Optimal model averaging forecasting in high-dimensional survival analysis

This article considers ultrahigh-dimensional forecasting problems with survival response variables. We propose a two-step model averaging procedure for improving the forecasting accuracy of the true conditional mean of a survival response variable. The first step is to construct a class of candidate models, each with low-dimensional covariates. For this, a feature screening procedure is developed to separate the active and inactive predictors through a marginal Buckley–James index, and to group covariates with a similar index size together to form regression models with survival response variables. The proposed screening method can select active predictors under covariate-dependent censoring, and enjoys sure screening consistency under mild regularity conditions. The second step is to find the optimal model weights for averaging by adapting a delete-one cross-validation criterion, without the standard constraint that the weights sum to one. The theoretical results show that the delete-one cross-validation criterion achieves the lowest possible forecasting loss asymptotically. Numerical studies demonstrate the superior performance of the proposed variable screening and model averaging procedures over existing methods.     

Nonparametric simultaneous testing for structural breaks

In this paper we consider a regression model with errors that are martingale differences. This modeling includes the regression of both independent and time series data. The aim is to study the appearance of structural breaks in both the mean and the variance functions, assuming that such breaks may occur simultaneously in both the functions. We develop nonparametric testing procedures that simultaneously test for structural breaks in the conditional mean and the conditional variance. The asymptotic distribution of an adaptive test statistic is established, as well as its asymptotic consistency and efficiency. Simulations illustrate the performance of the adaptive testing procedure. An application to the analysis of financial time series also demonstrates the usefulness of the proposed adaptive test in practice.     

G@RCH 2.2: An Ox Package for Estimating and Forecasting Various ARCH Models

This paper discusses and documents G@RCH 2.2, an Ox package dedicated to the estimation and forecast of various univariate ARCH–type models including GARCH, EGARCH, GJR, APARCH, IGARCH, FIGARCH, HYGARCH, FIEGARCH and FIAPARCH specifications of the conditional variance and an AR(FI)MA specification of the conditional mean.
These models can be estimated by Approximate (Quasi) Maximum Likelihood under four assumptions: normal, Student– t , GED or skewed Student errors. Explanatory variables can enter both the conditional mean and the conditional variance equations. h –step–ahead forecasts of both the conditional mean and the conditional variance are available as well as many mispecification tests.
We first propose an overview of the package's features, with the presentation of the different specifications of the conditional mean and conditional variance. Then further explanations are given about the estimation methods. Measures of the accuracy of the procedures are also given and the GARCH features provided by G@RCH are compared with those of nine other econometric softwares. Finally, a concrete application of G@RCH 2.2 is provided.     

Semiparametric quantile averaging in the presence of high-dimensional predictors

The paper proposes a method for forecasting conditional quantiles. In practice, one often does not know the “true” structure of the underlying conditional quantile function, and in addition, we may have a large number of predictors. Focusing on such cases, we introduce a flexible and practical framework based on penalized high-dimensional quantile averaging. In addition to prediction, we show that the proposed method can also serve as a predictor selector. We conduct extensive simulation experiments to asses its prediction and variable selection performances for nonlinear and linear time series model designs. In terms of predictor selection, the approach tends to select the true set of predictors with minimal false positives. With respect to prediction accuracy, the method competes well even with the benchmark/oracle methods that know one or more aspects of the underlying quantile regression model. We further illustrate the merit of the proposed method by providing an application to the out-of-sample forecasting of U.S. core inflation using a large set of monthly macroeconomic variables based on FRED-MD database. The application offers several empirical findings.     

Nonparametric identification and estimation of sample selection models under symmetry

Under a conditional mean restriction Das et al. (2003) considered nonparametric estimation of sample selection models. However, their method can only identify the outcome regression function up to a constant. In this paper we strengthen the conditional mean restriction to a symmetry restriction under which selection biases due to selection on unobservables can be eliminated through proper matching of propensity scores; consequently we are able to identify and obtain consistent estimators for the average treatment effects and the structural regression functions. The results from a simulation study suggest that our estimators perform satisfactorily.     

