It is well known that standard asymptotic theory is not applicable or is very unreliable in models with identification problems or weak instruments. One possible way out consists of using a variant of the Anderson–Rubin ((1949), AR) procedure. The latter allows one to build exact tests and confidence sets only for the full vector of the coefficients of the endogenous explanatory variables in a structural equation, but not for individual coefficients. This problem may in principle be overcome by using projection methods (Dufour (1997), Dufour and Jasiak (2001)). At first sight, however, this technique requires the application of costly numerical algorithms. In this paper, we give a general necessary and sufficient condition that allows one to check whether an AR-type confidence set is bounded. Furthermore, we provide an analytic solution to the problem of building projection-based confidence sets from AR-type confidence sets. The latter involves the geometric properties of “quadrics” and can be viewed as an extension of usual confidence intervals and ellipsoids. Only least squares techniques are needed to build the confidence intervals.

General characterizations of valid confidence sets and tests in problems which involve locally almost unidentified (LAU) parameters are provided and applied to several econometric models. Two types of inference problems are studied: (1) inference about parameters which are not identifiable on certain subsets of the parameter space, and (2) inference about parameter transformations with discontinuities. When a LAU parameter or parametric function has an unbounded range, it is shown under general regularity conditions that any valid confidence set with level 1 − α for this parameter must be unbounded with probability close to 1−α in the neighborhood of nonidentification subsets and will have a non-zero probability of being unbounded under any distribution compatible with

This article proposes a general method to build exact tests and confidence sets in linear regressions with first-order autoregressive Gaussian disturbances. Because of a nuisance parameter problem, we argue that generalized bounds tests and conservative confidence sets provide natural inference procedures in such a context. Given an exact confidence set for the autocorrelation coefficient, we describe how to obtain a similar simultaneous confidence set for the autocorrelation coefficient and any subvector of regression coefficient. Conservative confidence sets for the regression coefficients are then deduced by a projection method. For any hypothesis that specifies jointly the value of the autocorrelation coefficient and any set of linear restrictions on the regression coefficients, we get exact similar tests. For tesing linear hypotheses about the regression coefficients only, we suggest bounds-type procedures. Exact confidence sets for the autocorrelation coefficient are built by "inverting" autocorrelation tests. The method is illustrated with two examples.

In the context of the classical linear model, the problem of comparing two arbitrary hypotheses on the regression coefficients is considered. Problems involving nonlinear hypotheses, inequality restrictions, or non-nested hypotheses are included as special cases. Exact bounds on the null distribution of likelihood ratio statistics are derived. In an important special case, a bounds test similar to the Durbin-Watson test is proposed. Multiple testing problems are also studied

Causality in Granger's sense is defined in terms of predictibility one period ahead. The notion of causality is generalized by considering causality at any given horizon 1

We study the problem of measuring the uncertainty of computable general equilibrium (CGE) (or RBC)-type model simulations associated with parameter uncertainty. We describe two approaches for building confidence sets on model endogenous variables. The first uses a standard Wald-type statistic. The second approach assumes that a confidence set (sampling or Baycsian) is available for the free parameters, from which confidence sets are derived by a projection technique. The latter has two advantages: first, confidence set validity is not affected by model nonlinearities; second, we can easily build simultaneous confidence intervals for an unlimited number of variables. We study conditions under which these confidence sets take the form of intervals and show how they can be implemented using standard methods for solving CGE models. We present an application to a CGE model of the Moroccan economy to study the effects of policy-induced increases of transfers from Moroccan expatriates.

The hypothesis that a variable is independent of past information, such as its own past and past realizations of other observable variables, is a frequent implication of economic theory. Yet standard regression-based tests of orthogonality may not have the correct level if there is feedback from innovations to future values of the regressors. In this paper we develop nonparametric tests of orthogonality based on signs and signed ranks which are proved to reject at their nominal levels over a wide class of models admitting feedback. The tests are robust to problems of non-normality and heteroskedasticity. Further, in simulation studies of two specifications of feedback-a rational expectations model considered by Mankiw and Shapiro, and the random walk model-we find that the nonparametric tests display remarkable power. The paper concludes with an application which assesses the efficiency of survey data on interest rate expectations previously studied by Friedman.

The invariance properties of some well known asymptotic tests are studied. Three types of invariance are considered: invariance to the representation of the null hypothesis, invariance to one-to-one transformations of the parameter space (reparameterizations), and invariance to one-to-one transformations of the model variables such as changes in measurement units. Tests that are not invariant include the Wald test and generalized versions of it, a widely used variant of the Lagrange multiplier test, Neyman's C(a) test, and a generalized version of the latter. For all these tests, we show that simply changing measurement units can lead to vastly different answers even when equivalent null hypotheses are tested. This problem is illustrated by considering regression models with Box-Cox transformations on the variables. We observe, in particular, that various consistent estimators of the information matrix lead to test procedures with different invariance properties. General sufficient conditions are then established, under which the generalized C(a) test becomes invariant to reformulations of the null hypothesis and/or to one-to-one transformations of the parameter space as well as to transformations of the variables. In many practical cases where Wald-type tests lack invariance, we find that special formulations of the generalized C(a) test are invariant and hardly more costly to compute than Wald tests. This computational simplicity stands in contrast with other invariant tests such as the likelihood ratio test. We conclude that noninvariant asymptotic tests should be avoided or used with great care. Further, in many situations, the suggested implementation of the generalized C(a) test often yields an attractive substitute to the Wald test (which is not invariant) and to other invariant tests (which are more costly to perform).

To facilitate maximum likelihood estimation for Box-Cox models, several authors have suggested dividing the dependent variable by its sample geometric mean. This paper points out previously unmentioned drawbacks of this 'recalling.' First, the 'resealed' model is not actually equivalent to the untransformed one, so that the procedure involves more than a unit change. Second, there is no clear interpretation of the parameters after such resealing. The authors suggest an interpretation but find that the usual formulas for standard errors and confidence intervals are not asymptotically valid. Only tests for zero coefficients are valid. Thirdly, the authors discuss the appropriate way of measuring elasticities in such models.