OLS and IV estimation of regression models including endogenous interaction terms

We analyze a class of linear regression models including interactions of endogenous regressors and exogenous covariates. We show how to generate instrumental variables using the nonlinear functional form of the structural equation when traditional excluded instruments are unknown. We propose to use these instruments with identification robust IV inference. We furthermore show that, whenever functional form identification is not valid, the ordinary least squares (OLS) estimator of the coefficient of the interaction term is consistent and standard OLS inference applies. Using our alternative empirical methods we confirm recent empirical findings on the nonlinear causal relation between financial development and economic growth.     

Penalized angular regression for personalized predictions

Personalization is becoming an important aspect of many predictive applications. We introduce a penalized regression method which inherently implements personalization. Personalized angle (PAN) regression constructs regression coefficients that are specific to the covariate vector for which one is producing a prediction, thus personalizing the regression model itself. This is achieved by penalizing the normalized prediction for a given covariate vector. The method therefore penalizes the normalized regression coefficients, or the angles of the regression coefficients in a hyperspherical parametrization, introducing a new angle-based class of penalties. PAN hence combines two novel concepts: penalizing the normalized coefficients and personalization. For an orthogonal design matrix, we show that the PAN estimator is the solution to a low-dimensional eigenvector equation. Based on the hyperspherical parametrization, we construct an efficient algorithm to calculate the PAN estimator. We propose a parametric bootstrap procedure for selecting the tuning parameter, and simulations show that PAN regression can outperform ordinary least squares, ridge regression and other penalized regression methods in terms of prediction error. Finally, we demonstrate the method in a medical application.     

Use of non-normality in structural equation modeling: Application to direction of causation

Structural equation modeling (SEM) typically utilizes first- and second-order moment structures. This limits its applicability since many unidentified models and many equivalent models that researchers would like to distinguish are created. In this paper, we relax this restriction and assume non-normal distributions on exogenous variables. We shall provide a solution to the problems of underidentifiability and equivalence of SEM models by making use of non-normality (higher-order moment structures). The non-normal SEM is applied to finding the possible direction of a path in simple regression models. The method of (generalized) least squares is employed to estimate model parameters. A test statistic for examining a fit of a model is proposed. A simulation result and a real data example are reported to study how the non-normal SEM approach works empirically.     

Laplace based approximate posterior inference for differential equation models

Ordinary differential equations are arguably the most popular and useful mathematical tool for describing physical and biological processes in the real world. Often, these physical and biological processes are observed with errors, in which case the most natural way to model such data is via regression where the mean function is defined by an ordinary differential equation believed to provide an understanding of the underlying process. These regression based dynamical models are called differential equation models. Parameter inference from differential equation models poses computational challenges mainly due to the fact that analytic solutions to most differential equations are not available. In this paper, we propose an approximation method for obtaining the posterior distribution of parameters in differential equation models. The approximation is done in two steps. In the first step, the solution of a differential equation is approximated by the general one-step method which is a class of numerical numerical methods for ordinary differential equations including the Euler and the Runge-Kutta procedures; in the second step, nuisance parameters are marginalized using Laplace approximation. The proposed Laplace approximated posterior gives a computationally fast alternative to the full Bayesian computational scheme (such as Makov Chain Monte Carlo) and produces more accurate and stable estimators than the popular smoothing methods (called collocation methods) based on frequentist procedures. For a theoretical support of the proposed method, we prove that the Laplace approximated posterior converges to the actual posterior under certain conditions and analyze the relation between the order of numerical error and its Laplace approximation. The proposed method is tested on simulated data sets and compared with the other existing methods.     

LARS-type algorithm for group lasso

The least absolute shrinkage and selection operator (lasso) has been widely used in regression analysis. Based on the piecewise linear property of the solution path, least angle regression provides an efficient algorithm for computing the solution paths of lasso. Group lasso is an important generalization of lasso that can be applied to regression with grouped variables. However, the solution path of group lasso is not piecewise linear and hence cannot be obtained by least angle regression. By transforming the problem into a system of differential equations, we develop an algorithm for efficient computation of group lasso solution paths. Simulation studies are conducted for comparing the proposed algorithm to the best existing algorithm: the groupwise-majorization-descent algorithm.     

