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Fits two-step GMM, IVGMM(y, exog, endog, instruments, weight_type=…).fit().

public static IvSummary Gmm(IvDesign design, IvOptions options = null)

public static IvSummary Gmm(IvDesign design, ReadOnlySpan<int> clusters, IvOptions options)

The second overload takes the labels a clustered covariance or a clustered weight reads.

Parameters — design is the response, the regressors and the instruments. clusters is one label per row. options chooses the covariance, the second step's weight and the intercept; null takes the reference's defaults, a robust weight among them.

Returns — an IvSummary whose Overidentification is Hansen's J, and whose Kappa is null.

Exceptions — ArgumentOutOfRangeException when a column count is out of range. ArgumentException when a block's length is not its column count times the rows, when there are fewer instruments than endogenous regressors, when no residual degree of freedom is left, when the options ask for a cluster covariance or weight without labels, when Fuller is set, or when the regressors or instruments are collinear. ArgumentNullException when the clustered overload gets no options. ArgumentException naming design also when the rows times the larger of the parameters and the instruments, or that plus one where a rank test or LIML appends a column, are more cells than one array holds (#1614).

Example — the default robust weight, and Hansen's J.

using Lodestar.Stats.Regression;
using Lodestar.Stats.Regression.Instrumental;

double[] response = [3.1, 4.0, 5.2, 4.4, 6.9, 7.1, 6.0, 8.8, 9.1, 8.2, 10.7, 11.3];
double[] exogenous = [0.2, -1.0, 0.5, 1.3, -0.4, 0.9, -1.2, 0.1, 1.7, -0.6, 0.8, -0.3];
double[] endogenous = [1.0, 1.4, 2.1, 1.8, 3.0, 3.3, 2.6, 3.9, 4.2, 3.7, 4.9, 5.4];
double[] instruments =
    [0.9, 0.1, 1.5, -0.3, 2.2, 0.4, 1.7, 0.8, 3.1, -0.2, 3.3, 0.6,
     2.4, 1.1, 3.8, -0.5, 4.1, 0.9, 3.5, 0.2, 4.6, -0.1, 5.2, 0.7];

var design = new IvDesign(response, exogenous, 1, endogenous, 1, instruments, 2);

IvSummary gmm = InstrumentalVariables.Gmm(design);

double effect = gmm.Coefficients[2];                 // => 1.88441634…
double j = gmm.Overidentification!.Statistic;        // => 5.4959041…
double p = gmm.Overidentification.PValue;            // => 0.019061068…

Remarks — the first step weights by (ZᵀZ/n)⁻¹, which is two-stage least squares; the second by the inverse of the chosen score covariance at the first step's residuals — iter_limit=2, the reference's default. Iterated GMM is not written: it moves between tolerances, so no corpus pins it. A kernel weight or covariance without a bandwidth takes n − 2 here, as the reference's KernelWeightMatrix does, not Newey and West's rule.

Applies to — net10.0, netstandard2.0.

See also — InstrumentalVariables.TwoStageLeastSquares, IvOptions.