the normality assumption holds (that is, that ), then additional properties of the OLS estimators can be stated. The estimator is normally distributed, with mean and variance as given before: This estimator reaches the Cramér–Rao bound for the model, and thus is optimal in the class of all unbiased estimators. Note that unlike the Gauss–Markov theorem, this result establishes optimality among both linear and non-linear estimators, but only in the case of normally distributed error terms. The estimator s2 will be proportional to the chi-squared distribution: The variance of this estimator is equal to , which does not attain the Cramér–Rao bound of . However it was shown that there are no unbiased estimators of σ2 with variance smaller than that of the estimator s2. If we are willing to allow biased estimators, and consider the class of estimators that are proportional to the sum of squared residuals (SSR) of the model, then the best (in the sense of the mean squared error) estimator in this class will be , which even beats the Cramér–Rao bound in case when there is only one regressor (). Moreover, the estimators and s2 are independent, the fact which comes in useful when constructing the t- and F-tests for the regression. Influential observations As was mentioned before, the estimator is linear in y, meaning that it represents a linear combination of the dependent variables yi. The weights in this linear combination are functions of the regressors X, and generally are unequal. The observations with high weights are called influential because they have a more pronounced effect on the value of the estimator. To analyze which observations are influential we remove a specific j-th observation and consider how much the estimated quantities are going to change (similarly to the jackknife method). It can be shown that the change in the OLS estimator for β will be equal to where is the j-th diagonal element of the hat matrix P, and xj is the vector of regressors corresponding to the j-th observation. Similarly, the change in the predicted value for j-th observation resulting from omitting that observation from the dataset will be equal to From the properties of the hat matrix, , and they sum up to p, so that on average . These quantities hj are called the leverages, and observations with high hj are called leverage points. Usually the observations with high leverage ought