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Estimation and Inference in Factor Copula Models with Exogenous Covariates-.5cm
Keywords: factor analysis, simulation estimator, empirical process, dependence modeling. \newline JEL classifications: C13, C15, C22.
Factor copula models have been successfully introduced as a means to cope with data of high cross-sectional dimensionality; see, e.g., kj13, crts15, and ohpa17. The use of a latent factor structure offers an economically intuitive yet flexible way to multivariate modeling that parsimoniously handles commonly encountered characteristics of financial time series like, for example, the tail asymmetry and tail dependence described by hansen94. Recently, some research effort has been devoted to incorporate time variation and exogenous information to factor copula models; see, e.g., crts15, ohpa18, opetal20, and kj20. For example, ohpa18 and opetal20, by utilizing the generalized autoregressive score (GAS) framework of cretal13, consider specifications with latent factors and time-varying loadings that may depend on exogenous information. We contribute to this literature by introducing a class of factor copula models with exogenous, (partly) observable factors, an idea reminiscent of bertal05, boetal09, and stwa05. Contrary to the above cited factor copula models, we take a step back and treat the, possibly group-specific, loadings as time-invariant constants and the SMM estimator employed here is build on the unconditional copula---a concession in the name of tractability that frees us from the necessity of specifying parametric marginals [e.g., ohpa18] or a closed form likelihood of the copula [e.g., opetal20] and thereby allows for a large variety of `covariate-augmented' factor copulas, nesting the model ohpa17 as a special case.
Since the copula likelihood is rarely available in closed form for the model class considered here, an SMM framework for estimation and inference is proposed which uses the general principles outlined by ohpa13. Our main contribution is a novel distinction between simulable factors and factors that are estimable from exogenous information. Following the seminal SMM literature of mcf89, papo89, and lee92, we exploit the benefits from non-overlapping simulation draws. The incorporation of exogenous covariates considerably complicates the development of an asymptotic theory as many arguments made by ohpa13 do not apply. Nevertheless, we show that all technical hurdles can be overcome by combining recent developments from copula empirical process theory [see, e.g., buvo13, betal17, and neumetal19] with a seminal result for extremum estimation with nonsmooth objective function due to nemc94. In consequence, consistency, limiting normality, and validity of bootstrap standard errors are established. In doing so, we derive the stochastic equicontinuity of the objective function from primitive conditions on the distributional characteristics of the factor structure using the functional central limit theorem (FCLT) of anpo94 for \(\alpha\)-mixing triangular arrays. The theory developed here verifies earlier equicontinuity results from the literature that made use of high-level conditions; see, e.g., ohpa13, manetal19, and manetal20. Since stochastic equicontinuity is an essential ingredient of the asymptotic theory that links pointwise and uniform properties, more primitive conditions are of utmost interest. An application to dependence modeling of a cross-section of stock returns of eleven financial companies illustrates the theoretical results and highlights how the incorporation of estimable factors can help to achieve improvements in model performance.
The remainder of this paper is organized as follows. Section (ref) introduces the model. The main results for SMM estimation and inference are contained in Section (ref). A small Monte Carlo exercise is conducted in Section (ref) and an empirical application can be found in Section (ref). Section (ref) briefly summarizes and concludes the paper.
Our aim is to capture the dependence structure among the cross-sectional entities of the \(n \times 1\) vector of financial assets \(Y_{t} \coloneqq (Y_{1,t},\dots,Y_{n,t})'\) in time-period \(t \in \{1,\dots,T\}\), conditional on the available information \(\mathcal{F}_{t} \coloneqq \sigma(\{Y_{j}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}},Y_{j-1}: j \leq t\})\), where \(Y_{t}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}}\) represents a vector of exogenous regressors. The number of financial assets \(n\) might be large but is assumed finite. If the marginal conditional distributions \(Y_{i,t} \mid \mathcal{F}_{t} \sim \textsf{H}_{i,t}\) are continuous, we can follow pat06 and uniquely decompose the joint conditional distribution \(Y_t \mid \mathcal{F}_{t} \sim \textsf{H}_t\) into its \(n\) margins and a copula function \(\textsf{C}_t: [0,1]^n \mapsto [0,1]\), where \(\textsf{C}_t(\cdot)\) completely describes the dependence conditionally on \(\mathcal{F}_{t}\); i.e., \(\textsf{H}_t(x_1,\dots,x_n) = \textsf{C}_t\{\textsf{H}_{1,t}(x_1),\dots,\textsf{H}_{n,t}(x_n)\}\), \(x_i \in \mathbb{R},\) \(i \in \{1,\dots,n\}\). Following, among others, chef06, ohpa13, and fapa14, we assume for the \(n\) assets parametric location-scale specifications of the form
where, for each \(i \in \{1,\dots,n\}\), \(\{\eta_{i,t}: t \geq 1\}\) are i.i.d. innovations independent of \(\mathcal{F}_{t}\), while \(\mu_{1,i}\) and \(\mu_{2,i}\) are \(\mathcal{F}_{t}\)-measurable parametric specifications of the conditional mean \(\mu_{1,i}(\mathcal{F}_t,\lambda) = \textsf{E}[Y_{i,t} \mid \mathcal{F}_t]\) and the conditional standard deviation \(\mu_{2,i}(\mathcal{F}_t,\lambda) = \sqrt{\textsf{var}[Y_{i,t} \mid \mathcal{F}_t]}\) that are known up to the true \(r \times 1\) parameter vector \(\lambda = \lambda_{0} \in \Lambda_{0} \subset \mathbb{R}^r\). In particular, we assume that
where, for any \(\lambda \in \Lambda\), the \(p_{\textsf{R}} \times 1\) random vector \(R_t(\lambda)\) is \(\mathcal{F}_t\)-measurable with components that may parametrically depend on the entire history \(\mathcal{F}_t\) through \(\lambda\). For example, this model class includes nonlinear autoregressions (AR) with nonlinear autoregressive conditional heteroskedasticity (ARCH), in which case the dependence of \(R_{t}(\lambda)\) on \(\lambda\) is superfluous, as well as nonlinear generalized autoregressive conditional heteroskedasticity (GARCH) as illustrated below for the GJR model of glostetal93.
