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Hodrick-Prescott (HP) filtering of (most often, seasonally adjusted) quarterly series is analysed. Some of the criticism to the filter are adressed. It is seen that, while filtering strongly affects.autocorrelations, it has little effect on crosscorrelations. It is argued that the criticism that HP filtering induces a spurious cycle in the series is unwarranted. The filter, however, presents two serious drawbacks: First, poor perfor mance at the end periods, due to the size of the revisions in preliminary estimators) and, second, the amount of noise in the cyclical signal, which seriously disturbs its interpretation. We show how the addition of two model-based features (in particular, applying the filter to the series ex tended with proper ARIMA forecasts and backcasts, and using as input to the filter the trend-cycle component instead of the seasonally adjusted series) can considerably improve the filter performance. Throughout the discussion, we use a computationally and analytically convenient alter native derivation of the HP filter, and illustrate the results with an example consisting of 4 Spanish economic indicators.(1) Previously published Working Papers are Hsted in the Banco de Espana pubHcations catalogue.

The problem of estimating any sequence of missing observations in series with a nonstationary ARIMA model representation was solved by Kohn and Ansley (1986). In their approach, the likelihood is defined first by means of a transformation of the data; then, in order to obtain an efficient estimation procedure, a modified Kalman filter and a modified fixed point smoothing algorithm are used. In this paper we show how an alternative definition of the likelihood, based on the usual assumptions made in estimation of and forecasting with ARIMA models, permits a direct and standard state space representation of the nonstationary (original) data, so that the ordinary Kalman filter and fixed point smoother can be efficiently used for estimation, forecasting and interpolation. Our approach, like that of Kohn and Ansley (1986), can handle any arbitrary pattern of missing data and we show that the same results are obtained with both approaches. In this way, the problem of estimating missing values in nonstationary series is considerably simplified. When the available observations do not permit estimation of some of the missing values, the method indicates which are these values, and the forecasts that might be affected. Moreover, if linear combinations of the unestimable missing observations are estimable, the estimates are readily obtained. The method is illustrated using the same examples of Kohn and Ansley (1986), and an additional one for the case of unestimable missing values with estimable linear combinations thereof. It is shown that our likelihood is equal to that of Kohn and Ansley (1986); it also coincides with that of Harvey and Pierse (1984) when applicable, and to that of Box and Jenkins (1976) when no observation is missing. The results are extended to regression models with ARIMA errors, and a computer program, written in Fortran for MSDOS computers, is available from the authors.

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