北大暑期课程《回归分析》(linear regression analysis)讲义pku8.doc

Class 8: polynomial regression and dummy variables I. Polynomial Regression Polynomial regression is a minor topic. Because there is little that is new. What is new is that you may want to create a new variable from the same data set. This is necessary if you think that the true regression function is not linear but quadratic, you might want to try to use the quadratic function, that is, the first and the second order regressors. For example, we know that earnings increases as a function of age. But the relationship is not linear. Therefore, we regress earnings on age and age2. One important trick is that if you have polynomial regression, the regression line is no longer linear when you plot the dependent variable against independent variable. Use a hypothetical example, If we obtain has a linear effect, has a quadratic effect. If you are asked to plot against the line is linear. If you are asked to plot against , the line is quadratic. Say the sample mean of is 0.5 0 1 2 3 5 8 Interpretation of coefficients in quadratic equations Say important: there is no simple relationship between and . Sometime, the effect of on is positive, sometimes the effect of on is negative. In other words, the effect of on depends on the value of . Suggestion: plot the regression for the data range. One thing we can tell: When (2 0, the effect of on increases with ; When (2 0, the effect of on decreases with . [figure] II. Interpretation of Coefficients in Polynomial Regression Relationship between Y and the “polynomial” independent variable is no longer linear. Recall a special property of the linear function: the relationship between Y and an X (say Xk) is constant for all values of this X and other X variables: (1) . In a polynomial regression, this simple relationship no longer holds true. For a quadratic regression, for example, (2) we have (3) , which is dependent on the value of Xk. In general, the situation where a simple linear