Measure for Measure: Exact F Tests and the Mixed Models Controversy

We consider exact F tests for the hypothesis of null random factor effect in the presence of interaction under the two factor mixed models involved in the mixed models controversy. We show that under the constrained parameter ( CP ) model, even in unbalanced data situations, MSB/MSE (in the usual ANOVA notation) follows an exact F distribution when the null hypothesis holds. We also obtain an exact F test for what is generally (and erroneously) assumed to be an equivalent hypothesis under the unconstrained parameter ( UP ) model. For unbalanced data, such a corresponding test statistic does not coincide with MSB/MSAB (the test statistic advocated for balanced data cases). We compute the power of the exact test under different imbalance patterns and show that although the loss of power increases with the degree of imbalance, it still remains reasonable from a practical point of view.     

A score test under logistic regression models based on case–control data

We propose a score statistic to test the vector of odds ratio parameters under the logistic regression model based on case–control data. The proposed score test is based on the semiparametric profile loglikelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to the Wald test under the logistic regression model based on case–control data. In addition, we demonstrate that the proposed score statistic and its asymptotic distribution may be obtained by fitting the prospective logistic regression model to case–control data. We present some results on simulation and on the analysis of two real datasets.     

Testing the specification of multivariate models in the presence of alternative hypotheses

In this paper we propose a non-nested hypothesis test for testing the specification of a multivariate econometric model in the presence of an alternative model which purports to explain the same phenomenon. We demonstrate that the new test statistic tends to minus the same random variable as the CPD test statistic introduced by Pesaran and Deaton (1978), provided that the truth is ‘close’ to the null hypothesis. Since the new test is simpler to compute than the multivariate CPD test, it would seem to be the procedure of choice.     

A consistent nonparametric test of parametric regression functional form in fixed effects panel data models

We propose a consistent test for a linear functional form against a nonparametric alternative in a fixed effects panel data model. We show that the test has a limiting standard normal distribution under the null hypothesis, and show that the test is a consistent test. We also establish the asymptotic validity of a bootstrap procedure which is used to better approximate the finite sample null distribution of the test statistic. Simulation results show that the proposed test performs well for panel data with a large number of cross-sectional units and a finite number of observations across time.     

An adaptive empirical likelihood test for parametric time series regression models

We propose an adaptive empirical likelihood (EL) test for a parametric regression model against a class of alternatives for weakly dependent time series observations. The test is formulated by maximizing a standardized version of the EL statistic over a set of smoothing bandwidths. It is demonstrated that the proposed test is able to distinguish the null hypothesis from a series of local alternatives at an optimal rate.     

Model specification testing of time series regressions

Given the specification of the lag length and functional form of a (non)linear time series regression we shall propose a test of the null hypothesis that the expectation of the error conditional on the exogenous variables, all lagged exogenous variables and all lagged dependent variables equals zero with probability 1. In the case that the data-generating process is strictly stationary this test is consistent with respect to the alternative hypothesis that the null is false. The test is also applicable for a particular class of non-stationary time series regressions, although in that case consistency with respect to all possible alternatives is no longer guaranteed. The test involved is a generalization of a test proposed in Bierens (1982b). Moreover, we also present a similar but simpler test of the hypothesis that the errors are martingale differences.     

Kuhn-Tucker,likelihood ratio and Wald tests for nonlinear models with inequality constraints on the parameters

This paper considers the problem of testing statistical hypothesis in nonlinear regression models with inequality constraints on the parameters. First, the Kuhn-Tucker test procedure is defined. Next, it is shown that the distribution of the Kuhn-Tucker, the likelihood ratio and the Wald test statistics converges to the same mixture of chi-square distributions under the null hypothesis. To illustrate these results two examples are considered: (1) the problem of testing that individual effects are missing in an error component model, and (2) the problem of testing equilibrium for a model of markets in disequilibrium.     

Specification testing in nonparametric instrumental variable estimation

In nonparametric instrumental variable estimation, the function being estimated is the solution to an integral equation. A solution may not exist if, for example, the instrument is not valid. This paper discusses the problem of testing the null hypothesis that a solution exists against the alternative that there is no solution. We give necessary and sufficient conditions for existence of a solution and show that uniformly consistent testing of an unrestricted null hypothesis is not possible. Uniformly consistent testing is possible, however, if the null hypothesis is restricted by assuming that any solution to the integral equation is smooth. Many functions of interest in applied econometrics, including demand functions and Engel curves, are expected to be smooth. The paper presents a statistic for testing the null hypothesis that a smooth solution exists. The test is consistent uniformly over a large class of probability distributions of the observable random variables for which the integral equation has no smooth solution. The finite-sample performance of the test is illustrated through Monte Carlo experiments.     

