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Appendix 



For the convenience of the reader, we summarize here 

 Box et al.'s (1978) algorithm to calculate the coefficients 

 of the smoothing polynomial loiB). Recall from Equation 

 14 that 



coiB)-- 



ct; <p{B)(p{F) 

 (T^ n(B)ri(F) 



(A.l) 



where cp(B) is a polynomial of order P+D and i)(B) is a 

 polynomial of order Q. Because Q 2 P+D. one can write 



(p(B) = l-fiB^ -(P,B'-...-<PqB'^; 

 r]{B) = l-ri,B'-noB-...-nQB'^, 



(A.2) 



where (p=0 for j>P+D. 

 First, define 



CiB)^^^;CiB} = l + aB'+C,B'~ + .... (A.3) 



ri(B) ^ ^ 



One can solve for the coefficients of C using an iterative 

 process by recognizing that the coefficients of each power 

 of B in the following expression must be zero: 



= (p{B)-n(B)C(B). 

 Consequently, one obtains 



c, = ii^-<P,;C, = n_, -9, +Y,c,_,n„j = 2,...Q (A.4) 



and higher order coefficients of C (i.e., for j>Q) can be 

 computed recursively using the relation 



C,=IC,_,'7,. 



(A.5) 



Next, define 



X(B,F)^(B} 



<p(F) 

 ri(F) 



(A.6) 



where X( fi, F ) = X„ -h £ X/ B' +F'). 



,/=i 



From this definition, one obtains 



X(B,F)ri(F)^{B)(p(F). 



(A.7) 



