PANEL Procedure
Da Silva Method for Moving Average Models (DASILVA Option)
The Da Silva method assumes that the observed value of the dependent variable at the tth time point on the ith cross-sectional unit can be expressed as
where
is a vector of explanatory variables for the tth time point and ith cross-sectional unit 
is the vector of parameters 
is a time-invariant, cross-sectional unit effect 
is a cross-sectionally invariant time effect 
is a residual effect unaccounted for by the explanatory variables and the specific time and cross-sectional unit effects
Since the observations are arranged first by cross sections, then by time periods within cross sections, these equations can be written in matrix notation as
where
Here 1 
is an 
vector with all elements equal to 1, and 
denotes the Kronecker product.
The following conditions are assumed:
is a sequence of nonstochastic, known 
vectors in 
whose elements are uniformly bounded in . The matrix X has a full column rank p. 
is a 
constant vector of unknown parameters. a is a vector of uncorrelated random variables such that
and 
, 
. b is a vector of uncorrelated random variables such that
and 
where 
and 
. -
is a sample of a realization of a finite moving-average time series of order 
for each i ; hence, 
where
are unknown constants such that 
and 
, and 
is a white noise process for each i—that is, a sequence of uncorrelated random variables with 
, and 
. for 
are mutually uncorrelated. The sets of random variables
, 
, and 
for are mutually uncorrelated. The random terms have normal distributions
and 
for 
and 
.
If assumptions 1–6 are satisfied, then
and
where 
is a 
matrix with elements 
,
where 
for 
. For the definition of 
, 
, 
, and 
, see the section Fuller-Battese Method.
The covariance matrix, denoted by V, can be written in the form
where 
, and, for , 
is a band matrix whose kth off-diagonal elements are 1’s and all other elements are 0’s.
Thus, the covariance matrix of the vector of observations y has the form
where
The estimator of is a two-step GLS-type estimator—that is, GLS with the unknown covariance matrix replaced by a suitable estimator of V. It is obtained by substituting Seely estimates for the scalar multiples 
.
Seely (1969) presents a general theory of unbiased estimation when the choice of estimators is restricted to finite dimensional vector spaces, with a special emphasis on quadratic estimation of functions of the form 
.
The parameters 
(
) are associated with a linear model 
with covariance matrix 
where 
(
) are real symmetric matrices. The method is also discussed by Seely (1970b, 1970a); Seely and Zyskind (1971). Seely and Soong (1971) consider the MINQUE principle, using an approach along the lines of Seely (1969).