Sur les modèles non-linéaires autorégressifs à transition lisse et le calcul de leurs prévisions

This master’s thesis focuses on the study of dependent data. Classical literature has been widely focused on models described as linear. These models find great use when applied to macroeconomics data that are measured with relatively long time span (months, years, etc). When the data are measured using a shorter time span, and when the data are plentiful, it is then possible to describe the underlying stochastic process using more sophisticated models, which allows for proper modelling of the data’s nonlinear characteristics. It is in that contemporary setting that this master’s thesis takes place. It focuses on smooth transition autoregressive models, introduced and popularized by many authors including Teräsvirta (1994). We focus our study on modelling and forecasting for these models. As they are characterized by multiple regimes and a specific transition between those regimes, the smooth transition autoregressive models may allow for a more adequate modelling of a stochastic process using time series models that show nonlinear components. Our goal is to compare these models with the linear autoregressive models, and to study if they allow for more accurate forecasts. Thus, a main focus of this thesis is the elaboration of a forecasting system for these models.  Although the use of smooth transition autoregressive models may seem enticing, applying the models to data brings its own share of complications, especially regarding parameters estimation. The estimation of the model sometimes has trouble converging due to a nonlinear component in the model, therefore requiring non-linear optimisation which is more complex. Of course, this is also true for the class of autoregressive moving average models (ARMA), which rely on the choice of autoregressive order p and moving average q. When q > 0, it is well-known that optimisation is nonlinear as well. However, empirical evidence suggests that numerical problems are less difficult than for smooth transition autoregressive models. The estimation of standard deviations of the parameters may also be difficult to obtain since the calculation of the variance-covariance matrix may be affected by computing issues.  Forecasting for time series also poses issues for nonlinear time series. Classic linear theory is not applicable due to the nonlinear component in the model, and therefore point forecasting for smooth transition autoregressive models requires the use of other methods that vary in complexity, some which cause a bias for forecasts of lead times strictly larger than one which increase drastically in difficulty for longer lead times. For prediction intervals, because the linear theory is not applicable, they must be obtained using resampling methods. In fact, it can be asserted that an original contribution from this thesis is an elaborate and detailed study of point forecasting and forecast intervals for the models.  In the first chapter, we introduce time series and classic linear models that are used to model time series. We elaborate nonlinear models and the univariate smooth transition autoregressive models, as well as the main characteristics and stationarity conditions for these models. In the second chapter, we develop estimation methods for smooth transition autoregressive models. The modelling process includes specification, estimation and evaluation of the model. Specification includes linearity tests for the data, which is required and justifies the use of nonlinear time series models. Estimation is done by nonlinear optimisation, and by finding its starting values using a grid search. In the third chapter, we present forecasting methods for classic linear autoregressive models and smooth transition autoregressive models. We elaborate the problems encountered when applying conventional theory to nonlinear models, and the methods to overcome these issues. We also define prediction intervals and how to obtain them. In the fourth chapter, we apply the theory of prior chapters to empirical simulations, with the goal of comparing linear autoregressive models to smooth transition autoregressive ones, focusing on modelling and both point and interval forecasting. We discuss our results. In the fifth chapter, we apply the theory to a time series representing daily returns of the SPDR S&P 500 stock market index (SPY ). We compare our results with those available in the literature, for both estimation of the smooth transition autoregressive models and point forecasting and prediction intervals performance. We then conclude the thesis.

Keyphrases: Time series, Smooth transition autoregressive models, Forecasting, Prediction intervals