Example: Consider the AR(1) model \(Y_{i,t} = \gamma_i Y_{i,t-1} + \mu_{2,i,t}\eta_{i,t}\), with conditional heteroskedasticity \( \mu_{2,i,t}^2 = \omega_i + (\beta_i + \eta_i1\{\varepsilon_{i,t-1}(\gamma_i) < 0\})\varepsilon_{i,t-1}^2(\gamma_i) + \alpha_i \mu_{2,i,t-1}^2 \) for \(\varepsilon_{i,t}(\gamma_{i}) \coloneqq Y_{i,t} - \gamma_{i}Y_{i,t-1}\). In accordance with our notation, let \(\lambda = (\gamma_1,\dots,\gamma_n,\alpha_1,\dots,\alpha_n,\omega_1,\dots,\omega_n,\beta_1,\dots,\beta_n,\eta_1,\dots,\eta_n)'\) be the \(r\times 1\) parameter vector collecting all unknowns for \(r = 5n\). Following the discussion in yang96, by suitably restricting \(\Lambda_{0} \subset \mathbb{R}^{r}\), this specification can be cast in form of (ref) and (ref) if we set \(\mu_{1,i}(\mathcal{F}_t,\lambda) = \gamma_iY_{i,t-1}\), \(R_t(\lambda) = (Y_{t-1}',\tilde{R}_{t}(\lambda)')'\), where \(\tilde{R}_{t}(\lambda) = (\tilde{R}_{1,t}(\lambda),\dots,\tilde{R}_{n,t}(\lambda))'\), with \(\tilde{R}_{i,t}(\lambda) = \sum_{j \,=\, 1}^\infty \alpha_{i}^{j-1}g_i\{Y_{i,t-j}- \gamma_iY_{i,t-1},\eta_{i}\}\), \(g(x,a) = x^2+ a1\{x<0\}x^2\), and \(\mu_{2,i}(R_t(\lambda),\lambda) = \beta_i \tilde{R}_{i,t}(\lambda) + \omega_i/(1-\alpha_i)\).
Since we assume that individually \(\eta_{i,t}\) are independent of \(\mathcal{F}_t\) and, by Eq. (ref), \(\eta_{i,t} = \{Y_{i,t}-\mu_{1,i}(\mathcal{F}_t,\lambda_0)\}/\mu_{2,i}(\mathcal{F}_t,\lambda_0)\), we can --assuming continuous margins \(\textsf{F}_i(\eta_{i,t} \leq x_i) \coloneqq \textsf{P}(\eta_{i,t} \leq x_i)\), \(i \in \{1,\dots,n\}\)-- rephrase the conditional joint distribution of \(Y_t\) in terms of the innovation ranks \(V_{i,t} \coloneqq \textsf{F}_{i}(\eta_{i,t})\) as
\(u_i \in [0,1]\), \(i \in \{1,\dots,n\}\). While the effect of \(\mathcal{F}_{t}\) on the margins has been completely removed by the use of the location-scale model (ref), the joint cross-sectional distribution of \(\eta_{t} \coloneqq (\eta_{1,t},\dots,\eta_{n,t})'\) is allowed to depend on \(\mathcal{F}_{t}\) through some `exogenous' vector \(Z_t\). That is, we assume that
We do not model the conditional copula directly but we assume that the unconditional copula
can be generated from an auxiliary factor model via
where \(G_{i}(x_i) \coloneqq \textsf{P}(X_{i,t} \leq x_i)\) represents the \(i\)-th margin of
and \(\textsf{G}(x) \coloneqq \textsf{P}(X_{1,t}\leq x_1,\dots,X_{n,t} \leq x_n)\), \(x \coloneqq (x_1,\dots,x_n)' \in \mathbb{R}^n\), is the corresponding joint distribution. As defined below, \(F_t\) and \(\varepsilon_{i,t}\) denote latent factors and the idiosyncratic component, respectively. It is crucial to stress that the margins \(G_i\), \(i \in \{1,\dots,n\}\), can differ from the univariate distributions of the observed data and are not of interest here. Rather, Eq. (ref) serves as a means to generate the copula \(\textsf{C}\) that determines the joint distribution. Since any effect of \(\mathcal{F}_t\) on the margins has been filtered out while the conditional distribution is only affected by \(\mathcal{F}_t\) through the regressor, the unconditional copula obtains directly by taking the expectation with respect to \(Z_t\), i.e. \(\textsf{C}(\cdot) = \textsf{E}[\textsf{C}_t(\cdot)]\).
ohpa18 or opetal20 consider a related specification. Contrary to our approach, however, they model the conditional copula directly, i.e. they consider a conditional copula \(\textsf{C}_t = \textsf{C}(\theta_{0,t})\) indexed by a time-varying copula parameter \(\theta_{0,t} \coloneqq \theta_0(Z_t)\) that is driven by GAS-dynamics so that \(Z_t = (\eta_{1}',\dots,\eta_{t-1}')'\). We, on the other hand, assume that all components of the factor model including \(Z_t\) are \(i.i.d.\). To make these notions precise, Assumption (ref) formalizes the characteristics of the factor model. It constitutes a naturally extension of ohpa17.
The peculiar feature of Eq. (ref), and the main contribution of this paper, is the distinction between simulable and observable factors: while \(F_t \coloneqq (F_{t,1},\dots,F_{t,p_\alpha})'\) is a \(p_\alpha \times 1\) vector of latent random variables with known parametric distribution, the \(p_\beta \times 1\) vector \(Z_t \coloneqq (Z_{t,1},\dots,Z_{t,p_\beta})'\) can be recovered from observed data based on econometric tools. Therefore, \(Z_t\) is also referred to as the estimable factor. More specifically, both $F_t$ and \(\varepsilon_{i,t}\) are i.i.d. with parametric distributions \[ \textsf{D}_\varepsilon(x;\,\delta_0) \coloneqq \textsf{P}(\varepsilon_{i,t} \leq x),\;\;\;\textsf{D}_{F,j}(x;\,\gamma_{0,j}) \coloneqq \textsf{P}(F_{t,j} \leq x),\,j \in \{1,\dots, p_\alpha\}, \] which are partially known up to the \(p_\delta \times 1\) vector \(\delta_0 \coloneqq (\delta_{0,1},\dots,\delta_{0,p_\delta})'\) and the \(p_\alpha p_\gamma \times 1\) vector \(\gamma_0 \coloneqq (\gamma_{0,1}',\dots,\gamma_{0,p_\alpha}')'\), with \(\gamma_{0,j} \coloneqq (\gamma_{0,j,1},\dots,\gamma_{0,j,{p_\gamma}})'\), respectively. On the other hand, the distribution of the vector \(Z_{t}\) is unknown but we assume that its components \(Z_{t,j}\) can be represented as i.i.d. innovations of an observable \(\mathcal{F}_t\)-measurable processes \(W_{t,j}\) given by
where the measurable functions \(\sigma_j(\cdot,\nu)\) are known up to the \(m \times 1\) vector \(\nu = \nu_{0} \in \mathcal{V}_{0} \subset \mathbb{R}^m\); the i.i.d. innovations \(Z_{t,j}\) are independent of the history \(\mathcal{W}_t \subset \mathcal{F}_{t-1}\). Similar to the specification of Eq. (ref), we assume that \(\sigma_j(\mathcal{W}_t,\nu) = \sigma_j(M_t(\nu),\nu)\), where the \(p_{\textsf{M}} \times 1\) vector \(M_{t}(\nu)\) comprises short-range dependent covariates, possibly including lagged dependent variables, whose components may be parametrically generated from the complete history \(\mathcal{W}_t\). We cannot allow for both time-varying conditional means and variances to ensure that the limiting distribution of the SMM estimator is unaffected by the first step estimation of \(\nu_0\). Note, however, that several models with time-varying conditional variance that obey \(\tilde{W}_t = \tilde{\sigma}(M_t(\nu),\nu)\tilde{Z}_t\), \(\operatorname*{\textsf{sup}}_{x,y}\tilde{\sigma}(x,y) > 0\), can be cast in form of (ref) by setting \(W_t \coloneqq \textsf{log} |\tilde{W}_t|\), \(\sigma_t(M_t(\nu),\nu) \coloneqq \textsf{log}\,\tilde{\sigma}_t(M_t(\nu),\nu)\), and \(Z_t \coloneqq \textsf{log} |\tilde{Z}_t|\).\footnote{The argument is inspired by genestetal07, who propose this transformation in their Example 1 to ensure a nuisance-free distribution of the BDS-type test studied there; see also carpetal07 for a study of the BDS test based on the logarithm of absolute GARCH(1,1)-residuals.}
As in ohpa17 and opetal20, it is assumed that the factor loadings \(a_{0,i} \coloneqq (a_{0,i,1},\dots,a_{0,i,p_\alpha})'\) and \(b_{0,i} \coloneqq (b_{0,i,1},\dots,b_{0,i,p_\beta})'\) can be grouped into a small number of \(Q\) group-specific coefficients \(\alpha_{0,q} \coloneqq (\alpha_{0,q,1},\dots,\alpha_{0,q,p_\alpha})'\) and \(\beta_{0,q} \coloneqq (\beta_{0,q,1},\dots,\beta_{0,q,p_\beta})'\), \(q \in \{ 1,\dots,Q\}\). Put differently, there exists a finite collection of disjoint sets \(\{\mathcal{G}_1,\dots,\mathcal{G}_Q\}\) partitioning the cross-sectional index set \(\{1,\dots,n\}\) such that \(a_{0,i} = a_{0,j} = \alpha_{0,q}\) and \(b_{0,i} = b_{0,j} = \beta_{0,q}\) for any \(i,j \in \mathcal{G}_q\), \(q \in \{ 1,\dots,Q\}\). Importantly, this `block-equidependent' factor structure implies that the number of latent marginals needed to be specified reduces from \(n\) to \(Q\) distinct distributions \(\textsf{G}_1,\dots,\textsf{G}_Q\), say, so that \(G_{i}(x) = G_{j}(x) \eqqcolon \textsf{G}_q(x)\) for any \(i,j \in \mathcal{G}_q\), \(q \in \{ 1,\dots,Q\}\). Throughout, the group assignment is assumed to be known.
