Comparing climate time series – Part 6: Testing equality of autoregressive parameters without assuming equality of noise variances

Abstract. A critical question in climate science is whether climate model simulations are statistically consistent with observations. If simulations and observations are treated as realizations of Vector Autoregressive (VAR) models, then deciding that simulations and observations came from the same process is equivalent to deciding that the parameters of the respective VAR models are equal. This framework has been developed in parts 1–5 of this series of papers, including extensions to account for annual cycles and radiative forcing. However, the associated tests have been derived under the restriction of equal noise covariances. Previous studies have only allowed unequal noise variances in univariate settings. This paper presents a general test of parameter equality that applies to multivariate models, incorporates external forcing, and does not assume equal noise covariances. Monte Carlo experiments indicate that the test statistic is well approximated by a chi-squared distribution for large degrees of freedom, but that this distribution underestimates upper quantiles when the degrees of freedom are small. This bias can be partially compensated by adopting a more stringent significance level (e.g., using a 1 % level to achieve a nominal 5 % Type I error rate). Applying the method to monthly 2 m-temperature from an observational data set and climate model simulations aggregated over five regional domains reveals that most climate models tested differ significantly from the observational data set, both in their transfer coefficients for radiative forcing and in their AR coefficients, indicating differences in the representation of both internal and forced variability.