# gmmFit¶

## Purpose¶

Estimate parameters using generalized method of moments.

## Format¶

gOut = gmmFit(&fct, y[, ...[, gCtl]])
Parameters

&fct – Pointer to user specified moment equation function &fct. The function must have the parameter vector to be estimated as the first input and a data matrix as the second input. The data matrix y and all optional arguments are passed, untouched, directly to the moment function. The function fct should return the desired moments for the GMM objective function and should take the form:

m = fct(b, y, ...);

Parameters
• y (Nx1 matrix) – independent data vector.

• .. (Any) – Optional inputs. These arguments are passed untouched to the user-provided moment function by gmmFit().

• gCtl (struct) –

Optional argument, an instance of an gmmControl structure. The following members of gCtl* are referenced within the gmmFit() routine:

 gCtl.bStart column vector of parameter starting values. For all methods except continuous updating GMM default = 0.1. For continuous updating GMM default equals estimation from onestep GMM. Must be specified if data matrix syntax is used and gCtl.numParms* is not specified. For estimation stability it is highly recommended to speficy starting parameters. gCtl.method string, GMM method to be used. Default = "twostep". ”onestep” One-step GMM ”twostep” Two-step GMM ”iterative” Iterative GMM ”CU” Continuous updating GMM gCtl.vceType string, variance-covariance matrix type. Default = "robust". ”unadj” Unadjusted, non-robust SE. This option is only available if the formula string syntax is used. It assumes a moment function of the form $$m = f(Z, u)$$ or $$m = f(X, u)$$. The "unadj" vce is given by $$\sigma_{u}^2 (x'(z(z'z)^{-1}z)x)^{-1}$$. ”robust” Heteroscedastic robust SE. ”hac” Heteroscedastic-autocorrelation robust SE. gCtl.wType string, type of weight matrix used. Ignored for one-step case. Default = "robust". ”unadj” Unadjusted, non-robust SE. This option is only available if the formula string syntax is used. ”robust” Heteroscedastic robust SE. ”hac” Heteroscedastic-autocorrelation robust SE. gCtl.hacKernel string, type of kernel used for estimation of HAC robust weight matrix and/or variance-covariance matrix. Ignored if not using "hac" weight matrix and/or variance-covariance matrix. Bandwidth is determined using the Newey-West optimal lag length selection method. Default = "bartlett". ”bartlett” Bartlett kernel. ”parzen” Parzen kernel. ”quad” Quadraticspectral kernel. gCtl.gmmlags Scalar, Scalar, user specified lag truncation for HAC weight matrix and variance computations. gCtl.wInitMat data matrix, initial weight matrix to be used. If specified the matrix is used as initial weighting matrix and overrides specification of gCtl.wInit*. gCtl.wInit string or data matrix, type of initial weight matrix used. If data matrix, the specified matrix is used as initial weighting matrix. Default = "identity". ”identity” Identity matrix. ”unadj” Weight matrix $$\frac{1}{n}*inv(Z'Z)$$. Assumes moments are i.i.d. This option is only available if the formula string syntax is used. gCtl.gIter instance of gmmIterative structure. This structure houses the tolerances for convergence for iterative GMM. Ignored if iterative GMM is not specified. The members include: gCtl.gIter.maxIter scalar, maximum number of iterations. gCtl.gIter.parmTol scalar, tolerance level for convergence based on parameter estimates. Default = 1e-5. gCtl.gIter.wTol scalar, tolerance level for convergence based on weight matrix estimates. Default = 1e-5. gCtl.noconstant scalar, specified to indicate if constant is included in model. Only valid if data vector input method is used. Set to 1 to exclude constant from model. Constant is always first parameter in parameter vector. Default = 0 [constant included].For dataset and string formula method to remove constant from model specify "-1" as first dependent variable: e.g. : "y ~ -1 + X1 + X2" gCtl.varNames string array, dependent variable names. Only used for data vector input case. Default = "X1", "X2", ... gCtl.instNames string array, instrumental variable names. Only used for data vector input case. Default = "Z1", "Z2", ... gCtl.numParms scalar, number of parameters estimated in model. Must be specified if data matrix syntax is used and gCtl.bStart* is not specified. gCtl.A MxK matrix, linear equality constraint coefficients: gctl.A * p = gctl.B where p is a vector of the parameters. gCtl.B Mx1 vector, linear equality constraint constants: gctl.A * p = gctl.B where p is a vector of the parameters. gCtl.C MxK matrix, linear inequality constraint coefficients: gctl.C * p >= gctl.D where p is a vector of the parameters. gCtl.D Mx1 vector, linear inequality constraint constants: gctl.C * p >= gctl.D where p is a vector of the parameters. gCtl.bounds 1x2 or Kx2 matrix, bounds on parameters. If 1x2 all parameters have same bounds. Default = { -1e256 1e256 }.

Returns

gOut (struct) –

instance of gmmOut struct containing the following members:

 gOut.parEst column vector of final estimates. Constant, if included in model, is the first element. gOut.wFinal matrix, final weighting matrix. gOut.covPar matrix, estimated variance-covariance matrix. gOut.fct vector, mean value of the moment equations. gOut.hessian matrix, Hessian of mean of moment equation wrt parameters. gOut.gradient matrix, Gradient of mean of moment equation wrt parameters. gOut.numParms scalar, number of parameters estimated in model. gOut.numMoments scalar, number of moments. gOut.numObs scalar, number of observations. gOut.numInstruments scalar, number of instruments. gOut.JStat scalar, Hansen statistic of overidentification. gOut.df scalar, degrees of freedom.

## Examples¶

### Use data matrices¶

new;
rndseed 12576;

/*
** Simulate t distribution data
** degrees of freedom
*/
df = 10;

// Covariance matrix [columns are independent]
sigma = { 1 0,
0 1 };

// Number of observations>
num = 500;

// Generate data
y = rndMVt(num, sigma, df);

// Just use one of x's
yt = y[., 1];

struct gmmControl gctl;
gctl = gmmControlCreate();

/*
** Set starting values
** This or number of parameters must
** be specified if no x mats
*/
gctl.bStart = 7;

// Continuous estimation
struct gmmOut gOut1;
gOut1 = gmmFit(&meqn, yt, gctl);

/*
** User defined moment equation
** Use the y2 and y4 as moments
*/
proc (1) = meqn(b, yt);
local g1,g2;

g1 = yt.^2 - b/(b-2);
g2 = yt.^4 - (3*b^2)/((b-2)*(b-4));

retp(g1~g2);
endp;


## Remarks¶

The user defined moment equation function should be set up to take at least 2 inputs. The first input should always be the parameter vector and the second input should always be the dependent data vector. Additional optional arguments may be included. These arguments must be passed into gmmFit() in the order they are passed to the moment equation.

### Including four inputs¶

m = meqn(b, y, x, z);

proc meqn(b, yt, xt, zt);

local ut,dt;

// OLS residuals
ut = yt - b[1] - b[2]*xt[., 1] - b[3]*xt[., 2];

// Moment conditions
dt = ut.*zt;

retp( dt );

endp;


### Including two inputs¶

m = meqn(b, y);

proc meqn(b, yt);

local g1, g2;

g1= yt.^2 - b/(b-2);
g2 = yt.^4 - (3*b^2)/((b-2)*(b-4));

retp( g1~g2 );

endp;


The gmmFit() function does not support dataset and formula string syntax. Formula string syntax may be used for standard IV or ols relationships in the gmmFitIV() procedure.

Functions gmmFitControlCreate(), gmmFitIV()