多元回归分析其他问题.ppt

Economics 20 - Prof. Anderson Redefining Variables Changing the scale of the y variable will lead to a corresponding change in the scale of the coefficients and standard errors, so no change in the significance or interpretation Changing the scale of one x variable will lead to a change in the scale of that coefficient and standard error, so no change in the significance or interpretation 因变量或自变量以对数形式出现，改变度量单位不会影响斜率系数，只会改变截距项。 Beta Coefficients Occasional you’ll see reference to a “standardized coefficient” or “beta coefficient” which has a specific meaning Idea is to replace y and each x variable with a standardized version – i.e. subtract mean and divide by standard deviation Coefficient reflects standard deviation of y for a one standard deviation change in x Beta Coefficients (cont) Example 6.1 :Effects of Pollution on Housing Prices Functional Form OLS can be used for relationships that are not strictly linear in x and y by using nonlinear functions of x and y – will still be linear in the parameters Can take the natural log of x, y or both Can use quadratic forms of x Can use interactions of x variables Interpretation of Log Models ln(y) = b0 + b1ln(x) + u b1 is the elasticity of y with respect to x ln(y) = b0 + b1x + u b1 is approximately the percentage change in y given a 1 unit change in x y = b0 + b1ln(x) + u b1 is approximately the change in y for a 100 percent change in x Why use log models? Log models are invariant to the scale of the variables since measuring percent changes They give a direct estimate of elasticity For models with y 0, the conditional distribution is often heteroskedastic or skewed, while ln(y) is much less so The distribution of ln(y) is more narrow, limiting the effect of outliers. it can not be used if a variable takes on zero or negative values. It is more difficult to predict the original variable when using a dependent variable in log form. Some Rules of Thumb variables used in log form: Dollar amounts that