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This paper proposes a method for deriving the joint asymptotic distribution of sample proportions when the population and sample size increase jointly to infinity. The joint distribution of the sample proportions is derived by reducing the multi- variate to a univariate problem and applying the Cramer-Wold device. Knowing the asymptotic distribution we are in a position to conduct inference on linear or non-linear functions of the population proportions such as a ratio or the log-odds. We motivate and develop our method by means of a topical epidemiological application, namely the infection fatality rate of a virus such as SARS-CoV-2. We demonstrate that our method can be extended to more general settings of interest.
This paper develops techniques of estimation and inference in a prototypical macroeconomic adaptive learning model with slowly decreasing gains. A sequential three-step procedure based on a `super-consistent' estimator of the rational expectations equilibrium parameter is proposed. It is shown that this procedure is asymptotically equivalent to first estimating the structural parameters jointly via ordinary least-squares (OLS) and then using the so-obtained estimates to form a plug-in estimator of the rational expectations equilibrium parameter. In spite of failing Grenander's conditions for well-behaved data, a limiting normal distribution of the estimators centered at the true parameters is derived. Although this distribution is singular, it can nevertheless be used to draw inferences about joint restrictions by applying results from Andrews (1987) to show that Wald-type statistics remain valid when equipped with a pseudo-inverse. Monte-Carlo evidence confirms the accuracy of the asymptotic theory for the finite sample behaviour of estimators and test statistics discussed here.
Strong consistency of the least squares estimator in regression models with adaptive learning
(2017)
This paper looks at the strong consistency of the ordinary least squares (OLS) estimator in a stereotypical macroeconomic model with adaptive learning. It is a companion to Christopeit & Massmann (2017, Econometric Theory) which considers the estimator’s convergence in distribution and its weak consistency in the same setting. Under constant gain learning, the model is closely related to stationary, (alternating) unit root or explosive autoregressive processes. Under decreasing gain learning, the regressors in the model are asymptotically collinear. The paper examines, first, the issue of strong convergence of the learning recursion: It is argued that, under constant gain learning, the recursion does not converge in any probabilistic sense, while for decreasing gain learning rates are derived at which the recursion converges almost surely to the rational expectations equilibrium. Secondly, the paper establishes the strong consistency of the OLS estimators, under both constant and decreasing gain learning, as well as rates at which the estimators converge almost surely. In the constant gain model, separate estimators for the intercept and slope parameters are juxtaposed to the joint estimator, drawing on the recent literature on explosive autoregressive models. Thirdly, it is emphasised that strong consistency is obtained in all models although the near-optimal condition for the strong consistency of OLS in linear regression models with stochastic regressors, established by Lai & Wei (1982), is not always met.
Linear demand formulations for price competition in horizontally differentiated products are sometimes used to compare situations where additional varieties become available, e.g. due to market entry of new firms. We derive a consistent demand system to analyze such situations and highlight potential problems that can arise from an inconsistent approach.