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Journal ArticleDOI

An updated review of Goodness-of-Fit tests for regression models

25 Jul 2013-Test (Springer Berlin Heidelberg)-Vol. 22, Iss: 3, pp 361-411
TL;DR: In this article, the authors present a survey of the developments on Goodness-of-Fit for regression models during the last 20 years, from the very first origins with the idea of the tests for density and distribution, until the most recent advances for complex data and models.
Abstract: This survey intends to collect the developments on Goodness-of-Fit for regression models during the last 20 years, from the very first origins with the proposals based on the idea of the tests for density and distribution, until the most recent advances for complex data and models. Far from being exhaustive, the contents in this paper are focused on two main classes of tests statistics: smoothing-based tests (kernel-based) and tests based on empirical regression processes, although other tests based on Maximum Likelihood ideas will be also considered. Starting from the simplest case of testing a parametric family for the regression curves, the contributions in this field provide also testing procedures in semiparametric, nonparametric, and functional models, dealing also with more complex settings, as those ones involving dependent or incomplete data.
Citations
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Journal ArticleDOI

6,278 citations

Posted Content
TL;DR: In this paper, the authors test parametric models by comparing their implied parametric density to the same density estimated nonparametrically, and do not replace the continuous-time model by discrete approximations, even though the data are recorded at discrete intervals.
Abstract: Different continuous-time models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuous-time model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then mean-reverts strongly when far away from the mean. The volatility is higher when away from the mean.

830 citations

Journal ArticleDOI
TL;DR: This article considers identification and estimation of treatment effect parameters using DID with (i) multiple time periods, (ii) variation in treatment timing, and (iii) when the "parallel trends assumption" holds potentially only after conditioning on observed covariates.
Abstract: In this article, we consider identification, estimation, and inference procedures for treatment effect parameters using Difference-in-Differences (DID) with (i) multiple time periods, (ii) variation in treatment timing, and (iii) when the ``parallel trends assumption" holds potentially only after conditioning on observed covariates. We show that a family of causal effect parameters are identified in staggered DID setups, even if differences in observed characteristics create non-parallel outcome dynamics between groups. Our identification results allow one to use outcome regression, inverse probability weighting, or doubly-robust estimands. We also propose different aggregation schemes that can be used to highlight treatment effect heterogeneity across different dimensions as well as to summarize the overall effect of participating in the treatment. We establish the asymptotic properties of the proposed estimators and prove the validity of a computationally convenient bootstrap procedure to conduct asymptotically valid simultaneous (instead of pointwise) inference. Finally, we illustrate the relevance of our proposed tools by analyzing the effect of the minimum wage on teen employment from 2001--2007. Open-source software is available for implementing the proposed methods.

156 citations


Cites result from "An updated review of Goodness-of-Fi..."

  • ...These results build on many papers in the goodness-of-fit literature – see, e.g., Bierens (1982), Bierens and Ploberger (1997), Stute (1997), and Escanciano (2006b, 2008); for a recent overview, see González-Manteiga and Crujeiras (2013)....

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Posted Content
TL;DR: In this article, the authors proposed a new test for the comparison of two regression curves, which is based on a difference of two marked empirical processes based on residuals, and the large sample behaviour of the corresponding statistic is studied to provide a full nonparametric comparison of regression curves.
Abstract: We propose a new test for the comparison of two regression curves, which is based on a difference of two marked empirical processes based on residuals. The large sample behaviour of the corresponding statistic is studied to provide a full nonparametric comparison of regression curves. In contrast to most procedures suggested in the literature the new procedure is applicable in the case of different design points and heteroscedasticity. Moreover, it is demonstrated that the proposed test detects continuous alternatives converging to the null at a rate N-1/2. In the case of equal design points the fundamental statistic reduces to a test statistic proposed by Delgado (1993) and therefore resembles in spirit classical goodness-of-fit tests. As a byproduct we explain the problems of a related test proposed by Kulasekera (1995) and Kulasekera and Wang (1997) with respect to accuracy in the approximation of the level. These difficulties mainly originate from the comparison with the quantiles of an inappropriate limit distribution. A simulation study is conducted to investigate the finite sample properties of a wild bootstrap version of the new tests.

114 citations

Journal ArticleDOI
TL;DR: In this paper, a model adaptation concept in lack-of-fit testing is introduced and a dimension reduction model-adaptive test procedure is proposed for parametric single-index models, which behaves like a local smoothing test, as if the model were univariate.
Abstract: Summary Local smoothing testing based on multivariate non-parametric regression estimation is one of the main model checking methodologies in the literature. However, the relevant tests suffer from the typical curse of dimensionality, resulting in slow rates of convergence to their limits under the null hypothesis and less deviation from the null hypothesis under alternative hypotheses. This problem prevents tests from maintaining the level of significance well and makes tests less sensitive to alternative hypotheses. In the paper, a model adaptation concept in lack-of-fit testing is introduced and a dimension reduction model-adaptive test procedure is proposed for parametric single-index models. The test behaves like a local smoothing test, as if the model were univariate. It is consistent against any global alternative hypothesis and can detect local alternative hypotheses distinct from the null hypothesis at a fast rate that existing local smoothing tests can achieve only when the model is univariate. Simulations are conducted to examine the performance of our methodology. An analysis of real data is shown for illustration. The method can be readily extended to global smoothing methodology and other testing problems.

