Asymptotics and smoothing parameter selection for penalized spline regression with various loss functions

Penalized splines are used in various types of regression analyses, including non‐parametric quantile, robust and the usual mean regression. In this paper, we focus on the penalized spline estimator with general convex loss functions. By specifying the loss function, we can obtain the mean estimator, quantile estimator and robust estimator. We will first study the asymptotic properties of penalized splines. Specifically, we will show the asymptotic bias and variance as well as the asymptotic normality of the estimator. Next, we will discuss smoothing parameter selection for the minimization of the mean integrated squares error. The new smoothing parameter can be expressed uniquely using the asymptotic bias and variance of the penalized spline estimator. To validate the new smoothing parameter selection method, we will provide a simulation. The simulation results show that the consistency of the estimator with the proposed smoothing parameter selection method can be confirmed and that the proposed estimator has better behavior than the estimator with generalized approximate cross‐validation. A real data example is also addressed.     

SCAD‐penalized quantile regression for high‐dimensional data analysis and variable selection

The present penalized quantile variable selection methods are only applicable to finite number of predictors or do not have oracle property associated with estimator. This technique is considered as an alternative to ordinary least squares regression in case of the outliers and the heavy‐tailed errors existing in linear models. The variable selection through quantile regression with diverging number of parameters is investigated in this paper. The convergence rate of estimator with smoothly clipped absolute deviation penalty function is also studied. Moreover, the oracle property with proper selection of tuning parameter for quantile regression under certain regularity conditions is also established. In addition, the rank correlation screening method is used to accommodate ultra‐high dimensional data settings. Monte Carlo simulations demonstrate finite performance of the proposed estimator. The results of real data reveal that this approach provides substantially more information as compared with ordinary least squares, conventional quantile regression, and quantile lasso.     

BALANCED VARIABLE ADDITION IN LINEAR MODELS

This paper studies what happens when we move from a short regression to a long regression in a setting where both regressions are subject to misspecification. In this setup, the least‐squares estimator in the long regression may have larger inconsistency than the least‐squares estimator in the short regression. We provide a simple interpretation for the comparison of the inconsistencies and study under which conditions the additional regressors in the long regression represent a “balanced addition” to the short regression.     

The effect of a Durbin–Watson pretest on confidence intervals in regression

Consider a linear regression model and suppose that our aim is to find a confidence interval for a specified linear combination of the regression parameters. In practice, it is common to perform a Durbin–Watson pretest of the null hypothesis of zero first‐order autocorrelation of the random errors against the alternative hypothesis of positive first‐order autocorrelation. If this null hypothesis is accepted then the confidence interval centered on the ordinary least squares estimator is used; otherwise the confidence interval centered on the feasible generalized least squares estimator is used. For any given design matrix and parameter of interest, we compare the confidence interval resulting from this two‐stage procedure and the confidence interval that is always centered on the feasible generalized least squares estimator, as follows. First, we compare the coverage probability functions of these confidence intervals. Second, we compute the scaled expected length of the confidence interval resulting from the two‐stage procedure, where the scaling is with respect to the expected length of the confidence interval centered on the feasible generalized least squares estimator, with the same minimum coverage probability. These comparisons are used to choose the better confidence interval, prior to any examination of the observed response vector.     

An application of approximate finite sample results to parameter estimation in a linear errors-in-variables model

In the simple errors-in-variables model the least squares estimator of the slope coefficient is known to be biased towards zero for finite sample size as well as asymptotically. In this paper we suggest a new corrected least squares estimator, where the bias correction is based on approximating the finite sample bias by a lower bound. This estimator is computationally very simple. It is compared with previously proposed corrected least squares estimators, where the correction aims at removing the asymptotic bias or the exact finite sample bias. For each type of corrected least squares estimators we consider the theoretical form, which depends on an unknown parameter, as well as various feasible forms. An analytical comparison of the theoretical estimators is complemented by a Monte Carlo study evaluating the performance of the feasible estimators. The new estimator proposed in this paper proves to be superior with respect to the mean squared error.     

Another look at the naive estimator in a regression model

We consider the linear regression model where only a particular linear function of the dependent variables is observed, Stahlecker and Schmidt (1987) proposed a naive least squares (LS) estimator for regression coefficients in such a case. In this note we represent their estimator as a general ridge estimator. This observation leads to a view different from the previous work and provides an easy way of obtaining many important properties of the naive LS estimator. Our approach also gives some insight into the relationship between the naive LS estimator and the generalized least squares estimator.     

