $\tilde {\mathbb H}_n(\cdot,\cdot;\lambda_0) = \tilde {\mathbb H}^\circ_n(\cdot,\cdot;\lambda_0)$.
As discussed below, provided the mapping underlying Eq. (ref) is Hadamard-differentiability, the properties of $H_n$ can be linked to those of the functional input $\tilde H_n$. The preceding argument can be split into a stochastic part concerned with the weak convergence of the input processes and a deterministic part establishing the properties of the map. The following lemma summarizes the stochastic part:
Lemma (ref) shows how the infeasible process $\tilde {\mathbb H}_n$ behaves uniformly in a local neighbourhood around the true parameter $\lambda_0$ that governs the marginal location scale models. Lemma (ref) holds on the whole $u \in [0,1]^2$ and not just on the compact subset ${\cal U} \subset (0,1)^2$. Moreover, if the null hypothesis is strengthened to independence between $\{Z_s\}_{s \geq 1}$ and $\{U_s\}_{s \geq 1}$, then the covariance kernel of $\Sigma(u,v)$ is block-diagonal and the limiting processes ${\mathbb B}_C$ and ${\mathbb B}_Z$ are independent.
Turning to the deterministic part, we show in the appendix that, akin to bucher2013empirical and bucher2016dependent, the mapping defining Eq. (ref) is Hadamard differentiable. The weak limit of ${\mathbb H}_n(\cdot,\cdot;\hat\lambda)$ follows then from the preceding lemma and the extended continuous mapping theorem. Since Lemma (ref) analyses $\tilde{\mathbb H}_n(\cdot,\cdot;\lambda_0+h/\sqrt n)$ for deterministic $h$, deriving the asymptotics of the feasible statistic requires replacing $h$ by the random drift $h_n=\sqrt n(\hat\lambda-\lambda_0)$. To justify this substitution in the functional delta method, we impose the relatively weak assumption that Lemma (ref) (2) holds jointly with Assumption (ref) (1):
For a specific estimator of $\lambda$, the previous assumption can be derived under primitive assumptions.
Lemma (ref) implies that the feasible marked sequential empirical process converges uniformly to a tight Gaussian process. The score with respect to $\alpha$ depends only on the true underlying copula and is asymptotically unaffected by first-step estimation; in particular, ${\mathbb S}_{\alpha_0}(u,s)=\tau(u){\mathbb C}(u,s)\sum_{j=1}^4 1/p_j(u)$, where $p_j(u)$ are the four quadrant probabilities. This is in line with earlier results on sequential empirical copula processes based on AR-GARCH residuals (see, e.g. nasri2022change or bomw:24). In contrast, inference on $\beta$ generally inherits the additional drift term unless $\textnormal{\textsf{cov}}[Z_1,\xi_p(u_p,R_1)]=0$, $p\in\{1,2\}$. Finally, we can use these results to obtain the limiting distribution of the test statistic:
Three points are worth noting: Firstly, because the Gaussian bridge ${\mathbb S}_\beta(\cdot,s)-s{\mathbb S}_\beta(\cdot,1)$ is independent of the endpoint ${\mathbb S}_\beta(\cdot,1)$, the limiting distribution of the statistics $\Delta_1$ and $\Delta_2$ are independent, too. Secondly, if $\mu = 0$, then $\sqrt{n}\nabla_\theta \hat\ell (\cdot,\cdot,\hat\theta) = \sqrt{n}\nabla_\theta \hat\ell (\cdot,\cdot,\theta_0) + o_p(1)$. Finally, $\Delta_1$ behaves as if you knew both $\lambda_0$ and $\alpha_0$; i.e. \[ {\mathbb S}_\beta(u,s)-s{\mathbb S}_\beta(u,1) = \tau(u)(\varphi_u(\mathbb{G}_0(\cdot,s))-s\varphi_u(\mathbb{G}_0(\cdot,1))), \] using ${\mathbb G}(u,s) = {\mathbb G}_0(u,s)+s\Xi(u)\Lambda$. The estimation error does not, however, cancel in $\Delta_2$. For this reason we suggest the bootstrap scheme described in Section (ref).
In this section, we study the finite-sample properties of the tests introduced in the previous sections using Monte Carlo experiments. In particular, we consider our test $T = \omega(\Delta_1+\Delta_2)$, as well as its individual components, i.e. the aggregated CUSUM test $\Delta_1 \coloneqq \omega(\Delta_1)$ and the aggregated LM test $\Delta_2 \coloneqq \omega(\Delta_2)$. Following andpol:1994, the aggregation function $\omega: [0,1] \rightarrow {\mathbb R}_+$ is chosen as $\omega(f) = \operatorname*{\textnormal{\textsf{log}}} \int_{\cal U}{\sf exp}\{\frac1{2}f(u)\}\mathrm{d} u$.
