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Multinomial logistic regression is known by a variety of other names, including polytomous LR, [2] [3] multiclass LR, softmax regression, multinomial logit (mlogit), the maximum entropy (MaxEnt) classifier, and the conditional maximum entropy model.
The second term after the equal sign is the omitted-variable bias in this case, which is non-zero if the omitted variable z is correlated with any of the included variables in the matrix X (that is, if X′Z does not equal a vector of zeroes). Note that the bias is equal to the weighted portion of z i which is "explained" by x i.
Linear errors-in-variables models were studied first, probably because linear models were so widely used and they are easier than non-linear ones. Unlike standard least squares regression (OLS), extending errors in variables regression (EiV) from the simple to the multivariable case is not straightforward, unless one treats all variables in the same way i.e. assume equal reliability.
In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) [1] states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. [2]
There are several different possible causes of specification error; some are listed below. An inappropriate functional form could be employed. A variable omitted from the model may have a relationship with both the dependent variable and one or more of the independent variables (causing omitted-variable bias). [3]
Other estimation techniques besides FGLS were suggested for SUR model: [7] the maximum likelihood (ML) method under the assumption that the errors are normally distributed; the iterative generalized least squares (IGLS), where the residuals from the second step of FGLS are used to recalculate the matrix ^, then estimate ^ again using GLS, and ...
If the data exhibit a trend, the regression model is likely incorrect; for example, the true function may be a quadratic or higher order polynomial. If they are random, or have no trend, but "fan out" - they exhibit a phenomenon called heteroscedasticity. If all of the residuals are equal, or do not fan out, they exhibit homoscedasticity.
In R, White's Test can be implemented using the white function of the skedastic package. [5]In Python, White's Test can be implemented using the het_white function of the statsmodels.stats.diagnostic.het_white [6]