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:

1. is a sequence of nonstochastic, known vectors in whose elements are uniformly bounded in . The matrix X has a full column rank p.

2. is a constant vector of unknown parameters.

3. a is a vector of uncorrelated random variables such that and , .

4. b is a vector of uncorrelated random variables such that and where and .

5. 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—that is, a sequence of uncorrelated random variables with , and .

6. The sets of random variables , , and for are mutually uncorrelated.

7. 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 (1970a,1970b) and Seely and Zyskind (1971). Seely and Soong (1971) consider the MINQUE principle, using an approach along the lines of Seely (1969).