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REVIEW 221with the mean of t 1 estimated as 1973.3 and of t 2 as 1984.1. The prior ranges (withint ˆ 6, : : 116) for these two threshold parameters t 1 and t 2 are U(29, 61) and U(62, 98).These ranges are separated for identifiability and chosen by trial and error (Bauwens etal., 1999, p. 250).Estimates with this model may be sensitive to prior specifications on the single ordouble break points, t 1 and t 2 . Thus, uniform priors over the full range of times (6±116)may well give different estimates to priors restricted to an interior subinterval (e.g. 20±100). Similarly, a gamma prior such as G(0.6, 0.01) with average 60, approximately halfway through the periods, but with large variance ± and with sampling constrained to therange (6, 116)±might be used, combined with a constraint t 2 > t 1 . This may lead todifferent results than a uniform prior.There are also possible identification and convergence problems entailed in the nonlineareffects of r and n in Equation (5.56), when n is a free parameter. Here r is allowedto be outside the interval [1, 1]. One way to deal with the identifiability problem is tointroduce a conditional prior for n given r, or vice versa (see Bauwens et al., 1999,p.142), and so r is taken to be a linear function of n, namely a 1 a 2 n. Then N(0, 1)priors are adopted on a 1 and a 2 , and all regression coefficients with the exception of n,which is assigned an N(1, 1) prior, are constrained to positive values.We first fit Equation (5.56) with a single break point (i.e. a permanent shift in thepropensity to consume), n a free parameter, and a G(0.6, 0.01) prior on the breakpointt 1 . Convergence in all parameters in a three chain run occurs after 15 000 iterations andfrom iterations 15 000±20 000, a 95% credible interval for r of {1.02, 1.19} is obtained,and a pseudo-marginal likelihood of 390. The density for n is concentrated below unity,with 95% interval {0.86, 0.95}. The density for t 1 is negatively skewed and has someminor modes; however, there is a major mode at around t ˆ 85 to t ˆ 90, with theposterior median at 87 (i.e. 1984.3).To fit Equation (5.56) with n still a free parameter, and two breakpoints (i.e. atemporary shift in the propensity to consume), the intervals (29, 61) and (62, 98) ofBauwens et al. are used in conjunction with gamma priors (Model C in Program 5.17).Taking wider intervals within which sampled values may lie, such as (7, 61) for t 1 ,causes convergence problems. Even with the same intervals for the breakpoints asadopted by Bauwens et al., convergence on n is slow. The Gelman±Rubin scale reductionfactor on n remains at around 1.2 after iteration 8500 in a three chain run, and the95% interval from 5000 iterations thereafter is {0.88, 1.02}, including the equilibriumvalue of 1. On this basis the same model (5.56), but with n set to 1, may be fitted (seeexercises).It may be noted that an alternative methodology uses a smooth rather than abrupttransition function, such asD t ˆ 1=[1 exp ( w(t t 1 )(t t 2 ))]Other variations might include a stationarity constraint with jrj < 1.5.8 REVIEWBayesian time series analysis offers flexibility in several areas, and is now a major themein new time series developments. Among major modelling areas illustrating the benefitsof a Bayesian approach may be mentioned: