Double Dynamic Max-Copula Model with Application to Financial Time Series

Near universal consistency of the maximum pseudolikelihood estimator for discrete models

Partially ordered Markov models (POMMs) are Markov random fields (MRPs) with neigh­ borhood structures derivable from an associated partially ordered set. The most attractive feature of POMMs is that their joint distributions can be written in closed and product form. Therefore, simulation and maximum likelihood estimation for the models is quite straightfor­ ward, which is not the case in general for MRP models. In practice, one often has to modify the likelihood to account for edge components; the resulting composite likelihood for POMMs is similarly straightforward to maximize. In this article, we use a martingale approach to derive the asymptotic properties of maximum (composite) likelihood estimators for POMMs. One of our results establishes that, under regularity conditions that include Dobrushin's condition for spatial mixing, the maximum composite likelihood estimator is consistent, asymptotically normal, and also asymptotically efficient. AMS 1991 subject classifications: Primary 62M40; secondary 62F12.

A composite likelihood approach to (co)variance components estimation

We introduce the conditional maximum composite likelihood (MCL) estimation method for the stochastic factor ordered probit model of credit rating transitions of firms. This model is recommended for internal credit risk assessment procedures in banks and financial institutions under the Basel III regulations. Its exact likelihood function involves a high-dimensional integral, which can be approximated numerically before maximization. However, the estimated migration risk and required capital tend to be sensitive to the quality of this approximation, potentially leading to statistical regulatory arbitrage. The proposed conditional MCL estimator circumvents this problem and maximizes the composite log-likelihood of the factor ordered probit model. We present three conditional MCL estimators of different complexity and examine their consistency and asymptotic normality when n and T tend to infinity. The performance of these estimators at finite T is examined and compared with a granularity-based approach in a simulation study. The use of the MCL estimator is also illustrated in an empirical application.

We provide three new results concerning quasi-maximum likelihood (QML) estimators in generalized autoregressive conditional heteroskedastic in mean (GARCH-M) models. We first show that, depending on the functional form that we impose in the mean equation, the properties of the model may change and the conditional variance parameter space may be restricted, in contrast to the theory of traditional GARCH processes. Second, we also present a new test for GARCH effects in the GARCH-M context which is simpler to implement than alternative procedures such as in Beg et al. (2001). We propose a new way of dealing with parameters that are not identified by creating composites of parameters that are identified. Third, the finite sample properties of QML estimators are explored in a restricted ARCH-M model and bias and variance approximations are found which show that the larger the volatility of the process the better the variance parameters are estimated. The invariance properties that Lumsdaine (1995) proved for the traditional GARCH are shown not to hold in the GARCH-M. For those researchers who choose not to rely on the first order asymptotic approximation of our proposed test statistic, we also show how our bias expressions can be used to bias correct the QML estimates with a view to improving the finite sample performance of the test. Finally, we show how our new proposed test works in practice in an empirical economic application.

The most popular estimation approach is the maximum likelihood (ML) method. In this chapter, the ML estimator is defined first, and important asymptotic properties of the ML estimator are formulated in Sect. 4.2. Trans- formations of estimators, not only ML estimators, are discussed in Sect. 4.3. To illustrate the ML approach, we consider the ML method in the linear exponential family (Sect. 4.4) and in univariate GLM (Sect. 4.5). A crucial assumption of ML estimation is the correct specification of the underlying statistical model. Therefore, we discuss the consequences of using the ML method in misspecified models in Sect. 4.6. Even if the model is misspecified, it is based on a likelihood, and the resulting estimator is therefore called a quasi maximum likelihood (QML) estimator (for an in-depth discussion, see White, 1982, 1994). The reader should note that QML estimation is different from quasi likelihood (QL) estimation. The latter approach is a generalization of the generalized linear model (McCullagh and Nelder, 1989; Wedderburn, 1974) and requires the correct specification of the first two moments.