The object of interest is the \(p \times 1\) vector \(\theta_0 \coloneqq (\alpha_{0,1}',\beta_{0,1}',\dots,\alpha_{0,Q}',\beta_{0,Q}',\gamma_0',\delta_0')' \in \Theta \subseteq \mathbb{R}^p\), which, in view of the latent factor structure (ref), collects all \(p \coloneqq Q(p_\alpha + p_\beta) +p_\alpha p_\gamma + p_\delta\) unknown copula parameters. A different parameter vector \(\theta \coloneqq (\alpha_{1}',\beta_{1}',\dots,\alpha_{Q}',\beta_{Q}',\gamma',\delta')' \in \Theta\) gives rise to an alternative factor structure
say, where the notational conventions \(X_{i,t}(d_i)\), \(F_t(\gamma) \coloneqq (F_{t,1}(\gamma_1),\dots,F_{t,p_\alpha}(\gamma_{p_\alpha}))'\), and \(\varepsilon_{i,t}(\delta)\) are used to make the dependence of the various quantities on the \((p_\alpha + p_\beta + p_\alpha p_\gamma + p_\delta) \times 1\) vector \(d_i \coloneqq (a_i',b_i',\gamma',\delta')'\) explicit; e.g., \(\varepsilon_{i,t}(\delta) \sim \textsf{D}_\varepsilon(\delta)\) and \(F_{t,j}(\gamma_j) \sim \textsf{D}_{F,j}(\gamma_j)\), \(j \in \{1, \dots, p_\alpha\}\). The block-equidependent design ensures that
where the \((p_\alpha + p_\beta + p_\alpha p_\gamma + p_\delta) \times 1\) vector \(\theta_q \coloneqq (\alpha_q',\beta_q',\delta',\gamma')'\) contains the parameters specific to the \(q\)-th group. Thus, with a slight abuse of notation, \({\displaystyle\theta =\cup_{q \,=\, 1}^Q \theta_q}\). For each \(\theta \in \Theta\), Eq. (ref) generates a differently parametrized copula
Importantly, due to the block-equidependent design and Eq. (ref), we have for any two cross-sectional indices belonging to the same group \[ G_{i}(x;d_i) = G_{j}(x;d_j) \eqqcolon \textsf{G}_q(x;\theta_q), \quad i,j \in \mathcal{G}_q,\, q \in \{1,\dots,Q\}, \] say, so that \(G_{i}(X_{i,t}(d_i);d_i) = U_{i,t}(d_i) = U_{i,t}(\theta_q) = \textsf{G}_q(X_{i,t}(\theta_q); \theta_q)\).
The simulation-based estimation uses Assumption (ref) below to estimate the true value of \(\theta \in \Theta\) given by \(\theta_0 = \cup_{q \,=\, 1}^Q \theta_{0,q}\), \(\theta_{0,q} = (\alpha_{0,q}',\beta_{0,q}',\gamma_0',\delta_0')'\).
Assumption (ref) formalizes the introductory notion of a factor copula; i.e., the unknown copula \(\textsf{C}(u_1,\dots,u_n)\) can be generated by the latent factor structure for a suitable choice of \(\theta = \theta_0 \in \Theta\).
