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In linear mixed models the selection of fixed and random effects using a testing hypothesis approach brings up several problems. We deal with the boundary point problem emerging when no randomness is hypotesized and the confounding impact of randomness on the coefficients arising when fixed effects are tested. The test statistics are defined by a ratio of two quadratic forms derived from ordinary least squares, are simple, sufficiently general, easy to compute, with known finite sample properties. The test statistic on randomness has a known exact distribution, the density of the statistic on fixed effect is unknown and is approximated by a noncentral F−distribution. The goodness-of-approximation and the selection approach is examined in-depth by simulation. The method proposed in this paper must be seen as complementary to existing selection procedures widening and enriching all information necessary for taking a decision.

In longitudinal studies with subjects measured repeatedly across time, an important problem is how to select a model generating data by choosing between a linear regression model and a linear latent growth model. Approaches based both on information criteria and asymptotic hypothesis tests of the variances of “random” components are widely used but not completely satisfactory. We propose a test statistic based on the trace of the product of an estimate of a variance covariance matrix de?ned when data come from a linear regression model and a sample variance covariance matrix. We studied the sampling distribution of the test statistic giving a representation in terms of an in?nite series of generalized F-distributions. Knowledge about this distribution allows us to make inference within a classical hypothesis testing framework. The test statistic can be used by itself to discriminate between the two models and/or, if duly modi?ed, it can be used to test randomness on single components. Moreover, in conjunction with some model selection criteria, it gives additional information which can help in choosing the model. The test statistic proposed in this paper has been applied to two data sets. With the tourism data it is used by itself to discriminate between the two models, with the Cadralazine data it is used in conjunction with several indicators based on information criteria that give an estimate of the probability of accepting or rejecting the model chosen.

In linear mixed models the assessing of the signi?cance of all or a subset of the random effects is often of primary interest. Many techniques have been proposed for this purpose but none of them is completely satisfactory. One of the oldest methods for testing randomness is the F-test but it is often overlooked in modern applications due to poor statistical power and non-applicability in some important situations. In this work a two-step procedure is developed for generalizing an F-test and improving its statistical power. In the ?rst step, by comparing two covariance matrices of a least squares statistics, we obtain a “repeatable” F-type test. In the second step, by changing the projected matrix which de?nes the least squares statistic we apply the test repeteadly to the same data in order to have a set of correlated statistics analyzed within a multiple testing approach. The resulting test is suf?ciently general, easy to compute, with an exact distribution under the null and alternative hypothesis and, perhaps more importantly, with a strong increase of statistical power with respect to the F-test. In the light of these results we believe that our two-stage approach based on a combinnation of a ”repeatable” F-type test with a multiple testing approach may suggest a procedure for improving statistical power in linear mixed models.

In longitudinal studies with subjects measured repeatedly across time, an important problem is how to select a model generating data choosing between a linear regression model and a linear latent growth model. Approaches based both on information criteria and on asymptotic hypothesis test on the variances of "random" components are largely used but not completely satisfactory. In the paper we propose a finite sample parametric test based on the trace of the product of estimates of two variance covariance matrices, one defined when data come from a linear regression model, the other defined when data come from a linear latent growth model. The sampling distribution of the test statistic so defined depends on the model generating data. It can be a "standard" F -distribution or a linear combination of F -distributions. In the paper a unified sampling distribution based on a generalized F -distribution is proposed. The knowledge of this distribution allows us to make inference in a classical hypothesis testing framework. The test statistic can be used by itself to discriminate between the two models and/or, duly modified, it can be used to test randomness on single components of the linear latent growth model avoinding the boundary problem of the likelihood ratio test statistic. Moreover, it can be used in conjunction with some indicators based on information criteria giving estimates of probability of accepting or rejecting the model chosen.

In the paper we propose a way to test the null hypothesis h(θ)=0 when the information matrix is singular. The approach is based on the introduction of a quadratic penalty log-likelihood function with the penalty parameter close to zero (but not zero). This slight change of the log-likelihood allows to obtain a consistent estimator close to the maximum likelihood estimator and whose limiting distribution is Normal with variance-covariance matrix given by the Moore-Penrose pseudoinverse of the information matrix. This result allows to obtain an asymptoticx 2-distribution that may be used in hypothesis testing when the infonnation matrix is singular.

In this paper, we study a long-run disaggregated model of consumption following an approach based on an integrated cross-section and time-series demand system. The study consists of three steps. First, a cross-section analysis is performed on data from household budget surveys. At this stage, the problem of 'zero expenditures' is solved. The cross-section results are transformed into variables for use in the time-series system of demand. Then, this demand system is built and estimated. Some results for Italy concerning both the cross-section and the time-series analyses are presented.