﻿ Page 15 - Contributed Paper Session (CPS) - Volume 7
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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 , 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 10   11   12   13   14   15   16   17   18   19   20 