tailieunhanh - Train_Discrete Choice Methods with Simulation - Chapter 9

9 Drawing from Densities Introduction Simulation consists of drawing from a density, calculating a statistic for each draw, and averaging the results. In all cases, the researcher wants to ¯ calculate an average of the form t = t(ε) f (ε) dε, where t(·) is a statistic of interest and f (·) is a density. | P1 GEM IKJ P2 GEM IKJ QC GEM ABE T1 GEM August 20 2002 14 1 CharCount 0 CB495-09Drv CB495 Train KEY BOARDED 9 Drawing from Densities Introduction Simulation consists of drawing from a density calculating a statistic for each draw and averaging the results. In all cases the researcher wants to calculate an average of the form t f t s f s ds where t is a statistic of interest and f is a density. To approximate this average through simulation the researcher must be able to take draws from the density f . For some densities this task is simple. However in many situations it is not immediately clear how to draw from the relevant density. Furthermore even with simple densities there may be ways of taking draws that provide a better approximation to the integral than a sequence of purely random draws. We explore these issues in this chapter. In the first sections we describe the most prominent methods that have been developed for taking purely random draws from various kinds of densities. These methods are presented in a progressive sequence starting with simple procedures that work with a few convenient densities and moving to ever more complex methods that work with less convenient densities. The discussion culminates with the Metropolis-Hastings algorithm which can be used with practically any density. The chapter then turns to the question of whether and how a sequence of draws can be taken that provides a better approximation to the relevant integral than a purely random sequence. We discuss antithetics systematic sampling and Halton sequences and show the value that these types of draws provide in estimation of model parameters. Random Draws . Standard Normal and Uniform If the researcher wants to take a draw from a standard normal density that is a normal with zero mean and unit variance or a standard 208 P1 GEM IKJ P2 GEM IKJ QC GEM ABE T1 GEM August 20 2002 14 1 CharCount 0 CB495-09Drv CB495 Train KEY BOARDED Drawing from Densities 209 uniform .