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Fits limited-information maximum likelihood, IVLIML(y, exog, endog, instruments, fuller=α).fit().

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

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

The second overload is the cluster-robust fit, as for InstrumentalVariables.TwoStageLeastSquares.

Parameters — design is the response, the regressors and the instruments. clusters is one label per row. options chooses the covariance, Fuller's α and the intercept; null takes the reference's defaults.

Returns — an IvSummary whose Kappa is the k-class parameter the fit used.

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 not finite, 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 — LIML, then Fuller's correction at α = 1.

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 liml = InstrumentalVariables.Liml(design);
IvSummary fuller = InstrumentalVariables.Liml(design, new IvOptions { Fuller = 1.0 });

double kappa = liml.Kappa!.Value;          // => 2.4365392…
double lowered = fuller.Kappa!.Value;      // => 2.3115392…
double effect = liml.Coefficients[2];      // => 1.9021995…

Remarks — κ is the smallest eigenvalue of the LIML problem: E = [y, X_endog] purged of the exogenous regressors, against the same purged of the instruments. Fuller lowers it by α/(n − L), here 1/8. A just-identified model has κ = 1 and fits two-stage least squares exactly.

Applies to — net10.0, netstandard2.0.

See also — InstrumentalVariables.TwoStageLeastSquares, IvSummary.