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.