Akin to the `independent simulation' scheme known from classical SMM estimation [see, e.g. mcf89, papo89, and lee92], we generate for a given candidate value \(\theta \in \Theta \subset \mathbb{R}^p\) a random sample \(\{(F_{t,s}(\gamma)',\varepsilon_{i,t,s}(\delta))': i=1,\dots,n,\,t = 1,\dots,T,\,s = 1,\dots,S\}\) to obtain a version of the auxiliary factor model
where \(F_{t,s}(\gamma) \coloneqq (F_{t,s,1}(\gamma_1),\dots,F_{t,s,p_\alpha}(\gamma_{p_\alpha}))'\). Hence, we sample for each time period \(t\) a new batch of \(S\) random variables from \(\textsf{D}_\varepsilon\) and \(\textsf{D}_{F,j}\), \(j \in \{1,\dots,p_\alpha\}\). More specifically, let \(\textsf{D}_\varepsilon^{-1}\) and \(\textsf{D}_{F,j}^{-1}\) denote the inverse distribution functions of \(\varepsilon_t\) and \(F_{t,j}\), \(j \in \{1,\dots,p_\alpha\}\), respectively. We may then write \(\varepsilon_{i,t,s}(\delta) \coloneqq \textsf{D}_\varepsilon^{-1}(\varepsilon_{i,t,s}^{\raisebox{.1pt}{\textnormal{\footnotesize$*$}}};\delta)\) and \(F_{t,s}(\gamma) \coloneqq \textsf{D}_F^{-1}(F_{t,s}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}};\gamma)\), with \(\textsf{D}_F^{-1}(F_{t,s}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}};\gamma) \coloneqq (\textsf{D}_{F,1}^{-1}(F_{t,s,1}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}};\gamma_1),\dots,\textsf{D}_{F,p_\alpha}^{-1}(F_{t,s,p_\alpha}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}};\gamma_{p_\alpha}))'\), where \(F_{t,s}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}} \coloneqq (F_{t,s,1}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}}, \dots, F_{t,s,p_\alpha}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}})'\) and \(\varepsilon_{i,t,s}^{\raisebox{.1pt}{\textnormal{\footnotesize$*$}}}\) denote independent draws of i.i.d. standard uniform random variates which are drawn once. Note that for our estimation procedure we hold the underlying random draws \(F_{t,s}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}}\) and \(\varepsilon_{i,t,s}^{\raisebox{.1pt}{\textnormal{\footnotesize$*$}}}\) fixed while \(\theta\) is allowed to vary over the compact set \(\Theta\). This is important to ensure uniform convergence of simulated moments and to facilitate convergence of the numerical optimization routine used to find \(\theta_0\); see gofo97 and papo89 for further remarks. Since \(Z_{t}\) might be unobservable, we will replace the unknown innovation with \(\hat{Z}_{t}(\hat{\nu}_{T}) \coloneqq (\hat{Z}_{t,1}(\hat{\nu}_{T}),\dots,\hat{Z}_{t,p_\beta}(\hat{\nu}_{T}))'\), where \(\hat{Z}_{t,j}(\nu) \coloneqq W_{t,j}-\sigma_j(M_t(\nu),\nu)\), \(j \in \{1,\dots,p_\beta\}\), represents the generalized residual. The estimator \(\hat{\nu}_{T}\) is assumed to be \(\sqrt{T}\)-consistent for the \(m \times 1\) vector \(\nu_{0}\) satisfying certain mild regularity conditions outlined below; for example, in the empirical application maximum likelihood estimation is used. Therefore, a feasible counterpart of Eq. (ref) is obtained from
Throughout, the cross-sectional dimension \(n\) might be large but is considered fixed, while asymptotics are carried out as \(T \rightarrow \infty\); the number of simulation draws \(S\) can either be fixed or a function of \(T\) such that \(S \coloneqq S(T) \rightarrow \infty\) as \(T \rightarrow \infty\). For the sake of brevity we report only results for the latter case.
Similar to ohpa13, estimation aims at minimizing the difference between empirical and simulated rank dependence measures that only depend on the unknown bivariate marginal copulae. Importantly, Assumption (ref) implies also the equivalence at \(\theta = \theta_0\) between each of the \(n(n-1)/2\) bivariate marginals copulae of the joint copula from Eq. (ref), given by \(\textsf{C}_{i,j}(u_i,u_j) \coloneqq \textsf{P}(V_{i,t} \leq u_i, V_{j,t} \leq u_j)\), and the bivariate marginals of the joint factor copula from Eq. (ref), given by \(\textsf{C}_{i,j}(u_i,u_j;d_i,d_j) \coloneqq \textsf{P}(U_{i,t}(d_i) \leq u_i, U_{j,t}(d_j) \leq u_j)\), \(1 \leq i < j \leq n\). By block-equidependence, \[ \textsf{C}_{i,j}(u_i,u_j;d_i,d_j) \eqqcolon \textsf{C}_{q}(u_i,u_j;\theta_{q}), \quad i,j \in \mathcal{G}_q,\; q \in \{1,\dots,Q\}. \] Put differently, the number of distinct marginal copulae reduces from \(n(n-1)/2\) to \(Q\) block-specific copulae \(\textsf{C}_1(\theta_{1}),\dots,\textsf{C}_Q(\theta_{Q})\) for which \(\textsf{C}_{i,j}(u_i,u_j) = \textsf{C}_{q}(u_i,u_j;\theta_{q})\) if \(i,j \in \mathcal{G}_q\), \(q \in \{1,\dots,Q\}\), and \(\theta_q = \theta_{0,q}\). To illustrate the main idea behind the following SMM estimator, suppose \(i,j \in \mathcal{G}_q\) for some \(q \in \{1,\dots,Q\}\) and introduce, for some estimator \(\sqrt{T}\)-consistent estimator \(\hat{\lambda}_{T}\) of \(\lambda_0\) satisfying some regularity conditions outlined below, the following two \(\ell \times 1\) vectors
which collect bivariate dependence measures like, for example, Spearman's \(\rho\) , Blomqvist's \(\beta\), Gini's \(\gamma\), or the measures of quantile dependence used by ohpa13. Formally, these statistics can be expressed with the help of a suitable collection of bivariate functions \(\{\varphi_k: [0,1]^2 \mapsto \mathbb{R}, 1\leq k \leq \ell\}\) as follows
where $\hat{V}_{i,t}(\hat{\lambda}_{T})$ and $\hat{U}_{i,t,s}(\theta_q,\hat{\nu}_{T})$ represent the rank of $\hat{\eta}_{i,t}(\hat{\lambda}_{T}) \coloneqq \{Y_{i,t}-\mu_{1,i}(\mathcal{F}_t,\hat{\lambda}_{T})\}/\mu_{2,i}(\mathcal{F}_t,\hat{\lambda}_{T})$ among $\{\hat{\eta}_{i,t}(\hat{\lambda}_{T}): t = 1,\dots,T\}$ and the rank of $\hat{X}_{i,t,s}(\theta_q,\hat{\nu}_{T})$ among $\{\hat{X}_{i,t,s}(\theta_q,\hat{\nu}_{T}): t = 1,\dots,T;s=1,\dots,S\}$, respectively.
