Abstract:

This dissertation addresses theoretical and empirical issues in the econometrics of locally time varying parameters.

We consider parameter instabilities that are of the same magnitude as the local alternatives of efficient stability tests. The asymptotic thought experiment leads to a limit theory where there is only limited information about the form of the instability. In this way, the asymptotics reflect the difficulties of not being sure about the precise form or even presence of the instability in small samples in most econometric models of interest.

Throughout this dissertation, statistical procedures' performance (such as tests and parameter estimators) are evaluated by the explicit criterion of weighted average risk (or weighted average power in the case of tests). The weight function is proportional to the distribution of a Gaussian process, and focusses on local parameter instabilities.

The first chapter - a paper coauthored with Ulrich K. M?ller - investigates asymptotically efficient inference in general likelihood models with locally time varying parameters. It is shown that asymptotically, the sample information about the parameter path is efficiently summarized by a linear Gaussian pseudo model. This approximation leads to computationally convenient formulas for efficient path estimators and test statistics, and unifies the theory of stability testing and parameter path estimation.

However, much of econometric modelling by design eschews specific parametric distributional assumptions by imposing semiparametric restrictions. What is thereby won in breadth of applicability is lost in strength of results: The strong optimality result of Chapter 1 thus does not carry over to semiparametric models, such as GMM models. Moreover, classical concepts of semiparametric efficiency do not quite straightforwardly translate to the case of locally instable models.

Hence, the second chapter starts by considering the problem of inference procedures about local parameters in semiparametric models and proposes a criterion to assess their asymptotic efficiency. In particular, the general form of an asymptotic lower bound on the performance of convergent statistics is derived. The results are then applied to GMM models with local time variation in the parameters, where the semiparametric assumption is that the partial sums of the moment conditions converge in law. In models that satisfy such a semiparametric restriction, we provide an explicit form for the asymptotic lower bound on the performance of convergent statistics, as well as a general and computationally simple procedure to generate statistics sequences that achieve the lower bound.

The last chapter of the dissertation applies the theoretical results developed in the previous two chapters, and Chapter 1 in particular, to study the dynamic behaviour of the S&P 500's returns, which have a long history of being modeled by GARCH models. The standard GARCH model assumes global stationarity in that the long-run variance of the data is constant. We propose a version of the GARCH model with locally time varying parameters that isolates potential changes in the long-run variance of the data from the changes in the short-run variance. The optimal stability test for the long-run variance rejects the null hypothesis for the S&P 500's returns, which suggests that the assumption of global stationarity is inadequate to describe the volatility dynamics of stock returns. Moreover, the accuracy of sample information approximation provided by the Gaussian pseudo model of Chapter 1 is assessed.