By C. Chatfield (auth.)
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Extra info for The Analysis of Time Series: An Introduction
4; Box and Jenkins, 1970). For examp1e with monthly data one can emp10y the operator Vl2 where Alternative methods of seasonal adjustment are reviewed by Pierce (1980). These inc1ude the widely used X-lI method (C1eveland and Tiao, 1976; Shiskin and Plewes, 1978). The possib1e presence of ca1endar effects shou1d also be considered (C1eve1and and Durbin, 1982). 7 Autocorrelation An important guide to the properties of a time series is provided by aseries of quantities called sampie autocorrelation coefficients, which measure the correlation between observations at different distances apart.
An alternative dass of piece-wise polynomials, is the dass of spline functions (see Wold, 1974). Suppose we have N observations, Xl' . . , X N' F or a symmetrie filter, we can ca1culate Sm(x t ) for t ::: q + I up to t ::: N - q, but as a consequence lose 2q pieces of data. In some situations this is not important but in other situations it is particularly important to get smoothed values up to t ::: N. Kendall and Stuart (1966) discuss the end-effects problem. 2) takes q ::: 00, and aj ::: a(1 - a)i where a is a constant such that 0 < a < 1.
2) takes q ::: 00, and aj ::: a(1 - a)i where a is a constant such that 0 < a < 1. Having estimated the trend, we can look at the local fluctuations by examining Res(x t ) ::: residual from smoothed value ::: X t ::: +8 ~ - Sm(x t ) r=-q brXt+r' 18 SIMPLE DESCRIPTIVE TECHNIQUES This is alsp a linear filter, and if +9 Sm(x t )= L r=-q arXt+r. with ~ar = I, then ~br =0, b o = I - ao , and b r =-ar for r =1= O. How do we choose the appropriate filter? The answer to this question really requires considerable experience plus a knowledge of the frequency aspects of time-series analysis which will be discussed in later chapters.
The Analysis of Time Series: An Introduction by C. Chatfield (auth.)