The \(\ell\) different bivariate dependence measures are then aggregated according to the group-specific factor structure. To provide some intuition, note that \(\hat{\psi}_{T,i,j,k}(\hat{\lambda}_{T})\) and \(\hat{\psi}_{T,S,i,j,k}(\theta_q,\hat{\nu}_{T})\), \(k \in \{1,\dots,\ell\}\), can be viewed as sample estimates of the population statistics \(\textsf{E}[\varphi_k(V_{i,t},V_{j,t})]\) and \(\textsf{E}[\varphi_k(U_{i,t,s}(\theta_q),U_{j,t,s}(\theta_q))]\). These statistics depend only on the bivariate copulae \(\textsf{C}_{q}(u_i,u_j)\) and \(\textsf{C}_{q}(u_i,u_j;\theta_q)\), which, due to the block-equidependence, exhibit within-group homogeneity; i.e., each of the \(\ell\) statistics depends on the cross-sectional index set \(\{1,2,\dots,n\}\) only via the group-identifier \(q \in \{1,\dots,Q\}\):
Therefore, the following aggregation scheme of bivariate dependence measures is justified
We thus obtain the \(\bar{\ell} \times 1\), \(\bar{\ell} \coloneqq Q\ell\), vector of empirical dependence measures \[ \hat{\psi}_{T}(\hat{\lambda}_{T}) \coloneqq (\hat{\psi}_{T,1}(\hat{\lambda}_{T})',\dots,\hat{\psi}_{T,Q}(\hat{\lambda}_{T})')' \] and the \(\bar{\ell} \times 1\) vector of simulated dependence measures \[ \hat{\psi}_{T,S}(\theta,\hat{\nu}_{T}) \coloneqq (\hat{\psi}_{T,S,1}(\theta_1,\hat{\nu}_{T})',\dots,\hat{\psi}_{T,S,Q}(\theta_Q,\hat{\nu}_{T})')', \] respectively. Following the literature on extremum estimators [see, e.g., nemc94], we define, for some stochastically bounded and positive-definite weight matrix \(\hat{L}_{T,S}\), an SMM estimator \(\hat{\theta}_{T,S}\) as an estimator that minimizes the objective function \[\hat{A}_{T,S}(\theta,\hat{\lambda}_{T},\hat{\nu}_{T}) \coloneqq \hat{\Psi}_{T,S}(\theta,\hat{\lambda}_{T},\hat{\nu}_{T})'\hat{L}_{T,S}\hat{\Psi}_{T,S}(\theta,\hat{\lambda}_{T},\hat{\nu}_{T}),\] with \(\hat{\Psi}_{T,S}(\theta,\hat{\lambda}_{T},\hat{\nu}_{T}) \coloneqq \hat{\psi}_{T}(\hat{\lambda}_{T}) - \hat{\psi}_{T,S}(\theta,\hat{\nu}_{T})\), in the sense that
As pointed out by ohpa13, the objective function is non-differentiable and, in general, does not posses a population counterpart in known closed form. Thus, some care is required in deriving the asymptotic distribution of \(\hat{\theta}_{T,S}\). Due to the mutual dependence on the covariate, \(\hat{\psi}_{T}(\hat{\lambda}_{T})\) and \(\hat{\psi}_{T,S}(\theta,\hat{\nu}_{T})\) are not independent, which considerably complicates the analysis and precludes a direct application of the arguments developed by the aforementioned authors. To shed some light, let us recall from nemc94 that two crucial high-level conditions for asymptotic normality of \(\sqrt{T}(\hat{\theta}_{T,S}-\theta_0)\) are (1) limiting normality of the normalized sample `moments' \(\sqrt{T}\hat{\Psi}_{T,S}(\theta,\hat{\lambda}_{T},\hat{\nu}_{T})\) evaluated at \(\theta = \theta_0\) and (2) the stochastic equicontinuity of the map \(\theta \mapsto \sqrt{T}\hat{\Psi}_{T,S}(\theta,\hat{\lambda}_{T},\hat{\nu}_{T})\). Both conditions are shown to be closely tied to the limiting behaviour of the triangular-array empirical process
where $\textsf{H}^-(p) \coloneqq \operatorname*{\textsf{inf}}\{x \in \mathbb{R}: \textsf{H}(x) \geq p,\,p \in (0,1]\}$ denotes the left-continuous generalized inverse function of a distribution function $\textsf{H}$, and \[ \hat{\textsf{F}}_{T,k}(x;\hat{\lambda}_{T}) \coloneqq \frac{1}{T}\sum_{t \,=\, 1}^T1\{\hat{\eta}_{k,t}(\hat{\lambda}_{T}) \leq x\},\;\; \hat{\textsf{G}}_{T,S,k}(x;\theta_q,\hat{\nu}_{T}) \coloneqq \frac{1}{TS}\sum_{t \,=\, 1}^T\sum_{s \,=\, 1}^S1\{\hat{X}_{k,t,s}(\theta_q,\hat{\nu}_{T}) \leq x\}, \] for $x \in \mathbb{R}$, $k \in \{i,j\}$, with \(i,j \in \mathcal{G}_q\), \(q \in \{1,\dots,Q\}\). More specifically, taking fetal04 and buse13 into account, we can express the $k$-th entry of $\hat{\Psi}_{T,S,i,j}(\theta_q,\hat{\lambda}_{T},\hat{\nu}_{T}) \coloneqq \hat{\psi}_{T,i,j}(\hat{\lambda}_{T})-\hat{\psi}_{T,S,i,j}(\theta_q,\hat{\nu}_{T})$ as a Lebesgue-Stieltjes integral
for \(k \in \{1,\dots,\ell\}\). Hence, the limiting distribution of $\sqrt{T}\hat{\Psi}_{T,S}(\theta_0,\hat{\lambda}_{T},\hat{\nu}_{T})$ can be deduced from the weak convergence of the process $\{\hat{\mathbb{B}}_{T,S,i,j}(u_i,u_j;\theta_{0,q},\hat{\lambda}_{T},\hat{\nu}_{T}): u_i,u_j \in [0,1]\}$, which is readily recognized as the difference between two empirical copula processes. Since filtered data is used, an invariance result with respect to the corresponding statistics based on the unknown counterparts is desirable. If empirical and simulated rank statistics are independent, then it suffices to show that the empirical copula processes based on $\hat{V}_{i,t}(\hat{\lambda}_{T})$ and $\hat{U}_{i,t,s}(\theta_{0,q},\hat{\nu}_{T})$ share the same weak limit; a proof strategy employed by ohpa13 who argue along the lines of rem17. Here, we require the somewhat stronger notion of uniform asymptotic negligibility; i.e., we show, under sufficient regularity of the data, that
for each \(i,j \in \mathcal{G}_q\), \(q \in \{1,\dots,Q\}\). There exist already similar results in the literature for $\beta$-mixing processes [see neumetal19 and chetal20 who rely on detal09 and avk01]; the underlying stochastic equicontinuity result of dotal95 is, however, not directly applicable to the triangular-array case considered here. In order to overcome this difficulty, we resort to the FCLT of anpo94. The following regularity conditions are assumed to hold:
The smoothness condition (ref)---due to seg12---is needed to apply the functional delta method; see also buvo13. Assumptions (ref) and (ref) are similar to regularity conditions imposed by neumetal19; as discussed in cotetal19 and ometal20, this assumption can be relaxed at the expense of additional technicalities. Assumption (ref) summarizes conditions which, in conjunction with the remaining assumptions, ensure the asymptotic equicontinuity of \(\theta_q \mapsto \hat{\mathbb{B}}_{T,S,i,j}(\theta_q)\). When compared to similar conditions used by manetal20, Assumption (ref) is relatively primitive. The assumption is, for example, satisfied if factors and idiosyncratic errors are Gaussian. To give a less trivial example, suppose a scalar factor \(F_t\) follows a (standardized) Student's \(t\)-distribution with degrees of freedom parameter \(2 < \underaccent{\bar}{\gamma} \leq \gamma_0 \leq \bar{\gamma} < \infty\). Then, Assumption (ref) ($ii$) is satisfied by setting \(\dot{Q}_F(u) \coloneqq \sqrt{\bar{\gamma} \vee 1/(\underaccent{\bar}{\gamma}-2)}|\tilde{D}^{-1}(u,\underaccent{\bar}{\gamma})|\), where \(\tilde{D}^{-1}(u,\underaccent{\bar}{\gamma})\) is the inverse of the non-standardized Student's \(t\)-distribution so that \[ \frac{\underaccent{\bar}{\gamma}-1}{\underaccent{\bar}{\gamma}}\int_{[0,1]}\dot{Q}_F(u)\, \textsf{d} u \leq \sqrt{\frac{2}{\pi}\left(\bar{\gamma} \vee \frac{1}{\underaccent{\bar}{\gamma}-2}\right)} < \infty; \] see the online supplement for details. Assumption (ref) concerns the marginal time-series models: part (ref) is high-level and can be verified for many estimators of AR-GARCH type-models [see fraz04 for a more primitive underpinning]; part (ref) is similar to chef06 and means that the gradient vectors of the location and scale functions are locally dominated. The \(\alpha\)-mixing sizes in part (ref) are chosen as to match the conditions of the FCLT in anpo94. Assumption (ref) would, for instance, be satisfied by many stationary AR-GARCH processes with geometric mixing rate; see, e.g., cachen02, fryrao11, or liuyang16. Assumption (ref) is a regularity condition needed to establish the weak convergence of the `finite-dimensional distributions' of (ref) based on an argument borrowed from boist17. The assumption does not seem overly restrictive as similar results exist for univariate unconditional distributions; see, e.g., boist17. Bounded variation in the sense of Hardy-Krause, imposed by Assumption (ref) on the functions \(\varphi_k: [0,1]^2 \mapsto \mathbb{R}\), ensures an integration by parts formula for bivariate integrals; see fetal04 and radetal17. Since the identity map or indicators of axis-parallel boxes are of bounded Hardy-Krause variation [see, e.g., owen05], dependence measures used here and in ohpa13 like Spearman's \(\rho\) and quantile dependence can be expressed in terms of Eq. (ref) using functions that satisfy this assumption. Assumption (ref) implies Riemann-integrability and thus boundedness [see owenrud21], a strong assumption, admittedly, but one which completely suffices here and that could in principle be relaxed as pointed out by betal17.
