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``````CPS2014 Ma. S.B.P. et al.
a time series model as it produces consistent estimators of the variance of time
series statistics.
Nonparametric multivariate analysis simultaneously tracks the effects of
the array of explanatory variables on multiple variables of interest (response
variables) while letting the data speak for itself. Opsomer and Ruppert (1997)
came  up  with  a  bivariate  additive  model  and  fitted  it  by  local  polynomial
regression  and  showed  that  it  has  the  same  rate  of  convergence  as  its
univariate counterpart.
Friedman  and  Stuetzle  (1981)  suggested  the  use  of  an  additive  model
which  assumes  that  the  conditional  expectation  function  of  the  response
variable can be written as the sum of smooth terms of covariates. Buja et al.
(1989)  estimated  an  additive  nonparametric  regression  model  using  linear
smoothers  by  the  backfitting  algorithm  and  provided  proof  on  the
convergence of the method. The backfitting algorithm sequentially estimates
parameters of interest until convergence in an iterative manner. Hastie and
Tibshirani  (1986)  proposed  the  generalized  additive  model  (GAM)  which
replaces the sum of the linear covariates by a sum of smooth functions which
are  iteratively  estimated  by  the  local  scoring  algorithm.  Opsomer  (2000)
derived recursive asymptotic bias and variance expressions for the backfitting
estimators on local polynomial regression smoothers.
This paper focuses on estimating the short bivariate time series given the
contemporaneous  effects  of  predictors.  The  parameters  in  the  bivariate
additive model will be estimated using a backfitting framework.  The vector
autoregression  at  order  one  (VAR(1))  is  used  to  estimate  the  output
autocorrelation coefficient ρ. The Generalized Additive Model (GAM) is used
for regressing the sparse components with the bivariate output series . It is
of interest to characterize the underlying empirical distribution function of the
estimates. After achieving initial parameter estimates, the residuals is used for
the  sieve  bootstrap  procedure  to  give  the  final  parameter  estimates.  The
performance  of  the  model  is  evaluated  through  a  simulation  study  and
application  to  the  short  data  about  the  University  of  the  Philippines  on
teaching, research, and extension programs for the last two decades (1995 -
2015).

2.  Methodology
The postulated model [2] is compared to the (1) process defined as
=  −1  +  ,                  [1]

where    is  a  2  ×  2  output  autocorrelation  matrix  coefficient  of  the
immediate past  −1  of a given bivariate time series data  = ( ,  ) .   is

1

2
a two-dimensional white noise process with time invariant positive definite
covariance matrix (  ′) = Σ .

2 | I S I   W S C   2 0 1 9``````
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