Near-epoch Dependence Research Articles

Threshold Network GARCH Model

Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model and its variations have been widely adopted in the study of financial volatilities, while the extension of GARCH‐type models to high‐dimensional data is always difficult because of over‐parameterization and computational complexity. In this article, we propose a multi‐variate GARCH‐type model that can simplify the parameterization by utilizing the network structure that can be appropriately specified for certain types of high‐dimensional data. The asymmetry in the dynamics of volatilities is also considered as our model adopts a threshold structure. To enable our model to handle data with extremely high dimension, we investigate the near‐epoch dependence (NED) of our model, and the asymptotic properties of our quasi‐maximum‐likelihood‐estimator (QMLE) are derived from the limit theorems for NED random fields. Simulations are conducted to test our theoretical results. At last we fit our model to log‐returns of four groups of stocks and the results indicate that bad news is not necessarily more influential on volatility if the network effects are considered.

Nonlinear Spatial Dynamic Panel Data Models with Endogenous Dominant Units: An Application to Share Data

This article develops a nonlinear spatial dynamic panel data model with one particularly interesting application to a structural interaction model for share data. To account for effects from dominant (popular) units, the spatial weights matrix in our model can allow for unbounded column sums. To account for heterogeneity, our model includes two-way fixed effects and heteroscedastic errors. We further consider the potential endogeneity of the spatial weight matrix constructed from socioeconomic distance. We investigate the quasi-maximum likelihood estimator (QMLE), generalized methods of moments estimator (GMME), and root estimator (RTE), and establish their consistency and asymptotic normality based on the near epoch dependence (NED) framework. The RTE can derive a relatively computationally simple and closed-form solution without evaluating the QMLE’s Jacobian matrix as well as the iterations by GMME. We consider both n , T → ∞ , and the strength of the dominant units is equal to 1 when T → ∞ . For the purpose of empirical analysis, we derive the marginal effects and their limiting distributions based on the proposed estimators. In an empirical application, we apply our model to China’s prefecture city-level data, revealing significant spillover effects of the tertiary industry share. These findings suggest that the development of the tertiary sector in large cities can foster its growth in small cities.

Lasso inference for high-dimensional time series

In this paper we develop valid inference for high-dimensional time series. We extend the desparsified lasso to a time series setting under Near-Epoch Dependence (NED) assumptions allowing for non-Gaussian, serially correlated and heteroskedastic processes, where the number of regressors can possibly grow faster than the time dimension. We first derive an error bound under weak sparsity, which, coupled with the NED assumption, means this inequality can also be applied to the (inherently misspecified) nodewise regressions performed in the desparsified lasso. This allows us to establish the uniform asymptotic normality of the desparsified lasso under general conditions, including for inference on parameters of increasing dimensions. Additionally, we show consistency of a long-run variance estimator, thus providing a complete set of tools for performing inference in high-dimensional linear time series models. Finally, we perform a simulation exercise to demonstrate the small sample properties of the desparsified lasso in common time series settings.

Using Triples to Assess Symmetry Under Weak Dependence

ABSTRACT The problem of assessing symmetry about an unspecified center of the one-dimensional marginal distribution of a strictly stationary random process is considered. A well-known U-statistic based on data triples is used to detect deviations from symmetry, allowing the underlying process to satisfy suitable mixing or near-epoch dependence conditions. We suggest using subsampling for inference on the target parameter, establish the asymptotic validity of the method in our setting, and discuss data-driven rules for selecting the size of subsamples. The small-sample properties of the proposed inferential procedures are examined by means of Monte Carlo simulations. Applications to time series of output growth and stock returns are also presented.

On the local power of some tests of strict exogeneity in linear fixed effects models

The local asymptotic power of a variable addition test for strict exogeneity in a linear panel model is derived under near-epoch dependence and a maintained assumption of contemporaneously exogenous regressors. Local power is found to depend nontrivially on the relative panel size, on the width of the local neighborhood, and on the maintained notion of exogeneity under the alternative. Some (dis)similarities between this test and already existing test principles are explored.

A new consistency proof for HAC variance estimators

A consistency theorem for kernel HAC variance estimators was originally proposed by Hansen (1992) but corrected under stronger conditions on the order of existing moments by de Jong (2000). The present result restores and also generalizes the conditions of Hansen’s result by assuming the process to be adapted to a filtration. It allows for nonstationarity, and dependence is modelled by the assumption of near-epoch dependence on a mixing process.