Mixtures of Linear Regression with Measurement Errors

Existing research on mixtures of regression models are limited to directly observed predictors. The estimation of mixtures of regression for measurement error data imposes challenges for statisticians. For linear regression models with measurement error data, the naive ordinary least squares method, which directly substitutes the observed surrogates for the unobserved error-prone variables, yields an inconsistent estimate for the regression coefficients. The same inconsistency also happens to the naive mixtures of regression estimate, which is based on the traditional maximum likelihood estimator and simply ignores the measurement error. To solve this inconsistency, we propose to use the deconvolution method to estimate the mixture likelihood of the observed surrogates. Then our proposed estimate is found by maximizing the estimated mixture likelihood. In addition, a generalized EM algorithm is also developed to find the estimate. The simulation results demonstrate that the proposed estimation procedures work well and perform much better than the naive estimates.     

Is It Always Necessary to Take Sample Selection into Account?

We compare a simple ordinary least squares (OLS) with the maximum likelihood estimation of the Tobit I and Tobit II regression models, in the selected sample. We propose a new measure to quantify the performance of OLS.     

Error variance estimation via least squares for small sample nonparametric regression

In this paper we explore statistical properties of some difference-based approaches to estimate an error variance for small sample based on nonparametric regression which satisfies Lipschitz condition. Our study is motivated by Tong and Wang (2005), who estimated error variance using a least squares approach. They considered the error variance as the intercept in a simple linear regression which was obtained from the expectation of their lag-k Rice estimator. Their variance estimators are highly dependent on the setting of a regressor and weight of their simple linear regression. Although this regressor and weight can be varied based on the characteristic of an unknown nonparametric mean function, Tong and Wang (2005) have used a fixed regressor and weight in a large sample and gave no indication of how to determine the regressor and the weight. In this paper, we propose a new approach via local quadratic approximation to determine this regressor and weight. Using our proposed regressor and weight, we estimate the error variance as the intercept of simple linear regression using both ordinary least squares and weighted least squares. Our approach applies to both small and large samples, while most existing difference-based methods are appropriate solely for large samples. We compare the performance of our approach with other existing approaches using extensive simulation study. The advantage of our approach is demonstrated using a real data set.     

Selection of Predictors in Distance-Based Regression

Distance-based regression is a prediction method consisting of two steps: from distances between observations we obtain latent variables which, in turn, are the regressors in an ordinary least squares linear model. Distances are computed from actually observed predictors by means of a suitable dissimilarity function. Being generally nonlinearly related with the response, their selection by the usual F tests is unavailable. In this article, we propose a solution to this predictor selection problem by defining generalized test statistics and adapting a nonparametric bootstrap method to estimate their p-values. We include a numerical example with automobile insurance data.     

Kernel Density-Based Linear Regression Estimate

For linear regression models with non normally distributed errors, the least squares estimate (LSE) will lose some efficiency compared to the maximum likelihood estimate (MLE). In this article, we propose a kernel density-based regression estimate (KDRE) that is adaptive to the unknown error distribution. The key idea is to approximate the likelihood function by using a nonparametric kernel density estimate of the error density based on some initial parameter estimate. The proposed estimate is shown to be asymptotically as efficient as the oracle MLE which assumes the error density were known. In addition, we propose an EM type algorithm to maximize the estimated likelihood function and show that the KDRE can be considered as an iterated weighted least squares estimate, which provides us some insights on the adaptiveness of KDRE to the unknown error distribution. Our Monte Carlo simulation studies show that, while comparable to the traditional LSE for normal errors, the proposed estimation procedure can have substantial efficiency gain for non normal errors. Moreover, the efficiency gain can be achieved even for a small sample size.     

Ridge Regression for Estimation of Transition Probabilities from Aggregate Data

When data from several independent Markov chains are aggregated over each time point, least square estimation of transition probabilities faces the problem of multi-collinearity. We propose here an estimation procedure which involves use of ridge regression for the ordinary least square estimators. Performance of this estimator is then compared with that of the ordinary least squares.     

Second-order least squares estimation of censored regression models

This paper proposes the second-order least squares estimation, which is an extension of the ordinary least squares method, for censored regression models where the error term has a general parametric distribution (not necessarily normal). The strong consistency and asymptotic normality of the estimator are derived under fairly general regularity conditions. We also propose a computationally simpler estimator which is consistent and asymptotically normal under the same regularity conditions. Finite sample behavior of the proposed estimators under both correctly and misspecified models are investigated through Monte Carlo simulations. The simulation results show that the proposed estimator using optimal weighting matrix performs very similar to the maximum likelihood estimator, and the estimator with the identity weight is more robust against the misspecification.     

