This repository provides a julia module for estimation via Indirect Inference and Simulated Method of Moments. The module can be found in src/SMMIndirectInference.jl, a short tutorial can be found in indirect_inference_tutorial.jl, and some additional examples can be found in indirect_inference_examples.jl.
NOTE: If you want to use structs, etc. please use a wrapper function. See the provided examples for details.
TO DO:
- Allow weighting matrix to depend upon b or β
- Add in MCMC for bayesian estimation
- Standard errors for the case when the binding function is known
- Code cleanup
########################################### ########################################### Implements Indirect Inference and SMM
Indirect Inference Estimation Usage:
- indirect_inference(Y0=Y0, X0=X0, true_model=y_true, aux_estimation=est_aux, kwargs)
Required key word arguments:
- Y0 is the outcome variable from the data
- X0 is the list of covariates from the data
- true_model is a generic function Y=f(β, X) describing the structural model
- aux_estimation is a generic function b_hat = g(Y, X) describing the auxiliary model and estimation routine
Semi-Optional key word arguments:
- search ∈ {"NL", "grid"}, Default is "NL"
- β_init: initial value for the "NL" estimation routine (required for search="NL")
- β_grid: grid of βs for the "grid" search estimation routine (required for search="grid")
- J: number of times to simulate from the true model
- NLoptOptions: Options passed to the nonlinear optimizer NLopt (see NLopt_Options() below for details)
- W: the weighting matrix (I may change this later to get it from aux_estimation) Note that the
-
optimal weighting matrix depends upon the binding function in general. When the binding -
function is not known, results are not expected to be efficient.
-
- gradient: Must be =true if using search="NL" and a gradient based search algorithm.
Output:
- array of the structural parameter estimates
Examples: ii2 = indirect_inference(Y0=Y0, X0=X0, true_model=y_true, aux_estimation=est_aux, search="grid", β_grid=β_grid) ii2b = indirect_inference(Y0=Y0, X0=X0, true_model=y_true, aux_estimation=est_aux, search="NL", β_init=β_init, NLoptOptions=NLoptOptions) See indirect_inference_examples.jl for complete examples
########################################### ###########################################
Indirect Inference with Inference via the parameteric bootstrap Usage: iibootstrap(;β, X0, true_model, aux_estimation, J_bs=9, kwargs...)
Note: iibootstrap() should be called with the same arguments as indirect_inference()
Arguments: J_bs: the number of bootstrap samples β: the parameter used to simulate the model, usually the null hypothesis value β_0 if testing against a null or the estimates β_hat all other arguments are the same as indirect_inference()
Output: Array of size J_bs x K for the J_bs estimates
Examples: ii2 = indirect_inference(Y0=Y0, X0=X0, true_model=y_true, aux_estimation=est_aux, search="grid", β_grid=β_grid) ii2bs = iibootstrap(β=ii2, X0=X0, true_model=y_true, aux_estimation=est_aux, search="grid", β_grid=β_grid, J_bs=99)
Note:
- Not knowing the binding function that links the structural and auxiliary model parameters makes inference complicated.
- Hence, this implements a parameteric bootstrap.
- The trade-off is that this is computationally intensive.
########################################### ###########################################
Setting options for the NLopt Optimizer Usage:
- NLoptions = NLopt_options(lb=Nothing, ub=Nothing, cons_ineq=Nothing, alg=:LN_NELDERMEAD, xtol=1e-4)
Arguments:
- lb: array of lower bounds
- ub: array of upper bounds
- cons_ineq: array of generic functions representing inequality constraints
- alg: optimization algorithm (NelderMead is the default)
- xtol: x tolerance
Output:
- a struct of options to pass to the NLopt wrapper function
More details regarding NLopt can be found here: