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"The Lagrange Multiplier Test and Testing for Misspecification : An Extended Analysis". Misspecification Tests in Econometrics. New York: Cambridge University Press. pp. 69– 99. ISBN 0-521-26616-5. Ma, Jun; Nelson, Charles R. (2016). "The superiority of the LM test in a class of econometric models where the Wald test performs poorly".
The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then the gradient of the function (at that point) can be expressed as a linear combination of the gradients of the constraints (at that point), with the ...
Because the test is based on the idea of Lagrange multiplier testing, it is sometimes referred to as an LM test for serial correlation. [ 4 ] A similar assessment can be also carried out with the Durbin–Watson test and the Ljung–Box test .
The Lagrange multiplier (LM) test statistic is the product of the R 2 value and sample size: =. This follows a chi-squared distribution, with degrees of freedom equal to P − 1, where P is the number of estimated parameters (in the auxiliary regression). The logic of the test is as follows.
It was independently suggested with some extension by R. Dennis Cook and Sanford Weisberg in 1983 (Cook–Weisberg test). [2] Derived from the Lagrange multiplier test principle, it tests whether the variance of the errors from a regression is dependent on the values of the independent variables. In that case, heteroskedasticity is present.
Generally, when testing for heteroskedasticity in econometric models, the best test is the White test. However, when dealing with time series data, this means to test for ARCH and GARCH errors. Exponentially weighted moving average (EWMA) is an alternative model in a separate class of exponential smoothing models. As an alternative to GARCH ...
For the tests with very general and complicated alternatives, the formula of the test statistics might not have the exactly same representation as above. But we can still derive the formulas as well as its asymptotic distribution by Delta method [ 4 ] and implement Wald test , Score test or Likelihood-ratio test . [ 5 ]
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