Statistical inference on seemingly unrelated non‐parametric regression models with serially correlated errors

This article is concerned with the inference on seemingly unrelated non‐parametric regression models with serially correlated errors. Based on an initial estimator of the mean functions, we first construct an efficient estimator of the autoregressive parameters of the errors. Then, by applying an undersmoothing technique, and taking both of the contemporaneous correlation among equations and serial correlation into account, we propose an efficient two‐stage local polynomial estimation for the unknown mean functions. It is shown that the resulting estimator has the same bias as those estimators which neglect the contemporaneous and/or serial correlation and smaller asymptotic variance. The asymptotic normality of the resulting estimator is also established. In addition, we develop a wild block bootstrap test for the goodness‐of‐fit of models. The finite sample performance of our procedures is investigated in a simulation study whose results come out very supportive, and a real data set is analysed to illustrate the usefulness of our procedures.     

Inference on endogenously censored regression models using conditional moment inequalities

Under a quantile restriction, randomly censored regression models can be written in terms of conditional moment inequalities. We study the identified features of these moment inequalities with respect to the regression parameters where we allow for covariate dependent censoring, endogenous censoring and endogenous regressors. These inequalities restrict the parameters to a set. We show regular point identification can be achieved under a set of interpretable sufficient conditions. We then provide a simple way to convert conditional moment inequalities into unconditional ones while preserving the informational content. Our method obviates the need for nonparametric estimation, which would require the selection of smoothing parameters and trimming procedures. Without the point identification conditions, our objective function can be used to do inference on the partially identified parameter. Maintaining the point identification conditions, we propose a quantile minimum distance estimator which converges at the parametric rate to the parameter vector of interest, and has an asymptotically normal distribution. A small scale simulation study and an application using drug relapse data demonstrate satisfactory finite sample performance.     

Stock returns and volatility: Another look

Most empirical work examining the intertemporal mean-variance relationship in stock returns has tended to use relatively simple specifications of the mean and especially of the conditional variance. We augment the information set to include economic variables that other researchers have found to be important and use GARCH-M models to explore the relation between volatility and expected stock returns. We find that the additional variables have little impact on the conditional variance and that any intertemporal relationship between volatility and stock returns is weak or unstable. Our results signal the need for theoretical models of the intertemporal volatility-return relationship, and call for further studies of the determinants of the conditional variance of stock returns.     

A Test of the Conditional Independence Assumption in Sample Selection Models

Identification in most sample selection models depends on the independence of the regressors and the error terms conditional on the selection probability. All quantile and mean functions are parallel in these models; this implies that quantile estimators cannot reveal any—per assumption non‐existing—heterogeneity. Quantile estimators are nevertheless useful for testing the conditional independence assumption because they are consistent under the null hypothesis. We propose tests of the Kolmogorov–Smirnov type based on the conditional quantile regression process. Monte Carlo simulations show that their size is satisfactory and their power sufficient to detect deviations under plausible data‐generating processes. We apply our procedures to female wage data from the 2011 Current Population Survey and show that homogeneity is clearly rejected. Copyright © 2015 John Wiley & Sons, Ltd.     

Instrumental quantile regression inference for structural and treatment effect models

We introduce a class of instrumental quantile regression methods for heterogeneous treatment effect models and simultaneous equations models with nonadditive errors and offer computable methods for estimation and inference. These methods can be used to evaluate the impact of endogenous variables or treatments on the entire distribution of outcomes. We describe an estimator of the instrumental variable quantile regression process and the set of inference procedures derived from it. We focus our discussion of inference on tests of distributional equality, constancy of effects, conditional dominance, and exogeneity. We apply the procedures to characterize the returns to schooling in the U.S.     