The asymptotic behaviour of the residual sum of squares in models with multiple break points

ABSTRACT

Models with multiple discrete breaks in parameters are usually estimated via least squares. This paper, first, derives the asymptotic expectation of the residual sum of squares and shows that the number of estimated break points and the number of regression parameters affect the expectation differently. Second, we propose a statistic for testing the joint hypothesis that the breaks occur at specified points in the sample. Our analytical results cover models estimated by the ordinary, nonlinear, and two-stage least squares. An application to U.S. monetary policy rejects the assumption that breaks are associated with changes in the chair of the Fed.     

Ridge estimation in generalized linear models and proportional hazards regressions

Conventionally, a ridge parameter is estimated as a function of regression parameters based on ordinary least squares. In this article, we proposed an iterative procedure instead of the one-step or conventional ridge method. Additionally, we construct an indicator that measures the potential degree of improvement in mean squared error when ridge estimates are employed. Simulations show that our methods are appropriate for a wide class of non linear models including generalized linear models and proportional hazards (PHs) regressions. The method is applied to a PH regression with highly collinear covariates in a cancer recurrence study.     

The peculiar shrinkage properties of partial least squares regression

Partial least squares regression has been widely adopted within some areas as a useful alternative to ordinary least squares regression in the manner of other shrinkage methods such as principal components regression and ridge regression. In this paper we examine the nature of this shrinkage and demonstrate that partial least squares regression exhibits some undesirable properties.     

Differential equation model of carbon dioxide emission using functional linear regression

Carbon dioxide is one of the major contributors to Global Warming. In the present study, we develop a differential equation to model the carbon dioxide emission data in the atmosphere using functional linear regression approach. In the proposed method, a differential operator is defined as data smoother and we use the penalized least square fitting criteria to smooth the data. The profile error sum of squares is optimized to estimate the differential operators using functional regression. The solution of the developed differential equation estimates and predicts the rate of change of carbon dioxide in the atmosphere at a particular time. We apply the proposed model to fit the emission of carbon dioxide data in the continental United States. Numerical simulations of a number of test cases depict a satisfactory agreement with real data.     

Sparse partial least squares regression for simultaneous dimension reduction and variable selection

Summary.  Partial least squares regression has been an alternative to ordinary least squares for handling multicollinearity in several areas of scientific research since the 1960s. It has recently gained much attention in the analysis of high dimensional genomic data. We show that known asymptotic consistency of the partial least squares estimator for a univariate response does not hold with the very large p and small n paradigm. We derive a similar result for a multivariate response regression with partial least squares. We then propose a sparse partial least squares formulation which aims simultaneously to achieve good predictive performance and variable selection by producing sparse linear combinations of the original predictors. We provide an efficient implementation of sparse partial least squares regression and compare it with well-known variable selection and dimension reduction approaches via simulation experiments. We illustrate the practical utility of sparse partial least squares regression in a joint analysis of gene expression and genomewide binding data.     

Some small-sample results on a bounded influence rank regression method

Naranjo and HeUmansperger (1994) recently derved a bounded influence rank regression method and suggested how hypotheses about the regression coefficients might be tested. This brief note reports some simulation results on how their procedure performs when there is one predictor. Even when the error term is highly skewed, good control over the Type I error probability is obtained Power can be high relative to least squares regression when the error term has a heavy tailed distribution .and the predictor has a symmetric distribution However, if the predictor has a skewed distribution, power can be relatively low even when the distribution of the error term is heavy tailed. Despite this, it is argued that their method provides an important and useful alternative to ordinary least squares as well as other robust regression methods.     

Response and predictor folding to counter symmetric dependency in dimension reduction

In the regression setting, dimension reduction allows for complicated regression structures to be detected via visualisation in a low‐dimensional framework. However, some popular dimension reduction methodologies fail to achieve this aim when faced with a problem often referred to as symmetric dependency. In this paper we show how vastly superior results can be achieved when carrying out response and predictor transformations for methods such as least squares and sliced inverse regression. These transformations are simple to implement and utilise estimates from other dimension reduction methods that are not faced with the symmetric dependency problem. We highlight the effectiveness of our approach via simulation and an example. Furthermore, we show that ordinary least squares can effectively detect multiple dimension reduction directions. Methods robust to extreme response values are also considered.