to be scrutinized more carefully, in case they are erroneous, or outliers, or in some other way atypical of the rest of the dataset. Partitioned regression Sometimes the variables and corresponding parameters in the regression can be logically split into two groups, so that the regression takes form where X1 and X2 have dimensions n×p1, n×p2, and β1, β2 are p1×1 and p2×1 vectors, with . The Frisch–Waugh–Lovell theorem states that in this regression the residuals and the OLS estimate will be numerically identical to the residuals and the OLS estimate for β2 in the following regression: where M1 is the annihilator matrix for regressors X1. The theorem can be used to establish a number of theoretical results. For example, having a regression with a constant and another regressor is equivalent to subtracting the means from the dependent variable and the regressor and then running the regression for the de-meaned variables but without the constant term. Constrained estimation Suppose it is known that the coefficients in the regression satisfy a system of linear equations where Q is a p×q matrix of full rank, and c is a q×1 vector of known constants, where . In this case least squares estimation is equivalent to minimizing the sum of squared residuals of the model subject to the constraint A. The constrained least squares (CLS) estimator can be given by an explicit formula: This expression for the constrained estimator is valid as long as the matrix XTX is invertible. It was assumed from the beginning of this article that this matrix is of full rank, and it was noted that when the rank condition fails, β will not be identifiable. However it may happen that adding the restriction A makes β identifiable, in which case one would like to find the formula for the estimator. The estimator is equal to where R is a p×(p − q) matrix such that the matrix is non-singular, and . Such a matrix can always be found, although generally it is not unique. The second formula coincides with the first in case when XTX is invertible. Large sample properties The least squares estimators are point estimates of the linear regression model parameters β. However, generally we also want to know how close those estimates might be to the true values of parameters. In other words, we want to construct the interval estimates. Since we haven't made any