Time-varying Regression Coefficients Research Articles

Difference-based covariance matrix estimation in time series nonparametric regression with application to specification tests

Summary Long-run covariance matrix estimation is the building block of time series inference. The corresponding difference-based estimator, which avoids detrending, has attracted considerable interest due to its robustness to both smooth and abrupt structural breaks and its competitive finite sample performance. However, existing methods mainly focus on estimators for the univariate process, while their direct and multivariate extensions for most linear models are asymptotically biased. We propose a novel difference-based and debiased long-run covariance matrix estimator for functional linear models with time-varying regression coefficients, allowing time series nonstationarity, long-range dependence, state heteroscedasticity and combinations thereof. We apply the new estimator to (i) the structural stability test, overcoming the notorious nonmonotonic power phenomena caused by piecewise smooth alternatives for regression coefficients, and (ii) the nonparametric residual-based tests for long memory, improving the performance via the residual-free formula of the proposed estimator. The effectiveness of the proposed method is justified theoretically and demonstrated by superior performance in simulation studies, while its usefulness is elaborated via real data analysis. Our method is implemented in the R package mlrv.

Decadal modulation of ENSO and IOD impacts on the Indian Ocean upwelling

The decadal modulations are observed in impacts of El Niño and Southern Ocean (ENSO) and Indian Ocean Dipole (IOD) on the tropical Indian Ocean upwelling. Here, we explore important contributors to the decadal modulations by combining the observational data since 1958 and statistical model simulations. A Bayesian Dynamic Linear Model (BDLM), which represents the temporal modulations of the IOD and ENSO impacts, reproduces the timeseries of the eastern and western Indian Ocean (EIO and WIO) upwellings more realistically than a conventional Static Linear regression Model does. The time-varying regression coefficients in BDLM indicate that the observed shift of the IOD impact on the EIO upwelling around 1980 is mainly due to the changes of alongshore wind stress forcing and the sensitivity of the upper ocean temperature in the EIO through the surface warming tendency and the enhanced ocean stratification. In contrast, the impacts of ENSO and IOD on the WIO are modulated in relation to the decadal variability of the tropical Pacific Ocean. When the eastern tropical Pacific Ocean is observed warmer on decadal timescales, the accompanying changes of the dominant ENSO flavors contribute to modulating the strengths of the atmospheric convective activity over the Indian-Pacific warm pool and the easterly wind variations in the equatorial Indian Ocean.

A computationally efficient mixture innovation model for time-varying parameter regressions

The mixture innovation (MI) model places a spike-and-slab mixture distribution for the innovations of time-varying regression coefficients and permits flexible time variation patterns while allowing for dynamic shrinkage. Despite its appeal, the standard Bayesian algorithm to block sample the vector of 0/1 mixture indicators at each time t needs to evaluate the model likelihood over all its 2K scenarios for a regression model with K regressors and becomes impractical when K grows. As an alternative, a new specification of the MI model is proposed in which the 0/1 mixture indicators in the original MI model are approximated by a logistic function of latent continuous variables. As such the model likelihood only needs to be evaluated twice in an Metropolis-Hastings step to block update the latent variables and hence the approximated mixture indicators at each time t, offering large improvement in computational efficiency while keeping the benefits of the MI model. An efficient MCMC algorithm is developed to estimate the new model. A simulation study shows that the new model can achieve the same level of estimation accuracy as the original MI model but at a much smaller computation cost. The new model is further tested in two empirical applications where block sampling the mixture indicators at each time t in the original MI model is practically infeasible.

Functional coefficient quantile regression model with time-varying loadings

ABSTRACT This paper proposes a functional coefficient quantile regression model with heterogeneous and time-varying regression coefficients and factor loadings. Estimation of the model coefficients is done in two stages. First, we estimate the unobserved common factors from a linear factor model with exogenous covariates. Second, we plug-in an affine transformation of the estimated common factors to obtain the functional coefficient quantile regression model. The quantile parameter estimators are consistent and asymptotically normal. The application of this model to the quantile process of a cross-section of U.S. firms’ excess returns confirms the predictive ability of firm-specific covariates and the good performance of the local estimator of the heterogeneous and time-varying quantile coefficients.

An additive hazards frailty model with semi-varying coefficients.

Proportional hazards frailty models have been extensively investigated and used to analyze clustered and recurrent failure times data. However, the proportional hazards assumption in the models may not always hold in practice. In this paper, we propose an additive hazards frailty model with semi-varying coefficients, which allows some covariate effects to be time-invariant while other covariate effects to be time-varying. The time-varying and time-invariant regression coefficients are estimated by a set of estimating equations, whereas the frailty parameter is estimated by the moment method. The large sample properties of the proposed estimators are established. The finite sample performance of the estimators is examined by simulation studies. The proposed model and estimation are illustrated with an analysis of data from a rehospitalization study of colorectal cancer patients.

Estimation of covariate effects on net survivals in the relative survival progressive illness-death model.

Recently, there has been a lot of development in relative survival field. In the absence of data on the cause of death, the research has tended to focus on the estimation of survival probability of a cancer (as a disease of interest). In many cancers, one nonfatal event that decreases the survival probability can occur. There are a few methods that assess the role of prognostic factors for multiple types of clinical events while dealing with uncertainty about the cause of death. However, these methods require proportional hazard or Markov assumptions. In practice, one or both of these assumptions might be violated. Violation of the proportional hazard assumption can lead to estimates that are biased, and difficult to interpret and violation of Markov assumption results in inconsistent estimators. In this work, we propose a semi-parametric approach to estimate the possibly time-varying regression coefficients in the likely non-Markov relative survival progressive illness-death model. The performance of the proposed estimator is investigated through simulations. We illustrate our approach using data from a study on rectal cancer resected for cure conducted in two French population-based digestive cancer registries.

Predicting equity premium using dynamic model averaging. Does the state–space representation matter?

Dynamic model averaging (DMA) has become a very useful tool with regards to dealing with two important aspects of time-series analysis, namely, parameter instability and model uncertainty. An important component of DMA is the Kalman filter. It is used to filter out the latent time-varying regression coefficients of the predictive regression of interest, and produce the model predictive likelihood, which is needed to construct the probability of each model in the model set. To apply the Kalman filter, one must write the model of interest in linear state–space form. In this study, we demonstrate that the state–space representation has implications on out-of-sample prediction performance, and the degree of shrinkage. Using Monte Carlo simulations as well as financial data at different sampling frequencies, we document that the way in which the current literature tends to formulate the candidate time-varying parameter predictive regression in linear state–space form ignores empirical features that are often present in the data at hand, namely, predictor persistence and predictor endogeneity. We suggest a straightforward way to account for these features in the DMA setting. Results using the widely applied Goyal and Welch (2008) dataset document that modifying the DMA framework as we suggest has a bearing on equity premium point prediction performance from a statistical as well as an economic viewpoint.

Sequential Poisson Regression in Diffusion Networks

The Poisson regression is a popular model for positive integer random variables determined by known explanatory variables. This letter studies the problem of its collaborative Bayesian sequential estimation under potentially slowly time-varying regression coefficients. We assume networks where agents share their information about the inferred quantities with adjacent neighbors in order to improve the overall estimation performance. The communication strategy is the information diffusion, i.e., only one information exchange per time instant is allowed.

Time-Varying Panel Data Models with an Additive Factor Structure

Motivated by many key features of real data from economics and finance, we study a semiparametric panel data model with time-varying regression coefficients associated with an additive factor structure. In our model, factor loadings are unknown functions of observable variables which can capture time-variant and heterogeneous covariate information. A profile marginal integration (PMI) method is proposed to estimate unknown coefficient functions, factors and their loadings jointly in a single step, which can result in estimators with closed forms. Asymptotic distributions for the proposed profile estimators are established. Two empirical applications on US stock returns and OECD health care expenditure are provided. Thorough numerical results demonstrate the finite sample performance of our estimation and its advantage over traditional models in the relevant literature.

Time-varying relationships between land use and crime: A spatio-temporal analysis of small-area seasonal property crime trends

Neighborhood land use composition influences the geographical patterns of property crime. Few studies, however, have investigated if, and how, the relationships between land use and crime change over time. This research applies a Bayesian spatio-temporal regression model to analyze 12 seasons of property crime at the small-area scale. Time-varying regression coefficients estimate the seasonally varying relationships between land use and crime and distinguish both time-constant and season-specific effects. Seasonal property crime trends are commonly hypothesized to be associated with fluctuating routine activity patterns around specific land uses, but past studies do not quantify the time-varying effects of neighborhood characteristics on small-area crime risk. Results show that, accounting for sociodemographic contexts, parks are more positively associated with property crime during spring and summer seasons, and eating and drinking establishments are more positively associated during autumn and winter seasons. Land use is found to have a more substantial impact on spatial, rather than spatio-temporal, crime patterns. Proposed explanations for results focus on seasonal activity patterns and corresponding spatio-temporal interactions with the built environment. The theoretical and analytical implications of this modeling approach are discussed. This research advances past cross-sectional spatial analyses of crime by identifying built environment characteristics that simultaneously shape both where and when crime occurs.