Proposition (ref) distills the main ingredients needed to derive the asymptotic properties of the SMM estimator.
The stochastic equicontinuity result of Proposition (ref) provides the link between the pointwise properties of $\theta \mapsto \hat{\Psi}_{T,S}(\theta,\hat{\lambda}_{T},\hat{\nu}_{T})$ and the asymptotic behavior of $\hat{\theta}_{T,S}$. To make the argument precise, an additional set of regularity conditions is imposed.
Assumption (ref) is common for extremum estimators with non-smooth objective function; see, e.g., nemc94. Analogously to ohpa13, we make use of nemc94 to derive the asymptotic normality of the SMM estimator.
We follow ohpa13 by resorting to numerical derivatives and the bootstrap to estimate \(\dot{\psi}_0\) and the limiting variance-covariance matrix \(\Sigma_{0}\), respectively. The former estimator is almost completely analogous to the one used by ohpa13; i.e., for a step size \(\pi_T \rightarrow 0^+\) define \(\hat{\dot{\psi}}_{T,S}\), whose \(k\)-th column is given by \[ \hat{\dot{\psi}}_{T,S,k} \coloneqq \frac{\hat{\psi}_{T,S}(\hat{\theta}_{T,S}+e_k\pi_{T},\hat{\nu}_{T})-\hat{\psi}_{T,S}(\hat{\theta}_{T,S}-e_k\pi_{T},\hat{\nu}_{T})}{2\pi_{T}},\;\; k \in \{1,\dots,p\}, \] where \(e_k\) denotes the \(p\)-dimensional vector of zeros with one at \(k\)-th position.
Contrary to the aforementioned authors, however, the estimator of \(\Sigma_{0}\) needs to account for the dependence structure induced by the exogenous regressor. Therefore, we propose standard errors based on bootstrap replications of both empirical and simulated statistics. More specifically, draw for each \(i,j \in \mathcal{G}_q\), \(q \in \{1,\dots,Q\}\), with replacement \(B\) bootstrap samples \(\big\{\mathcal{Z}_{S,t,i,j}^{(b)}(\hat{\theta}_{T,S,q},\hat{\lambda}_{T},\hat{\nu}_{T})\big\}_{t = 1}^T\), \(b \in \{1,\dots,B\}\), from \(\big\{\mathcal{Z}_{S,t,i,j}(\hat{\theta}_{T,S,q},\hat{\lambda}_{T},\hat{\nu}_{T})\big\}_{t = 1}^T\), where
with \(\hat{\theta}_{T,S,q}\) representing the \((p_\alpha + p_\beta + p_\alpha p_\gamma + p_\delta) \times 1\) sub-vector of \(\hat{\theta}_{T,S}\) that contains the SMM estimates pertaining to group \(\mathcal{G}_q\). Next, denote the corresponding ranks by \(\hat{V}_{k,t}^{(b)}(\hat{\lambda}_{T})\), \(\hat{U}_{k,t,s}^{(b)}(\hat{\theta}_{T,S},\hat{\nu}_{T})\), \(k \in \{i,j\}\). In view of Eq. (ref), introduce the bootstrap rank-based dependence measures
Akin to the discussion surrounding (ref), let \(\hat{\Psi}_{T,S}^{(b)}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T})\) denote the \(\bar{\ell} \times 1\) vector of group-averages of \(\hat{\Psi}_{T,S,i,j}^{(b)}(\hat{\theta}_{T,S,q},\lambda,\lambda,\hat{\nu}_{T})\coloneqq \hat{\psi}_{T,i,j}^{(b)}(\hat{\nu}_{T})-\hat{\psi}_{T,S,i,j}^{(b)}(\hat{\theta}_{T,S,q},\hat{\lambda}_{T})\). Following the arguments made by fetal04, we can then show that the conditional distribution of \(\sqrt{T}(\hat{\Psi}_{T,S}^{(b)}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T})-\hat{\Psi}_{T,S}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T}))\) consistently estimates the limiting distribution of \(\sqrt{T}\hat{\Psi}_{T,S}(\theta_0,\hat{\lambda}_{T},\hat{\nu}_{T})\). Importantly, since the first-step estimation of the location-scale parameters does not contaminate the limiting distribution of \(\sqrt{T}\hat{\Psi}_{T,S}(\theta_0,\hat{\lambda}_{T},\hat{\nu}_{T})\), there is no need to adjust for this source of uncertainty as is, for example, done in gonetal19. Hence, the limiting variance-covariance \(\Sigma_{0,S}\) is consistently estimable by the bootstrap second moment\footnote{Throughout, `${\raisebox{.1pt}{\textnormal{\footnotesize$*$}}}$' indicates that the given probability/moment has been computed under the bootstrap distribution conditional on the original sample.} \(\textsf{cov}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}}[\sqrt{T}\hat{\Psi}_{T,S}^{(b)}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T})]\), provided \(\sqrt{T}(\hat{\Psi}_{T,S}^{(b)}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T})-\hat{\Psi}_{T,S}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T}))\) is uniformly square integrable; see, e.g., brwe02 or cheng15. Since an estimator of \(\textsf{cov}^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}}[\sqrt{T}\hat{\Psi}_{T,S}^{(b)}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T})]\) is given by \[ \hat{\Sigma}_{T,S,B} \coloneqq \frac{T}{B}\sum_{b \,=\, 1}^B (\hat{\Psi}_{T,S}^{(b)}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T})-\hat{\Psi}_{T,S}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T}))(\hat{\Psi}_{T,S}^{(b)}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T})-\hat{\Psi}_{T,S}(\hat{\theta}_{T,S},\hat{\lambda}_{T},\hat{\nu}_{T}))', \] we can introduce the following consistent estimator of \(\Omega_{0,S}\)
Corollary (ref) allows to conduct inference about \(\theta_0\) and to obtain the two-step SMM estimator with optimal weight matrix \(\hat{L}_{T,S} = \hat{\Sigma}_{T,S,B}^{-1}\). More primitive conditions under which the the uniform square integrability holds are not readily available. However, as discussed by hahnliao21, if this assumption fails, bootstrap standard errors based on \(\hat{\Sigma}_{T,S,B}\) are likely to yield conservative tests. Provided \(\bar{\ell} > p\), the preceding result can also be used to ascertain overidentifying restrictions based on the Sargan-Hansen type \(J\)-statistic
where \(\mathcal{A}_{0} \coloneqq L_{0}^{1/2}\Sigma_{0}^{1/2}R_{0}\), with \(R_{0} \coloneqq I_{\bar{\ell} \times \bar{\ell}} -\Sigma_{0}^{-1/2}\dot{\psi}_0(\dot{\psi}_0'L_{0} \dot{\psi}_0)^{-1}\dot{\psi}_0'L_{0}\Sigma_{0}^{1/2}.\) Critical values for \(J_{T,S}\) need to be simulated (using estimators of \(\Sigma_{0}\) and \(\dot{\psi}_0\)) unless the optimal weight matrix is used, in which case the common result \(J_{T,S} \stackrel{d}{\longrightarrow} \chi^2(\bar{\ell}-p)\) obtains; see also ohpa13.
