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A continuous Euler model has time-varying coefficients. Through a logarithmic time transformation, a continuous Euler model can be transformed to a continuous autoregressive (AR) model. By using the continuous Kalman filtering through the Laplace method, this article explores the data application of a continuous Euler process. This time deformation of an Euler process deforms specific time-variant (non-stationary) behaviour to time-invariant (stationary) data on the deformed time scale. With these time-invariant data on the transformed time scale, one may use traditional tools to conduct parameter estimation and forecasts. The obtained results can then be transformed back to the original time scale. Simulated data and actual data such as bat echolocation and the US residential investment growth are used to demonstrate the usefulness of time deformation in forecasting. The results indicate that fitting a traditional autoregressive moving-average (ARMA) model on an Euler data set without imposing time transformation leads to forecasts that are out of phase while the forecasts of an Euler model stay mostly in phase. [ABSTRACT FROM AUTHOR]

In this article we discuss methodology for analyzing nonstationary time series whose periodic nature changes approximately linearly with time. We make use of the M-stationary process to describe such data sets, and in particular we use the discrete Euter(p) model to obtain forecasts and estimate the spectral characteristics. We discuss the use of the M-spectrum for displaying linear time-varying periodic content in a time series realization in much the same way that the spectrum shows periodic content within a realization of a stationary series. We also introduce the instantaneous frequency and spectrum of an M-stationary process for purposes of describing how frequency changes with time. To illustrate our techniques we use one simulated data set and two bat echolocation signals that show time varying frequency behavior. Our results indicate that for data whose periodic content is changing approximately linearly in time, the Euler model serves as a very good model for spectral analysis, filtering, and forecasting. Additionally, the instantaneous spectrum is shown to provide better representation of the time-varying frequency content in the data than window-based techniques such as the Gabor and wavelet transforms. Finally, it is noted that the results of this article can he extended to processes whose frequencies change like atα, a > 0, -∞ < α < - ∞. [ABSTRACT FROM AUTHOR]

This paper uses three different approaches to investigate whether the declining provision of public capital is a major cause of declining labor productivity. The juxtaposition of approaches removes the variability in estimates due to dissimilar variable definitions and econometric methodologies. Estimates are based on U.S. time - series data and are evaluated by the implied elasticities of substitution, the prediction of labor productivity trends, and the impact of public capital on productivity. As the three approaches yield very different estimates, it will be hard to ever settle the debate about the effect of public capital on private productivity. [ABSTRACT FROM AUTHOR]