Marginal likelihood methods in econometrics
2017-04-27T04:55:02Z (GMT) by
This thesis is concerned with the application of the method of marginal likelihood to certain problems in econometrics; namely those concerning estimation and inference in the setting established by the linear regression model. Particular attention is given to the problem of estimation and inference in the dynamic linear regression model, in which lags of the dependent variable are included as additional explanatory variables. The problem of selecting the most appropriate model for the random disturbances of the standard linear model is also investigated. <br><br>The literature survey of Chapter 2 draws attention to the extent to which the presence of so-called nuisance, or incidental, parameters in a given model can bias estimation procedures and render hypothesis tests unreliable in finite samples. Also described are some of the strategies devised for dealing with nuisance parameters; with attention focused on likelihood-based approaches; in particular those that factorize the model likelihood function so that one part depends only on the parameter or parameters of interest. We find that at least some of the wide variety of modified likelihoods are equivalent to the marginal or conditional likelihood. <br><br>Chapter 2 also reviews previous work on estimation and testing in the context of the dynamic linear model; paying particular attention to the estimation bias that inevitably arises in such models, and the problem of testing for disturbance autocorrelation. <br><br>Chapter 3 examines the marginal likelihood (MGL) for a particular subset of the parameters of a generalised regression model, in which both regressor and covariance matrix depend on one or more unknown parameters. An alternative derivation of the marginal likelihood in this model is presented; and the precise form of the likelihood for the general order dynamic model with iid normal disturbances, and the first order dynamic model with first order serial correlation in the disturbances, is derived.<br><br>In Chapter 4 we derive the marginal likelihood-based score, Hessian, and information matrix for parameters appearing in both the regressor and covariance matrices of the generalised regression model. We find that, in contrast to the linear model considered in earlier applications of marginal likelihood, no completely satisfactory closed form solution exists for the MGL-based information; for which we therefore suggest an alternative "quasi-expected" estimator.<br><br>Chapter 5 considers marginal likelihood-based Wald, LM, and LR tests of the parameters of the generalised regression model; and subsequently derives formulae for MGL-based tests of the coefficients of the lagged dependent variables in the first and second order dynamic models; and for MGL-based tests of first order disturbance autocorrelation in the first order dynamic model. The finite sample performance of these tests, applied to the coefficient of the lag of the dependent variable in the first order dynamic model with iid normal disturbances, is compared, via a simulation experiment, to that of conventional equivalents based on the profile, or concentrated, likelihood. In broad outline, we find that the MGL-based tests almost always outperform profile likelihood-based equivalents with respect to both size and power, particularly for positive values of the coefficient. We also find that, for this half of the parameter space, the best performer among all the tests considered was almost invariably the MGL-based LR test. <br><br>Chapter 6 investigates the finite sample properties of estimates of all the coefficients of the dynamic model, based on the MGL-based estimator of the coefficient of the lags of the dependent variable. The specific example considered is once again the first order dynamic model with iid normal disturbances; in which it is found that use of the MGL leads to greatly reduced estimation bias in the coefficients of both the lag of the dependent variable, and the exogenous regressors. <br><br>Finally, we return, in Chapter 7, to the standard linear regression model, and here investigate the worth of the marginal likelihood as the basis for an information criteria (IC)-style procedure for the problem of selecting between competing models for the covariance structure of the disturbances. The problem chosen for investigation in this case is that of choosing between a first order autoregression and a first order moving average. A simulation study shows conventional IC to be quite unsatisfactory in this situation, and the use of the MGL is strongly recommended. <br><br>Overall, although much of the comparative finite sample work is of necessity performed via simulation, and can therefore only consider a limited set of example models, the results obtained strongly support the use of the marginal likelihood in models in which either the covariance matrix, or the regressor matrix, depend on unknown parameters.