Analysis of binary longitudinal data with time-varying effects

This paper considers the analysis of longitudinal data where a binary response variable is observed repeatedly for each subject over time. In analyzing such data, regression coefficients are commonly assumed constant over time, which may not properly account for the time-varying effects of some subject characteristics on a sequence of binary outcomes. This paper proposes a Bayesian method for the analysis of binary longitudinal data with time-varying regression coefficients and random effects to account for nonlinear subject-specific effects over time as well as between-subject variation. The proposed method facilitates posterior computation via the method of partial collapse and accommodates spatially inhomogeneous smoothness of nonparametric functions without overfitting via a basis search technique. The proposed method is illustrated with a simulated study and the binary longitudinal data from the German socioeconomic panel study.

Forecasting aggregate stock market volatility using financial and macroeconomic predictors: Which models forecast best, when and why?

Abstract This paper revisits the topic of forecasting aggregate stock market volatility using financial and macroeconomic predictors in a comprehensive Bayesian model averaging framework. Candidate models include time-varying (with various degrees of dynamics) and constant-coefficient autoregressions based on the logarithm of monthly realized volatility augmented with exogenous predictors capturing risk premia, leverage, bond rates and proxies for credit risk. Thus, we simultaneously address parameter instability and model uncertainty that unavoidably impact volatility predictions. Applied to monthly S&P 500 volatility from 1926 to 2010, we find that Bayesian model averaging with time-varying regression coefficients provides very competitive density and modest improvements in point forecasts compared to rival approaches.

An Application of Heterogeneous Bayesian Regression Models with Time Varying Coefficients to Explore the Relationship between Customer Satisfaction and Shareholder Value

The authors propose new Bayesian models to obtain individual-level and time-varying regression coefficients in longitudinal data involving a single observation per response unit at each time period. An application to explore the association between customer satisfaction and shareholder value is included in the paper. The Bayesian models allow the flexibility of incorporating industry and firm factors in the context of the application to help explain variations of the regression coefficients. Results from the analysis indicate that the effect of customer satisfaction on shareholder value is not homogeneous over time. The proposed methodology provides a powerful tool to explore the relationship between two important business concepts.

Marginal Distribution Plots for Proportional Hazards Models with Time-Dependent Covariates or Time-Varying Regression Coefficients

Given a sample of regression data from (Y, Z), a new diagnostic plotting method is proposed for checking the hypothesis H0: the data are from a given Cox model with the time-dependent covariates Z. It compares two estimates of the marginal distribution FY of Y. One is an estimate of the modified expression of FY under H0, based on a consistent estimate of the parameter under H0, and based on the baseline distribution of the data. The other is the Kaplan-Meier-estimator of FY, together with its confidence band. The new plot, called the marginal distribution plot, can be viewed as a test for testing H0. The main advantage of the test over the existing residual tests is in the case that the data do not satisfy any Cox model or the Cox model is mis-specified. Then the new test is still valid, but not the residual tests and the residual tests often make type II error with a very large probability.

Direct modeling of regression effects for transition probabilities in the progressive illness-death model.

In this work, we present direct regression analysis for the transition probabilities in the possibly non-Markov progressive illness-death model. The method is based on binomial regression, where the response is the indicator of the occupancy for the given state along time. Randomly weighted score equations that are able to remove the bias due to censoring are introduced. By solving these equations, one can estimate the possibly time-varying regression coefficients, which have an immediate interpretation as covariate effects on the transition probabilities. The performance of the proposed estimator is investigated through simulations. We apply the method to data from the Registry of Systematic Lupus Erythematosus RELESSER, a multicenter registry created by the Spanish Society of Rheumatology. Specifically, we investigate the effect of age at Lupus diagnosis, sex, and ethnicity on the probability of damage and death along time. Copyright © 2017 John Wiley & Sons, Ltd.

Forecasting Aggregate Stock Market Volatility Using Financial and Macroeconomic Predictors: Which Models Forecast Best, When and Why?

Abstract This paper revisits the topic of forecasting aggregate stock market volatility using financial and macroeconomic predictors in a comprehensive Bayesian model averaging framework. Candidate models include time-varying (with various degrees of dynamics) and constant-coefficient autoregressions based on the logarithm of monthly realized volatility augmented with exogenous predictors capturing risk premia, leverage, bond rates and proxies for credit risk. Thus, we simultaneously address parameter instability and model uncertainty that unavoidably impact volatility predictions. Applied to monthly S&P 500 volatility from 1926 to 2010, we find that Bayesian model averaging with time-varying regression coefficients provides very competitive density and modest improvements in point forecasts compared to rival approaches.