The Monte Carlo experiment uses a data generating process similar to that in ohpa13; that is, we consider an AR(1)-GARCH(1,1) process to describe the evolution of each of \(n\) assets over time:
where \(\eta_t \coloneqq (\eta_{1,t},\dots,\eta_{n,t}) \sim \textsf{C}(\Phi_1,\dots,\Phi_n),\) with \(\Phi_i\), \(i\in \{1,\dots,n\}\), denoting the marginal (Gaussian) distribution function of \(\eta_{i,t}\). The copula \(\textsf{C}\) is generated by the following `block-equidependent' factor model
We consider three groups \(\mathcal{G}_1, \mathcal{G}_2,\) and \(\mathcal{G}_3\), of equal size partitioning the cross-sectional index set; i.e., \(\{1,2,\dots,n\} = \mathcal{G}_1 \cup \mathcal{G}_2 \cup \mathcal{G}_3\), with \(|\mathcal{G}_q| = n/3, q \in \{1,2,3\}\) for \(n \in \{15,\,30\}\). The factor \(F_t\) and the idiosyncratic component \(\varepsilon_{i,t}\) are latent but simulable. It is assumed that \(F_t \stackrel{\textsf{IID}}{\sim} t(\zeta_0,\xi_0)\), with \(\zeta_0 = 1/4,\,\xi_0 = -1/2\), where \(t(\zeta_0,\xi_0)\) denotes Hansen's standardized skewed t-distribution with tail thickness parameter \(2 < 1/\zeta < \infty\) and skewness parameter \(-1 < \xi < 1\). For the idiosyncratic component, we either consider \(\varepsilon_{i,t} \stackrel{\textsf{IID}}{\sim} t(\zeta_0,0)\) (called the skew-\(t\)/\(t\) specification) or \(\varepsilon_{i,t} \stackrel{\textsf{IID}}{\sim} \mathcal{N}(0,1)\) (called the skew-\(t\)/normal specification). Turning to the loadings and the estimable factors, we consider two `block-equidependent' specifications similar to the multilevel model of bw15; see the summary of Table (ref) . {1pt}
Under design 1, loadings on \(F_t\) are group specific while the loading on the single estimable factor is common; the latter factor has to be estimated from an AR(1) model. Under design 2, the loading on \(F_t\) is common while three estimable factors have group-specific loadings; the estimable factors have to be estimated from an AR(1) model, from a GARCH(1,1) model, or follow observable white noise. Hence, design 1 and design 2 imply 6 unknown copula parameters \(\theta_0 = (\alpha_{0,1},\alpha_{0,2},\alpha_{0,3},\beta_0,\zeta_0,\xi_0)'\) and \(\theta_0 = (\alpha_{0},\beta_{0,1},\beta_{0,2},\beta_{0,3},\zeta_0,\xi_0)'\), respectively. Estimation of \(\theta_0\) is in both cases based on Spearman's rank correlation \[ \hat{\varphi}_{T,i,j,1} \coloneqq \frac{12}{T}\sum_{t \,=\, 1}^T \hat{V}_{i,t}(\hat{\lambda}_{T})\hat{V}_{j,t}(\hat{\lambda}_{T})-3,\; \hat{\varphi}_{T,S,i,j,1} \coloneqq \frac{12}{ST}\sum_{t \,=\, 1}^T\sum_{s \,=\, 1}^S \hat{U}_{i,t,s}(\theta_q,\hat{\nu}_{T})\hat{U}_{j,t,s}(\theta_q,\hat{\nu}_{T})-3 \] and quantile dependence
Throughout, we set \(T \in \{\)500,\, 1,000,\, 2,000\(\}\), \(S = 25\), and use the identity weight matrix \(L_{T,S} = I_{\bar{\ell}}\). We make use of quantile dependence for \(\tau \in \{0.05,0.10,0.90,0.95\}\) alongside Spearman's rank correlation, which yields \(\bar{\ell} = 3 \times 5 = 15\) rank-based dependence measures for estimation. Numerical optimization employs a derivative-free simplex search based on MATLAB's (2019a) fminsearchbnd routine; see derr21. The starting values are obtained from a first-step surrogate minimization using MATLAB's (2019a) surrogateopt optimization for time-consuming objective functions and the individual time-series models are estimated using maximum likelihood.
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Tables (ref), (ref), (ref), and (ref) contain Monte Carlo estimates of mean, median, and variance of the SMM estimator using 500 Monte Carlo iterations\footnote{The computations were implemented in Matlab, parallelized and performed using CHEOPS, the DFGfunded (Funding number: INST 216/512/1FUGG) High Performance Computing (HPC) system of the Regional Computing Center at the University of Cologne (RRZK).}. Moreover, we report rejection frequencies of two-sided $t$-tests under the null \(\theta = \theta_0\) and rejection frequencies of the test of overidentifying restrictions (ref). Both hypothesis tests are investigated at a nominal significance level of five percent and the test statistics are equipped with the bootstrap standard error (ref). We use \(B =\) 500 bootstrap replications and set the tuning parameter for the numerical derivative to \(\pi_T = 0.05\); moreover, we use 1,000 random draws to obtain critical values for the test of overidentifying restrictions (ref). We report results for the feasible and the unfeasible SMM estimator that differ with respect to whether the parameters governing the estimable factors reported in Table (ref) are estimated (feasible) or treated as known constants (unfeasible). The simulation evidence reveals that, in accordance with the theory, the estimation accuracy increases with \(T\). Although some size distortions can be observed for \(T = 500\), rejection frequencies are close to the nominal significance level when \(T \geq\) 1,000. Moreover, the feasible estimator performs almost equally well as its unfeasible counterpart.
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We apply the above to study the cross-sectional dependence between \(n = 43\) companies matching the first four largest groups of the S&P 100 found by ohpa21. Specifically, we consider daily close prices, adjusted for stock splits and dividends from January 2014 to January 2020 resulting in \(T = \) 1,461 trading days.
