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example_unem_text - 124 5 Time Series Modeling and...

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124 5 Time Series Modeling and Forecasting given period d . For identifiability, the periodic component is assumed to be centered, i.e., d t =1 s ( t ) = 0. If d is odd, let q = ( d - 1) / 2 and define y t by the moving average in (5.19), noting that the window size in this case is 2 q +1 = d . If d is even, let q = d/ 2 and define y t = 1 2 q y t - q 2 + y t - q +1 + · · · + y t + q - 1 + y t + q 2 , q + 1 t n - q. (5.21) For k = 1 , . . . , d , let ¯ Δ k be the mean of the sample { y k + jd - y k + jd : q +1 - k jd n - q - k } . The method-of-moments estimate , which replaces population moments by their sample counterparts, of the periodic function s ( t ) is s ( k ) = ¯ Δ k - d - 1 d i =1 ¯ Δ i for 1 k d, s ( t ) = s ( k ) for t = k + jd, (5.22) in which j is some integer. From the deseasonalized series y t - s ( t ), m ( t ) can be estimated by parametric regression or moving average methods. More refined nonparametric regression techniques than simple averaging, as in (5.20) and (5.22), will be described in Chapter 7. The function stl in R (or S ) uses these more refined techniques to estimate the decomposition y t = m ( t ) + s ( t ) + w t of a time series into a trend m ( t ), a seasonal component s ( t ) and a stationary disturbance w t ; see Venables and Ripley (2002, pp. 403–404) for details and illustrations. 5.2.2 An empirical example Figure 5.4 plots the monthly unemployment rates in Dallas County, Arizona, from January 1980 to June 2005. The data are obtained from the Website www.Economagic.com . The ACF and PACF of the unemployment rates are plotted in Figure 5.5. Note that all rates vary from 4% to 12% except those in the first five months, which are above 15%. We split the time series into
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