Entropy is a classical statistical concept with appealing properties. Establishing asymptotic distribution theory for smoothed nonparametric entropy measures of dependence has so far proved challenging. In this paper, we develop an asymptotic theory for a class of kernel-based smoothed nonparametric entropy measures of serial dependence in a time series context. We use this theory to derive the limiting distribution of Granger and Lins (1994) normalized entropy measure of serial dependence, which was previously not available in the literature. We also apply our theory to construct a new entropy-based test for serial dependence, providing an alternative to Robinsons (1991) approach. To obtain accurate inferences, we propose and justify a consistent smoothed bootstrap procedure. The naive bootstrap is not consistent for our test. Our test is useful in, for example, testing the random walk hypothesis, evaluating density forecasts, and identifying important lags of a time series. It is asymptotically locally more powerful than Robinsons (1991) test, as is confirmed in our simulation. An application to the daily S&P 500 stock price index illustrates our approach.

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Given that it is quite impractical to use standard model selection criteria in a nonlinear modeling context, the builders of nonlinear models often choose lag length by setting it equal to the lag length chosen for a linear autoregression of the data. This paper studies the performance of this procedure in a variety of circumstances, andthenproposessomenewandsimplemodelselectionprocedures,basedonlinear approximations of the nonlinear forms. The idea here is to apply standard selection criteria to these linear approximations, rather than to autoregressions that make no provision for nonlinear behavior. A simulation study compares the properties of these proposed procedures with the properties of linear selection procedures.

Phase space reconstruction is investigated as a diagnostic tool for determining the structure of detected nonlinear processes in regression residuals. Empirical evidence supporting this approach is provided using simulations from an Ikeda mapping and the S&P 500. Results in the form of phase portraits (e.g., scatter plots of reconstructed dynamical systems) provide qualitative information to discern structural components from apparent randomness and provide insights categorizing structural components into functional classes to enhance econometric/time series modeling efforts.