Near-epoch dependence in Riesz spaces

The abstraction of the study of stochastic processes to Banach lattices and vector lattices has received much attention by Grobler, Kuo, Labuschagne, Stoica, Troitsky and Watson over the past fifteen years. By contrast mixing processes have received very little attention. In particular mixingales were generalized to the Riesz space setting in Kuo et al. (2013) [12]. The concepts of strong and uniform mixing as well as related mixing inequalities were extended to this setting in Kuo et al. (2017) [11]. In the present work we formulate the concept of near-epoch dependence for Riesz space processes and show that if a process is near-epoch dependent and either strong or uniform mixing then the process is a mixingale, giving access to a law of large numbers. The above is applied to autoregressive processes of order 1 in Riesz spaces.

WEAK CONVERGENCE TO STOCHASTIC INTEGRALS UNDER PRIMITIVE CONDITIONS IN NONLINEAR ECONOMETRIC MODELS

Limit theory with stochastic integrals plays a major role in time series econometrics. In earlier contributions on weak convergence to stochastic integrals, the literature commonly uses martingale and semi-martingale structures. Liang, Phillips, Wang, and Wang (2016) (see also Wang (2015), Chap. 4.5) currently extended weak convergence to stochastic integrals by allowing for a linear process or a α-mixing sequence in innovations. While these martingale, linear process and α-mixing structures have wide relevance, they are not sufficiently general to cover many econometric applications that have endogeneity and nonlinearity. This paper provides new conditions for weak convergence to stochastic integrals. Our frameworks allow for long memory processes, causal processes, and near-epoch dependence in innovations, which have applications in a wide range of econometric areas such as TAR, bilinear, and other nonlinear models.

QML estimation of spatial dynamic panel data models with endogenous time varying spatial weights matrices

In spatial panel data models, when a spatial weights matrix is constructed from economic or social distance, spatial weights could be endogenous and also time varying. This paper presents model specification and proposes QMLE estimation of spatial dynamic panel data models with endogenous time varying spatial weights matrices. Asymptotic properties of the proposed QMLE are rigorously established. We extend the notion of spatial near-epoch dependence to allow time dependence. By using spatial-time LLN for near-epoch dependence process and CLT for martingale difference sequence, we establish the consistency and asymptotic normality of QMLE. Monte Carlo experiments show that the proposed estimators have satisfactory finite sample performance.

Distribution Free Estimation of Spatial Autoregressive Binary Choice Panel Data Models

This paper proposes a semiparametric estimator for spatial autoregressive (SAR) binary choice models in the context of panel data with fixed effects. The estimation procedure is based on the observational equivalence between distribution free models with a conditional median restriction and parametric models (such as Logit/Probit) exhibiting (multiplicative) heteroskedasticity and autocorrelation. Without imposing any parametric structure on the error terms, we consider the semiparametric nonlinear least squares (NLLS) estimator for this model and analyze its asymptotic properties under spatial near-epoch dependence. The main advantage of our method over the existing estimators is that it consistently estimates choice probabilities. The finite-dimensional estimator is shown to be consistent and root-n asymptotically normal under some reasonable conditions. Finally, a Monte Carlo study indicates that the estimator performs quite well in finite samples.

Distribution Free Estimation of Spatial Autoregressive Binary Choice Panel Data Models

This paper proposes a semiparametric estimator for spatial autoregressive (SAR) binary choice models in the context of panel data with fixed effects. The estimation procedure is based on the observational equivalence between distribution free models with a conditional median restriction and parametric models (such as Logit/Probit) exhibiting (multiplicative) heteroskedasticity and autocorrelation. Without imposing any parametric structure on the error terms, we consider the semiparametric nonlinear least squares (NLLS) estimator for this model and analyze its asymptotic properties under spatial near-epoch dependence. The main advantage of our method over the existing estimators is that it consistently estimates choice probabilities. The finite-dimensional estimator is shown to be consistent and root-n asymptotically normal under some reasonable conditions. Finally, a Monte Carlo study indicates that the estimator performs quite well in finite samples.

Maximum likelihood estimation of a spatial autoregressive Tobit model

This paper examines a Tobit model with spatial autoregressive interactions. We consider the maximum likelihood estimation for this model and analyze asymptotic properties of the estimator based on the spatial near-epoch dependence of the dependent variable process generated from the model structure. We show that the maximum likelihood estimator is consistent and asymptotically normally distributed. Monte Carlo experiments are performed to verify finite sample properties of the estimator.