64 citations

References
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Book ChapterDOI
TL;DR: The analysis of censored failure times is considered in this paper, where the hazard function is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time.
Abstract: The analysis of censored failure times is considered. It is assumed that on each individual arc available values of one or more explanatory variables. The hazard function (age-specific failure rate) is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time. A conditional likelihood is obtained, leading to inferences about the unknown regression coefficients. Some generalizations are outlined.

28,264 citations


"An updated review of Goodness-of-Fi..." refers methods in this paper

  • ...The model in H0PH is the well-known Cox regression model, introduced by Cox (1972)....

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Journal ArticleDOI
TL;DR: In this article, the authors discuss the problem of estimating the sampling distribution of a pre-specified random variable R(X, F) on the basis of the observed data x.
Abstract: We discuss the following problem given a random sample X = (X 1, X 2,…, X n) from an unknown probability distribution F, estimate the sampling distribution of some prespecified random variable R(X, F), on the basis of the observed data x. (Standard jackknife theory gives an approximate mean and variance in the case R(X, F) = \(\theta \left( {\hat F} \right) - \theta \left( F \right)\), θ some parameter of interest.) A general method, called the “bootstrap”, is introduced, and shown to work satisfactorily on a variety of estimation problems. The jackknife is shown to be a linear approximation method for the bootstrap. The exposition proceeds by a series of examples: variance of the sample median, error rates in a linear discriminant analysis, ratio estimation, estimating regression parameters, etc.

14,483 citations


"An updated review of Goodness-of-Fi..." refers methods in this paper

  • ...Under these circumstances, calibration can be done by means of resampling procedures, such as bootstrap (see Efron 1979)....

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Journal ArticleDOI
TL;DR: In this paper, the problem of the estimation of a probability density function and of determining the mode of the probability function is discussed. Only estimates which are consistent and asymptotically normal are constructed.
Abstract: : Given a sequence of independent identically distributed random variables with a common probability density function, the problem of the estimation of a probability density function and of determining the mode of a probability function are discussed. Only estimates which are consistent and asymptotically normal are constructed. (Author)

10,114 citations


"An updated review of Goodness-of-Fi..." refers background or methods in this paper

  • ...The test statistic for a density can be written as Tn = T (fnh, f̂θ ), or more exactly as Tn = T ( fnh,Êθ (fnh) ) ≡ T (α̃n) (2) where fnh(x) = n−1 ∑ni=1 Kh(x −Xi) is the kernel density estimator (see Rosenblatt 1956 and Parzen 1962), with K a kernel density function and h the smoothing parameter or bandwidth....

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  • ...Based on model (35), Paparoditis (2000) proposed a GoF for a parametric model for the spectral density, following the ideas in Bickel and Rosenblatt (1973)....

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  • ...This alternative route for GoF is related to the seminal paper by Bickel and Rosenblatt (1973), whose ideas were extended to the p-dimensional setting in the 1990s....

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  • ...…for a density can be written as Tn = T (fnh, f̂θ ), or more exactly as Tn = T ( fnh,Êθ (fnh) ) ≡ T (α̃n) (2) where fnh(x) = n−1 ∑ni=1 Kh(x −Xi) is the kernel density estimator (see Rosenblatt 1956 and Parzen 1962), with K a kernel density function and h the smoothing parameter or bandwidth....

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  • ...Tests for marginal density functions have been also studied by Gao and King (2004) and Hong and Li (2005), involving kernel estimators, as well as Arapis and Gao (2006) and the Bickel and Rosenblatt test was adapted by Lee (2006) for diffusion processes....

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Book
01 Jan 1991
TL;DR: In this paper, the authors present a survey of statistics for spatial data in the field of geostatistics, including spatial point patterns and point patterns modeling objects, using Lattice Data and spatial models on lattices.
Abstract: Statistics for Spatial Data GEOSTATISTICAL DATA Geostatistics Spatial Prediction and Kriging Applications of Geostatistics Special Topics in Statistics for Spatial Data LATTICE DATA Spatial Models on Lattices Inference for Lattice Models SPATIAL PATTERNS Spatial Point Patterns Modeling Objects References Author Index Subject Index.

8,631 citations

Book ChapterDOI
01 Jan 2008

6,615 citations