Stata 15 help for mf_ghk

[M-5] ghk() -- Geweke-Hajivassiliou-Keane (GHK) multivariate normal simulator

Syntax

S = ghk_init(real scalar npts)

(varies) ghk_init_method(S [, string scalar method])

(varies) ghk_init_start(S [, real scalar start])

(varies) ghk_init_pivot(S [, real scalar pivot])

(varies) ghk_init_antithetics(S [, real scalar anti])

real scalar ghk_query_npts(S)

real scalar ghk(S, real vector x, V)

real scalar ghk(S, real vector x, V, real rowvector dfdx, dfdv)

where S, if declared, should be declared

transmorphic S

and where method, optionally specified in ghk_init_method(), is

method Description ------------------------------------------------------------ "halton" Halton sequences "hammersley" Hammersley's variation of the Halton set "random" pseudorandom uniforms ------------------------------------------------------------

Description

The ghk*() set of functions provide a Geweke-Hajivassiliou-Keane (GHK) multivariate normal simulator.

S = ghk_init(npts) initializes the simulator with the desired number of simulation points and returns a transmorphic object S, which is a handle that should be used in subsequent calls to the other ghk*() functions. Calls to ghk_init_method(S, method), ghk_init_start(S, start), ghk_init_pivot(S, pivot), and ghk_init_antithetics(S, anti) prior to calling ghk(S, ...) allow you to modify the simulation algorithm through the object S.

ghk(S, x, V) returns a real scalar containing the simulated value of the multivariate normal (MVN) distribution with variance-covariance V at the point x. First, code S = ghk_init(npts) and then use ghk(S, ...) to obtain the simulated value based on npts simulation points.

ghk(S, x, V, dfdx, dfdv) does the same thing but also returns the first-order derivatives of the simulated probability with respect to x in dfdx and the simulated probability derivatives with respect to vech(V) in dfdv. See vech() in [M-5] vec() for details of the half-vectorized operator.

The ghk_query_npts(S) function returns the number of simulation points, the same value given in the construction of the transmorphic object S.

Remarks

Halton and Hammersley point sets are composed of deterministic sequences on [0,1] and, for sets of dimension less than 10, generally have better coverage than that of the uniform pseudorandom sequences.

Antithetic draws effectively double the number of points and reduce the variability of the simulated probability. For draw u, the antithetic draw is 1 - u. To use antithetic draws, call ghk_init_antithetic(S, 1) prior to executing ghk().

By default, ghk() will pivot the wider intervals of integration (and associated rows/columns of the covariance matrix) to the interior of the multivariate integration. This improves the accuracy of the quadrature estimate. When ghk() is used in a likelihood evaluator for [R] ml or [M-5] optimize(), discontinuities may result in the computation of numerical second-order derivatives using finite differencing (for the Newton-Raphson optimization technique) when few simulation points are used, resulting in a non-positive-definite Hessian. To turn off the interval pivoting, call ghk_init_pivot(S, 0) prior to executing ghk().

If you are using ghk() in a likelihood evaluator, be sure to use the same sequence with each call to the likelihood evaluator. For a uniform pseudorandom sequence, ghk_init_method("random"), set the seed of the uniform random-variate generator -- see runiform() in [M-5] runiform() -- to the same value with each call to the likelihood evaluator.

If you are using the Halton or Hammersley point sets, you will want to keep the sequences going with each call to ghk() within one likelihood evaluation. This can be done in one expression executed after each call to ghk(): ghk_init_start(S, ghk_init_start(S)+ghk_query_npts(S)). With each call to the likelihood evaluator, you will need to reset the starting index to 1. This last point assumes that the transmorphic object S is not re-created on each call to the likelihood evaluator.

Unlike ghkfast_init() (see [M-5] ghkfast()), the transmorphic object S created by ghk_init() is inexpensive to create, so it is possible to re-create it with each call to your likelihood evaluator instead of storing it as external global and reusing the object with each likelihood evaluation. Alternatively, the initialization function for optimize(), optimize_init_arguments(), can be used.

Conformability

All initialization functions have 1 x 1 inputs and have 1 x 1 or void outputs except

ghk_init(npts) input: npts: 1 x 1 output: S: transmorphic

ghk_query_npts(S) input: S: transmorphic output: result: 1 x 1

ghk(S, x, V): input: S: transmorphic x: 1 x m or m x 1 V: m x m (symmetric, positive definite) output: result: 1 x 1

ghk(S, x, V, dfdx, dfdv): input: S: transmorphic x: 1 x m or m x 1 V: m x m (symmetric, positive definite) output: result: 1 x 1 dfdx: 1 x m dfdv: 1 x m(m+1)/2

Diagnostics

The maximum dimension, m, is 20.

V must be symmetric and positive definite. ghk() will return a missing value when V is not positive definite. When ghk() is used in an ml (or optimize()) likelihood evaluator, return a missing likelihood to ml and let ml take the appropriate action (that is, step halving).

Source code

ghk.mata


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