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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 -

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