linear regression, there are several independent variables or functions of independent variables. Adding a term in to the preceding regression gives: parabola: This is still linear regression; although the expression on the right hand side is quadratic in the independent variable , it is linear in the parameters , and In both cases, is an error term and the subscript indexes a particular observation. Returning our attention to the straight line case: Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model: The residual, , is the difference between the value of the dependent variable predicted by the model, , and the true value of the dependent variable, . One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals, SSR: Minimization of this function results in a set of normal equations, a set of simultaneous linear equations in the parameters, which are solved to yield the parameter estimators, . In the case of simple regression, the formulas for the least squares estimates are where is the mean (average) of the values and is the mean of the values. Under the assumption that the population error term has a constant variance, the estimate of that variance is given by: This is called the mean square error (MSE) of the regression. The denominator is the sample size reduced by the number of model parameters estimated from the same data, for regressors or if an intercept is used. In this case, so the denominator is . The standard errors of the parameter estimates are given by Under the further assumption that the population error term is normally distributed, the researcher can use these estimated standard errors to create confidence intervals and conduct hypothesis tests about the population parameters. General linear model In the more general multiple regression model, there are independent variables: where is the -th observation on the -th independent variable. If the first independent variable takes the value 1 for all , , then is called the regression intercept. The least squares parameter estimates are obtained from normal equations. The residual can be written as The normal equations are In matrix notation, the normal equations are written as where the element of is , the element of the column vector is , and the element of is . Thus is , is