Abstract
Random coefficient regression (RCR) models are the regression versions of random effects models in analysis of variance and panel data analysis. Optimal detection of the presence of random coefficients (equivalently, optimal testing of the hypothesis of constant regression coefficients) has been an open problem for many years. The simple regression case has been solved recently and the multiple regression case is considered here. The latter poses several theoretical challenges: (a) a nonstandard ULAN structure, with log-likelihood gradients vanishing at the null; (b) cone-shaped alternatives under which traditional optimality concepts are no longer adequate; (c) nuisance parameters that are not identified under the null but have a significant impact on local powers. We propose a new (local and asymptotic) concept of optimality for this problem and, for specified error densities, derive parametrically optimal procedures. A suitable modification of the Gaussian version of the latter is shown to qualify as a pseudo-Gaussian test. The asymptotic performances of those pseudo-Gaussian tests, however, are quite poor under skewed and heavy-tailed densities. We therefore also construct rank-based tests, possibly based on data-driven scores, the asymptotic relative efficiencies of which are remarkably high with respect to their pseudo-Gaussian counterparts.
Highlights
Akharif A., Fihri M., Hallin M. and Mellouk A./Random Coefficients Detection in Multiple Regression[3] derivatives) involving second-order derivatives of the density; (b) alternatives, which were one-sided in the simple regression context, are cone-shaped, so that the usual tests (such as the Neyman C(α), Lagrange multiplier, or Rao efficient score tests), based on quadratic statistics are inappropriate; (c) the correlation matrix of the random regression coefficients, which is not identified under the null, has a very significant impact on local powers
: (i) as far as λ2 is concerned, uniform local asymptotic normality (ULAN) is “cone-shaped”—namely, only those local perturbations of λ2 that belong to C+ are meaningful—so that the classical maximin and stringency arguments, leading to quadratic test statistics with asymptotic Neyman C(α) structure, do not apply; (ii) the correlation matrix P of the random regression parameter, which plays an essential role under the alternative, is not identified, cannot be estimated, under the null
As well as any asymptotically equivalent sequence of tests. This intuitively appealing concept of directionally maximin test has been proposed for C = C+ by Novikov (2011) in a broader context, where densities are not necessarily Gaussian, and optimality is described in terms of derivatives of power functions
Summary
Random coefficients regression (RCR) models are the regression versions of random effects models in analysis of variance and panel data analysis. Adopting a more general Le Cam approach, Fihri et al (2017) derive locally asymptotically optimal, optimal pseudo-Gaussian, and optimal rank-based tests for the same problem; based on local power evaluations, they provide evidence of the poor performance of pseudo-Gaussian procedures in the presence of non-Gaussian (skew or leptokurtic densities), and recommend the use of rank-based tests with data-driven scores which significantly outperform the pseudo-Gaussian test All those methods, including the analysis of Fihri et al (2017), are limited to simple regression models, which in practice is quite restrictive. Akharif A., Fihri M., Hallin M. and Mellouk A./Random Coefficients Detection in Multiple Regression[3] derivatives) involving second-order derivatives of the density; (b) alternatives, which were one-sided in the simple regression context, are cone-shaped, so that the usual tests (such as the Neyman C(α), Lagrange multiplier, or Rao efficient score tests), based on quadratic statistics are inappropriate; (c) the correlation matrix of the random regression coefficients, which is not identified under the null, has a very significant impact on local powers. P and h are not identified under the null hypothesis
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