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This paper develops a method for determining whether two vector time series originate from a common stochastic process. The stochastic process considered incorporates both serial correlations and multivariate annual cycles. Specifically, the process is modeled as a vector autoregressive model with periodic forcing, referred to as a VARX model (where X stands for exogenous variables). The hypothesis that two VARX models share the same parameters is tested using the likelihood ratio method. The resulting test can be further decomposed into a series of tests to assess whether disparities in the VARX models stem from differences in noise parameters, autoregressive parameters, or annual cycle parameters. A comprehensive procedure for compressing discrepancies between VARX models into a minimal number of components is developed based on discriminant analysis. Using this method, the realism of climate model simulations of monthly mean North Atlantic sea surface temperatures is assessed. As expected, different simulations from the same climate model cannot be distinguished stochastically. Similarly, observations from different periods cannot be distinguished. However, every climate model differs stochastically from observations. Furthermore, each climate model differs stochastically from every other model, except when they originate from the same center. In essence, each climate model possesses a distinct fingerprint that sets it apart stochastically from both observations and models developed by other research centers. The primary factor contributing to these differences is the difference in annual cycles. The difference in annual cycles is often dominated by a single component, which can be extracted and illustrated using discriminant analysis. [ABSTRACT FROM AUTHOR]