Breusch-Pagan LM statistic test
The Breusch-Pagan LM statistic tests the random effects model against the
pooled OLS model.
The specific hypothesis under investigation is the following:
H0: σ2α = 0
H1: σ2α ≠ 0
The random effects model reduces to a pooled OLS regression under the null (when σ2α = 0).
The test is based on OLS residuals built from the pooled regression model.
Under the null, this statistic should be distributed as a χ-square.
Breusch–Godfrey test
The Breusch–Godfrey test is a test for autocorrelation in the errors in a regression model. It makes use of the residuals from the model being considered in a regression analysis, and a test statistic is derived from these. The null hypothesis is that there is no serial correlation of any order up to p.
Breusch-Pagan test
Breush-Pagan test measures how errors increase across the explanatory variable, Y. The test assumes the error variances are due to a linear function of one or more explanatory variables in the model. That means heteroskedasticity could still be present in your regression model, but those errors (if present) are not correlated with the Y-values.