Inference in Unstable Time Series Models

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Instabilities in the parameters of econometric time series models are a theoretically plausible and empirically widespread phenomenon. Past research has largely focused on testing the stability of models. This proposal is concerned with the natural next step: How does one do inference in unstable time series models? The first project concerns efficient inference on the time varying path of the parameters in general likelihood models, including the special case of efficient estimation of the parameter path. Knowledge of the path is of primary interest for many purposes: (i) it helps understand the source of the instability, (ii) the endpoint of the path is intimately linked to efficient forecasting and (iii) economic theory sometimes implies specific forms of instability, such that inference on the path becomes a test of economic theory. The main result is that asymptotically efficient inference on the path may be obtained by treating the sequence of score vectors, evaluated at the usual maximum likelihood estimator that ignores the instability, as a Gaussian local level model. This in particular implies that an efficient path estimator can be obtained by applying a simple Kalman smoother to the sequence of scores. The second project addresses the question of how to conduct valid inference on a subset of stable parameters in a Generalized Methods of Moments (GMM) framework when other parameters are time varying. Partially stable models arise naturally when optimizing agents adapt to policy changes, which induces time variation in their reduced form behavioral equations, such as Euler equations. One of many possible applications of this research is how to conduct inference on the stable structural parameters, describing technology and preferences. The main-and maybe surprising-result is that standard GMM inference, ignoring the time variation, remains asymptotically valid on the stable subset of parameters. A third and unrelated project develops 'robust' long-run variance estimators. The projects address first order problems for applied time series econometricians. The approaches are innovative, and careful arguments are employed to solve the technical difficulties. From an econometric theory point of view, the results arguably represent substantial advances in the understanding of unstable time series models. Some of the results are generic and may be used in other contexts. Broader Impacts: While based on technically sophisticated arguments, the main results of this research are very straightfoward to apply. Given the prevalence of parameter instability in practice, this research should therefore have a major impact on empirical research. This includes tests of economic theory, forecasting and policy analysis. Dissemination of the project's ideas and results will be facilitated through presentations at universities and conferences, as well as freely available computer code. Finally, through collaborations with graduate students and research assistance, the proposal has a direct impact on student training.