GMRESm

GMRESm, or Generalized Multivariate Residual Error Sum of Squares, is a statistical method used to assess the fit of a model in the context of multivariate regression analysis. It is an extension of the traditional residual sum of squares (RSS) used in univariate regression, adapted to handle multiple dependent variables simultaneously.

In multivariate regression, the goal is to model the relationship between a set of dependent variables and one or more independent variables. GMRESm provides a measure of the overall discrepancy between the observed values and the values predicted by the model. It is calculated by summing the squared differences between the observed and predicted values for each dependent variable, across all observations, and then summing these values across all dependent variables.

The formula for GMRESm is as follows:

GMRESm = Σ(i=1 to n) Σ(j=1 to m) (Yij - Ŷij)^2

where:

- n is the number of observations,

- m is the number of dependent variables,

- Yij is the observed value of the jth dependent variable for the ith observation,

- Ŷij is the predicted value of the jth dependent variable for the ith observation.

A smaller GMRESm value indicates a better fit of the model to the data, as it implies that the predicted values are closer to the observed values. However, it is important to note that a low GMRESm value does not guarantee that the model is the best or most appropriate for the data, as it does not account for the complexity of the model or the potential for overfitting.

GMRESm is a useful tool for comparing the fit of different models in multivariate regression analysis, but it should be used in conjunction with other diagnostic tools and statistical measures to assess the overall performance and validity of a model.