Handbook of Economic Forecasting part 7. Research on forecasting methods has made important progress over recent years and these developments are brought together in the Handbook of Economic Forecasting. The handbook covers developments in how forecasts are constructed based on multivariate time-series models, dynamic factor models, nonlinear models and combination methods. The handbook also includes chapters on forecast evaluation, including evaluation of point forecasts and probability forecasts and contains chapters on survey forecasts and volatility forecasts. Areas of applications of forecasts covered in the handbook include economics, finance and marketing | 34 J. Geweke and C. Whiteman There is a simple two-step argument that motivates the convergence of the sequence 0 m generated by the Metropolis-Hastings algorithm to the distribution of interest. This approach is due to Chib and Greenberg 1995 . First note that if a transition probability density function p 0 m 0 m-1 T satisfies the reversibility condition p 0 m-1 I p 0 m 0 m-1 T p 0 m I p 0 m-1 0 m T with respect to p 0 I then f p 0 m-1 I p 0 m 0 m-1 T d0 m-1 Ja . i p 0 m I p 0 m-1 0 m T d0 m-1 p 0 m I p 0 m-1 0 m T d0 m-1 p 0 m I . 41 Expression 41 indicates that if 0 m-1 p 0 I then the same is true of 0 m . The density p 0 I is an invariant density of the Markov chain with transition density p 0 m 0 m-1 T . The second step in this argument is to consider the implications of the requirement that the Metropolis-Hastings transition density p 0 m 0 m-1 H be reversible with respect to p 0 I p 0 m-1 I p 0 m 0 m-1 H p 0 m I p 0 m-1 0 m H . For 0 m-1 0 m the requirement holds trivially. For 0 m-1 0 m it implies that p 0 m-1 I p 0 0 m-1 H a 0 0 m-1 H p 0 I p 0 m-1 0 H a 0 m-1 0 H . 42 Suppose without loss of generality that p 0 m-1 I p 0 0 m-1 H p 0 I p 0 m-1 0 H . If a 0 m-1 0 H 1 and ai0 0 m-1 H p 0 p 0 m-1 0 H a 0 H p 0 m-1 I p 0 0 m-1 H then 42 is satisfied. . Metropolis within Gibbs Different MCMC methods can be combined in a variety of rich and interesting ways that have been important in solving many practical problems in Bayesian inference. One of the most important in econometric modelling has been the Metropolis within Gibbs algorithm. Suppose that in attempting to implement a Gibbs sampling algorithm Ch. 1 Bayesian Forecasting 35 a conditional density p 0 b 0 a a b is intractable. The density is not of any known form and efficient acceptance sampling algorithms are not at hand. This occurs in the stochastic volatility example for the volatilities h1 . hT. This problem can be addressed by applying the Metropolis-Hastings algorithm in block b of the Gibbs .
