This paper is structured as follows. The following section describes the statistical basis of the ES filter along with its interpretation. There, we emphasize the role played by the smoothing constant required by the filter and show how to choose that constant, based on a measure of smoothness proposed by Guerrero [9]. In Section 3 we show how to extend the applicability of the method to series with different frequencies of observation. Section 4 focuses on the empirical basis of the method and argues in favor of using daily time series data as the standard of reference. Section 5 provides some illustrative numerical examples. In Section 6 we conclude with some final remarks.
This section presents results that are similar in nature to those appearing in Guerrero [10] for the HP filter. However, the results shown here pertain specifically to the ES filter and we consider them of such importance that they must be shown here for completeness and to make this paper self-contained. The remaining sections of this work are completely new, in comparison with the material appearing in papers previously published by one of us (i.e. Guerrero [9, 10]).
Let us assume that an observed time series {Y1,...,YN} can be represented by the following unobserved component model:
Yt =t +t for t =1,..., N (1)
where t denotes the (unobserved) trend and t the (unobserved) noise at time t. We will assume that the time series {Yt} is I(1) in such a way that its trend will also be I(1) and the noise process will be stationary. By using this model we do not pretend to say that it represents the true data generating process. It is only an easy-to-use representation that captures some stylized facts of the time series under study. There are other representations equally useful in practice to capture some other features of the data. For instance, conditional heteroscedastic models play a central role in the literature on financial volatility, as indicated by Tsay [7, Chapter 3]. Therefore, we can choose different models for the same series, depending on the problem we want to attack. In many cases we can use a time series model for the mean together with a volatility model for the variance. We shall use the term filter very frequently, with a filter defined here as any operation on the observed series that yields a new series, which in the present case will be the estimated trend. Gomez´ [11] proved that the HP filter produces results equivalent to those obtained with any of the following three methods: (i) smoothing by penalized least squares (PLS), (ii) Kalman filtering plus smoothing and (iii) Wiener–Kolmogorov (WK) filtering. Such a result can be extended in a straightforward manner to the ES filter, since the ES and the HP filters share the same statistical basis.
The PLS smoothing approach that leads to the ES filter establishes that the trend must minimize the function M() defined as
NN
M
(Yt 
t)2 (2) t=2
where ∇ denotes the difference operator given by ∇Zt = Zt −Zt−1 for every variable Z and index t, and the constant >0 is a smoothing parameter. By writing
N N
F
t)2 and S
(3)
we see that F measures the goodness
of fit of the trend to the data, while S is related to trend smoothness.
The constant controls
the emphasis given to F and S in the minimization of M(), hence, it plays an
important role in expression (2). On the one hand, when →0 the fit of the trend to the observed series is
emphasized over its smoothness, consequently t →Yt for all t.
On the other hand, if →∞ the
smoothness constraint dominates the solution and we obtain the constant trend
model
Уважаемый посетитель!
Чтобы распечатать файл, скачайте его (в формате Word).
Ссылка на скачивание - внизу страницы.