Econometric Modelling with Time Series

Specification, Estimation and Testing

Organisation of the Book

Part ONE of the book is an exposition of the basic maximum likelihood framework. To implement this approach, three conditions are required: the probability distribution of the stochastic process must be known and specified correctly, the parametric specifications of the moments of the distribution must be known and specified correctly and the likelihood must be tractable. The properties of maximum likelihood estimators are presented and three fundamental testing procedures - the Likelihood Ratio test, the Wald test and the Lagrange Multiplier test - are discussed in detail. There is also a comprehensive treatment of iterative algorithms to compute maximum likelihood estimators when no analytical expressions are available.

Part TWO is the usual regression framework taught in standard econometric courses but presented within the maximum likelihood framework. Both nonlinear regression models and non-spherical models exhibiting either autocorrelation or heteroskedasticity or both, are presented. A further advantage of the maximum likelihood strategy is that it provides a mechanism for deriving new estimators and new test statistics, which are designed specifically for non-standard problems.

Part THREE provides a coherent treatment of a number of alternative estimation procedures which are applicable when the conditions to implement maximum likelihood estimation are not satisfied. For the case where the probability distribution is incorrectly specified, quasi-maximum likelihood is appropriate. If the joint probability distribution of the data is treated as unknown, then a generalised method of moments estimator is adopted. This estimator has the advantage of circumventing the need to specify the distribution and hence avoids any potential misspecification from an incorrect choice of the distribution. An even less restrictive approach is not to specify either the distribution or the parametric form of the moments of the distribution and use nonparametric procedures to model either the distribution of variables or the relationships between variables. Simulation estimation methods are used for models where the likelihood is intractable arising, for example, from the presence of latent variables. Indirect inference, efficient methods of moments and simulated methods of moments are presented and compared.

Part FOUR examines stationary time series models with a special emphasis on using maximum likelihood methods to estimate and test these models. Both single equation models, including the autoregressive moving average class of models, and multiple equation models, including vector autoregressions and structural vector autoregressions, are dealt with in detail. Also discussed are linear factor models where the factors are treated as latent. The presence of the latent factor means that the full likelihood is generally not tractable. However, if the models are specified in terms of the normal distribution with moments based on linear parametric representations, a Kalman filter is used to rewrite the likelihood in terms of the observable variables, making estimation and testing by maximum likelihood feasible.

Part FIVE focusses on nonstationary time series models and in particular tests for unit roots and cointegration. Some important asymptotic results for nonstationary time series are presented followed by a comprehensive discussion of testing for unit roots. Cointegration is tackled from the perspective that the well-known Johansen estimator may be usefully interpreted as a maximum likelihood estimator based on the assumption of a normal distribution applied to a system of equations that is subject to a set of cross-equation restrictions arising from the assumption of common long-run relationships. Further, the trace and maximum eigenvalue tests of cointegration are shown to be likelihood ratio tests.

Part SIX is concerned with nonlinear time series models. Models that are nonlinear in mean include the threshold class of model, bilinear models and also artificial neural network modelling, which, contrary to many existing treatments, is again addressed from the econometric perspective of estimation and testing based on maximum likelihood methods. Nonlinearities in variance are dealt with in terms of the GARCH class of models. The final chapter focusses on models that deal with discrete or truncated time series data.