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.

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: When volatility feedback is taken into account, there is strong evidence of a positive tradeoff between stock market volatility and expected returns on a market portfolio. In this paper, we ask whether this intertemporal tradeoff between risk and return is responsible for the reported evidence of mean reversion in stock prices. There are two relevant findings. First, price movements not related to the effects of Markov-switching market volatility are largely unpredictable over long horizons. Second, time-varying parameter estimates of the long-horizon predictability of stock returns reject any inherent mean reversion in favour of behaviour implicit in the historical tradeoff between risk and return. JEL classification: G12; G14 Keywords: Volatility Feedback; Mean Reversion; Markov Switching; TimeVarying Parameter 1 1. Introduction More than a decade has passed since Fama and French (1988) and Poterba and Summers (1988) reported that price movements for market portfolios...

It is a well accepted empirical result that forward exchange rate unbiasedness is rejected in tests using the “differences regression ” of the change in the logarithm of the spot exchange rate on the forward discount. The result is referred to in the International Finance literature as the forward discount puzzle. Competing explanations of the negative bias of the forward discount coefficient include the possibilities of a time-varying risk premium or the existence of “peso problems. ” We offer an alternative explanation for this anomaly. One of the stylized facts about the forward discount is that it is highly persistent. We model the forward discount as an AR(1) process and argue that its persistence is exaggerated due to the presence of structural breaks. We document the temporal variation in persistence, using a time-varying parameter specification for the AR(1) model, with Markov-switching disturbances. We also show, using a stochastic multiple break model, suggested recently by Bai and Perron (1998), that for the G-7 countries, with the exception of Japan, the forward discount persistence is substantially less, if one allows for multiple structural breaks in the mean of the process. These breaks could be identified as monetary shocks to the central bank’s reaction function, as discussed in Eichenbaum and Evans (1995). Using Monte Carlo simulations we show that if we do not account for structural breaks which are present in the forward discount process, the forward discount coefficient in the “differences regression ” is severely biased downward, away from its true value of 1. The authors would like to thank Charles Engel and the participants of the macroeconomics seminar at the New York Federal Reserve Bank for helpful comments and suggestions and Jushan Bai for generously providing the GAUSS code to estimate the multiple break models. The usual disclaimer applies

We develop a theoretical model to compare forecast uncertainty estimated from time series models to those available from survey density forecasts. The sum of the average variance of individual densities and the disagreement is shown to approximate the predictive uncertainty from well-specified time series models when the variance of the aggregate shocks is relatively small compared to that of the idiosyncratic shocks. Due to grouping error problems and compositional heterogeneity in the panel, individual densities are used to estimate aggregate forecast uncertainty. During periods of regime change and structural break, ARCH estimates tend to diverge from survey measures.

cial econometrics, for several reasons. First, Engle's Nobel citation was explicitly "for methods of analyzing economic time series with time-varying volatility (ARCH)," whereas Granger's was for "for methods of analyzing economic time series with common trends (cointegration)." Second, the credit for See also Engle's recent interview in Econometric Theory (Diebold, 2003) for additional insights into the development and impact of his work. -2creating the ARCH model goes exclusively to Engle, whereas the original cointegration idea was Granger's, notwithstanding Engle's powerful and well-known contributions to the development. Third, volatility models are a key part of the financia l econometrics theme that defines Engle's broader contributions, whereas cointegration has found wider application in macroeconomics than in finance. Last and not least, David Hendry's insightful companion review of Granger's work, also in this issue of the Scandinavian Journal, discusses the origins and deve

framework for the conduct of monetary policy consists of a policy instrument, an intermediate policy target and a long-run policy objective. The policy instrument is a lever which the central bank can manipulate to achieve its intermediate target. Possible choices for the policy instrument include the quantity of bank reserves, the monetary base (hank reserves plus currency in circulation) or a short-term interest rate. Monetary policymakers aim at a value of the intermediate target variable that will make current monetary policy consistent with a longrun policy objective, such as price stability.1 Potential intermediate target variables include nominal gross domestic product (GUP) and monetary

Engle’s footsteps range widely. His major contributions include early work on band-spectral regression, development and unification of the theory of model specification tests (particularly Lagrange multiplier tests), clarification of the meaning of econometric exogeneity and its relationship to causality, and his later stunningly influential work on common trend modeling (cointegration) and volatility modeling (ARCH, short for AutoRegressive Conditional Heteroskedasticity). 2 More generally, Engle’s cumulative work is a fine example of best-practice applied time-series econometrics: he identifies important dynamic economic phenomena, formulates precise and interesting questions about those phenomena, constructs sophisticated yet simple econometric models for measurement and testing, and consistently obtains results of widespread substantive interest in the scientific, policy, and financial communities. Although many of Engle’s contributions are fundamental, I will focus largely on the two most important: the theory and application of cointegration, and the theory and application of dynamic volatility models. Moreover, I will discuss much more extensively Engle’s volatility models and their role in financial econometrics, for several reasons. First, Engle’s Nobel citation was explicitly “for methods of analyzing economic time series with time-varying volatility (ARCH), ” whereas Granger’s was for “for methods of analyzing economic time series with common trends (cointegration). ” Second, the credit for 1 Prepared at the invitation of the Scandinavian Journal of Economics. Financial support from the