Since Kitagawa (1987) and Kramer and Sorenson (1988) proposed the filter and smoother using numerical integration, nonlinear and/or non-Gaussian state estimation problems have been developed. Numerical integration becomes extremely computer-intensive in the higher dimensional cases of the state vector. Therefore, to improve the above problem, the sampling techniques such as Monte Carlo integration with importance sampling, resampling, rejection sampling, Markov chain Monte Carlo and so on are utilized, which can be easily applied to multi-dimensional cases. Thus, in the last decade, several kinds of nonlinear and non-Gaussian filters and smoothers have been proposed using various computational techniques. The objective of this paper is to introduce the nonlinear and non-Gaussian filters and smoothers which can be applied to any nonlinear and/or non-Gaussian cases. Moreover, by Monte Carlo studies, each procedure is compared by the root mean square error criterion.

Recent evidence suggests that many economic time series are subject to structural breaks, yet little is known about the properties of alternative forecasting methods for such data. This paper proposes a new method for determining the window size that explores the trade-off between bias and forecast error variance to minimize the mean squared forecast error in the presence of breaks in autoregressive models. An application to output growth in the OECD compares the performance of the proposed method to that of several existing approaches to forecasting under model instability.

This paper derives analytical results for determination of the window size that explores the trade-off between bias and forecast error variance to minimize the mean squared forecast error in the presence of breaks. We show analytically how to determine the estimation window optimally for the case with strictly exogenous regressors. Through Monte Carlo simulations the paper compares the performance of the proposed method to that of several existing approaches to forecasting under breaks.

In many applications involving time-varying parameter VARs, it is desirable to restrict the VAR coefficients at each point in time to be non-explosive. This is an example of a problem where inequality restrictions are imposed on states in a state space model. In this paper, we describe how existing MCMC algorithms for imposing such inequality restrictions can work poorly (or not at all) and suggest alternative algorithms which exhibit better performance. Furthermore, previous algorithms involve an approximation relating to a key integrating constant. Our algorithms are exact, not involving this approximation. In an application involving a commonly-used U.S. data set, we show how

CAPM betas are widely used in practice. While they are estimated from historical data, they are generally applied to a future period. This is appropriate only if the betas are stable over time. However, there is widespread evidence that the CAPM betas vary considerably over time. This raises two questions: is the time-variation in the betas systematic and can systematic variation in the betas be used to generate forecasts which improve on the usual practice of using the betas estimated over the most recent five-year period (the “five-year rule of thumb”)? We address both of these questions. We estimate time-varying betas using both overlapping sub-periods (five-year rolling regressions) and non-overlapping two-year periods. We proceed to explain the time-variation in the betas using two different regression models which we subsequently use for forecasting. We find that, despite the forecasting equations having relatively high within-sample explanatory power, the forecasts generated by these equations are dominated, on average, by the five-year rule of thumb. Further experimentation shows that the five-year rule of thumb is itself dominated by two-, three-, four-, six- and seven-year rules of thumb, with the three-year rule being optimal for our sample. 2 1

This paper studies the excess returns on stocks, associated to various company fundamentals on a panel of US stocks from 1979 to 2008. The returns premia are measured using a random coefficient panel data model on the individual stock level. We show that the HML and SMB factors in the Fama and French model probably have no particular economic meaning as sources of systematic risk other than being proxies for the impact of the book-to-price and size characteristics. While the book-to-price ratio, market capitalization, past year sales growth and the share of reinvested profits generate significant premia, earnings history and forecasts are of little predictive power. We statistically confirm the time-varying nature of the style premia but find no strong evidence for the value and growth momentum in a multivariate setting when the systematic risk is controlled for. Some of the premia are positively correlated with the market return and between each other, while others seem to be unrelated. Variations in premia associated with companies’ high internal growth and growth of sales are positively correlated between each other, with the market return and with the value premium. Variations of the size premium are probably driven by different factors.