Page 16 - Contributed Paper Session (CPS) - Volume 7

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CPS2014 Ma. S.B.P. et al.
This yields the residual matrix = − −1 − that will be used for the
sieve bootstrap procedure in the succeeding algorithm.
Algorithm 3: Sieve Bootstrap on the Bivariate Residual Matrix
The Gram-Schmidt process is the generation of a series of quantities by
means of scalar products of vectors. Winch (1996) proved that these series of
quantities are identical with those that arise in the solution of the normal
equations by compact elimination methods. The Gram-Schmidt
̂
orthogonalization is applied on the covariance matrix Σ of the residual matrix
and the orthogonal transformation is derived.
1. From the residual matrix = − −1 − in Algorithm 2, derive the
̂
orthogonal transformation of the covariance matrix Σ of by the
Gram-Schmidt orthogonalization [refer to Appendix]. is the square
root matrix of Σ. That is, Σ = .
̂
′
̂
2. Generate a bivariate matrix of length such that = [ 1 ], where
2
, ~ (0,1).
1
2
3. Resample rows of for times ( ≥ 200) each with length . Multiply
the orthogonal transformation to each of the resampled , with
̂
( () ) = ( () ) = = = Σ, = 1, … , .
′
′
′
4. From each of (), recreate time series = ̂ + + () ,
() −1 ()
for = 1, … , , where is the initial estimate of and is the estimate
of the smooth functions of SPC obtained in Step 1.
5. Estimate the parameters until convergence on each recreated time
series as in Algorithm 2.
6. Obtain ̂ (1) , ̂ (2) , … , ̂ () on each of the recreated time series
(1) , (2) , … , () .
( )
7. Average of , = 1, … , is the final estimate of .
Lagged values up to of each of the generated , = 1, … , , is
ℎ
,
included in the covariate matrix − , = 1, … , and = 1, … , . That is,
Each of the 1, , 2, , … , , baseline input time series has the same mean of
20 and has the same variance of 4. The baseline input series is simulated by
the AR(1) process:
(1 − 0.5)( − 20) = , where ~(0,1). [3]
The bivariate time series data is be generated using its immediate past value
−1 with the output autocorrelation matrix , the function of the covariate
2
matrix − , and with the error matrix ~ (0, Σ ), where Σ = [ 11 12 ]
2
12 22 2
and 12 ≠ 0. A burn-in period of 1000 is considered for the initialization of
values.
To test the sensitivity of the model with the misspecification error, the
variance of the error matrix is made 3 or 6 times larger. To achieve robust
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