Approximation Methods

In this chapter, we deal with a very important problem that we will encounter in a wide variety of economic problems: approximation of functions. Such a problem commonly occurs when it is too costly either in terms of time or complexity to compute the true function or when this function is unknown and we just need to have a rough idea of its main properties. Usually the only thing that is required then is to be able to compute this function at one or a few points and formulate a guess for all other values. This leaves us with some choice concerning either the local or global character of the approximation and the level of accuracy we want to achieve. As we will see in different applications, choosing the method is often a matter of efficiency and ease of computing. Following Judd [1998], we will consider 3 types of approximation methods 1. Local approximation, which essentially exploits information on the value of the function in one point and its derivatives at the same point. The idea is then to obtain a (hopefully) good approximation of the function in a neighborhood of the benchmark point. 2. L p approximations, which actually find a nice function that is close to the function we want to evaluate in the sense of a L p norm. Ideally, we would need information on the whole function to find a good approximation , which is usually infeasible – or which would make the problem 1