Stata 15 help for mf_ghkfast

[M-5] ghkfast() -- Geweke-Hajivassiliou-Keane (GHK) multivariate normal simulator using pregenerated points

Syntax

S = ghkfast_init(real scalar n, npts, dim, string scalar method)

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

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

real scalar ghkfast_query_n(S)

real scalar ghkfast_query_npts(S)

real scalar ghkfast_query_dim(S)

string scalar ghkfast_query_method(S)

string scalar ghkfast_query_rseed(S)

real matrix ghkfast_query_pointset_i(S, i)

real colvector ghkfast(S, real matrix X, V)

real colvector ghkfast(S, real matrix X, V, dfdx, dfdv)

real scalar ghkfast_i(S, real matrix X, V, i)

real scalar ghkfast_i(S, real matrix X, V, i, dfdx, dfdv)

where S, if it is declared, should be declared

transmorphic S

and where method specified in ghkfast_init() is

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

Description

Please see [M-5] ghk(). The routines documented here do the same thing, but ghkfast() can be faster at the expense of using more memory. First, code S = ghkfast_init(...) and then use ghkfast(S, ...) to obtain the simulated values. There is a time savings because the simulation points are generated once in ghkfast_init(), whereas for ghk() the points are generated on each call to ghk(). Also, ghkfast() can generate simulated probabilities from the generalized Halton sequence; see [M-5] halton().

ghkfast_init(n, npts, dim, method) computes the simulation points to be used by ghkfast(). Inputs n, npts, and dim are the number of observations, the number of repetitions for the simulation, and the maximum dimension of the multivariate normal (MVN) distribution, respectively. Input method specifies the type of points to generate and can be one of "halton", "hammersley", "random", or "ghalton".

ghkfast(S, X, V) returns an n x 1 real vector containing the simulated values of the MVN distribution with dim x dim variance-covariance matrix V at the points stored in the rows of the n x dim matrix X.

ghkfast(S, X, V, dfdx, dfdv) does the same thing as ghkfast(S, X, V) but also returns the first-order derivatives of the simulated probability with respect to the rows of 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_n(S), ghk_query_npts(S), ghk_query_dim(S), and ghk_query_method(S) functions extract the number of observations, number of simulation points, maximum dimension, and method of point-set generation that is specified in the construction of the transmorphic object S. Use ghk_query_rseed(S) to retrieve the uniform random-variate seed used to generate the "random" or "ghalton" point sets. The ghkfast_query_pointset_i(S, i) function will retrieve the ith point set used to simulate the MVN probability for the ith observation.

The ghkfast_i(S, X, V, i, ...) function computes the probability and derivatives for the ith observation, i = 1, ..., n.

Remarks

For problems where repetitive calls to the GHK algorithm are required, ghkfast() might be a preferred alternative to ghk(). Generating the points once at the outset of a program produces a speed increase. For problems with many observations or many simulation points per observation, ghkfast() will be faster than ghk() at the cost of requiring more memory.

If ghkfast() is used within a likelihood evaluator for ml or optimize(), you will need to store the transmorphic object S as an external global and reuse the object with each likelihood evaluation. Alternatively, the initialization function for optimize(), optimize_init_argument(), can be used.

Prior to calling ghkfast(), call ghkfast_init_npivot(S,1) to turn off the integration interval pivoting that takes place in ghkfast(). By default, ghkfast() pivots the wider intervals of integration (and associated rows/columns of the covariance matrix) to the interior of the multivariate integration to improve quadrature accuracy. This option may be useful when ghkfast() is used in a likelihood evaluator for [R] ml or [M-5] optimize() and few simulation points are used for each observation. Here the pivoting may cause discontinuities when computing numerical second-order derivatives using finite differencing (for the Newton-Raphson technique), resulting in a non-positive-definite Hessian.

Also the sequences "halton", "hammersley", and "random", ghkfast() will use the generalized Halton sequence, "ghalton". Generalized Halton sequences have the same uniform coverage (low discrepancy) as the Halton sequences with the addition of a pseudorandom uniform component. Therefore, "ghalton" sequences are like "random" sequences in that you should set the random-number seed before using them if you wish to replicate the same point set; see [M-5] runiform().

Conformability

All initialization functions have 1 x 1 inputs and have 1 x 1 or void outputs, and all query functions have the transmorphic input and 1 x 1 outputs except

ghkfast_init(n, npts, dim, method): input: n: 1 x 1 npts: 1 x 1 dim: 1 x 1 method: 1 x 1 output: result: transmorphic

ghkfast_query_pointset_i(S, i): input: S: transmorphic i: 1 x 1 output: result: npts x dim

ghkfast(S, X, V): input: S: transmorphic X: n x dim V: dim x dim (symmetric, positive definite) output: result: n x 1

ghkfast(S, X, V, dfdx, dfdv): input: S: transmorphic X: n x dim V: dim x dim (symmetric, positive definite) output: result: n x 1 dfdx: n x dim dfdv: n x dim(dim+1)/2

ghkfast_i(S, X, V, i, dfdx, dfdv): input: S: transmorphic X: n x dim or 1 x dim V: dim x dim (symmetric, positive definite) i: 1 x 1 (1 < i < n) output: result: n x 1 dfdx: 1 x dim dfdv: 1 x dim(dim+1)/2

Diagnostics

ghkfast_init(n, npts, dim, method) aborts with error if the dimension, dim, is greater than 20.

ghkfast(S, X, V, ...) and ghkfast_i(S, X, V, i, ...) require that V be symmetric and positive definite. If V is not positive definite, then the returned vector (scalar) is filled with missings.

Source code

ghkfast.mata


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