and regard it as the key element of the following all prediction pooling methods.

Recently, statistical approaches to model combination for forecasting and decision making have been paid attentions, and one of them is Bayesian model averaging (BMA) proposed by Raftery et al. (1997) and Hoeting et al. (1999), in which a predictive distribution for future observation $ y_{t} $ is obtained by averaging a set of competing models and given as

$$ p^{B M A}\left(y_{t}\mid Y_{t-1}\right)=\sum_{i=1}^{m}p(y_{t}^{O};Y_{t}^{O},\mathcal{M}_{i})p\left(\mathcal{M}_{i}\mid Y_{t-1},S_{t-1},\varTheta\right),\mathrm{f o r}i=1,\cdots,m, $$

where $ p(\mathcal{M}_{i} \mid Y_{t-1}, S_{t-1}, \Theta) $ is a posterior model probability (or a posterior model weights). The key of the BMA is that the posterior model probability is derived from $ \frac{p^{\text{Prior}}(y_{t} \mid Y_{t-1}^{O}, \mathcal{M}_{\backslash})}{\sum_{i=1}^{m} p^{\text{Prior}}(y_{t} \mid Y_{t-1}^{O}, \mathcal{M}_{i})} $ using the marginal likelihood. However, this value makes one problem such that the posterior probability of one of competing models is often set as a extreme value such as almost 100% and those of others almost 0%. It indicates that forecasting of BMA is likely to be same as that of one model selected from the marginal likelihood, and that it is not preferable to use the marginal likelihood as the weighting coefficient of forecasting future observations. Instead, we will consider what model weights or functions should be used as the posterior model weights to improve performance of forecasting empirically in the optimal prediction pooling. The aim of this study is to examine various model choices and combinations in terms of the macro-finance term structure modeling using new model weighting criteria. To this end, we will conduct three proposed approaches of prediction pool which we explain as follows.

### 2.2 Static prediction pooling

Firstly, we discuss the static pooling based on a constant weighting as in Geweke and Amisano (2011). Given two prediction models $ M_{1} $ and $ M_{2} $ , the pool of the prediction density can be constructed as the convex combination

$$ p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}\right)=\lambda p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}_{1}\right)+\left(1-\lambda\right)p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}_{2}\right),\qquad0\leq\lambda\leq1, $$

with $ \mathcal{M} = (\mathcal{M}_{1}, \mathcal{M}_{2}) $ is the collection of models and $ p\left(y_{t}^{O}; Y_{t-1}^{O}, \mathcal{M}_{i}\right) = p\left(y_{t}^{O}; Y_{t-1}^{O}, \hat{\Theta}_{i}\right) $ is the prediction density with the posterior estimates $ \hat{\Theta}_{i} $ of the model $ M_{i} $ . The optimal prediction pooling is then obtained by maximizing the cumulative log prediction score, $ LPS^{SP} $ , as

$$ L P S^{S P}(\lambda)\equiv\sum_{t=1}^{T}\log\left[\lambda p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}_{1}\right)+\left(1-\lambda\right)p\left(y_{t}^{O};Y_{t-1}^{O},\mathcal{M}_{2}\right)\right] $$

by choosing $ \lambda^{*} = \arg\max LPS^{SP}(\lambda) $ . An important assumption as noted in Geweke and Amisano (2011) is that the two candidate prediction models have to be substantially different in terms of the functional form of their predictive densities (i.e., non-nested models). In our case, the Gaussian ATSM generates a prediction density of bond yield close to a normal