Random Sample Generation and Simulation of Probit Choice ...
Random Sample Generation and Simulation of Probit Choice Probabilities
Based on sections 9.1-9.2 and 5.6 of Kenneth Train's
Discrete Choice Methods with Simulation
Presented by Jason Blevins Applied Microeconometrics Reading Group
Duke University 21 June 2006
Anyone attempting to generate random numbers by deterministic means is, of course, living in a state of sin. John Von Neumann, 1951
Outline
Density simulation and sampling
Univariate Truncated univariate Multivariate Normal Accept-Reject Method for truncated densities Importance sampling Gibbs sampling The Metropolis-Hastings Algorithm
Simulation of Probit Choice Probabilities
Accept-Reject Simulator Smoothed AR Simulators GHK Simulator
1
Simulation in Econometrics
Goal: approximate a conditional expectation which lacks a closed form.
Statistic of interest: t@A, where $ F .
Want to approximate E t@A a t@Af @Ad.
Basic idea: calculate t@A for R draws of and take the average.
Unbiased: E
I
R
@ A RraI t r a E t@A
Consistent:
I
R
@ A RraI t r 3p E t@A
This is straightforward if we can generate
draws
from
F.
In discrete choice models we want to simulate the probability that agent n
chooses alternative i.
a C $ @ A Utility: Un;j Vn;j n;j with n F n .
a f j C C V Ta g Bn;i
. n Vn;i n;i > Vn;j n;j j i
a @ A @ A Pn;i
1 . Bn;i n f n d n
2
Random Number Generators
True Random Number Generators:
Collect entropy from system (keyboard, mouse, hard disk, ) etc. Unix: /dev/random, /dev/urandom
Pseudo-Random Number Generators:
Linear Congruential Generators (xnCI a C axn b mod c): fast but
predictable, good for Monte Carlo Nonlinear: more dicult to determine parameters, used in cryptography
Desirable properties for Monte Carlo work:
Portability Long period Computational simplicity
DIEHARD Battery of Tests of Randomness, Marsaglia (1996)
3
Uniform and Standard Normal Generators
Canned:
Matlab: rand(), randn() Stata: uniform(), invnormal(uniform())
Known algorithms:
Box-Muller algorithm Marsaglia and Zaman (1994): mzran Numerical Recipes, Press et al. (2002): ran1, ran2, ran3, gasdev
4
Simulating Univariate Distributions
Direct vs. indirect methods. Transformation
Let u $ N @H; IA. Then v a C u $ N ; P and a $ w eCu Lognormal ; P . Inverse CDF transformation: Let u $ N @H; IA. If F @A is invertible, then a F I@uA $ F @A.
Only works for univariate distributions
5
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