Economics 20 - Prof. Anderson,1,Multiple Regression Analysis,y = b0 + b1x1 + b2x2 + . . . bkxk + u 4. Further Issues,Economics 20 - Prof. Anderson,2,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,Economics 20 - Prof. Anderson,3,Beta Coefficients,Occasional youll 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,Economics 20 - Prof. Anderson,4,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,Economics 20 - Prof. Anderson,5,Interpretation of Log Models,If the model is ln(y) = b0 + b1ln(x) + u b1 is the elasticity of y with respect to x If the model is ln(y) = b0 + b1x + u b1 is approximately the percentage change in y given a 1 unit change in x If the model is y = b0 + b1ln(x) + u b1 is approximately the change in y for a 100 percent change in x,Economics 20 - Prof. Anderson,6,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,Economics 20 - Prof. Anderson,7,Some Rules of Thumb,What types of variables are often used in log form? Dollar amounts that must be positive Very large variables, such as population What types of variables are often used in level form? Variables measured in years Variables that are a proportion or percent,Economics 20 - Prof. Anderson,8,Quadratic Models,For a model of the form y = b0 + b1x + b2x2 + u we cant interpret b1 alone as measuring the change in y with respect to x, we need to take into account b2 as well, since,Economics 20 - Prof. Anderson,9,More on Quadratic Models,Suppose that the coefficient on x is positive and the coefficient on x2 is negative Then y is increasing in x at first, but will eventually turn around and be decreasing in x,Economics 20 - Prof. Anderson,10,More on Quadratic Models,Suppose that the coefficient on x is negative and the coefficient on x2 is positive Then y is decreasing in x at first, but will eventually turn around and be increasing in x,Economics 20 - Prof. Anderson,11,Interaction Terms,For a model of the form y = b0 + b1x1 + b2x2 + b3x1x2 + u we cant interpret b1 alone as measuring the change in y with respect to x1, we need to take into account b3 as well, since,Economics 20 - Prof. Anderson,12,Adjusted R-Squared,Recall that the R2 will always increase as more variables are added to the model The adjusted R2 takes into account the number of variables in a model, and may decrease,Economics 20 - Prof. Anderson,13,Adjusted R-Squared (cont),Its easy to see that the adjusted R2 is just (1 R2)(n 1) / (n k 1), but most packages will give you both R2 and adj-R2 You can compare the fit of 2 models (with the same y) by comparing the adj-R2 You cannot use the adj-R2 to compare models with different ys (e.g. y vs. ln(y),Economics 20 - Prof. Anderson,14,Goodness of Fit,Important not to fixate too much on adj-R2 and lose sight of theory and common sense If economic theory clearly predicts a variable belongs, generally leave it in Dont want to include a variable that prohibits a sensible interpretation of the variable of interest remember ceteris paribus interpretation of multiple regression,Economics 20 - Prof. Anderson,15,Standard Errors for Predictions,Suppose we want to use our estimates to obtain a specific prediction? First, suppose that we want an estimate of E(y|x1=c1,xk=ck) = q0 = b0+b1c1+ + bkck This is easy to obtain by substituting the xs in our estimated model with cs , but what about a standard error? Really just a test of a linear combination,Economics 20 - Prof. Anderson,16,Predictions (cont),Can rewrite as b0 = q0 b1c1 bkck Substitute in to obtain y = q0 + b1 (x1 - c1) + + bk (xk - ck) + u So, if you regress yi on (xij - cij) the intercept will give the predicted value and its standard error Note that the standard error will be smallest when the cs equal the means of the xs,Economics 20 - Prof. Anderson,17,Predictions (cont),This standard error for the expected value is not the same as a standard error for an outcome on y We need to also take into