Robust feature screening for varying coefficient models via quantile partial correlation

This article is concerned with feature screening for varying coefficient models with ultrahigh-dimensional predictors. We propose a new sure independence screening method based on quantile partial correlation (QPC-SIS), which is quite robust against outliers and heavy-tailed distributions. Then we establish the sure screening property for the QPC-SIS, and conduct simulations to examine its finite sample performance. The results of simulation study indicate that the QPC-SIS performs better than other methods like sure independent screening (SIS), sure independent ranking and screening, distance correlation-sure independent screening, conditional correlation sure independence screening and nonparametric independent screening, which shows the validity and rationality of QPC-SIS.     

Nonparametric estimation of conditional distribution functions with longitudinal data and time-varying parametric models

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies. We propose in this paper an estimation approach based on time-varying parametric models. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model, but the parameters are smooth functions of time. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Asymptotic properties, including the asymptotic biases, variances and mean squared errors, are derived for the local polynomial smoothed estimators. Applicability of our two-step estimation method is demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through simulation study.     

Semiparametric time series models driven by latent factor

We introduce a class of semiparametric time series models (SemiParTS) driven by a latent factor process. The proposed SemiParTS class is flexible because, given the latent process, only the conditional mean and variance of the time series are specified. These are the primary features of SemiParTS: (i) no parametric form is assumed for the conditional distribution of the time series given the latent process; (ii) it is suitable for a wide range of data: non-negative, count, bounded, binary, and real-valued time series; (iii) it does not constrain the dispersion parameter to be known. The quasi-likelihood inference is employed in order to estimate the parameters in the mean function. Here, we derive explicit expressions for the marginal moments and for the autocorrelation function of the time series process so that a method of moments can be employed to estimate the dispersion parameter and also the parameters related to the latent process. Simulated results that aim to check the proposed estimation procedure are presented. Forecasting procedures are proposed and evaluated in simulated and real data. Analyses of the number of admissions in a hospital due to asthma and a total insolation time series illustrate the potential for practical situations that involve the proposed models.     

Forecasting the volatility of asset returns: The informational gains from option prices

A new class of forecasting models is proposed that extends the realized GARCH class of models through the inclusion of option prices to forecast the variance of asset returns. The VIX is used to approximate option prices, resulting in a set of cross-equation restrictions on the model’s parameters. The full model is characterized by a nonlinear system of three equations containing asset returns, the realized variance, and the VIX, with estimation of the parameters based on maximum likelihood methods. The forecasting properties of the new class of forecasting models, as well as a number of special cases, are investigated and applied to forecasting the daily S&P500 index realized variance using intra-day and daily data from September 2001 to November 2017. The forecasting results provide strong support for including the realized variance and the VIX to improve variance forecasts, with linear conditional variance models performing well for short-term one-day-ahead forecasts, whereas log-linear conditional variance models tend to perform better for intermediate five-day-ahead forecasts.     

On a double-threshold autoregressive heteroscedastic time series model

Tong's threshold models have been found useful in modelling nonlinearities in the conditional mean of a time series. The threshold model is extended to the so-called double-threshold ARCH(DTARCH) model, which can handle the situation where both the conditional mean and the conditional variance specifications are piecewise linear given previous information. Potential applications of such models include financial data with different (asymmetric) behaviour in a rising versus a falling market and business cycle modelling. Model identification, estimation and diagnostic checking techniques are developed. Maximum likelihood estimation can be achieved via an easy-to-use iteratively weighted least squares algorithm. Portmanteau-type statistics are also derived for checking model adequacy. An illustrative example demonstrates that asymmetric behaviour in the mean and the variance could be present in financial series and that the DTARCH model is capable of capturing these phenomena.     

Nonparametric independence feature screening for ultrahigh-dimensional survival data

With the explosion of digital information, high-dimensional data is frequently collected in prevalent domains, in which the dimension of covariates can be much larger than the sample size. Many effective methods have been developed to reduce the dimension of such data recently, however, few methods might perform well for survival data with censoring. In this article, we develop a novel nonparametric feature screening procedure based on ultrahigh-dimensional survival data by incorporating the inverse probability weighting scheme to tackle the issue of censoring. The proposed method is model-free and hence can be implemented for extensive survival models. Moreover, it is robust to heterogeneity and invariant to monotone increasing transformations of the response. The sure screening property and ranking consistency property are also established under mild conditions. The competence and robustness of our method is further confirmed through comprehensive simulation studies and an analysis of a real data example.     