Goodness-of-fit test for monotone proportional subdistribution hazards assumptions based on weighted residuals.

Recently goodness-of-fit tests have been proposed for checking the proportional subdistribution hazards assumptions in the Fine and Gray regression model. Zhou, Fine, and Laird proposed weighted Schoenfeld-type residuals tests derived under an assumed model with specific form of time-varying regression coefficients. Li, Sheike, and Zhang proposed an omnibus test based on cumulative sums of Schoenfeld-type residuals. In this article, we extend the class of weighted residuals tests by allowing random weights of Schoenfeld-type residuals at ordered event times. In particular, it is demonstrated that weighted residuals tests using monotone weight functions of time are consistent against monotone proportional subdistribution hazards assumptions. Extensive Monte Carlo studies were conducted to evaluate the finite-sample performance of recent goodness-of-fit tests. Results from simulation studies show that weighted residuals tests using monotone random weight functions commonly used in non-proportional hazards regression settings tend to be more powerful for detecting monotone departures than other goodness-of-fit tests assuming no specific time-varying effect or misspecified time-varying effects. Two examples using real data are provided for illustrations. Copyright © 2016 John Wiley & Sons, Ltd.

Analysis of two-phase sampling data with semiparametric additive hazards models

Under the case-cohort design introduced by Prentice (Biometrica 73:1-11, 1986), the covariate histories are ascertained only for the subjects who experience the event of interest (i.e., the cases) during the follow-up period and for a relatively small random sample from the original cohort (i.e., the subcohort). The case-cohort design has been widely used in clinical and epidemiological studies to assess the effects of covariates on failure times. Most statistical methods developed for the case-cohort design use the proportional hazards model, and few methods allow for time-varying regression coefficients. In addition, most methods disregard data from subjects outside of the subcohort, which can result in inefficient inference. Addressing these issues, this paper proposes an estimation procedure for the semiparametric additive hazards model with case-cohort/two-phase sampling data, allowing the covariates of interest to be missing for cases as well as for non-cases. A more flexible form of the additive model is considered that allows the effects of some covariates to be time varying while specifying the effects of others to be constant. An augmented inverse probability weighted estimation procedure is proposed. The proposed method allows utilizing the auxiliary information that correlates with the phase-two covariates to improve efficiency. The asymptotic properties of the proposed estimators are established. An extensive simulation study shows that the augmented inverse probability weighted estimation is more efficient than the widely adopted inverse probability weighted complete-case estimation method. The method is applied to analyze data from a preventive HIV vaccine efficacy trial.

Spatio-temporal regression on compositional covariates: modeling vegetation in a gypsum outcrop

Investigating the relationship between vegetation cover and substrate typologies is important for habitat conservation. To study these relationships, common practice in modern ecological surveys is to collect information regarding vegetation cover and substrate typology over fine regular lattices, as derived from digital ground photos. Information on substrate typologies is often available as compositional measures, e.g., the area proportion occupied by a certain substrate. Two primary issues are of interest for ecologists: first, how much substrate typologies differ in terms of relative suitability for vegetation cover and, second, whether suitability varies over time. This paper develops a procedure for managing compositional covariates within a Bayesian hierarchical framework to effectively address the aforementioned issues. A spatio-temporal model is adopted to estimate the temporal pattern characterizing substrate relative suitability for vegetation cover and, at the same time, to account for spatio-temporal correlation. Relative suitability is modeled by time-varying regression coefficients, and spatial, temporal and spatio-temporal random effects are modeled using Gaussian Markov Random Field models.

Nonparametric Shrinkage Estimation for Aalen's Additive Hazards Model

Aalen's nonparametric additive model in which the regression coefficients are assumed to be unspecified functions of time is a flexible alternative to Cox's proportional hazards model when the proportionality assumption is in doubt. In this paper, we incorporate a general linear hypothesis into the estimation of the time-varying regression coefficients. We combine unrestricted least squares estimators and estimators that are restricted by the linear hypothesis and produce James-Stein-type shrinkage estimators of the regression coefficients. We develop the asymptotic joint distribution of such restricted and unrestricted estimators and use this to study the relative performance of the proposed estimators via their integrated asymptotic distributional risks. We conduct Monte Carlo simulations to examine the relative performance of the estimators in terms of their integrated mean square errors. We also compare the performance of the proposed estimators with a recently devised LASSO estimator as well as with ridge-type estimators both via simulations and data on the survival of primary billiary cirhosis patients.