Determinacy,indeterminacy and dynamic misspecification in linear rational expectations models

This paper proposes a testing strategy for the null hypothesis that a multivariate linear rational expectations (LRE) model may have a unique stable solution (determinacy) against the alternative of multiple stable solutions (indeterminacy). The testing problem is addressed by a misspecification-type approach in which the overidentifying restrictions test obtained from the estimation of the system of Euler equations of the LRE model through the generalized method of moments is combined with a likelihood-based test for the cross-equation restrictions that the model places on its reduced form solution under determinacy. The resulting test has no power against a particular class of indeterminate equilibria, hence the non rejection of the null hypothesis can not be interpreted conclusively as evidence of determinacy. On the other hand, this test (i) circumvents the nonstandard inferential problem generated by the presence of the auxiliary parameters that appear under indeterminacy and that are not identifiable under determinacy, (ii) does not involve inequality parametric restrictions and hence the use of nonstandard inference, (iii) is consistent against the dynamic misspecification of the LRE model, and (iv) is computationally simple. Monte Carlo simulations show that the suggested testing strategy delivers reasonable size coverage and power against dynamic misspecification in finite samples. An empirical illustration focuses on the determinacy/indeterminacy of a New Keynesian monetary business cycle model of the US economy.     

Testing parameter constancy in linear models against stochastic stationary parameters

This paper considers testing parameter constancy in a linear model when the alternative is that a subset of the parameters follows a stationary vector autoregressive process of known finite order. This kind of a linear model is only identified under the alternative, which usually precludes finding a test statistic with an analytic null distribution. In the present situation, however, it is still possible to derive a test statistic with an asymptotic chi-squared distribution under the null hypothesis and this is done in the paper. The small-sample properties of the test statistic are investigated by simulation and found statisfactory. The test retains its power when the alternative to parameter constancy is a random walk parameter process.     

Narrow Replication of Yogo (2004) Estimating the Elasticity of Intertemporal Substitution when Instruments are Weak

This narrow replication exercise of Yogo (Review of Economics and Statistics 2004; 86 (3): 797–810) finds results identical to the original paper, and provides results on overidentification tests for specifications that do not suffer from the weak instruments problem. The null hypothesis of the Sargan test is rejected in several cases, in particular for US and UK quarterly data. When other combinations of instruments are used the null of the Sargan test is rejected again in several cases. These rejections cast doubts on either instrument validity or model specification. Copyright © 2011 John Wiley & Sons, Ltd.     

Bootstrap testing for the null of no cointegration in a threshold vector error correction model

We develop a test for the linear no cointegration null hypothesis in a threshold vector error correction model. We adopt a sup-Wald type test and derive its null asymptotic distribution. A residual-based bootstrap is proposed, and the first-order consistency of the bootstrap is established. A set of Monte Carlo simulations shows that the bootstrap corrects size distortion of asymptotic distribution in finite samples, and that its power against the threshold cointegration alternative is significantly greater than that of conventional cointegration tests. Our method is illustrated with used car price indexes.     

The Bierens test for certain nonstationary models

We adapt the Bierens (1990) test to the I-regular models of Park and Phillips (2001). Bierens (1990) defines the test hypothesis in terms of a conditional moment condition. Under the null hypothesis, the moment condition holds with probability one. The probability measure used is that induced by the variables in the model, that are assumed to be strictly stationary. Our framework is nonstationary and this approach is not always applicable. We show that the Lebesgue measure can be used instead in a meaningful way. The resultant test is consistent against all I-regular alternatives.     

Kalman filtering estimation of unobserved rational expectations with an application to the German hyperinflation

The assumption that rational expectations always lie on a convergent path is subject to an empirical test using the German hyperinflation data. The estimation technique employs a Kalman filtering algorithm. After presenting a brief background for the convergent expectations problem and a derivation of the various model specifications, a generalized expectations model and its attendant Kalman filtering estimation technique are discussed. Additional estimation details and empirical results are then presented. Based on an assumption of normally distributed errors, the null hypothesis of convergent paths is rejected in all situations involving a deterministic specification of the evolution of the unobserved parameter which characterizes the convergent path. The same null hypothesis is rejected in four of the six cases corresponding to a stochastic specification of the evolution of the unobserved parameter which characterizes the convergent path. A discussion of these findings, their economic significance, and suggestions for further research concludes the paper.     

