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 as follows:
where for . For the definition of , , , and , see the section Fuller and Battese’s Method.
The covariance matrix, denoted by V, can be written in the form
where , and, for k =1,, m, 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 (i =1,, n) are associated with a linear model E(y )=X with covariance matrix where (i =1, , n) 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).