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### 15.16 REGRESSION

The REGRESSION procedure fits linear models to data via least-squares estimation. The procedure is appropriate for data which satisfy those assumptions typical in linear regression:

• The data set contains n observations of a dependent variable, say Y_1,…,Y_n, and n observations of one or more explanatory variables. Let X_{11}, X_{12}, …, X_{1n} denote the n observations of the first explanatory variable; X_{21},…,X_{2n} denote the n observations of the second explanatory variable; X_{k1},…,X_{kn} denote the n observations of the kth explanatory variable.
• The dependent variable Y has the following relationship to the explanatory variables: Y_i = b_0 + b_1 X_{1i} + ... + b_k X_{ki} + Z_i where b_0, b_1, …, b_k are unknown coefficients, and Z_1,…,Z_n are independent, normally distributed noise terms with mean zero and common variance. The noise, or error terms are unobserved. This relationship is called the linear model.

The REGRESSION procedure estimates the coefficients b_0,…,b_k and produces output relevant to inferences for the linear model.