The maximum approximate composite marginal likelihood (MACML) estimation of multinomial probit-based unordered response choice models

Cluster-type point process models are the popular model for modeling the arrangement of locations of earthquake occurrences. When a spatial trend is presence due to e.g. geological factors, the log-linear intensity of the point process is often considered to exploit inhomogeneity due to such factors. However, this could be a major drawback, especially in seismology when the relation between the intensity of earthquake occurrences and environmental covariates is not log-linear. In this paper, we consider the Cauchy cluster process with a log-additive intensity model to quantify two effects in modeling the distribution of locations of major earthquakes in Sulawesi-Maluku: (1) spatial trend due to geological covariates such as subduction zones, faults, and volcanoes and (2) clustering effect due to seismic activities. The Cauchy cluster process could detect the clustering effect even when the aftershocks are extremely distant to the mainshocks while log-additive intensity is a more flexible model to study inhomogeneity due to the environment. To estimate the parameters, we apply two-step estimation procedure, in the first step, we estimate the regression parameter corresponding to effects of the geological variables by maximum composite likelihood by involving penalized iteratively re-weighted least squares (PIRLS) technique, and in the second step, we obtain the cluster estimates by maximum second order composite likelihood. The results show that the active faults and volcanoes are significant covariates that trigger earthquakes, the estimated mainshock intensity is around 78, and the aftershocks are distributed around it with a distance of 15.2 km due to mainshock activity.

ABSTRACTThis article establishes the almost sure convergence and asymptotic normality of levels and differenced quasi maximum likelihood (QML) estimators of dynamic panel data models. The QML estimators are robust with respect to initial conditions, conditional and time-series heteroskedasticity, and misspecification of the log-likelihood. The article also provides an ECME algorithm for calculating levels QML estimates. Finally, it compares the finite-sample performance of levels and differenced QML estimators, the differenced generalized method of moments (GMM) estimator, and the system GMM estimator. The QML estimators usually have smaller— typically substantially smaller—bias and root mean squared errors than the panel data GMM estimators.

The class of nonlinear time series models known as autoregressive conditional duration [ACD] models plays a central role for modeling durations, also known as waiting-times. These waiting-times/durations are positive random variables which are defined in relation to an irregularly observed time series of events occurring at random points in time. For example, the waiting time for the change in the share-price of an asset to exceed a certain threshold, or the duration between consecutive financial transactions such as share trading. This thesis focusses on the ACD family of models for such durations or waiting times. The ACD model describes a duration process {Zi } by Zi = Ψiei , where Ψi is the expected duration conditional on the past information, and ei is an independent and identically distributed [iid] positive error term with E(ei ) = 1. An ACD model is characterized by two specifications: (i) the specification of the conditional mean functionΨi , and (ii) the specification of the distribution of the error term ei . A number of parametric models have been proposed for (i) and (ii). A large proportion of these models have complicated probabilistic structures. As a consequence, assessing the goodness-of-fit of such models is a non-trivial task. However, the methodology for inference in ACD models is still in the early stages of development, and hence the literature on testing for the goodness or lack of fit of such parametric models is relatively scant. Thus, there is a need for developing new methodologies for testing parametric specifications for the ACD class of models. This thesis contributes to this broad area of statistical inference. Our results and methodology are presented in detail in Chapters 3, 4 and 5 of this thesis. The main objectives and contributions of this thesis can be summarized as follows. (a) Martingale transformation techniques have been successfully adapted in the literature to develop asymptotically distribution free specification tests for regression and autoregressive models. We extend this transformation technique to multiplicative models in Chapter 3, in order to develop an asymptotically distribution free test for the specification of the conditional mean of an ACD model. (b) In Chapter 4 we introduce a new parametric form which nests all linear specifications of the conditional mean function of an ACD model, and develop a class of tests for checking its goodness-of-fit. In the process of deriving the asymptotic distributions of the proposed test statistics, we show that a quasi maximum likelihood estimator [QMLE] is root-n consistent and asymptotically normal for the proposed family of models. Further, a residual-based bootstrap procedure is proposed for computing the critical values of the test statistics, and its asymptotic validity is established. (c) Finally, in Chapter 5 a new class of tests is developed for fitting a parametric form for the error distribution of an ACD model. The critical values of the test statistics are obtained via a parametric bootstrap algorithm and its asymptotic validity is established. The tests proposed in this thesis were evaluated via simulation studies, comparing them with some of the best available tests from the existing literature. In these simulations, the tests proposed in this thesis exhibited a good overall performance. In each chapter, the proposed testing procedures are illustrated using empirical examples.

A simulation evaluation of the maximum approximate composite marginal likelihood (MACML) estimator for mixed multinomial probit models

For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.

