Time-varying Coefficient Models Research Articles

Robust Inference for Time-Varying Coefficient Models with Longitudinal Data

Time-varying coefficient models are useful in longitudinal data analysis. Various efforts have been invested for the estimation of the coefficient functions, based on the least squares principle. Related work includes smoothing spline and kernel methods among others, but these methods suffer from the shortcoming of non-robustness. In this paper, we introduce a local M-estimation method for estimating the coefficient functions and develop a robustified generalized likelihood ratio (GLR) statistic to test if some of the coefficient functions are constants or of certain parametric forms. The robustified GLR test is robust against outliers and the error distribution. This provides a useful robust inference tool for the models with longitudinal data. The bandwidth selection issue is also addressed to facilitate the implementation in practice. Simulations show that the proposed testing method is more powerful in some situations than its counterpart based on the least squares principle. A real example is also given for illustration.

Unemployment forecasts, time varying coefficient models and the Okun’s law in Spanish regions

During the Great Recession, output and unemployment responses have differed markedly across Spanish regions. Our objective is to evaluate the relative accuracy of forecasting models based on the Okun’s law compared to alternative approaches. In particular, we want to analyse if a time varying coefficient specification of the Okun’s law provide better forecasts than alternative models in two different periods: a first period from 2002 to 2007 characterized by sustained economic growth in all provinces, and a second period from 2008 to 2013 characterized by the impact of the Great Recession. The obtained results allow us to conclude that, in general, the use of these models improve the forecasting capacity in most regions, but do not provide reliable forecast.

Modeling and testing smooth structural changes with endogenous regressors

Modeling and detecting parameter stability of econometric models is a long standing problem. Most existing estimation and testing methods are designed for models without endogeneity. Little attention has been paid to models with endogeneous regressors, which may arise in many scenarios in economics. In this paper, we first consider a time-varying coefficient time series model with potential time-varying endogeneity. A local linear two stage least squared estimation is developed to estimate coefficient functions. The consistency and asymptotic normality of the estimator are derived. Furthermore, a nonparametric test is proposed to check smooth structural changes as well as abrupt structural breaks with possibly unknown change points in regression models with potential endogeneity. The idea is to compare the fitted values of the unrestricted nonparametric time-varying coefficient model and the restricted constant parameter model. The test has an asymptotic N(0,1) distribution and does not require any prior information about the alternatives. A simulation study highlights the merits of the proposed estimator and test. In an application, we estimate the New Keynesian Phillips Curve for the US nonparametrically and find strong evidence against its stability.

Time-varying linkages between tourism receipts and economic growth in South Africa

The causal link between tourism receipts and GDP has recently become a major focus in the tourism economics literature. Results obtained in recent studies about the causal link appear to be sensitive with respect to the countries analysed, sample period and methodology employed. Considering the sensitivity of the causal link, we use rolling window and time-varying coefficient estimation methods to analyse the parameter stability and Granger causality based on a vector error correction model (VECM). When applied to South Africa for the period 1960–2011, the findings are as follows: results from the full-sample VECM indicate that there is no Granger causality between tourism receipts and GDP, while the findings from the time-varying coefficients model based on the state-space representation show that tourism receipts have positive-predictive content for GDP for the entire period, with the exception of the period between 1985 and 1990. Full-sample time-varying causality tests show bidirectional strong causality between tourism receipts and GDP.

Time-varying coefficient models with ARMA–GARCH structures for longitudinal data analysis

Time-varying coefficient models with autoregressive and moving-average–generalized autoregressive conditional heteroscedasticity structure are proposed for examining the time-varying effects of risk factors in longitudinal studies. Compared with existing models in the literature, the proposed models give explicit patterns for the time-varying coefficients. Maximum likelihood and marginal likelihood (based on a Laplace approximation) are used to estimate the parameters in the proposed models. Simulation studies are conducted to evaluate the performance of these two estimation methods, which is measured in terms of the Kullback–Leibler divergence and the root mean square error. The marginal likelihood approach leads to the more accurate parameter estimates, although it is more computationally intensive. The proposed models are applied to the Framingham Heart Study to investigate the time-varying effects of covariates on coronary heart disease incidence. The Bayesian information criterion is used for specifying the time series structures of the coefficients of the risk factors.

