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In this paper, a two-dimensional panel data model of economic policy uncertainty is investigated based on the individual fixed effects of panel quantile regression, and a nonparametric panel model with individual fixed effects is established. The unfolding of nonparametric penalized spline and the introduction of Bayesian in stratified quantile are utilized to construct regression models applicable to accounting robustness, respectively. In the empirical study, the economic policy uncertainty index, accounting robustness and commercial credit supply are measured respectively. The annual data of China’s Shenzhen and Shanghai A-share listed companies during the period from 2012 to 2021 were selected as the research basis, and Bayesian quantile regression was made on the basis of correlation analysis. The coefficient of commercial credit supply is found to be -0.0821, and the variable RD1 is negatively correlated with economic policy uncertainty. This regression result confirms hypothesis H1 of this paper, suggesting that private firms invest less in innovation when economic policy uncertainty is higher. In the test of economic policy uncertainty by type, the regression coefficients of RD2, EPU, and SIZE are negative, respectively -0.0368, −0.2124, and -0.1458, which indicates that fiscal policy, monetary policy, and exchange rate and capital account policy uncertainty are negatively correlated with the supply of business credit to enterprises. Based on this correlation, this study provides guidance for the development of business credit for enterprises.

Semiparametric regression offers a flexible framework for modeling nonlinear relationships between a response and covariates. A prime example are generalized additive models (GAMs) where splines (say) are used to approximate nonlinear functional components in conjunction with a quadratic penalty to control for overfitting. Estimation and inference are then generally performed based on the penalized likelihood, or under a mixed model framework. The penalized likelihood framework is fast but potentially unstable, and choosing the smoothing parameters needs to be done externally using cross-validation, for instance. The mixed model framework tends to be more stable and offers a natural way for choosing the smoothing parameters, but for nonnormal responses involves an intractable integral. In this article, we introduce a new framework for semiparametric regression based on variational approximations (VA). The approach possesses the stability and natural inference tools of the mixed model framework, while achieving computation times comparable to using penalized likelihood. Focusing on GAMs, we derive fully tractable variational likelihoods for some common response types. We present several features of the VA framework for inference, including a variational information matrix for inference on parametric components, and a closed-form update for estimating the smoothing parameter. We demonstrate the consistency of the VA estimates, and an asymptotic normality result for the parametric component of the model. Simulation studies show the VA framework performs similarly to and sometimes better than currently available software for fitting GAMs. Supplementary materials for this article are available online.

We study additive function-on-function regression where the mean response at a particular time point depends on the time point itself, as well as the entire covariate trajectory. We develop a computationally efficient estimation methodology based on a novel combination of spline bases with an eigenbasis to represent the trivariate kernel function. We discuss prediction of a new response trajectory, propose an inference procedure that accounts for total variability in the predicted response curves, and construct pointwise prediction intervals. The estimation/inferential procedure accommodates realistic scenarios, such as correlated error structure as well as sparse and/or irregular designs. We investigate our methodology in finite sample size through simulations and two real data applications.

A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y, as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton-Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.

Hidden Markov models (HMMs) are flexible time series models in which the distribution of the observations depends on unobserved serially correlated states. The state-dependent distributions in HMMs are usually taken from some class of parametrically specified distributions. The choice of this class can be difficult, and an unfortunate choice can have serious consequences for example on state estimates, and more generally on the resulting model complexity and interpretation. We demonstrate these practical issues in a real data application concerned with vertical speeds of a diving beaked whale, where we demonstrate that parametric approaches can easily lead to overly complex state processes, impeding meaningful biological inference. In contrast, for the dive data, HMMs with nonparametrically estimated state-dependent distributions are much more parsimonious in terms of the number of states and easier to interpret, while fitting the data equally well. Our nonparametric estimation approach is based on the idea of representing the densities of the state-dependent distributions as linear combinations of a large number of standardized B-spline basis functions, imposing a penalty term on non-smoothness in order to maintain a good balance between goodness-of-fit and smoothness.

