Estimation of a Linear Model in Terms of Intra-Class Correlations of the Residual Error and the Regressors ()
ABSTRACT
Objectives: The objective is to analyze the
interaction of the correlation structure and values
of the regressor variables in the estimation of a linear model when there is a
constant, possibly negative, intra-class correlation of residual errors and the
group sizes are equal. Specifically: 1) How does the variance of the
generalized least squares (GLS) estimator (GLSE) depend on the regressor values?
2) What is the bias in estimated
variances when ordinary least squares (OLS) estimator is used? 3) In what cases
are OLS and GLS equivalent. 4) How can the best linear unbiased estimator (BLUE) be constructed
when the covariance matrix is singular? The purpose is to make general matrix
results understandable. Results: The effects of the regressor values can
be expressed in terms of the intra-class correlations of the regressors. If the
intra-class correlation of residuals is large, then it is beneficial to have
small intra-class correlations of the regressors, and vice versa. The algebraic
presentation of GLS shows how the GLSE gives different weight to the
between-group effects and the within-group effects, in what cases OLSE is equal
to GLSE, and how BLUE can be constructed when the residual covariance matrix is
singular. Different situations arise when the intra-class correlations of the
regressors get their extreme values or intermediate values. The derivations
lead to BLUE combining OLS and GLS weighting in an estimator, which can be obtained
also using general matrix theory. It is indicated how the analysis can be
generalized to non-equal group sizes. The analysis gives insight to models
where between-group effects and within-group
effects are used as separate regressors.
Share and Cite:
Lappi, J. (2022) Estimation of a Linear Model in Terms of Intra-Class Correlations of the Residual Error and the Regressors.
Open Journal of Statistics,
12, 188-199. doi:
10.4236/ojs.2022.122013.
Cited by
No relevant information.