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CPS2014 Ma. S.B.P. et al.
            through a backfitting algorithm) incorporating the VAR(1) model estimation
            illustrated  in  Algorithm  1  and  the  GAM  procedure  which  involves  the
            implementation of the backfitting algorithm and the General Local Scoring
            Algorithm,  and  (2)  Algorithm  3  (sieve  bootstrap  on  the  bivariate  residual
            matrix).

            Algorithm  1:  ()  to  estimate  the  output  vector  autocorrelation
            coefficient 
                The  vector  autoregression  (VAR)  model  is  one  of  the  most  successful,
            flexible, and easy to use models for the analysis of multivariate time series. The
            VAR  model  is  useful  in  describing  the  dynamic behavior  of  economic  and
            financial time series (Zivot and Wang, 2006).
                                                                         ′
              1.  From the proposed bivariate additive model in [2], let (  1  ) =
                                                                        2 ′
                  ∑   ∑    ( ,− ) +   and consider the bivariate (1) model
                                          
                             ,
                    =1
                         =1
                                     
                                    (  1 ) = (  11   12 ) (  1−1 ) + (  1 )
                                             
                                                        
                                                                  
                                                  
                                      2
                                              21
                                                         2−1
                                                   22
                                                                   2
                  That is,
                                       1  =    +    +  1
                                                       12 2−1
                                             11 1−1
                                       2  =    +    +  2
                                                       22 2−1
                                             21 1−1
                  where  ,  21  ≠ 0 and ( ,  ) =  12  ≠ 0.
                                              1
                                                 2
                          12
              2.  Estimate   by  fitting  (1)  model  separately  for  each  of  1−1  and
                            
                   2−1  and   , ,  = 1,2,  ≠   by  the  correlation  of   1−1  with   2−1  .
                              
                  Substitute these preliminary estimates in the VAR model to compute for
                  the residual vector per time period.
                Suppose that the model is expressed as  = ∑    ( ) + , for ℎ = 1,2,
                                                         ℎ
                                                                  
                                                                     
                                                              =1
            where  ,  ,  =  1,  …  ,    are  the  smooth  functions  of  the  sparse  principal
            components of exogenous variables. The backfitting algorithm adapted from
            Hastie and Tibshirani (1986) is performed as detailed in the following.

            Algorithm 2: Initial Parameter Estimation through Backfitting Algorithm
                1.  Estimate  by VAR(1) as in Algorithm 1:  =  −  +  , where  =
                                                             
                                                                                    
                                                                          
                   ∑   ∑    ( ,− ) +  .
                                            
                              ,
                          =1
                     =1
                2.  Compute  residual  vector    =    −  − .  This  contains  information
                   about the covariate effects.
                3.  Implement the GAM on the SPCs produced in Step 1 with . Obtain
                   the bivariate fitted values .
                4.  A new response vector is computed,
                                                ∗   =  −  .
                                                      
                                                          
                   This will set aside the covariate effects to focus on estimating .
                5.  Repeat Steps (1) to (4) until minimal changes at 0.01% (convergence)
                   are observed in the values of the estimates in ̂ and  .
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