In data analysis using a nonparametric regression approach, we are often faced with the problem of analyzing a set of data that has mixed patterns, namely, some of the data have a certain pattern and the rest of the data have a different pattern. To handle this kind of datum, we propose the use of a mixed estimator. In this study, we theoretically discuss a developed estimation method for a nonparametric regression model with two or more response variables and predictor variables, and there is a correlation between the response variables using a mixed estimator. The model is called the multiresponse multipredictor nonparametric regression (MMNR) model. The mixed estimator used for estimating the MMNR model is a mixed estimator of smoothing spline and Fourier series that is suitable for analyzing data with patterns that partly change at certain subintervals, and some others that follow a recurring pattern in a certain trend. Since in the MMNR model there is a correlation between responses, a symmetric weight matrix is involved in the estimation process of the MMNR model. To estimate the MMNR model, we apply the reproducing kernel Hilbert space (RKHS) method to penalized weighted least square (PWLS) optimization for estimating the regression function of the MMNR model, which consists of a smoothing spline component and a Fourier series component. A simulation study to show the performance of proposed method is also given. The obtained results are estimations of the smoothing spline component, Fourier series component, MMNR model, weight matrix, and consistency of estimated regression function. In conclusion, the estimation of the MMNR model using the mixed estimator is a combination of smoothing spline component and Fourier series component estimators. It depends on smoothing and oscillation parameters, and it has linear in observation and consistent properties.

We study generalized additive partial linear models, proposing the use of polynomial spline smoothing for estimation of nonparametric functions, and deriving quasi-likelihood based estimators for the linear parameters. We establish asymptotic normality for the estimators of the parametric components. The procedure avoids solving large systems of equations as in kernel-based procedures and thus results in gains in computational simplicity. We further develop a class of variable selection procedures for the linear parameters by employing a nonconcave penalized quasi-likelihood, which is shown to have an asymptotic oracle property. Monte Carlo simulations and an empirical example are presented for illustration.

The development goal of a country must be focused on the quality of human life to achieve prosperity. One important indicator for measuring the success of a country's development is the Human Development Index (HDI). In 2018, Papua was the province with the lowest HDI in Indonesia. Special attention is needed to improve HDI in Papua Province, one of them is by paying attention to the variables that affect HDI such as population growth rate, percentage of poor population, and economic growth. The relationships between HDI and the predictor variables do not have a clear pattern and tend to change at certain subintervals. This case can be approached using Spline Smoothing in multivariable nonparametric regression. Spline Smoothing is a type of estimator in nonparametric regression that has an excellent ability to handle data that tend to change at certain subintervals. Therefore, the purposes of this study are to obtain the form of Spline Smoothing estimator function in multivariable nonparametric regression, estimate the function and apply it to the HDI in Papua Province. The empirical results of modeling HDI in Papua Province show that it can be adequately applied which gives GCV = 58.108, R2 = 99.77% and RMSE = 0.0505.

Structured additive regression (STAR) provides a general framework for complex Gaussian and non-Gaussian regression models, with predictors comprising arbitrary combinations of nonlinear functions and surfaces, spatial effects, varying coefficients, random effects, and further regression terms. The large flexibility of STAR makes function selection a challenging and important task, aiming at (1) selecting the relevant covariates, (2) choosing an appropriate and parsimonious representation of the impact of covariates on the predictor, and (3) determining the required interactions. We propose a spike-and-slab prior structure for function selection that allows to include or exclude single coefficients as well as blocks of coefficients representing specific model terms. A novel multiplicative parameter expansion is required to obtain good mixing and convergence properties in a Markov chain Monte Carlo simulation approach and is shown to induce desirable shrinkage properties. In simulation studies and with (real) benchmark classification data, we investigate sensitivity to hyperparameter settings and compare performance to competitors. The flexibility and applicability of our approach are demonstrated in an additive piecewise exponential model with time-varying effects for right-censored survival times of intensive care patients with sepsis. Geoadditive and additive mixed logit model applications are discussed in an extensive online supplement.

Nonparametric smoothing methods are used to model longitudinal data, but the challenge remains to incorporate correlation into nonparametric estimation procedures. In this article, we propose an efficient estimation procedure for varying‐coefficient models for longitudinal data. The proposed procedure can easily take into account correlation within subjects and deal directly with both continuous and discrete response longitudinal data under the framework of generalized linear models. The proposed approach yields a more efficient estimator than the generalized estimation equation approach when the working correlation is misspecified. For varying‐coefficient models, it is often of interest to test whether coefficient functions are time varying or time invariant. We propose a unified and efficient nonparametric hypothesis testing procedure, and further demonstrate that the resulting test statistics have an asymptotic chi‐squared distribution. In addition, the goodness‐of‐fit test is applied to test whether the model assumption is satisfied. The corresponding test is also useful for choosing basis functions and the number of knots for regression spline models in conjunction with the model selection criterion. We evaluate the finite sample performance of the proposed procedures with Monte Carlo simulation studies. The proposed methodology is illustrated by the analysis of an acquired immune deficiency syndrome (AIDS) data set.