Improved Holt Metho d for Irregular Time Series (2008)

The paper suggests an improvement of Holt method for irregular time series as was presented by Wright. The modiflcation deals with problem of time-close observations, i.e. two subsequent observations with time distance much shorter than in average. If this situation occurs in time series when using the original Wright's formulae, one can obtain seriously wrong results. Simulation study is provided to compare the performance of the original and improved method. At the end of 50's, Holt and Winters proposed an ad hoc method for smoothing and forecasting time series with locally linear trend (Winters, 1960) nowadays referred usually as Holt method. They employed the exponentially weighted moving average approach to estimate the level and newly also the slope of the time series. The extension of this method for the case of seasonal time series (Holt-Winters method) was provided at the same time. Holt method has achieved a broad popularity among forecasters due to its simplicity and good performance. Several modiflcations of this original method have appeared in literature and are used in practice. The version with damped linear trend which is said to give better results in longer forecasting horizons is well known (Gardner and McKenzie, 1985). Wright (1986) has suggested a straightforward generalization of Holt method for the case of time series observed at irregular time intervals. He has extended the original smoothing formulae to this situation and has tested his method on several real time series. The formulae of classical Holt method and its extension for irregular time series by Wright are reminded in Section 2. Concerning time series with missing observations, the time distance between two consequent obser- vations is bounded from below by the concerned time unit. This bound is not dramatically lower than the average time spacing in the series. In contrast to this, if a general time irregularity is allowed, we can come to a situation which we call as a problem of time-close observations: two consequent observations have much shorter time distance, compared to the average time spacing in the series. Due to the measurement noise or due to any other special character of data, these observations can have their values signiflcantly difierent even if their times are almost the same. Thus a one-step slope between these two observations can be theoretically arbitrarily large. Using the Wright's extension of Holt method, this phenomenon will then appear more or less also in the time series of corresponding smoothed levels and can seriously afiect the slope estimate and consequently cause a bias in forecasts for a quite long time period. Moreover, the slope estimate bias can occur even if the values of time-close observations are close to each other. Detailed discussion of this impact is provided in Section 3. In Section 4 we suggest a reasonable modiflcation of Wright's formulae to overcome these di-culties. This modiflcation is not less intuitive or easy to understand than the original formulae. In author's opinion it has even a better interpretation. Moreover, as far as the resulting formulae are concerned, the modiflcation means just adding one term in an updating formula for slope smoothing coe-cient. In Section 5 a simulation study is provided to check the impact of the suggested modiflcation in practice. Several settings of time series generating scheme are employed to flnd out how the improvement depends on the conflguration of time-close observations in data. Section 6 brings the conclusions of the paper.