﻿ Page 14 - Contributed Paper Session (CPS) - Volume 7
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``````CPS2014 Ma. S.B.P. et al.
The  postulated  model  accounts  the  additional  contribution  of  the
contemporaneous  effects  of  some  input  series. This  study  is  proposing an
estimation procedure for an additive bivariate model

=  −1  + ∑ ∑  ( ,− ) +   ,       = 1, … , ,  = 1, … , ,  = 1, … ,    

,
=1  =1
where
∑   ∑    ( ,− ) = ∑   ∑
,−
,
=1
=1
=1
=1
is the bivariate response variable with columns  = ( ,  , … ,  ) , ℎ = 1,2
′
ℎ
ℎ1
ℎ2
ℎ

is the autocorrelation coefficient of  − , i.e.  = [  11   12 ] , 0 ≤  ≤ 1, ,  = 1,2

22
21
ℎ
−  is the covariate matrix up to the   lag, i.e. for each  X variable,
′
,−1
′
,−  =     ,−2     ,  = 1, … ,

⋮
[  ′ ,−]
,  = 1, … , ,  = 1, … , , is the coefficient matrix of  −

2
is the error matrix with columns  ,  ~(0,  ),   2  > 0,  = 1,2 and
1

2
( ,  ) =  12  ≠ 0
2
1
The postulated model states that at a given time point t, the variation in
the  bivariate  output  series  can  be  attributed  to  a  combination  of  additive
components  and  some  error  component   .  The  first  component  is  the

VAR(1) or the linear function of the first lag of the bivariate output series. The
contribution  of  the  input  series  is  summarized  by  the  second  component
which is a function of the contemporaneous effects of the input series. The
wide array of the m lagged values of p input series contained in the covariate
matrix  −  is of dimension,  ×  ∗ , where  <  ∗ .
The  proposed  bivariate  additive  model    illustrates  how  the
contemporaneous effects of input variables could simultaneously affect a pair
of response variable at time . This is commonly encountered on analyzing
economic indices that are usually of limited length and are affected by the
contemporaneous effects of numerous factors.
The  proposed  two-step estimation  procedure  for  the  bivariate  additive
model includes (1) identifying the sparse principal components using SPCA,
and  (2)  performing  a  nonparametric  regression  with  the  sieve  bootstrap
procedure.  The  sparse  principal  component  analysis  (SPCA)  procedure  by
Witten  et  al.  (2009)  will  be  used  for  dimension  reduction,  and  the  sieve
bootstrap is incorporated in the algorithm to produce consistent parameter
estimates.  The  SPCA  includes:  selection  of  the  tuning  parameter  for  PMD,
computation of single-factor PMD model, and computation of  factors of
PMD.  The  estimation  of  the  model  with  the  sparse  principal  components
(SPCs)  derived  from  the  first  step  is  detailed  in  the  second  step  of  the
procedure  which  includes:  (1)  Algorithm  2  (initial  parameter  estimation
3 | I S I   W S C   2 0 1 9`````` 9   10   11   12   13   14   15   16   17   18   19 