The state variable $Z_t$ is generated as a stationary AR(1) process, $Z_t = 0.85 Z_{t-1} + v_t$, where the error term is ${\sf IID}$ with $v_t \sim \mathcal{N}(0,1-0.85^2)$. Each marginal time series $Y_{j,t}$, $j\in\{1,2\}$, follows an AR(1)--GARCH(1,1) model: \[ Y_{j,t} = \mu + \phi Y_{j,t-1} + \gamma Z_t + \eta_{j,t},\quad \sigma_{j,t}^2 = \omega + \alpha\eta_{j,t-1}^2 + \beta\,\sigma_{j,t-1}^2, \] where $\eta_{j,t}=\sigma_{j,t}\varepsilon_{j,t}$ and $\varepsilon_{j,t}$ are {\sf IID} standard Gaussian innovations. We set $\mu=0$, $\phi=0.1$, $\gamma=1.5$, $\omega=0.01$, $\alpha=0.1$, and $\beta=0.85$. The AR(1)-GARCH(1,1)-parameters are estimated using the (quasi) Maximum Likelihood estimator from {\sf R}-package {\sf rugarch} (rugarch). After this, the ranks of the residuals are obtained in order to calculate the test statistics.
We consider three copula specifications under the null: ($i$) a Gaussian copula with constant correlation, ($ii$) a Frank copula with constant parameter, and ($iii$) a Gaussian patchwork copula Durante:2009 with correlation $\rho(u)=\tanh(\alpha(u))$, where $\alpha(u)=1/4$ for $u\in[0,0.5]^2$ and $\alpha(u)=1/2$ otherwise. Intuitively, the copula in case ($iii$) is a rectangular patchwork that allows for different dependence in different subrectangles. Under the alternative, we consider a Gaussian patchwork copula with time-varying correlation $\rho_t(u)={\sf tanh}(\alpha(u)+\beta_t(u)Z_t)$, where $\alpha(u)$ is as above. For $u\in[0,0.5]^2$ we set:
We set $\beta_t(u)=0$ for $u\notin[0,0.5]^2$.
We focus on the quantile dependencies (e.g. patton:12,patton2013copula or patton:13,oh2017modeling), i.e. we restrict the analysis to the main diagonal $u=(u_1,u_2)^\top$ with $u_1=u_2=u$ and report results for the lower and upper diagonal regions ${\cal U}_l=[0.05,0.50]$ and ${\cal U}_u=[0.50,0.95]$. Following gonccalves2023, the MBB block length $l$ is set equal to the next integer of the bandwidth from the automatic procedure for the Bartlett kernel of andrews1991heteroskedasticity.
As $n$ increases, the empirical rejection rate approaches $5\%$ in all cases, the power properties reflect the design of the tests in all cases. In the case of the upper tail, none of the tests has non-trivial power as expected. In the case of the lower tail, the LM has highest power in the first alternative scenario (constant GC). In the mid-break scenario, the test still has power, but there is no non-trivial power in the offset scenario. The CUSUM test has highest power in the offset scenario, some power in the mid-break scenario and no power in the first alternative scenario. The combination of both test statistics always has non-trivial power, whereas the ranking is similar to the ranking of the LM test.
We now revisit the preliminary evidence from the introductory example, summarized in Figure (ref) of Section (ref). Our goal is to examine how the dependence between the market return ($r_{m,t}$) and an individual asset return ($r_t$) varies with lagged market conditions. Specifically, let $r_{m,t}$ denote the daily CRSP value-weighted market index return. For $r_t$, we consider daily CRSP returns (Jan 2000-Dec 2019) of JP Morgan Chase ({\sf JPM}), Morgan Stanley ({\sf MS}), and {\sf AIG}, representing the Depositories, Broker-Dealers, and Insurance groups, respectively; see also han:2016.
