This thesis consists of four essays in theoretical and empirical econometrics. The essays are presented in the form of chapters. Chapter 1 and the first part of Chapter 2 consider estimation and testing problems in econometric models involving time-varying coefficients. The methodology developed is then used in two macroeconomic applications, namely, the investment dynamics of the U.S. presented in the second part of Chapter 2, and the inflation dynamics of the U.S. investigated in Chapter 3. Chapter 4 considers the problem of small-sample inference in rational expectations models with persistent data.; The first chapter, which is co-authored with Ulrich Muller, investigates how to conduct valid inference on a stable subset of parameters in an econometric model with time-varying parameters. The answer turns out to be more straightforward than it might seem: For a very wide range of unstable parameter paths, and for a large class of Hansen's (1982) GMM models, standard inference ignoring the partial instability remains asymptotically valid for the set of stable parameters, as long as the instability is of moderate magnitude in the sense of not being detectable with probability one.; The second chapter contributes to a line of research that follows from the Lucas (1976) critique. It considers estimation and testing of Euler equation models with time-varying reduced-form coefficients. The chapter formalizes the necessity of structural stability assessment of Euler equations, arguably one of the most relevant (but often overlooked) criterion for model validation by the standards of the Lucas critique. As an application, standard investment Euler equations are submitted to examination using U.S. quarterly data. The empirical outcomes appear to suggest that the standard models have not been a success thus far, at least for aggregate investment.; The third chapter investigates the empirical relevance of a broad range of price adjustment models in which price inflation is driven by the movement of real marginal cost. Applying the econometric methodology developed in Chapter 1 and Chapter 2, and based on U.S. quarterly data, this chapter leads to several important empirical findings. In particular, it finds that a second lead and a second lag of inflation are statistically important and enter with negative coefficients, which is not predicted by any leading theories of price adjustment. Also, stability is not rejected for some of the simpler models suggested in the literature, while larger models with additional leads and lags of inflation often suffer from instability.; Chapter 4 proposes a method to conduct accurate small sample inference in rational expectations models in the presence of highly persistent variables. It shows that when rational expectations models are tested on vector autoregressions, the methodology developed in Sims, Stock and Watson (1990) for linear regression models can be extended to study the unit-root or near unit-root problem in rational expectations models. Based on the derived analytical small sample distributions, two measures are proposed to quantify small sample effects of persistent variables in rational expectations models.
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