Small samples and collateral information: An application of the hyperparameter model

When time-series and cross-section data are available for estimating a linear relationship, the data may be pooled using a number of frameworks of varying degrees of restrictiveness. One possible framework which appears not to have been exploited in previous applied econometric work, but which may have some merit especially when the number of observations in the time-series or cross-section dimension is not very large, is the hyperparameter model of Lindley-Smith (1972). Under appropriate conditions the use of this framework enables one to obtain ‘improved’ Bayesian point estimates which yield predictions with smaller mean-square-error than those from ordinary least squares. The paper uses the hyperparameter framework to estimate a Cobb-Douglas production function using U.K. cross-sectional data. A cross-validation exercise is carried out to see how much of the potential of the Bayesian method is realised in practice.