As discussed in Section (ref), for each $u=(u_1,u_2)\in{\cal U}$, these co-movements are summarized by the local dependence measure $\varrho_t(u)$, which determines the conditional copula of $(r_t,r_{m,t})$. We distinguish between two predetermined market states $Z_t\in\{0,1\}$ corresponding to bear and bull market conditions. Specifically, we define the predetermined down market indicator \( Z_t \coloneqq \textbf{1}\{r_{m,t-1}<0\}, \) so that $Z_t=1$ represents a bear (down market) state and $Z_t=0$ a bull (up market) state. Hence, for a given $u\in{\cal U}$ and in the absence of structural breaks, dependence is described by the state-specific local dependence measures \[ \varrho^{\text{bear}}(u) = {\sf tanh}(\alpha(u)+\beta(u)) \quad \text{if } Z_t=1, \qquad \varrho^{\text{bull}}(u) = {\sf tanh}(\alpha(u)) \quad \text{if } Z_t=0. \]
For both series, we fit GJR--AR(1)--GARCH(1,1) models glosten1993relation. In the conditional mean of the individual asset return, we additionally include the market-state indicator $Z_t$. Estimation is carried out by quasi-maximum likelihood using the {\sf R}-package {\sf rugarch} (rugarch).
As motivated by the systemic-risk application, we focus on quantile dependence along the main diagonal of the conditional copula, i.e., $u=(u_1,u_2)^\top$ with $u_1=u_2$. As in the simulation study, we consider the lower and upper regions \({\cal U}_{l}=[0.05,0.50],\) \({\cal U}_{u}=[0.50,0.95]. \) Finally, the block length is set equal to the optimal bandwidth selected by the automatic procedure of andrews1991heteroskedasticity for the Bartlett kernel.
If $T$ rejects, we apply the procedure of mwt:25 to determine whether the rejection is driven by $H_{1,1}$ or by $H_{1,2}$. We begin with a significance level of $\alpha=0.10$. If $T$ rejects at level $\alpha$, we compute the CUSUM detector $\Delta_1$. If the CUSUM test rejects at the level \( 1-(1-\alpha)^{1/2}=0.051, \) we declare a break $s \in [0,1]$. We then set the adjusted significance level to \( 1-(1-\alpha)^{1/4}=0.026 \) and recompute $T$ on the subsamples $[0,s]$ and $[s,1]$. This Šidák correction ensures that the size corresponds to an overall significance level $\alpha$ (see also galeano:2017).
Overall, the results reported in Table (ref) suggest that state-dependence becomes stronger during crisis periods. For {\sf MS}, we detect significant dependence predictability in both the lower and upper tails after 2005, but not before. For {\sf JPM}, we detect significant dependence predictability in the lower tail after 2002, but not before. For {\sf AIG}, we detect significant dependence predictability in the upper tail after 2002, but not before. Hence, across all three institutions, the evidence points to pronounced dependence predictability around 2008, often regarded as the climax of the global financial crisis associated with real-estate frictions (see wied:2012).
A possible economic interpretation is that crises amplify negative signals. This view is consistent with evidence that dependence and correlation tend to strengthen in downturns, in particular in the lower tail and in bear-market regimes (e.g., longin2001extreme, ang2002asymmetric). During crisis periods, a single negative market return may carry greater informational content and is less likely to be interpreted as a transitory fluctuation; instead, it is more readily viewed as confirmation of systemic stress. This interpretation is also in line with the systemic-risk literature, which emphasizes increases in tail co-movement under financial distress (adrian:16).
This paper proposes a flexible semiparametric test for dependence predictability that is designed to accommodate a broad class of marginal dynamics, remain agnostic about the copula family, and provide local information on how predetermined covariates affect the dependence structure in different parts of the distribution and over different periods. Although the primary application here is financial, related challenges are frequent in economics and beyond. For example, similar questions of Granger causality arise frequently in environmental econometrics (e.g. Runge2019InferringCausation).
Several technical extensions constitute promising directions for future research: First, it would be of interest to generalize the framework to multivariate settings with $J\ge 2$ variates $Y_{1,t},\dots,Y_{J,t}$, which would require addressing the increased complexity of the local Gaussian approximation. Secondly, in the spirit of patton:06, one could let the state vector be constructed from lagged pseudo-observations, for example \[ \hat Z_{t,s}(\lambda)\coloneqq \Phi^{-1}(\hat U_{1,t-1,s})\Phi^{-1}(\hat U_{2,t-1,s}) \] thereby allowing for more flexible dependence dynamics driven by the history of the system. Thirdly, replacing the parametric marginal filtering step with nonparametric methods as in neumeyer2019copula and chen2021efficient could be an attractive alternative. Their results suggest that a $\sqrt{n}$-consistent DR estimation of $\theta$ might be possible. Finally, it may be of interest to relax Assumption (ref) by allowing for additional control variables $W_t$ when assessing the effect of $Z_t$, i.e., to consider conditional copulas of the form ${\sf C}(\cdot \mid {\cal F}_{t-1}) = {\sf C}(\cdot \mid W_t, Z_t)$.