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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 identifability, the periodic component is assumed to be centered, i.e., d t =1 s ( t ) = 0. I± d is odd, let q =( d - 1) / 2 and defne ± y t by the moving average in (5.19), noting that the window size in this case is 2 q +1 = d . d is even, let q = d/ 2 and defne ± y t = 1 2 q ² y t - q 2 + y t - q +1 + ··· + y t + q - 1 + y t + q 2 ³ ,q +1 t n - q. (5.21) For k =1 , . . ., d , let ¯ Δ k be the mean o± the sample { y k + jd - ± y k + : q - k n - q - k } . The method-of-moments estimate , which replaces population moments by their sample counterparts, o± the periodic ±unction s ( t ) is ± s ( k )= ¯ Δ k - d - 1 d ´ i =1 ¯ Δ i ±or 1 k d, ± s ( t ± s ( k ) ±or 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 refned nonparametric regression techniques than simple averaging, as in (5.20) and (5.22), will be described in Chapter 7. The ±unction stl in R (or S ) uses these more refned techniques to estimate the decomposition y t = m ( t )+ s ( t w t o± 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) ±or details and illustrations. 5.2.2 An empirical example Figure 5.4 plots the monthly unemployment rates in Dallas County, Arizona, ±rom January 1980 to June 2005. The data are obtained ±rom the Website www.Economagic.com . The ACF and PACF o± the unemployment rates are plotted in Figure 5.5. Note that all rates vary ±rom 4% to 12% except those in the frst fve months, which are above 15%. We split the time series into
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This note was uploaded on 12/06/2011 for the course AMS 316 taught by Professor Xing during the Fall '09 term at SUNY Stony Brook.

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example_unem_text - 124 5 Time Series Modeling and...

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