Weighted least squares estimators in possibly misspecified nonlinear regression

The behavior of estimators for misspecified parametric models has been well studied. We consider estimators for misspecified nonlinear regression models, with error and covariates possibly dependent. These models are described by specifying a parametric model for the conditional expectation of the response given the covariates. This is a parametric family of conditional constraints, which makes the model itself close to nonparametric. We study the behavior of weighted least squares estimators both when the regression function is correctly specified, and when it is misspecified and also involves possible additional covariates.     

Chi-squared tests for evaluation and comparison of asset pricing models

This paper presents a general statistical framework for estimation, testing and comparison of asset pricing models using the unconstrained distance measure of Hansen and Jagannathan (1997). The limiting results cover both linear and nonlinear models that could be correctly specified or misspecified. We propose modified versions of the existing model selection tests and new pivotal specification and model comparison tests with improved finite-sample properties. In addition, we provide formal tests of multiple model comparison. The excellent size and power properties of the proposed tests are demonstrated using simulated data from linear and nonlinear asset pricing models.     

How to Choose a Working Model for Measuring the Statistical Evidence About a Regression Parameter

Consider using a likelihood ratio to measure the strength of statistical evidence for one hypothesis over another. Recent work has shown that when the model is correctly specified, the likelihood ratio is seldom misleading. But when the model is not, misleading evidence may be observed quite frequently. Here we consider how to choose a working regression model so that the statistical evidence is correctly represented as often as it would be under the true model. We argue that the criteria for choosing a working model should be how often it correctly represents the statistical evidence about the object of interest (regression coefficient in the true model). We see that misleading evidence about the object of interest is more likely to be observed when the working model is chosen according to other criteria (e.g., parsimony or predictive accuracy).     

Parametric Identification of Multiplicative Exponential Heteroscedasticity

Harvey ( 1976 ) first proposed multiplicative exponential heteroscedasticity in the context of linear regression. These days it is more commonly seen in latent variable models such as Probit or Logit where correctly modelling the heteroscedasticity is imperative for consistent parameter estimates (Yatchew and Griliches, 1985 ). However, it appears the literature lacks a formal proof of point identification for the parametric model. This paper presents several examples that show the conditions presumed throughout the literature are not sufficient for identification. As a contribution, this paper discusses when identification can and cannot be easily obtained and provides proofs of point identification in common specifications.     

最优组合预测在四川省人才需求预测中的应用

本文依据相关资料数据,构建灰色GM(1,1)模型和二元线性回归模型,分别对四川省2005-2015年从业人才需求进行预测,然后使用二模型最优组合预测对预测结果进行修正。在此基础上,提出了实现预测目标的策略建议,为四川省人才培养决策提供参考。     

Projection‐Based Inference in Randomised Clinical Trials

Covariate information is often available in randomised clinical trials for each subject prior to treatment assignment and is commonly utilised to make covariate adjustment for baseline characteristics predictive of the outcome in order to increase precision and improve power in the detection of a treatment effect. Motivated by a nonparametric covariance analysis, we study a projection approach to making objective covariate adjustment in randomised clinical trials on the basis of two unbiased estimating functions that decouple the outcome and covariate data. The proposed projection approach extends a weighted least‐squares procedure by projecting one of the estimating functions onto the linear subspace spanned by the other estimating function that is E‐ancillary for the average treatment effect. Compared with the weighted least‐squares method, the projection method allows for objective inference on the average treatment effect by exploiting the treatment specific covariate–outcome associations. The resulting projection‐based estimator of the average treatment effect is asymptotically efficient when the treatment‐specific working regression models are correctly specified and is asymptotically more efficient than other existing competitors when the treatment‐specific working regression models are misspecified. The proposed projection method is illustrated by an analysis of data from an HIV clinical trial. In a simulation study, we show that the proposed projection method compares favourably with its competitors in finite samples.     

Applications of subsampling,hybrid, and size-correction methods

This paper analyzes the properties of subsampling, hybrid subsampling, and size-correction methods in two non-regular models. The latter two procedures are introduced in Andrews and Guggenberger (2009a). The models are non-regular in the sense that the test statistics of interest exhibit a discontinuity in their limit distribution as a function of a parameter in the model. The first model is a linear instrumental variables (IV) model with possibly weak IVs estimated using two-stage least squares (2SLS). In this case, the discontinuity occurs when the concentration parameter is zero. The second model is a linear regression model in which the parameter of interest may be near a boundary. In this case, the discontinuity occurs when the parameter is on the boundary.     

