Refined Truncation

Main idea

The unifrom truncation method (presented here) is simple to implement, but it has the drawback of considering many histories, some of them being very unlikely to be experienced by agents. By the law of large number, these histories concern a very small number of agents.

The main idea of the refined truncation is to consider different truncation lengths for different histories. Histories more likely to be experienced (i.e., with a bigger size) can be “refined”, which means that they can be substituted by a set of histories with higher truncation lengths.

For instance, a truncated history \(y_{1},y_{1}\) \(N=2\) can be refined into \(k\) histories of lenght \(N=3\) : \({(y_{1},y_{1},y_{1})}\),…,\({(y_{k},y_{1},y_{1})}\), where \(k\) is the number of idiosyncratic states, whereas the other histories are unchanged.

A benefit of this construction is that the number of histories is a linear function of the maximum truncation length, instead of an exponential function.

Construction of the refinment

The construction of the refinements is detailed here. The constraint of the construction is that the set of refined histories must form a well-defined partition of the set of idiosyncratic histories in each period.

Algorithm

The algorithm is the same as for the uniform truncation, the \(\xi s\) being now constructed for each refined history.