Variable selection via composite quantile regression with dependent errors

We propose composite quantile regression for dependent data, in which the errors are from short‐range dependent and strictly stationary linear processes. Under some regularity conditions, we show that composite quantile estimator enjoys root‐n consistency and asymptotic normality. We investigate the asymptotic relative efficiency of composite quantile estimator to both single‐level quantile regression and least‐squares regression. When the errors have finite variance, the relative efficiency of composite quantile estimator with respect to the least‐squares estimator has a universal lower bound. Under some regularity conditions, the adaptive least absolute shrinkage and selection operator penalty leads to consistent variable selection, and the asymptotic distribution of the non‐zero coefficient is the same as that of the counterparts obtained when the true model is known. We conduct a simulation study and a real data analysis to evaluate the performance of the proposed approach.     

Nonlinear regression models with single‐index heteroscedasticity

We consider nonlinear heteroscedastic single‐index models where the mean function is a parametric nonlinear model and the variance function depends on a single‐index structure. We develop an efficient estimation method for the parameters in the mean function by using the weighted least squares estimation, and we propose a “delete‐one‐component” estimator for the single‐index in the variance function based on absolute residuals. Asymptotic results of estimators are also investigated. The estimation methods for the error distribution based on the classical empirical distribution function and an empirical likelihood method are discussed. The empirical likelihood method allows for incorporation of the assumptions on the error distribution into the estimation. Simulations illustrate the results, and a real chemical data set is analyzed to demonstrate the performance of the proposed estimators.     

Mean estimation bias in least squares estimation of autoregressive processes

Estimation of the parameters of an autoregressive process with a mean that is a function of time is considered. Approximate expressions for the bias of the least squares estimator of the autoregressive parameters that is due to estimating the unknown mean function are derived. For the case of a mean function that is a polynomial in time, a reparameterization that isolates the bias is given. Using the approximate expressions, a method of modifying the least squares estimator is proposed. A Monte Carlo study of the second-order autoregressive process is presented. The Monte Carlo results agree well with the approximate theory and, generally speaking, the modified least squares estimators performed better than the least squares estimator. For the second-order process we also considered the empirical properties of the estimated generalized least squares estimator of the mean function and the error made in predicting the process one, two and three periods in the future.     

Estimation in monotone single‐index models

Single‐index models are popular regression models that are more flexible than linear models and still maintain more structure than purely nonparametric models. We consider the problem of estimating the regression parameters under a monotonicity constraint on the unknown link function. In contrast to the standard approach of using smoothing techniques, we review different “non‐smooth” estimators that avoid the difficult smoothing parameter selection. For about 30 years, one has had the conjecture that the profile least squares estimator is an ‐consistent estimator of the regression parameter, but the only non‐smooth argmin/argmax estimators that are actually known to achieve this ‐rate are not based on the nonparametric least squares estimator of the link function. However, solving a score equation corresponding to the least squares approach results in ‐consistent estimators. We illustrate the good behavior of the score approach via simulations. The connection with the binary choice and current status linear regression models is also discussed.     

Binary response panel data models with sample selection and self‐selection

We consider estimating binary response models on an unbalanced panel, where the outcome of the dependent variable may be missing due to nonrandom selection, or there is self‐selection into a treatment. In the present paper, we first consider estimation of sample selection models and treatment effects using a fully parametric approach, where the error distribution is assumed to be normal in both primary and selection equations. Arbitrary time dependence in errors is permitted. Estimation of both coefficients and partial effects, as well as tests for selection bias, are discussed. Furthermore, we consider a semiparametric estimator of binary response panel data models with sample selection that is robust to a variety of error distributions. The estimator employs a control function approach to account for endogenous selection and permits consistent estimation of scaled coefficients and relative effects.     

Statistical inference on seemingly unrelated varying coefficient partially linear models

This paper is concerned with the statistical inference on seemingly unrelated varying coefficient partially linear models. By combining the local polynomial and profile least squares techniques, and estimating the contemporaneous correlation, we propose a class of weighted profile least squares estimators (WPLSEs) for the parametric components. It is shown that the WPLSEs achieve the semiparametric efficiency bound and are asymptotically normal. For the non‐parametric components, by applying the undersmoothing technique, and taking the contemporaneous correlation into account, we propose an efficient local polynomial estimation. The resulting estimators are shown to have mean‐squared errors smaller than those estimators that neglect the contemporaneous correlation. In addition, a class of variable selection procedures is developed for simultaneously selecting significant variables and estimating unknown parameters, based on the non‐concave penalized and weighted profile least squares techniques. With a proper choice of regularization parameters and penalty functions, the proposed variable selection procedures perform as efficiently as if one knew the true submodels. The proposed methods are evaluated using wide simulation studies and applied to a set of real data.     

On the relative efficiency of estimators which include the initial observations in the estimation of seemingly unrelated regressions with first-order autoregressive disturbances

The generalized least squares estimator for a seemingly unrelated regressions model with first-order vector autoregressive disturbances is outlined, and its efficiency is compared with that of an approximate generalized least squares estimator which ignores the first observation. A scalar index for the loss of efficiency is developed and applied to a special case where the matrix of autoregressive parameters is diagonal and the regressors are smooth. Also, for a more general model, a Monte Carlo study is used to investigate the relative efficiencies of various estimators. The results suggest that Maeshiro (1980) has overstated the case for the exact generalized least squares estimator, because, in many circumstances, it is only marginally better than the approximate generalized least squares estimator.     