Spline-based quasi-likelihood estimation of mixed Poisson regression with single-index models

We consider spline-based quasi-likelihood estimation for mixed Poisson regression with single-index models. The unknown smooth function is approximated by B-splines, and a modified Fisher scoring algorithm is employed to compute the estimates. The spline estimate of the nonparametric component is shown to achieve the optimal rate of convergence, and the asymptotic normality of the regression parameter estimates is still valid even if the variance function is misspecified. The semiparametric efficiency of the model can be established if the variance function is correctly specified. The variance of the regression parameter estimates can be consistently estimated by a simple procedure based on the least-squares estimation. The proposed method is evaluated via an extensive Monte Carlo study, and the methodology is illustrated on an air pollution study.     

Linear models, smooth transition autoregressions, and neural networks for forecasting macroeconomic time series: A re-examination

In this paper, we examine the forecast accuracy of linear autoregressive, smooth transition autoregressive (STAR), and neural network (NN) time series models for 47 monthly macroeconomic variables of the G7 economies. Unlike previous studies that typically consider multiple but fixed model specifications, we use a single but dynamic specification for each model class. The point forecast results indicate that the STAR model generally outperforms linear autoregressive models. It also improves upon several fixed STAR models, demonstrating that careful specification of nonlinear time series models is of crucial importance. The results for neural network models are mixed in the sense that at long forecast horizons, an NN model obtained using Bayesian regularization produces more accurate forecasts than a corresponding model specified using the specific-to-general approach. Reasons for this outcome are discussed.     

Forecast Encompassing and Parameter Estimation*

A desirable property of a forecast is that it encompasses competing predictions, in the sense that the accuracy of the preferred forecast cannot be improved through linear combination with a rival prediction. In this paper, we investigate the impact of the uncertainty associated with estimating model parameters in‐sample on the encompassing properties of out‐of‐sample forecasts. Specifically, using examples of non‐nested econometric models, we show that forecasts from the true (but estimated) data generating process (DGP) do not encompass forecasts from competing mis‐specified models in general, particularly when the number of in‐sample observations is small. Following this result, we also examine the scope for achieving gains in accuracy by combining the forecasts from the DGP and mis‐specified models.     

Inference in a nearly integrated autoregressive model with nonnormal innovations

Robust tests and estimators based on nonnormal quasi-likelihood functions are developed for autoregressive models with near unit root. Asymptotic power functions and power envelopes are derived for point-optimal tests of a unit root when the likelihood is correctly specified. The shapes of these power functions are found to be sensitive to the extent of nonnormality in the innovations. Power loss resulting from using least-squares unit-root tests in the presence of thick-tailed innovations appears to be greater than in stationary models.     

What went wrong with the Hungarian consumption function? An econometric investigation of the time series aggregate consumption function for Hungary for 1960–1989

The paper focuses on the time series aggregate consumption function for the Hungarian economy. The empirical econometric analysis presented produces several new results. First, it shows that the income and consumption variables used in this type of model by previous studies are I(2) variables. Consequently, error correction models formulated in terms of their first differences are mis-specified. Second, it provides a strong empirical evidence supporting the view that consumption (and thus saving) was (real) interest rate elastic during the period under investigation, having impact both on the long run and on the short relationships between income and consumption. Third, it provides empirical evidence on choosing the proper income variable in the consumption function. The model selection results clearly supports the model with unadjusted total real money income variable. Fourth, it shows that for the period 1960–1986 a correctly specified and stable error correction model can be established. Finally, the analysis shows that when used for the period beyond 1986, this model suffers from a structural break.     

Nowcasting and forecasting GDP in emerging markets using global financial and macroeconomic diffusion indexes

This paper contributes to the nascent literature on nowcasting and forecasting GDP in emerging market economies using big data methods. This is done by analyzing the usefulness of various dimension-reduction, machine learning and shrinkage methods, including sparse principal component analysis (SPCA), the elastic net, the least absolute shrinkage operator, and least angle regression when constructing predictions using latent global macroeconomic and financial factors (diffusion indexes) in a dynamic factor model (DFM). We also utilize a judgmental dimension-reduction method called the Bloomberg Relevance Index (BRI), which is an index that assigns a measure of importance to each variable in a dataset depending on the variable’s usage by market participants. Our empirical analysis shows that, when specified using dimension-reduction methods (particularly BRI and SPCA), DFMs yield superior predictions relative to both benchmark linear econometric models and simple DFMs. Moreover, global financial and macroeconomic (business cycle) diffusion indexes constructed using targeted predictors are found to be important in four of the five emerging market economies that we study (Brazil, Mexico, South Africa, and Turkey). These findings point to the importance of spillover effects across emerging market economies, and underscore the significance of characterizing such linkages parsimoniously when utilizing high-dimensional global datasets.     

