The linear regression model

The linear regression model
In the linear regression model, the dependent variable is assumed to be a linear function of one or more independent variables plus an error introduced to account for all other factors: In the above regression equation, y_i is the dependent variable, x_i1, ...., x_iK are the independent or explanatory variables, and u_i is the disturbance or error term. The goal of regression analysis is to obtain estimates of the unknown parameters Beta_1, ..., Beta_K which indicate how a change in one of the independent variables affects the values taken by the dependent variable.
Applications of regression analysis exist in almost every field. In economics, the dependent variable might be a family's consumption expenditure and the independent variables might be the family's income, number of children in the family, and other factors that would affect the family's consumption patterns. In political science, the dependent variable might be a state's level of welfare spending and the independent variables measures of public opinion and institutional variables that would cause the state to have higher or lower levels of welfare spending. In sociology, the dependent variable might be a measure of the social status of various occupations and the independent variables characteristics of the occupations (pay, qualifications, etc.). In psychology, the dependent variable might be individual's racial tolerance as measured on a standard scale and with indicators of social background as independent variables. In education, the dependent variable might be a student's score on an achievment test and the independent variables characteristics of the student's family, teachers, or school.
The common aspect of the applications described above is that the dependent variable is a quantitative measure of some condition or behavior. When the dependent variable is qualitative or categorical, then other methods (such as logit or probit analysis, described in Chapter 7) might be more appropriate.