A natural generalization of the ARCH (Autoregressive Conditional Heteroskedastic) process introduced in Engle (1982) to allow for past conditional variances in the current conditional variance equation is proposed. Stationarity conditions and autocorrelation structure for this new class of parametric models are derived. Maximum likelihood estimation and testing are also considered. Finally an empirical example relating to the uncertainty of the inflation rate is presented.

We consider the CUSUM of squares test in a linear regression model with general mixing assumptions on the regressors and the errors. We derive its limit distribution and show how it depends on the nature of the error process. We suggest a corrected version that has a limit distribution free of nuisance parameters. We also discuss how it provides an improvement over the standard approach to testing for a change in the variance in a univariate times series. Simulation evidence is presented to support this. We illustrate the usefulness of our method by analyzing changes in the variance of stock returns and a variety of macroeconomic time series, as well as by testing for change in the variance of the residuals in a typical four-variable VAR model. Our results show the widespread prevalence of changes in the variance of such series and the fact that the variability of shocks affecting the U.S. economy has decreased. Keywords: Change-point, Variance shift, Recursive residuals, Dynamic models, Conditional heteroskedasticity.

We consider the power properties of the CUSUM and CUSUM of squares tests in the presence of a one-time change in the parameters of a linear regression model. A result due to Ploberger and Krämer (1990) is that the CUSUM of squares test has only trivial asymptotic local power in this case, while the CUSUM test has non-trivial local asymptotic power unless the change is orthogonal to the mean regressor. The main theme of the paper is that such conclusions obtained from a local asymptotic framework are not reliable guides to what happens in finitesamples. Theapproachwetakeisto derive expansions of the test statistics that retain terms related to the magnitude of the change under the alternative hypothesis. This enables us to analyze what happens for non-local to zero breaks. Our theoretical results are able to explain how the power function of the tests can be drastically different depending on whether one deals with a static regression with uncorrelated errors, a static regression with correlated errors, a dynamic regression with lagged dependent variables, or whether a correction for non-Normality is applied in the case of the CUSUM of squares. We discuss in which cases the tests are subject to a non-monotonic power function that goes to zero as the magnitude of the change increases, and uncover some curious properties. All theoretical results are verified to yield good guides to the finite sample power through simulation experiments. We finally highlight the practical importance of our results.

: 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

Abstract: Attention to visual motion can increase the responsiveness of the motion-selective cortical area V5 and the posterior parietal cortex. We addressed attentional modulation of effective connectivity using variable parameter regression and functional magnetic resonance imaging. We present data from a single subject scanned under identical stimulus conditions (visual motion) while varying only the attentional component of the task. Variable parameter regression of the influence of V5 on PP revealed increased effective connectivity during attention to visual motion. With this dynamic measure of effective connectivity we were able to make inferences about the source of modulation by looking for regions that predicted the observed changes in connectivity. Using an ordinary regression analysis, we showed that activity in the prefrontal cortex could explain these changes and was sufficient to account for these modulatory influences on connections in the dorsal visual pathway. Hum. Brain Mapping 6:403–408, 1998. � 1998 Wiley-Liss, Inc. Key words: effective connectivity; fMRI; attention; Kalman filter; variable parameter regression

Estimates of time-varying term premia

Quarterly Review vol. 4, no. 3 This publication primarily presents economic research aimed at improving policymaking by the Federal Reserve System and other governmental authorities.