In specifying a regression equation, we need to determine which regressors to include, but also how these regressors are measured. This gives rise to two levels of uncertainty: concepts (level 1) and measurements within each concept (level 2). Multiple measurements within each concept can lead to ambiguity in explanation, and conventional approaches may produce misleading estimates due to multicollinearity and dimensionality problem. In this paper we propose a hierarchical weighted least squares (HWALS) method to address the problems caused by multiple measurements.The proposed estimates can fully reflect three sources of uncertainty, uncertainty represented by the error term, uncertainty about which concepts to include, and uncertainty about which measurements to include in each concept.
We examine the effects of different growth theories taking into account the measurement problem in the growth regression, and our empirical results provide new insights. We find that estimates produced by HWALS provide more intuitive and robust explanations compared with existing methods. Besides, in contrast to the current literature we find that education and government intervention are not robust, because some of the measurements in these groups have poor explanatory power in the growth regression. We also consider approximation techniques when the number of variables is large or when computing time is limited. We propose and compare four approximation methods, and we show that each methods have their own advantages in different situations. Also provided are extensive sensitivity analysis with respect to prior specification and grouping.