mation is accomplished by replacing \u03b2 0 by \u03b2 1 in 3126 This process is repeated

Mation is accomplished by replacing β 0 by β 1 in

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mation is accomplished by replacing β (0) by β (1) in ( 3.126 ). This process is repeated by calculating, at iteration j = 2 , 3 , . . . , β ( j ) = β ( j - 1) + ( β ( j - 1) ) until convergence. Example 3.31 Gauss–Newton for an MA( 1 ) Consider an invertible MA(1) process, x t = w t + θw t - 1 . Write the truncated errors as w t ( θ ) = x t - θw t - 1 ( θ ) , t = 1 , . . . , n, (3.127) where we condition on w 0 ( θ ) = 0. Taking derivatives, - ∂w t ( θ ) ∂θ = w t - 1 ( θ ) + θ ∂w t - 1 ( θ ) ∂θ , t = 1 , . . . , n, (3.128) where ∂w 0 ( θ ) /∂θ = 0 . Using the notation of ( 3.123 ), we can also write ( 3.128 ) as z t ( θ ) = w t - 1 ( θ ) - θz t - 1 ( θ ) , t = 1 , . . . , n, (3.129) where z 0 ( θ ) = 0 . Let θ (0) be an initial estimate of θ , for example, the estimate given in Ex- ample 3.28 . Then, the Gauss–Newton procedure for conditional least squares is given by θ ( j +1) = θ ( j ) + n t =1 z t ( θ ( j ) ) w t ( θ ( j ) ) n t =1 z 2 t ( θ ( j ) ) , j = 0 , 1 , 2 , . . . , (3.130) where the values in ( 3.130 ) are calculated recursively using ( 3.127 ) and ( 3.129 ). The calculations are stopped when | θ ( j +1) - θ ( j ) | , or | Q ( θ ( j +1) ) - Q ( θ ( j ) ) | , are smaller than some preset amount.
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3.6 Estimation 135 0 5 10 15 20 25 30 35 0.4 0.0 0.4 0.8 LAG ACF 0 5 10 15 20 25 30 35 0.4 0.0 0.4 0.8 LAG PACF Fig. 3.8. ACF and PACF of transformed glacial varves. Example 3.32 Fitting the Glacial Varve Series Consider the series of glacial varve thicknesses from Massachusetts for n = 634 years, as analyzed in Example 2.6 and in Problem 2.8 , where it was argued that a first-order moving average model might fit the logarithmically transformed and differenced varve series, say, log( x t ) = log( x t ) - log( x t - 1 ) = log x t x t - 1 , which can be interpreted as being approximately the percentage change in the thickness. The sample ACF and PACF, shown in Figure 3.8 , confirm the tendency of log( x t ) to behave as a first-order moving average process as the ACF has only a significant peak at lag one and the PACF decreases exponentially. Using Table 3.1, this sample behavior fits that of the MA(1) very well. The results of eleven iterations of the Gauss–Newton procedure, ( 3.130 ), starting with θ (0) = - . 10 are given in Table 3.2 . The final estimate is b θ = θ (11) = - . 773; interim values and the corresponding value of the condi- tional sum of squares, S c ( θ ) given in ( 3.121 ), are also displayed in the table. The final estimate of the error variance is b σ 2 w = 148 . 98 / 632 = . 236 with 632 degrees of freedom (one is lost in differencing). The value of the sum of the squared derivatives at convergence is n t =1 z 2 t ( θ (11) ) = 369 . 73, and con-
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136 3 ARIMA Models 1.0 0.8 0.6 0.4 0.2 0.0 150 160 170 180 190 200 210 S c Fig. 3.9. Conditional sum of squares versus values of the moving average parameter for the glacial varve example, Example 3.32 . Vertical lines indicate the values of the parameter obtained via Gauss–Newton; see Table 3.2 for the actual values. Table 3.2. Gauss–Newton Results for Example 3.32 j θ ( j ) S c ( θ ( j ) ) n t =1 z 2 t ( θ ( j ) ) 0 - 0 . 100 195.0010 183.3464 1 - 0 . 250 177.7614 163.3038 2 - 0 . 400 165.0027 161.6279 3 - 0 . 550 155.6723 182.6432 4 - 0 . 684 150.2896 247.4942 5 - 0 . 736 149.2283 304.3125 6 - 0 . 757 149.0272 337.9200 7 - 0 . 766 148.9885 355.0465 8 - 0 . 770 148.9812 363.2813 9 - 0 . 771 148.9804 365.4045 10 - 0 . 772 148.9799 367.5544 11 - 0 . 773 148.9799 369.7314 sequently, the estimated standard error of b θ is p . 236 / 369 . 73 = . 025; 7 this leads to a t -value of - . 773 /. 025 = - 30 . 92 with 632 degrees of freedom.
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