A NOTE ON GENERALIZING THE CONCEPT OF COINTEGRATION

Building on the time-varying-coefficient (TVC) model, we propose a generalization of the concept of cointegration, allowing for the possibility that a set of variables measured with error entails a nonlinear relationship with unknown functional form. Both the dependent and explanatory variables of this relationship may be nonstationary (not necessarily of unit-root type), but there exists a nonlinear combination of all these explanatory variables that completely explains all the variation in the dependent variable. The TVC model allows us to test for the presence of this generalized cointegration in the absence of knowledge of the true nonlinear functional form and the full set of explanatory variables. We present the basic stages of the technique and discuss in detail how the issues of nonstationarity and cointegration affect each stage of the TVC estimation procedure.

Semivarying coefficient models for capture–recapture data: Colony size estimation for the little penguin Eudyptula minor

To accommodate seasonal effects that change from year to year into models for the size of an open population we consider a time-varying coefficient model. We fit this model to a capture–recapture data set collected on the little penguin Eudyptula minor in south-eastern Australia over a 25year period using Jolly–Seber type estimators and nonparametric P-spline techniques. The time-varying coefficient model identified strong changes in the seasonal pattern across the years which we further examined using functional data analysis techniques. To evaluate the methodology we also conducted several simulation studies that incorporate seasonal variation.

Nonparametric specification for non-stationary time series regression

We investigate the behavior of the Generalized Likelihood Ratio Test (GLRT) (Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153–193]) for time varying coefficient models where the regressors and errors are non-stationary time series and can be cross correlated. It is found that the GLRT retains the minimax rate of local alternative detection under weak dependence and non-stationarity. However, in general, the Wilks phenomenon as well as the classic residual bootstrap are sensitive to either conditional heteroscedasticity of the errors, non-stationarity or temporal dependence. An averaged test is suggested to alleviate the sensitivity of the test to the choice of bandwidth and is shown to be more powerful than tests based on a single bandwidth. An alternative wild bootstrap method is proposed and shown to be consistent when making inference of time varying coefficient models for non-stationary time series.

Functional Coefficient Models for Nearly (Possibly Weakly) <i>I</i> (1) Processes

The focus of this article is on nonlinear time varying coefficient models when the covariates and coefficient components are weakly, nearly, or possibly purely integrated time series processes. Local linear fitting is used to derive coefficient estimators along with their asymptotic distributions. The rates of convergence for the estimators is shown to differ based on whether stationary, weakly, nearly, or purely integrated covariates are being modelled. Similar conclusions also hold for the derived optimal bandwidth parameters.

Tutorial: Survival Estimation for Cox Regression Models with Time-Varying Coefficients UsingSASandR

Survival estimates are an essential compliment to multivariable regression models for time-to-event data, both for prediction and illustration of covariate effects. They are easily obtained under the Cox proportional-hazards model. In populations defined by an initial, acute event, like myocardial infarction, or in studies with long-term followup, the proportional-hazards assumption of constant hazard ratios is frequently violated. One alternative is to fit an interaction between covariates and a prespecified function of time, implemented as a time-dependent covariate. This effectively creates a time-varying coefficient that is easily estimated in software such as SAS and R. However, the usual programming statements for survival estimation are not directly applicable. Unique data manipulation and syntax is required, but is not well documented for either software. This paper offers a tutorial in survival estimation for the time-varying coefficient model, implemented in SAS and R. We provide a macro coxtvc to facilitate estimation in SAS where the current functionality is more limited. The macro is validated in simulated data and illustrated in an application.

Analyzing Disparities Trends for Health Care Insurance Coverage Among Non-Elderly Adults in the US: Evidence from the Behavioral Risk Factor Surveillance System, 1993-2009

Access to health care in the United States remains greatly disproportionate across socioeconomic groups. It is not known, however, whether the disparities between the socioeconomic categories are increasing or decreasing. This analysis used a well-established non-parametric technique, employing time-varying coefficient models applied to data from the 1993 to 2009 US Behavioral Risk Factor Surveillance System (BRFSS). The analysis was able to show the changes in the odds ratios of having no health insurance plan for variables of interest over time, therefore highlighting the changes in the disparities between the categories of a variable over time. While other studies have attempted to show the changes in health insurance coverage by socioeconomic groups in different time periods, there is no study to date that has shown these changes as a smooth function with time, therefore providing a clearer picture of the changes in these disparities. The results of this analysis show, for instance, that when compared with individuals with a college education or greater, those with less than a high school education showed a steady increase in the odds ratios for having no health insurance. The same trend seems applicable although in a less-clear way to Hispanics and Non-Hispanic black race-ethnicities, compared with non-Hispanic whites (the reference race category). As measures of the Affordable Care Act are being gradually implemented, studies are needed to provide baseline information about health care access disparity, in order to gauge any changes in health care access over time; BRFSS can be a useful data source in accomplishing this task.

