Variance estimation in integrated assessment models and its importance for hypothesis testing

Variance in likelihood functions for multiple normally distributed data sets can be reliably estimated in integrated assessment models, and their values are important for accurate hypothesis tests. Commonly, assessment models are fitted to multiple types of observations by constructing a joint likelihood function that is then maximized. When a model contains no random effects and all random variables in the likelihood function represent errors in the prediction of measurements, then variances for each of the error distributions are estimable provided that no likelihood component has zero degrees of freedom. Theory for estimation of variances is reviewed. We show the relationship between concentrated likelihood based on the normal distribution and weighted least squares. Concentrated likelihood and weighted least squares are equivalent when the likelihood is made of normally distributed errors with constant variances, and the least squares weights are inversely proportional to the maximum likelihood estima...