In many spatial and spatial-temporal models, and more generally in models with complex dependencies, it may be too difficult to carry out full maximum-likelihood (ML) analysis. Remedies include the use of pseudo-likelihood (PL) and quasi-likelihood (QL) (also called the composite likelihood). The present paper studies the ML, PL and QL methods for general Markov chain models, partly motivated by the desire to understand the precise behaviour of the PL and QL methods in settings where this can be analysed. We present limiting normality results and compare performances in different settings. For Markov chain models, the PL and QL methods can be seen as maximum penalized likelihood methods. We find that QL is typically preferable to PL, and that it loses very little to ML, while sometimes earning in model robustness. It has also appeal and potential as a modelling tool. Our methods are illustrated for consonant-vowel transitions in poetry and for analysis of DNA sequence evolution-type models. Copyright (c) 2007 Board of the Foundation of the Scandinavian Journal of Statistics..