Tests of seasonal integration and cointegration in multivariate unobserved component models

This paper considers tests of seasonal integration and cointegration for multivariate unobserved component models. First, the locally best invariant (LBI) test of the null hypothesis of a deterministic seasonal pattern against the alternative of seasonal integration is derived for a model with Gaussian i.i.d. disturbances and deterministic trend. Then the null hypothesis of seasonal cointegration is considered and a test for common nonstationary components at the seasonal frequencies is proposed. The tests are subsequently generalized to account for stochastic trends, weakly dependent errors and unattended unit roots. Asymptotic representations and critical values of the tests are provided, while the finite sample performance is evaluated by Monte Carlo simulation experiments. Finally, the tests are applied to the series of industrial production of the four largest countries of the European Monetary Union. It is found that Germany does not appear to cointegrate with the other countries at most seasonal frequencies, while there seems to exist a common nonstationary seasonal component between France, Italy and Spain. Copyright © 2006 John Wiley & Sons, Ltd.     

Distribution-free test procedures in two-way layouts; a concept of rank-interaction

Summary  In this paper the concept of 'rank-interaction' is introduced and a distribution-free method for testing against the presence of 'rank-interaction' is suggested in the case of a two-way layout (classification) with m (> 1) observations per cell. Roughly speaking rank-interaction can be understood as the phenomenon at which the ranks of the levels of some relevant variable are different for different classes of the other factor. The exact null distribution of the test statistic has been computed in some cases. The asymptotic distribution under the null hypothesis has been derived. A test suggested by J.V. B radley in his book 'Distribution-free Statistical Tests' [2] is discussed. In the opinion of the authors it is doubtful whether the asymptotic distribution of the test statistic under the null hypothesis, as given by B radley , is correct. The test of B radley was intended to be sensitive to the presence of interactions defined in the usual way and hence not only to 'rank-interaction'. The same applies to methods proposed by some other authors. We claim that situations exist where one should test against rank-interaction and not against the usual more general alternative.     

Alternative bootstrap procedures for testing cointegration in fractionally integrated processes

This paper considers alternative methods of testing cointegration in fractionally integrated processes, using the bootstrap. The investigation focuses on (a) choice of statistic, (b) use of bias correction techniques, and (c) designing the simulation of the null hypothesis. Three residual-based tests are considered, two of the null hypothesis of non-cointegration, the third of the null hypothesis that cointegration exists. The tests are compared in Monte Carlo experiments to throw light on the relative roles of issues (a)–(c) in test performance.     

Posterior predictive p-values for classical null hypotheses

We show that for the purpose of testing a classical null hypothesis the posterior predictive check of Rubin (1984) may be inadequate. This inadequacy is caused by the estimation of the nuisance parameters under the null hypothesis. We show that this problem can be solved if the parameters are estimated under the encompassing model.     

Much Ado About Nothing: the Mixed Models Controversy Revisited

We consider a well-known controversy that stems from the use of two mixed models for the analysis of balanced experimental data with a fixed and a random factor. It essentially originates in the different statistics developed from such models for testing that the variance parameter associated to the random factor is null. The corresponding hypotheses are interpreted as that of null random factor main effects in the presence of interaction. The controversy is further complicated by different opinions regarding the appropriateness of such hypothesis. Assuming that this is a sensible option, we show that the standard test statistics obtained under both models are really directed at different hypotheses and conclude that the problem lies in the definition of the main effects and interactions. We use expected values as in the fixed effects case to resolve the controversy showing that under the most commonly used model, the test usually associated to the inexistence of the random factor main effects addresses a different hypothesis. We discuss the choice of models, and some further problems that occur in the presence of unbalanced data.     

The new approach in the analysis of 2×2 tables

Conventional statistical analysis of contingency tables is logically untenable. It rejects the knowledge of the prior model of cause, and it tests the null hypothesis of independence which cannot be proved to be true or false. Our approach is based on the knowledge of the prior model of cause, and we test the hypothesis of significant dependence. This hypothesis can be proved to be true or false if data are representative of the homogeneous invisible population.