The importance of providing structural validity evidence for test score(s) derived from psychometric test instruments is highlighted by several institutions; for example, the American Psychological Association (2014) demands that evidence for the validity of an instruments' internal structure and its underlying measurement model must be provided before it is applied in psychological assessment. The knowledge about the latent structure of data obtained with tests addressing the major question "What is/are the construct[s] being measured" by psychological tests under investigation (Ziegler, 2014 (Ziegler, , 2020)) . The study of structural validity is typically addressed with factor analyses when the test scores reflect continuous latent traits. As most submissions to Psychological Test Adaptation and Development (PTAD) deal with the adaptation and further development of existing measures, authors typically test a measurement model that is based on theoretical considerations and prior findings on original versions (or adaptations) of the test under investigation. Our literature review of PTAD's publications showed that more than 90% of the articles contain at least one confirmatory factor analysis (CFA). As editor and reviewers of PTAD, we appreciate that authors are rigorous in providing evidence on the structural validity of their tests' data. However, since PTAD's inception in 2019, we experience that one comment is frequently communicated to authors during the review process, namely, the request to adjust the analytic approach in CFA from maximum likelihood (ML) estimation toward using the mean-and variance-adjusted weighted least squares (WLSMV; Muthén et al., 1997) estimator to account for the ordinal nature of the data that psychological instruments typically generate on the item level. In this editorial, we discuss the rationale behind choosing the WLSMV estimator when analyzing test adaptations and developments that are based on ordinal categorical data and concisely illustrate the problems associated with using the ML estimator (potentially in combination with robust tests of model fit) for such data.

It is demonstrated that the sampling distributions of the maximum likelihood (ML) estimator and its Studentized statistic for the generalized Gaussian distribution do not pass the most powerful normality tests even for fairly large sample sizes. This disagreement with what the standard large sample ML theory predicts and the computational burden of having to deal with its associated polygamma functions motivate the consideration of a competing convexity-based estimator. The asymptotic normality of this estimator is derived. It is shown that the competing estimator is almost as efficient as the ML estimator and its asymptotic relative efficiency to the ML estimator is equal to 1 in the limit as the shape parameter approaches zero. More important, its asymptotic distribution admits an exact variance stabilizing transformation, whereas the asymptotic variance function of the ML estimator does not have a closed-form variance stabilizing transformation. The exact transformation is a composition of the inverse hyperbolic cotangent and square root functions. Besides stabilizing the variance, the inverse hyperbolic cotangent and square root transformation is remarkably effective for symmetrizing and normalizing the sampling distribution of the estimator and hence improving the standard normal approximation. Furthermore, this simple transformation provides a quite accurate approximation to the non-closed-form variance stabilizing transformation of the ML estimator.

The Canadian Study of Health and Aging (CSHA) employed a prevalent cohort design to study survival after onset of dementia, where patients with dementia were sampled and the onset time of dementia was determined retrospectively. The prevalent cohort sampling scheme favors individuals who survive longer. Thus, the observed survival times are subject to length bias. In recent years, there has been a rising interest in developing estimation procedures for prevalent cohort survival data that not only account for length bias but also actually exploit the incidence distribution of the disease to improve efficiency. This article considers semiparametric estimation of the Cox model for the time from dementia onset to death under a stationarity assumption with respect to the disease incidence. Under the stationarity condition, the semiparametric maximum likelihood estimation is expected to be fully efficient yet difficult to perform for statistical practitioners, as the likelihood depends on the baseline hazard function in a complicated way. Moreover, the asymptotic properties of the semiparametric maximum likelihood estimator are not well-studied. Motivated by the composite likelihood method (Besag 1974), we develop a composite partial likelihood method that retains the simplicity of the popular partial likelihood estimator and can be easily performed using standard statistical software. When applied to the CSHA data, the proposed method estimates a significant difference in survival between the vascular dementia group and the possible Alzheimer's disease group, while the partial likelihood method for left-truncated and right-censored data yields a greater standard error and a 95% confidence interval covering 0, thus highlighting the practical value of employing a more efficient methodology. To check the assumption of stable disease for the CSHA data, we also present new graphical and numerical tests in the article. The R code used to obtain the maximum composite partial likelihood estimator for the CSHA data is available in the online Supplementary Material, posted on the journal web site.