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Fits two-stage least squares, IV2SLS(y, exog, endog, instruments).fit().

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

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

The second overload is the cluster-robust fit: its options must ask for IvCovarianceType.Clustered, and no other covariance reads the labels.

Parameters — design is the response, the exogenous and endogenous regressors and the excluded instruments, as an IvDesign. clusters is one label per row, any integers, naming at least two clusters. options chooses the covariance, its kernel and scaling, and the intercept; null takes the reference's defaults.

Returns — an IvSummary: the table, R², the model test, the first-stage diagnostics and Sargan's overidentification test.

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 without labels or set a value this estimator does not read, 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 robust default, and the unadjusted covariance beside it.

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 robust = InstrumentalVariables.TwoStageLeastSquares(design);
IvSummary unadjusted = InstrumentalVariables.TwoStageLeastSquares(
    design, new IvOptions { CovarianceType = IvCovarianceType.Unadjusted });

double robustError = robust.StandardErrors[2];          // => 0.0310248100…
double unadjustedError = unadjusted.StandardErrors[2];  // => 0.0345804417…
double sargan = robust.Overidentification!.Statistic;   // => 7.0786454…

Remarks — the estimate is (X̂ᵀX̂)⁻¹X̂ᵀy with X̂ = P_zX, and the covariance is V⁻¹SV⁻¹/n over the projected regressors, S the chosen estimator's. Sargan's test above rejects the two instruments' agreement at 1%: one of them, at least, is not excluded from the response.

Applies to — net10.0, netstandard2.0.

See also — InstrumentalVariables.Liml, InstrumentalVariables.Gmm, IvOptions.