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Fits the random-effects model, RandomEffects(y, x).fit(): Swamy and Arora's variance components, then least squares on quasi-demeaned rows.

public static PanelSummary RandomEffects(PanelDesign design, PanelOptions options = null)

public static PanelSummary RandomEffects(PanelDesign design, ReadOnlySpan<int> clusters, PanelOptions options)

The second overload clusters by the caller's labels, alone or beside the entity or the period.

Parameters — design is the panel. clusters is one label per row. options chooses the covariance and the intercept; null takes the reference's defaults.

Returns — a PanelSummary whose Theta holds each entity's quasi-demeaning weight, and whose variance components are the estimated ones.

Exceptions — ArgumentOutOfRangeException when ExogenousCount is below one or the confidence level is outside (0, 1). ArgumentException when a block's length is not its width times the rows, when two rows share an entity and a period, when no residual degree of freedom is left, when a Bartlett or Parzen bandwidth reaches past the periods, or when the options set a value this estimator does not read. ArgumentNullException when the clustered overload gets no options. ArgumentException too when there are no more rows than entities plus coefficients, or no more entities than coefficients: the variance components' degrees of freedom, where the reference divides by zero. ArgumentException naming design also when the rows times the parameters plus one are more cells than one array holds (#1614).

Example — the slope, and how much of each entity's mean it removes.

using Lodestar.Stats.Regression;
using Lodestar.Stats.Regression.Panel;

int[] entities = [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4];
int[] periods = [2020, 2021, 2022, 2023, 2020, 2021, 2022, 2023, 2020, 2021, 2022, 2023, 2020, 2021, 2022, 2023];
double[] x = [0.5, 1.1, 1.9, 2.4, 1.2, 1.8, 2.9, 3.1, -0.3, 0.4, 0.8, 1.6, 2.0, 2.2, 3.1, 3.9];
double[] y = [2.1, 3.0, 4.2, 4.9, 4.4, 5.1, 6.8, 7.0, 0.2, 1.3, 1.7, 3.1, 6.1, 6.3, 7.9, 9.2];

var design = new PanelDesign(y, x, 1, entities, periods);

PanelSummary summary = PanelRegression.RandomEffects(design);

double slope = summary.Coefficients[1];      // => 1.653028044…
double theta = summary.Theta![0];            // => 0.6691202669…
double share = summary.Rho!.Value;           // => 0.6703470171…

Remarks — σ²ₑ comes from the within regression over n − k − N + 1 degrees of freedom and σ²ᵤ from the between one, less σ²ₑ over the harmonic mean of the entity sizes, floored at zero. θᵢ = 1 − √(σ²ₑ/(Tᵢσ²ᵤ + σ²ₑ)): zero reduces to pooled least squares, one to fixed effects.

Applies to — net10.0, netstandard2.0.

See also — PanelRegression.FixedEffects, PanelSummary.