Strategic Interaction Model with Censored Strategies

In this paper, we develop a new model of a static game of incomplete information with a large number of players. The model has two key distinguishing features. First, the strategies are subject to threshold effects, and can be interpreted as dependent censored random variables. Second, in contrast to most of the existing literature, our inferential theory relies on a large number of players, rather than a large number of independent repetitions of the same game. We establish existence and uniqueness of the pure strategy equilibrium, and prove that the censored equilibrium strategies satisfy a near-epoch dependence property. We then show that the normal maximum likelihood and least squares estimators of this censored model are consistent and asymptotically normal. Our model can be useful in a wide variety of settings, including investment, R&D, labor supply, and social interaction applications.

A spatial autoregressive model with a nonlinear transformation of the dependent variable

This paper develops a nonlinear spatial autoregressive model. Of particular interest is a structural interaction model for share data. We consider possible instrumental variable (IV) and maximum likelihood estimation (MLE) for this model, and analyze asymptotic properties of the IV and MLE based on the notion of spatial near-epoch dependence. We also design a statistical test to compare the nonlinear transformation against alternatives. Monte Carlo experiments are designed to investigate finite sample performance of the proposed estimates and the sizes and powers of the test.

On spatial processes and asymptotic inference under near-epoch dependence

The development of a general inferential theory for nonlinear models with cross-sectionally or spatially dependent data has been hampered by a lack of appropriate limit theorems. To facilitate a general asymptotic inference theory relevant to economic applications, this paper first extends the notion of near-epoch dependent (NED) processes used in the time series literature to random fields. The class of processes that is NED on, say, an α-mixing process, is shown to be closed under infinite transformations, and thus accommodates models with spatial dynamics. This would generally not be the case for the smaller class of α-mixing processes. The paper then derives a central limit theorem and law of large numbers for NED random fields. These limit theorems allow for fairly general forms of heterogeneity including asymptotically unbounded moments, and accommodate arrays of random fields on unevenly spaced lattices. The limit theorems are employed to establish consistency and asymptotic normality of GMM estimators. These results provide a basis for inference in a wide range of models with spatial dependence.

Strong near-epoch dependence

We introduce a new class of dependent sequences of random variables, which is a subclass of near-epoch dependent sequences, but can also be approximated by mixing sequences. For this kind of sequences of random variables, we call them strong near-epoch dependent sequences, ap-order,p > 2, (maximum) moment inequality is established under weaker dependence sizes.

Moment and Memory Properties of Linear Conditional Heteroscedasticity Models, and a New Model

This article analyses the statistical properties of that general class of conditional heteroscedasticity models in which the conditional variance is a linear function of squared lags of the process. GARCH, IGARCH, FIGARCH, and a newly proposed generalization, the HYGARCH model, belong to this class. Conditions are derived for the existence of second and fourth moments, and for the limited memory condition of near-epoch dependence. The HYGARCH model is applied to 10 daily dollar exchange rates, and also to data for Asian exchange rates over the 1997 crisis period. In the latter case, the model exhibits notable stability across the pre-crisis and post-crisis periods.

Nonlinear error correction models

The relationship between cointegration and error correction (EC) models is well characterized in a linear context, but the extension to the nonlinear context is still a challenge. Few extensions of the linear framework have been done in the context of nonlinear error correction (NEC) or asymmetric and time varying error correction models. In this paper, we propose a theoretical framework based on the concept of near epoch dependence (NED) that allows us to formally address these issues. In particular, we partially extend the Granger Representation Theorem to the nonlinear case.

Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes

This paper considers methods of deriving sufficient conditions for the central limit theorem and functional central limit theorem to hold in a broad class of time series processes, including nonlinear processes and semiparametric linear processes. The common thread linking these results is the concept of near-epoch dependence on a mixing process, since powerful limit results are available under this limited-dependence property. The particular case of near-epoch dependence on an independent process provides a convenient framework for dealing with a range of nonlinear cases, including the bilinear, GARCH, and threshold autoregressive models. It is shown in particular that even SETAR processes with a unit root regime have short memory, under the right conditions. A simulation approach is also demonstrated, applicable to cases that are analytically intractable. A new FCLT is given for semiparametric linear processes, where the forcing processes are of the NED-on-mixing type, under conditions that are evidently close to necessary.