Null hypothesis

Abstract: Using the result that under the null hypothesis of no misspecification an asymptotically efficient estimator must have zero asymptotic covariance with its difference from a consistent but asymptotically inefficient estimator, specification tests are devised for a number of model specifications in econometrics. Local power is calculated for small departures from the null hypothesis. An instrumental variable test as well as tests for a time series cross section model and the simultaneous equation model are presented. An empirical model provides evidence that unobserved individual factors are present which are not orthogonal to the included right-hand-side variable in a common econometric specification of an individual wage equation.

Abstract: A framework for hypothesis testing and power analysis in the assessment of fit of covariance structure models is presented. We emphasize the value of confidence intervals for fit indices, and we stress the relationship of confidence intervals to a framework for hypothesis testing. The approach allows for testing null hypotheses of not-good fit, reversing the role of the null hypothesis in conventional tests of model fit, so that a significant result provides strong support for good fit. The approach also allows for direct estimation of power, where effect size is defined in terms of a null and alternative value of the root-mean-square error of approximation fit index proposed by J. H. Steiger and J. M. Lind (1980). It is also feasible to determine minimum sample size required to achieve a given level of power for any test of fit in this framework. Computer programs and examples are provided for power analyses and calculation of minimum sample sizes.

Abstract: We consider the null hypothesis that a time series has a unit root with possibly nonzero drift against the alternative that the process is «trend-stationary». The interest is that we allow under both the null and alternative hypotheses for the presence for a one-time change in the level or in the slope of the trend function. We show how standard tests of the unit root hypothesis against trend stationary alternatives cannot reject the unit root hypothesis if the true data generating mechanism is that of stationary fluctuations around a trend function which contains a one-time break

Abstract: Many econometric models are susceptible to analysis only by asymptotic techniques and there are three principles, based on asymptotic theory, for the construction of tests of parametric hypotheses. These are: (i) the Wald (W) test which relies on the asymptotic normality of parameter estimators, (ii) the maximum likelihood ratio (LR) procedure and (iii) the Lagrange multiplier (LM) method which tests the effect on the first order conditions for a maximum of the likelihood of imposing the hypothesis. In the econometric literature, most attention seems to have been centred on the first two principles. Familiar " t-tests " usually rely on the W principle for their validity while there have been a number of papers advocating and illustrating the use of the LR procedure. However, all three are equivalent in well-behaved problems in the sense that they give statistics with the same asymptotic distribution when the null hypothesis is true and have the same asymptotic power characteristics. Choice of any one principle must therefore be made by reference to other criteria such as small sample properties or computational convenience. In many situations the W test is attractive for this latter reason because it is constructed from the unrestricted estimates of the parameters and their estimated covariance matrix. The LM test is based on estimation with the hypothesis imposed as parametric restrictions so it seems reasonable that a choice between W or LM be based on the relative ease of estimation under the null and alternative hypotheses. Whenever it is easier to estimate the restricted model, the LM test will generally be more useful. It then provides applied researchers with a simple technique for assessing the adequacy of their particular specification. This paper has two aims. The first is to exposit the various forms of the LM statistic and to collect together some of the relevant research reported in the mathematical statistics literature. The second is to illustrate the construction of LM tests by considering a number of particular econometric specifications as examples. It will be found that in many instances the LM statistic can be computed by a regression using the residuals of the fitted model which, because of its simplicity, is itself estimated by OLS. The paper contains five sections. In Section 2, the LM statistic is outlined and some alternative versions of it are discussed. Section 3 gives the derivation of the statistic for