Lagrange multiplier test

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One of the three tests of restrictions on an unknown parameter, or a vector of unknown parameters, θ, based on the maximum likelihood estimation of θ (along with the likelihood ratio test and the Wald test). The null hypothesis is H0: λ = 0, where λ is the vector of Lagrange multipliers of the constrained maximization problem, in which the objective function is the log-likelihood function and the constraint is the set of restrictions. In an equivalent, simpler formulation, the LM test statistic is based on the derivatives of the log-likelihood function evaluated at θ̂R, the maximum likelihood estimator of θ obtained under restrictions. Under the null hypothesis the distribution of the LM test statistic in a large sample is chi-square, with degrees of freedom equal to the number of restrictions.

Subjects: Economics.

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