Threshold Models in Non linear Time Series Analysis In the last two years or so I was most fortunate in being given opportunities of lecturing on a new methodology to a variety of audiences in Britain China Finland France and Spain. Despite my almost Confucian attitude of preferring talking (i.e. a transient record) to writing (i.e. a permanent record) the warm encouragement of friends has led to the ensuing notes. I am also only too conscious of the infancy of the methodology introduced in these notes. However it is my sincere hope that exposure to a wider audience will accelerate its maturity. Readers are assumed to be familiar with the basic theory of time series analysis. The book by Professor M.B. Priestley (1981) may be used as a general reference. Chapter One is addressed to the general question: why do we need non linear time series models After describing some significant advantages of linear models it singles out several major limitations of linearity. Of course the selection reflects my personal view on the subject which is only at its very beginning although there does seem to be a general agreement in the literature that time irreversibility and limit cycles are among the most obvious. One Introduction 1. Time Series Model Building 2. Stationarity 3. Linear Gaussian Models 4. Some Advantages and Some Limitations of Arma Models 5. What Next Two Some Basic Concepts 1. Orientation 2. Limit Cycles 3. Some Examples of Threshold Models 4. Time Delay 5. Discussion Three Threshold Models 1. A Canonical Form 2. Generality of Setar Models 3. Non Linear Difference Equations 4. Threshold Models and Discrete Time Non Linear Vibrations 5. Ergodicity 6. Stationary Distributions and Moments 7. Cyclical Structure and Multi Step Ahead Forecasting Four Identification 1. A General Principle 2. Estimation of Parameters 3. Sampling Properties 4. Diagnostics and Graphical Methods 5. Miscellanea Five Some Case Studies 1. Analysis of Some Ecological Data 2. Analysis of the Sunspot Numbers 3. Analysis of Some Riverflow Data 4. A Case Study with Laboratory Data 5. A Fuzzy Extension 6. Concluding Remarks Appedices References Author Index.