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Since gold often acts as an hedge and/or a safe haven for stock markets, its price may convey information about the inter-dependencies between stock returns; see, e.g., Baurmc10. Hence, we examine the extent to which information on gold prices can help us to describe the cross-sectional dependence structure among the 43 companies. The conditional mean of the (percentage) logarithmic return of the \(i\)-th stock price \(Y_{i,t}\), \(i \in \{1,\dots,43\}\), is modeled as an AR(1) process augmented with the first lag of the (percentage) logarithmic change of the three p.m. gold fixing price in London bullion market \(W_t\)
while, similar to ohpa13, ohpa17, ohpa21, the conditional variance \(\mu_{i,t}^2\) is assumed to follow a GJR-GARCH(1,1) model
As we clearly fail to reject the null hypothesis\footnote{The $p$-value of a Wald test with Newey-West standard errors is about \(0.77\).} of a zero conditional mean based on an AR(1) specification with unrestricted constant, a GJR-GARCH(1,1) model is considered for the gold price
Thus, the use of \(\textsf{log}|\hat{Z}_{t-1}|\) as an estimable factor is justified because, as mentioned earlier, the logarithmic transformation fits into the location specification (ref). Table (ref) summarizes descriptive statistics alongside the results from quasi maximum-likelihood estimation with skewed Student's $t$-distributed innovations. The stock returns are left-skewed and leptokurtic with conditional mean and variance dynamics that are similar to findings from the literature; see, e.g., boletal94. Note that the distribution of \(W_t\) (gold), while also leptokurtic, is right-skewed.
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Turning to the specification of the cross-sectional distribution of the 43 companies, we use various skewed-t factor models, inspired by ohpa13, ohpa17, as our benchmark specifications for the copula. Based on Table (ref) we consider the following block-equidependent design
where \(\mathcal{G}_1 = \{1,\dots,13\}\), \(\mathcal{G}_2 = \{14,\dots,24\}\), \(\mathcal{G}_3 = \{25,\dots,35\}\), and \(\mathcal{G}_4 = \{36,\dots,43\}\). We assume that \(F_t \stackrel{\textsf{IID}}{\sim} t(\zeta,\xi)\), \(F_{j,t} \stackrel{\textsf{IID}}{\sim}t(\zeta)\), and \(\varepsilon_{i,t} \stackrel{\textsf{IID}}{\sim} t(\zeta)\), while factors and idiosyncratic errors are mutually independent. We use the following four versions of Eq. (ref), labeled A1, A2, A3, and A4, imposing certain restrictions on the loadings: specification A1 is a one-factor equidependent model so that \(\alpha_{1,j} = \alpha\) and \(\alpha_{2,j} = 0\); specification A2 is a one-factor block-equidependent model with \(\alpha_{2,j} = 0\); specification \textsf{A3} has a common factor with common loading \(\alpha_{1,j} = \alpha\) and group-specific factors with group-specific loadings; specification \textsf{A4} does not impose any restrictions and allows for a common factor and group-specific factors, both with group-specific loadings. The following competitor for the copula is generated from a factor model with estimable gold factor:
where \(\varepsilon_{i,t} \stackrel{\textsf{IID}}{\sim} t(\zeta)\). Similar to Eq. (ref), four versions of Eq. (ref), labeled B1, B2, B3, and B4, are considered: specification B1 allows for group-specific loadings on a common simulable factor \(F_{j,t} = F_t \stackrel{\textsf{IID}}{\sim}t(\zeta,\xi)\) and imposes \(\beta_j = \beta\); specification B2 imposes \(\beta_j = \beta\) but allows for group-specific simulable factors \(F_{j,t} \stackrel{\textsf{IID}}{\sim}t(\zeta,\xi)\) with group-specific loadings; specifications \textsf{B3} and \textsf{B4} allow for group-specific loadings assuming a common simulable factor \(F_{j,t} = F_{t} \stackrel{\textsf{IID}}{\sim}t(\zeta,\xi)\), \(\xi = 0\) (\textsf{B3}) and group-specific simulable factors \(F_{j,t} \stackrel{\textsf{IID}}{\sim}t(\zeta,\xi)\), \(\xi = 0\) (\textsf{B4}), respectively.
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Table (ref) summarizes the SMM estimation results for the benchmark specification (ref) as well as for the counterpart with estimable gold factor (ref) based on the same rank-based dependence measures used in the Monte Carlo study. We set \(S = 25\), \(B = \) 2,000, and \(\pi_T = 0.05\) and report estimation results using identity weighting \(L_{T,S} = I_{\bar{p}}\) and `optimal' weighting \(L_{T,S} = \hat{\Sigma}^{-1}_{T,S}\). The point estimates for the benchmark specifications in the upper panel are in line with the values reported in ohpa17 and suggest significant (negative) asymmetric dependence and significant tail dependence. As can be deduced from the results of the overidentifying restrictions test (ref), all specifications but A3 are clearly rejected by the data. Moving to the lower panel, we find results for our competitors with estimable factor. The specifications improve the performance of the benchmark models and cannot (at least for the identity weight matrix) be rejected by the data. Interestingly, conditionally on the estimable gold factor, the asymmetry parameter is smaller in size and no longer statistically significant different from zero. This can be explained by the fact that the estimable gold factor already accounts for (some) asymmetry. To illustrate, Figure (ref) depicts the distribution of the logarithmic absolute residuals, which is seen to be left-skewed; for comparison, the density of \(\textsf{log}|Z_t^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}}|\), \(Z_t^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}} \sim \mathcal{N}(0,1)\), is depicted as the solid line in panel A.\footnote{If \(Z^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}} \sim\mathcal{N}(0,1)\), then \(Z \coloneqq \textsf{log}|Z^{\raisebox{1pt}{\textnormal{\footnotesize$*$}}}|\) has absolutely continuous density \(\textsf{f}\) \[\textsf{f}(z) \coloneqq \sqrt{\frac{2\,\textsf{exp}\{2z - \textsf{exp}\{2z\}\}}{\pi}}, \;\;z \in \mathbb{R}, \] with mean \(\textsf{E}[Z] = (-\gamma-\textsf{log}\, 2)/2\) and variance \(\textsf{var}[Z] = \pi^2/8\), where \(\gamma \approx 0.5772\) is the Euler‐Mascheroni constant. The density given by the preceding display is depicted as the solid line in Figure (ref).}
We derive the asymptotic properties of an SMM estimator of the unknown parameter vector governing a factor copula model with estimable factors and show how to estimate its limiting variance-covariance matrix consistently. The asymptotic theory is derived from primitive conditions, thereby complementing also the earlier work of ohpa13, whose model is nested in our framework. One avenue for future resarch that can be pursued is to consider quasi-Bayesian estimation to alleviate the difficulties of having to deal with a non-smooth objective function. For example, the sample criterion function considered may be shown to fulfill the regularity conditions needed for Laplace-type estimation in cheho03; see also hongetal21 for a recent application of this idea to an SMM objective function with overlapping simulation draws.
\addcontentsline{toc}{section}{References}