Functional Coefficient Cointegration Models Subject to Time–Varying Volatility with an Application to the Purchasing Power Parity

This paper analyses functional coefficient cointegration models with both stationary and non‐stationary covariates, allowing time‐varying (unconditional) volatility of a general form. The conventional kernel weighted least squares (KLS) estimator is subject to potential efficiency loss, and can be improved by an adaptive kernel weighted least squares (AKLS) estimator that adapts to heteroscedasticity of unknown form. The AKLS estimator is shown to be as efficient as the oracle generalized kernel weighted least squares estimator asymptotically, and can achieve significant efficiency gain relative to the KLS estimator in finite samples. An illustrative example is provided by investigating the Purchasing Power Parity hypothesis.     

Optimal inference for instrumental variables regression with non-Gaussian errors

This paper is concerned with inference on the coefficient on the endogenous regressor in a linear instrumental variables model with a single endogenous regressor, nonrandom exogenous regressors and instruments, and i.i.d. errors whose distribution is unknown. It is shown that under mild smoothness conditions on the error distribution it is possible to develop tests which are “nearly” efficient in the sense of Andrews et al. (2006) when identification is weak and consistent and asymptotically optimal when identification is strong. In addition, an estimator is presented which can be used in the usual way to construct valid (indeed, optimal) confidence intervals when identification is strong. The estimator is of the two stage least squares variety and is asymptotically efficient under strong identification whether or not the errors are normal.     

To pool or not to pool: What is a good strategy for parameter estimation and forecasting in panel regressions?

This paper considers estimating the slope parameters and forecasting in potentially heterogeneous panel data regressions with a long time dimension. We propose a novel optimal pooling averaging estimator that makes an explicit trade‐off between efficiency gains from pooling and bias due to heterogeneity. By theoretically and numerically comparing various estimators, we find that a uniformly best estimator does not exist and that our new estimator is superior in nonextreme cases and robust in extreme cases. Our results provide practical guidance for the best estimator and forecast depending on features of data and models. We apply our method to examine the determinants of sovereign credit default swap spreads and forecast future spreads.     

Practical Problems with Reduced‐rank ML Estimators for Cointegration Parameters and a Simple Alternative*

Johansen's reduced‐rank maximum likelihood (ML) estimator for cointegration parameters in vector error correction models is known to produce occasional extreme outliers. Using a small monetary system and German data we illustrate the practical importance of this problem. We also consider an alternative generalized least squares (GLS) system estimator which has better properties in this respect. The two estimators are compared in a small simulation study. It is found that the GLS estimator can indeed be an attractive alternative to ML estimation of cointegration parameters.     

The asymptotic behaviour of the estimated generalized least squares method in the linear regression model

Abstract  In the linear regression model the generalized least squares (GLS) method is only applicable if the covariance matrix of the errors is known but for a scalar factor. Otherwise an estimator for this matrix has to be used. Then we speak of the estimated generalized least squares (EGLS) method. In this paper the asymptotic behaviour of both methods is compared. Results are applied to some standard models commonly used in econometrics     

Quadratic risk domination of restricted least squares estimators via Stein-ruled auxiliary constraints

This paper develops an estimator that under the standard assumption of the General Linear Model, including normality of disturbances, can be designed to dominate the Restricted Least Squares estimator in quadratic risk under very general conditions. The domination is achieved for any choice of symmetric positive definite weighting matrix used in defining the quadratic risk function, regardless of the correctness of the constraints used to define the restricted least squares estimator. The general problem conditions under which the estimator exists, and the risk behavior of the estimator over the parameter space are identified.     

Application of one‐step method to parameter estimation in ODE models

In this paper, we study application of Le Cam's one‐step method to parameter estimation in ordinary differential equation models. This computationally simple technique can serve as an alternative to numerical evaluation of the popular non‐linear least squares estimator, which typically requires the use of a multistep iterative algorithm and repetitive numerical integration of the ordinary differential equation system. The one‐step method starts from a preliminary ‐consistent estimator of the parameter of interest and next turns it into an asymptotic (as the sample size n →∞ ) equivalent of the least squares estimator through a numerically straightforward procedure. We demonstrate performance of the one‐step estimator via extensive simulations and real data examples. The method enables the researcher to obtain both point and interval estimates. The preliminary ‐consistent estimator that we use depends on non‐parametric smoothing, and we provide a data‐driven methodology for choosing its tuning parameter and support it by theory. An easy implementation scheme of the one‐step method for practical use is pointed out.