Data in which each observation is a curve occur in many applied problems. This paperexplores prediction in time series in which the data is generated by a curve-valuedautoregression process. It develops a novel technique, the predictive factor decomposition, forestimation of the autoregression operator, which is designed to be better suited for predictionpurposes than the principal components method. The technique is based on finding a reducedrankapproximation to the autoregression operator that minimizes the norm of the expectedprediction error.Implementing this idea, we relate the operator approximation problem to an eigenvalueproblem for an operator pencil that is formed by the cross-covariance and covarianceoperators of the autoregressive process. We develop an estimation method based onregularization of the empirical counterpart of this eigenvalue problem, and prove that with acertain choice of parameters, the method consistently estimates the predictive factors. Inaddition, we show that forecasts based on the estimated predictive factors converge inprobability to the optimal forecasts.The new method is illustrated by an analysis of the dynamics of the term structure ofEurodollar futures rates. We restrict the sample to the period of normal growth and find that inthis subsample the predictive factor technique not only outperforms the principal components method but also performs on par with the best available prediction methods.
