Estimated parameters of the MSIH
In LY317615 turning point prediction exercise we follow Canova and Ciccarelli (2004) and use the full predictive densities. More specifically we set K = 1 as in Canova and Ciccarelli (2004) and calculate the expected value of Dt(K). We use the MCMC approximation of the predictive densities p(yt+1 y1:t, Mj), j = AR, MS − AR, to evaluate the following downward turn probabilities
We evaluate turning point forecasting ability of the different models by the concordance statistics given byequation(20)ηj,t+1CS=∑s=1t+1−K(zj,szR,s)−(1−zj,s)(1−zR,s)Although the concordance statistics could be used to compute BMA weights similarly to Eq. (16) and to combine the predictive densities, we follow an alternative route and use it to combine the phase indicator from the different models. The phase indicator variable that results from the combination must be a binary variable. Therefore, we propose combining the phased indicators by using weights that take value 0 or 1. In fact, for the concordance statistics, we adopt a model selection approach, which can be viewed as a very special case of model averaging. The model with the highest concordance with the reference cycle has a weight of 1, and the other models have null weights. In formula we haveequation(21)wj,t+1CS=I kt* (j)where kt*=argmax ηj,tCS,j=AR,MS−AR .