Variable selection in Cox regression models with varying coefficients

We deal with Cox regression models with varying coefficients. In this paper we concentrate on time-varying coefficient models and just give a brief comment on another kind of varying coefficient model. When we have p-dimensional covariates and p increases with the sample size, it is often the case that only a small part of the covariates are relevant. Therefore we consider variable selection and estimation of the coefficient functions by using the group SCAD-type estimator and the adaptive group Lasso estimator. We examine the theoretical properties of the estimators, especially the L2 convergence rate, the sparsity, and the oracle property. Simulation studies and a real data analysis show the performance of these procedures.

Inference on stochastic time-varying coefficient models

Recently, there has been considerable work on stochastic time-varying coefficient models as vehicles for modelling structural change in the macroeconomy with a focus on the estimation of the unobserved paths of random coefficient processes. The dominant estimation methods, in this context, are based on various filters, such as the Kalman filter, that are applicable when the models are cast in state space representations. This paper introduces a new class of autoregressive bounded processes that decompose a time series into a persistent random attractor, a time varying autoregressive component, and martingale difference errors. The paper examines, rigorously, alternative kernel based, nonparametric estimation approaches for such models and derives their basic properties. These estimators have long been studied in the context of deterministic structural change, but their use in the presence of stochastic time variation is novel. The proposed inference methods have desirable properties such as consistency and asymptotic normality and allow a tractable studentization. In extensive Monte Carlo and empirical studies, we find that the methods exhibit very good small sample properties and can shed light on important empirical issues such as the evolution of inflation persistence and the purchasing power parity (PPP) hypothesis.

Leverage and Alpha: The Case of Funds of Hedge Funds

In this paper, we develop a theoretical model of fund of hedge fund net leverage and alpha where the cost of borrowing is increasing with net leverage, thereby impacting the performance. We use this model to determine the conditions under which the leverage has a negative or a positive impact on investors' alpha. Moreover, we show that the manager of a fund of hedge fund has an incentive to take a leverage that hurts the investors' alpha. Next, we develop a statistical method combining the Sharpe style analysis and a time-varying coefficient model. This method allows the weights of the regression to vary over time while being constrained to sum up to one. Subsequently, we use this method to get estimates of the leverages of a sample of funds of hedge funds. The estimates of leverages are then used in predictive regression analyses to confirm the negative impact of leverage on fund of hedge fund alphas and appraisal ratios. Finally, our results being robust to various robustness checks, we argue that this effect may be an explanation for the disappointing alpha delivered by funds of hedge funds.

Analysis of Term Structure in Dry Bulk Freight Market

Based on the monthly data set from January 1992 to May 2012, this paper applies a VAR model and two time-varying coefficient models to analyze term structure in dry bulk freight market. The resulting three main empirical findings are as follows: 1) The response of long-term rate to short- term structural shock is relatively small and not statistically significant but the response of short-term rate to long-term structural shock is large and statistically significant. 2) Compared with the presumption that in backwardation the short-term rate is inclined to decrease but the long-term rate to increase, but in contango the opposite pressure would function, there is lack of evidence for the stable adjustment speed in both equations for the short- and long-term freight rate. 3) When the market is relatively above the normal level (more backwardation), the effect of the implied forward time charter (IFTC) rate on the actual TC rate becomes larger. When predicting the future TC rate based on IFTC rate, it recommends giving more weight to the IFTC rate in backwardation than in contango.

Bayesian Bandwidth Selection in Nonparametric Time-Varying Coefficient Models

Bandwidth plays an important role in determining the performance of local linear estimators. In this paper, we propose a Bayesian approach to bandwidth selection for local linear estimation of time-varying coefficient time series models, where the errors are assumed to follow the Gaussian kernel error density. A Markov chain Monte Carlo algorithm is presented to simultaneously estimate the bandwidths for local linear estimators in the regression function and the bandwidth for the Gaussian kernel error-density estimator. A Monte Carlo simulation study shows that: 1) our proposed Bayesian approach achieves better performance in estimating the bandwidths for local linear estimators than normal reference rule and cross-validation; 2) compared with the parametric assumption of either the Gaussian or the mixture of two Gaussians, Gaussian kernel error-density assumption is a data-driven choice and helps gain robustness in terms of different specification of the true error density. Moreover, we apply our proposed Bayesian sampling method to the estimation of bandwidth for the time-varying coefficient models that explain Okun's law and the relationship between consumption growth and income growth in the U.S. For each model, we also provide calibrated parametric form of its time-varying coefficients.

Gray’s Time-Varying Coefficients Model for Posttransplant Survival of Pediatric Liver Transplant Recipients with a Diagnosis of Cancer

Transplantation is often the only viable treatment for pediatric patients with end-stage liver disease. Making well-informed decisions on when to proceed with transplantation requires accurate predictors of transplant survival. The standard Cox proportional hazards (PH) model assumes that covariate effects are time-invariant on right-censored failure time; however, this assumption may not always hold. Gray's piecewise constant time-varying coefficients (PC-TVC) model offers greater flexibility to capture the temporal changes of covariate effects without losing the mathematical simplicity of Cox PH model. In the present work, we examined the Cox PH and Gray PC-TVC models on the posttransplant survival analysis of 288 pediatric liver transplant patients diagnosed with cancer. We obtained potential predictors through univariable (P < 0.15) and multivariable models with forward selection (P < 0.05) for the Cox PH and Gray PC-TVC models, which coincide. While the Cox PH model provided reasonable average results in estimating covariate effects on posttransplant survival, the Gray model using piecewise constant penalized splines showed more details of how those effects change over time.

All-cause and cause-specific mortality associated with diabetes in prevalent hemodialysis patients

BackgroundDiabetes is the most common risk factor for end-stage renal disease (ESRD) and has been associated with increased risk of death. In order to better understand the influence of diabetes on outcomes in hemodialysis, we examine the risk of death of diabetic participants in the HEMODIALYSIS (HEMO) study.MethodsIn the HEMO study, 823 (44.6%) participants were classified as diabetic. Using the Schoenfeld residual test, we found that diabetes violated the proportional hazards assumption. Based on this result, we fit two non-proportional hazard models: Cox’s time varying covariate model (Cox-TVC) that allows the hazard for diabetes to change linearly with time and Gray’s time-varying coefficient model.ResultsUsing the Cox-TVC, the hazard ratio (HR) for diabetes increased with each year of follow up (p = 0.02) for all cause mortality. Using Gray’s model, the HR for diabetes ranged from 1.41 to 2.21 (p <0.01). The HR for diabetes using Gray’s model exhibited a different pattern, being relatively stable at 1.5 for the first 3 years in the study and increasing afterwards.ConclusionRisk of death associated with diabetes in ESRD increases over time and suggests that an increasing risk of death among diabetes may be underappreciated when using conventional survival models.

Trending time-varying coefficient market models

In this paper we study time-varying coefficient (beta coefficient) models with a time trend function to characterize the nonlinear, non-stationary and trending phenomenon in time series and to explain the behavior of asset returns. The general local polynomial method is developed to estimate the time trend and coefficient functions. More importantly, a graphical tool, the plot of the kth-order derivative of the parameter versus time, is proposed to select the proper order of the local polynomial so that the best estimate can be obtained. Finally, we conduct Monte Carlo experiments and a real data analysis to examine the finite sample performance of the proposed modeling procedure and compare it with the Nadaraya–Watson method as well as the local linear method.

Functional random effect time-varying coefficient model for longitudinal data.

We propose a functional random effect time-varying coefficient model to establish the dynamic relationship between the response and predictor variables in longitudinal data. This model allows us not only to interpret time-varying covariate effects, but also to depict random effects via time-varying profiles that are characterized by functional principal components. We develop the functional profiling-backfitting method to estimate model components, which includes the profiling and backfitting procedures via a set of least squares type estimating equations. Asymptotic properties of the resulting estimator are obtained. Furthermore, we investigate the finite sample performance of the proposed method through simulation studies and present an application to primary biliary cirrhosis data.