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.

In this paper we construct prediction intervals for ARCH models using the bootstrap. We use both a parametric and non-parametric bootstrap, which take account of parameter uncertainty. We compare our prediction intervals to traditional asymptotic prediction intervals, and find that the bootstrap leads to improved accuracy. The accuracy of the bootstrap is empirically demonstrated with the Yen/$US exchange rate.

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.

In this paper, we propose a bounded influence estimation (BIE) and outlier detection procedure for GARCH models. Previous studies show that maximum likelihood estimates of GARCH models are sensitive to outliers and financial time series present a heavy tail due to outliers. The proposed BIE limits the influence of a small subset of the data and is asymptotically normal. Its robustness against outliers and model misspecification is examined and supported. We further use BIE with GARCH models to develop a method for detection of additive outliers. An application to the exchange rates of major currencies is provided.

Stuber. The views expressed in this paper are those of the authors and should not be attributed to the Bank of Canada. ISSN 1192-5434 ISBN The purpose of this study is to test the hypothesis that inflation uncertainty increases at higher levels of inflation. Our analysis is based on the generalized autoregressive conditional heteroscedasticity (GARCH) class of models, which allow the conditional variance of the error term to be time-varying. Since this variance is a proxy for inflation uncertainty, a positive relationship between the conditional variance and inflation would be interpreted as evidence that inflation uncertainty increases with the level of inflation. We apply GARCH techniques to two models of the inflation process in Canada: a simple autoregressive model and a reduced-form Phillips-curve model. Our findings concerning the link between inflation and its uncertainty are somewhat model-dependent. In the autoregressive case, there is a significant positive relationship between inflation and

This paper examines the determinants of inflation forecast uncertainty using a panel of density forecasts from the Survey of Professional Forecasters (SPF). Based on a dynamic heterogeneous panel data model, we find that the persistence in forecast uncertainty is much less than what the aggregate time series data would suggest. In addition, the strong link between past forecast errors and current forecast uncertainty, as often is noted in the ARCH literature, is largely lost in a multiperiod context with varying forecast horizons. We propose a novel way of estimating “news ” and its variance using the Kullback-Leibler Information, and show that the latter is an important determinant of forecast uncertainty. Our evidence suggests a strong relationship of forecast uncertainty with level of inflation, but not with forecaster discord or with the volatility of a number of other macroeconomic indicators.

“A nickel ain’t worth a dime any more ” [Yogi Berra]

“A nickel ain’t worth a dime any more ” [Yogi Berra]

GMM estimation has been frequently applied to rational expectations models. This chapter discusses examples of these applications. The main purpose is not to provide a survey of the literature but to illustrate applications. Problems that researchers have encountered in applying GMM are discussed as well as procedures they have used to address these problems. In this chapter, the notations for the NLIV model of Section 9.2 will be used. 10.1 Euler Equation Approach Hansen and Richard (1987) show that virtually all asset pricing models can be written as (10.1) vt = E[mt+1dt+1|It] where vt is the asset price at date t, mt+1 is the intertemporal marginal rate of substitution (IMRS) between date t and date t+1, and dt+1 is the payoff of an asset at date t+1. Each asset pricing model specifies different IMRS.

This paper considers forecasting the conditional mean and variance from an ARMA model with GARCH in mean e ects. Expressions for the optimal predictors and their conditional and unconditional MSE's are presented. We also derive the formula for the covariance structure of the process and its conditional variance.