Explained Sum of Squares - Partitioning in The General OLS Model

Partitioning in The General OLS Model

The general regression model with n observations and k explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is

where y is an n × 1 vector of dependent variable observations, each column of the n × k matrix X is a vector of observations on one of the k explanators, is a k × 1 vector of true coefficients, and e is an n× 1 vector of the true underlying errors. The ordinary least squares estimator for is

The residual vector is, so the residual sum of squares is, after simplification,

Denote as the constant vector all of whose elements are the sample mean of the dependent variable values in the vector y. Then the total sum of squares is

The explained sum of squares, defined as the sum of squared deviations of the predicted values from the observed mean of y, is

Using in this, and simplifying to obtain, gives the result that TSS = ESS + RSS if and only if . The left side of this is times the sum of the elements of y, and the right side is times the sum of the elements of, so the condition is that the sum of the elements of y equals the sum of the elements of, or equivalently that the sum of the prediction errors (residuals) is zero. This can be seen to be true by noting the well-known OLS property that the k × 1 vector : since the first column of X is a vector of ones, the first element of this vector is the sum of the residuals and is equal to zero. This proves that the condition holds for the result that TSS = ESS + RSS.