Optimal weighted average power similar tests for the covariance structure in the linear regression model

This paper suggests a procedure for the construction of optimal weighted average power similar tests for the error covariance matrix of a Gaussian linear regression model when the alternative model belongs to the exponential family. The paper uses a saddlepoint approximation to construct simple test statistics for a large class of problems and overcomes the computational burden of evaluating the complicated integrals arising in the derivation of optimal weighted average power tests. Extensions to panel data models are considered. Applications are given to tests for error autocorrelation in the linear regression model and in a panel data framework.     

On applied econometric research

Summary This paper discusses criteria by which one may investigate whether stochastic relations and models are correctly specified. In this connection, the concept of the systematic coincidence arises. A systematic coincidence is a specified relationship, which, on closer analysis, appears to be the result of simultaneous interdependence via the model as a whole, and not a valid relation in its own right at all. The problem is then surveyed from the point of view of the practical research worker, and the emphasis is on time-series analysis.     

Coarsened Propensity Scores and Hybrid Estimators for Missing Data and Causal Inference

In the areas of missing data and causal inference, there is great interest in doubly robust (DR) estimators that involve both an outcome regression (RG) model and a propensity score (PS) model. These DR estimators are consistent and asymptotically normal if either model is correctly specified. Despite their theoretical appeal, the practical utility of DR estimators has been disputed (e.g. Kang and Schaffer, Statistical Science 2007; 22: 523–539). One of the major concerns is the possibility of erratic estimates resulting from near‐zero denominators due to extreme values of the estimated PS. In contrast, the usual RG estimator based on the RG model alone is efficient when the RG model is correct and generally more stable than the DR estimators, although it can be biased when the RG model is incorrect. In light of the unique advantages of the RG and DR estimators, we propose a class of hybrid estimators that attempt to strike a reasonable balance between the RG and DR estimators. These hybrid estimators are motivated by heuristic arguments that coarsened PS estimates are less likely to take extreme values and less sensitive to misspecification of the PS model than the original model‐based PS estimates. The proposed estimators are compared with existing estimators in simulation studies and illustrated with real data from a large observational study on obstetric labour progression and birth outcomes.     

Bayesian averaging,prediction and nonnested model selection

This paper studies the asymptotic relationship between Bayesian model averaging and post-selection frequentist predictors in both nested and nonnested models. We derive conditions under which their difference is of a smaller order of magnitude than the inverse of the square root of the sample size in large samples. This result depends crucially on the relation between posterior odds and frequentist model selection criteria. Weak conditions are given under which consistent model selection is feasible, regardless of whether models are nested or nonnested and regardless of whether models are correctly specified or not, in the sense that they select the best model with the least number of parameters with probability converging to 1. Under these conditions, Bayesian posterior odds and BICs are consistent for selecting among nested models, but are not consistent for selecting among nonnested models and possibly overlapping models. These findings have important bearing for applied researchers who are frequent users of model selection tools for empirical investigation of model predictions.     

Testing for functional misspecification in regression analysis

Recursive residuals may be used to detect functional misspecification in a regression equation. A simple t-statistic and a related Sign test may be constructed from the residuals. The powers of these tests compare favourably with the Durbin–Watson and other tests commonly used to detect functional misspecification from residuals. In addition the tests are relatively robust to serial correction in an otherwise correctly specified model, and this is a further point in their favour.     

Regression Discontinuity Applications with Rounding Errors in the Running Variable

Many empirical applications of regression discontinuity (RD) models use a running variable that is rounded and hence discrete, e.g. age in years, or birth weight in ounces. This paper shows that standard RD estimation using a rounded discrete running variable leads to inconsistent estimates of treatment effects, even when the true functional form relating the outcome and the running variable is known and is correctly specified. This paper provides simple formulas to correct for this discretization bias. The proposed approach does not require instrumental variables, but instead uses information regarding the distribution of rounding errors, which is easily obtained and often close to uniform. Bounds can be obtained without knowing the distribution of the rounding error. The proposed approach is applied to estimate the effect of Medicare on insurance coverage in the USA, and to investigate the retirement‐consumption puzzle in China, utilizing the Chinese mandatory retirement policy. Copyright © 2014 John Wiley & Sons, Ltd.     

Higher order properties of the wild bootstrap under misspecification

We examine the higher order properties of the wild bootstrap (Wu, 1986) in a linear regression model with stochastic regressors. We find that the ability of the wild bootstrap to provide a higher order refinement is contingent upon whether the errors are mean independent of the regressors or merely uncorrelated with them. In the latter case, the wild bootstrap may fail to match some of the terms in an Edgeworth expansion of the full sample test statistic. Nonetheless, we show that the wild bootstrap still has a lower maximal asymptotic risk as an estimator of the true distribution than a normal approximation, in shrinking neighborhoods of properly specified models. To assess the practical implications of this result we conduct a Monte Carlo study contrasting the performance of the wild bootstrap with a normal approximation and the traditional nonparametric bootstrap.