Asymptotic theory for nonparametric regression with spatial data

Nonparametric regression with spatial, or spatio-temporal, data is considered. The conditional mean of a dependent variable, given explanatory ones, is a nonparametric function, while the conditional covariance reflects spatial correlation. Conditional heteroscedasticity is also allowed, as well as non-identically distributed observations. Instead of mixing conditions, a (possibly non-stationary) linear process is assumed for disturbances, allowing for long range, as well as short-range, dependence, while decay in dependence in explanatory variables is described using a measure based on the departure of the joint density from the product of marginal densities. A basic triangular array setting is employed, with the aim of covering various patterns of spatial observation. Sufficient conditions are established for consistency and asymptotic normality of kernel regression estimates. When the cross-sectional dependence is sufficiently mild, the asymptotic variance in the central limit theorem is the same as when observations are independent; otherwise, the rate of convergence is slower. We discuss the application of our conditions to spatial autoregressive models, and models defined on a regular lattice.     

Complete subset regressions

This paper proposes a new method for combining forecasts based on complete subset regressions. For a given set of potential predictor variables we combine forecasts from all possible linear regression models that keep the number of predictors fixed. We explore how the choice of model complexity, as measured by the number of included predictor variables, can be used to trade off the bias and variance of the forecast errors, generating a setup akin to the efficient frontier known from modern portfolio theory. In an application to predictability of stock returns, we find that combinations of subset regressions can produce more accurate forecasts than conventional approaches based on equal-weighted forecasts (which fail to account for the dimensionality of the underlying models), combinations of univariate forecasts, or forecasts generated by methods such as bagging, ridge regression or Bayesian Model Averaging.     

Robust Hierarchical Bayes Small Area Estimation for the Nested Error Linear Regression Model

Standard model‐based small area estimates perform poorly in presence of outliers. Sinha & Rao ( 2009 ) developed robust frequentist predictors of small area means. In this article, we present a robust Bayesian method to handle outliers in unit‐level data by extending the nested error regression model. We consider a finite mixture of normal distributions for the unit‐level error to model outliers and produce noninformative Bayes predictors of small area means. Our modelling approach generalises that of Datta & Ghosh ( 1991 ) under the normality assumption. Application of our method to a data set which is suspected to contain an outlier confirms this suspicion, correctly identifies the suspected outlier and produces robust predictors and posterior standard deviations of the small area means. Evaluation of several procedures including the M‐quantile method of Chambers & Tzavidis ( 2006 ) via simulations shows that our proposed method is as good as other procedures in terms of bias, variability and coverage probability of confidence and credible intervals when there are no outliers. In the presence of outliers, while our method and Sinha–Rao method perform similarly, they improve over the other methods. This superior performance of our procedure shows its dual (Bayes and frequentist) dominance, which should make it attractive to all practitioners, Bayesians and frequentists, of small area estimation.     

Reparameterized inverse gamma regression models with varying precision

In this article, we propose a mean linear regression model where the response variable is inverse gamma distributed using a new parameterization of this distribution that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive response variable, as usual in the context of generalized linear models. The variance function of the proposed model has a quadratic form. The inverse gamma distribution is a member of the exponential family of distributions and has some distributions commonly used for parametric models in survival analysis as special cases. We compare the proposed model to several alternatives and illustrate its advantages and usefulness. With a generalized linear model approach that takes advantage of exponential family properties, we discuss model estimation (by maximum likelihood), black further inferential quantities and diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the obtained results. A real application using minerals data set collected by Department of Mines of the University of Atacama, Chile, is considered to demonstrate the practical potential of the proposed model.