Page 110 - Contributed Paper Session (CPS) - Volume 4

CPS2135 Sumonkanti Das et al.
′
= ( , . . . , 1 . . . ) are taken to be normally distributed as ∼
11
(0, ) where Σ = Φ = ⨁ Φ with Φ the covariance matrix for the
=1

transformed direct estimates observed in year t. Here covariances between
direct domain estimates are ignored and so Φ is assumed to be diagonal.

Equations (1) with the distribution of e define the likelihood function as
( |, Σ) = ( |, Σ), (2)
̂
̂

where = + ∑ () () , called the linear predictor. The vector β of

fixed effects is assigned a normal prior () = (0,100), which is very
weakly informative relative to the scales of the transformed direct estimates
and the covariates. The random effect vectors () for different α are assumed
to be independent, but the components within vector () are possibly
correlated to accommodate temporal or cross-sectional correlation. The
superscript α is suppressed in what follows for notational convenience. Each
random effects vector v is assumed to be distributed as
∼ (0, ⊗ ), (3)
where V and A are d × d and l × l covariance matrices, respectively, and
⊗ denotes the Kronecker product of A with V . The total length of v is
= , and these coefficients may be thought of as corresponding to d
effects allowed to vary over l levels of a factor variable.The covariance matrix
A describes the covariance structure between the levels of the factor variable,
and is assumed to be known. Instead of covariance matrices, precision
matrices QA = A −1 are actually used for computational efficiency. The
covariance matrix V is allowed to be parameterized as (i) fully parameterized
(unstructured) covariance matrix, (ii) a diagonal matrix with different elements
(diagonal), and (iii) a diagonal matrix with equal elements (scalar). A
generalisation of (3) to non-normal distributions of random effects are
considered assuming a Student-t distribution, horseshoe prior, or Laplace
distribution.
The models are fitted using MCMC sampling, in particular the Gibbs
sampler (Gelfand and Smith, 1990). See Boonstra and van den Brakel (2018)
for a specification of the full conditional distributions. Model selection is based
on WAIC (Watanabe, 2010) and DIC (Spiegelhalter et al., 2002) criteria. The
model, a longer run of 1000 burn-in plus 10000 iterations of which the draws
of every fifth iteration are stored, giving 3 ∗ 2000 = 6000 draws to compute
estimates and standard errors.

3. Result
Some additional covariates are constructed in order to model the MON
level break in 2004 (br mon taking values 1 for 2004-2009 years), the OViN
level break in 2010 (br ovin taking values 1 for 2010-2017 years), and the
influence of some lesser quality 2009 input estimates (as a dummy variable

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