Page 14 - Contributed Paper Session (CPS) - Volume 7

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, … , [2]

,
=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 [2] 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
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