Applied stochastic models in business and industry. Trend estimation of financial time series, страница 6

of Table II. These values are similar, but the minimum values in Table I are generally greater than or equal to the values of N obtained with the regression model. Thus, Table III allows us to see the distorting effect of using the fitting model rather than Equation (18).

It is important to say that another way of looking at the smoothness property of a trend produced by a filter is by means of spectral analysis. To assess smoothing in the frequency domain we should look at squared gain plots, as did Kaiser and Maravall [14] when analyzing the HP filter. That kind of approach was also adopted by King and Rebelo [8] to compare the properties of the ES and HP filters within the context of cyclical analysis. Moreover, several tools have been developed in the frequency domain for performing cyclical analysis. For instance, Kaiser and Maravall [14] suggests to choose the value for the HP filter by fixing the length of the period over which the

Table II. Estimation results of fitting models that relate with N and S% (daily series).

Table III. Minimum values of N required to produce at most S%. From Tables I and II.

analyst wishes to measure cyclical activity. The same reasoning could be applied with the ES filter, but we did not follow that approach in this paper, because we are not interested in performing cyclical analysis in particular, but in obtaining a trend that provides a nice visual representation of the data. That is, our aim is to obtain an appropriate estimate of the trend (in terms of the percentage of smoothness attained) not in detrending the series in order to study the underlying cyclical activity.

3. EXTENSION TO OTHER THAN DAILY FREQUENCIES

To estimate the trend of a series with frequency of observation different than daily, it is not adequate to use the same obtained for a daily series. To see why let us assume that the observation period spans years 2000–2002. Thus, we either have 1095 daily data, or 781 daily data by considering 5-day weeks, or only 156 weekly data, so that the sample size changes for each type of periodicity under consideration. Therefore, for each of those series and the same S% Table I would lead to different values. However, we should bear in mind that the long term behavior of the series must be essentially the same, no matter what the periodicity of the series is. Moreover, it is important to realize that a series with lower frequency of observation is related to that with higher frequency by means of some type of aggregation mechanism. This fact has been recognized by Maravall and del R´ıo [16], who proposed different solutions to find values that produce equivalent results on series with different periodicities, from a frequency domain perspective.