# Handbook of financial econometrics. Volume 2, Applications by Yacine Ait-Sahalia, Lars Peter Hansen

By Yacine Ait-Sahalia, Lars Peter Hansen

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Additional info for Handbook of financial econometrics. Volume 2, Applications

Example text

Consider the following algorithm. Given two initial values, (0) and X (0) , draw X (1) ∼ p X | (0) , Y and then (1) ∼ p |X (1) , Y . Continuing in this fashion, the algorithm generates a sequence of random variables, G X (g ) , (g ) g=1 . , but instead forms a Markov Chain with attractive properties: under a number of metrics and mild conditions, the distribution of the chain converges to p( , X |Y ), the target distribution. The key to MCMC is that it is typically easier to characterize the complete conditional distributions, p( |X , Y ) and p(X | , Y ), then to directly analyze the higher-dimensional joint distribution, p( , X |Y ).

In general, the transition densities are not known in closed form and our MCMC approach relies on a time-discretization and data augmentation. Given the state variables, arbitrage and equilibrium arguments provide the prices of other assets. We assume there are two types of prices. The ﬁrst, denoted by a vector St are the prices whose dynamics we model. Common examples include equity prices, equity index values, or exchange rates. The second case are derivatives such as option or bond prices, which can be viewed as derivatives on the short rate.

The Griddy Gibbs Sampler The Griddy Gibbs sampler is an approximation that can be applied to approximate the conditional distribution by a discrete set of points. Suppose that is continuously distributed and univariate and that p( |X , Y ) can be evaluated on a point-by-point basis, but that the distribution p( |X , Y ) is nonstandard and direct draws are not possible. The Griddy Gibbs sample approximates the continuously distributed with a discrete mass N of N -points, j j=1 . 1) j=1 Step 2 : Approximate the inverse CDF of p( |X , Y ); Step 3 : Generate a uniform on [0, 1] and invert the approximate CDF.