Multiple regression analysis y-Bo+BixI+Bx2+.Bixk+u ◆4. Further Issues Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = b0 + b1 x1 + b2 x2 + . . . bk xk + u 4. Further Issues
● Redefining v ariables e 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 o 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
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
Beta Coefficients Occasional you ll see reference to a standardized coefficient or beta coefficient which has a specific meaning A Idea is to replace y and each x variable with a standardized version -i.e. subtract mean and divide by standard deviation e Coefficient reflects standard deviation of y for a one standard deviation change in x Economics 20- Prof anderson
Economics 20 - Prof. Anderson 3 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
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 o Can take the natural log ofx, y or bot Can use quadratic forms of x Can use interactions of x variables Economics 20- Prof anderson 4
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
Interpretation of Log models o If the model is In()=Bo+BIn(x)+u o, is the elasticity of y with respect to x o If the model is In()=Bo+Bx+u e B, is approximately the percentage change in y given a l unit change in x o If the model is y=Bo+ BIn(x)+u B, is approximately the change in y for a 100 percent change in x Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 Interpretation of Log Models If the model is ln(y) = b0 + b1 ln(x) + u b1 is the elasticity of y with respect to x If the model is ln(y) = b0 + b1 x + u b1 is approximately the percentage change in y given a 1 unit change in x If the model is y = b0 + b1 ln(x) + u b1 is approximately the change in y for a 100 percent change in x
Why use log me odels? o Log models are invariant to the scale of the variables since measuring percent changes e They give a direct estimate of elasticity e For models with y >0, the conditional distribution is often heteroskedastic or skewed, while In(y) is much less so o The distribution of In() is more narrow limiting the effect of outliers Economics 20- Prof anderson 6
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
Some rules of thumb e What types of variables are often used in log form? o Dollar amounts that must be positive Very large variables such as population o What types of variables are often used in level form? Variables measured in years o Variables that are a proportion or percent Economics 20- Prof anderson 7
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
Quadratic models o For a model of the form y-Bo+ B-x+ B2x2+ u we cant interpret B, alone as measuring the change in y with respect to x, we need to take into account B, as well, since 4≈(B1+2B2 x|x、SO △ B+2B2x Economics 20- Prof anderson 8
Economics 20 - Prof. Anderson 8 Quadratic Models For a model of the form y = b0 + b1x + b2x 2 + u we can’t interpret b1 alone as measuring the change in y with respect to x, we need to take into account b2 as well, since ( ) x x y y x x 1 2 1 2 ˆ 2 ˆ ˆ ,so ˆ 2 ˆ ˆ b b b b + +
More on quadratic models o Suppose that the coefficient on x is positive and the coefficient on x2 is negative o Then y is increasing in x at first, but w eventually turn around and be decreasing in x For b>0 and B2<o the turning point will be at x= B/2B, Economics 20- Prof anderson 9
Economics 20 - Prof. Anderson 9 More on Quadratic Models Suppose that the coefficient on x is positive and the coefficient on x 2 is negative Then y is increasing in x at first, but will eventually turn around and be decreasing in x ( ) 1 2 * 1 2 ˆ 2 ˆ will be at 0 the turning point ˆ 0 and ˆ For b b b b = x
More on quadratic models o Suppose that the coefficient on x is negative and the coefficient on x2 is positive e Then y is decreasing in x at first, but will eventually turn around and be increasing in x For,0 the turning point will be at x=B /22 Which IS the same as when B, >0 and d B <0 Economics 20- Prof anderson 10
Economics 20 - Prof. Anderson 10 More on Quadratic Models Suppose that the coefficient on x is negative and the coefficient on x 2 is positive Then y is decreasing in x at first, but will eventually turn around and be increasing in x ( ) 0 ˆ 0 and ˆ the same as when , which is ˆ 2 ˆ will be at 0 the turning point ˆ 0 and ˆ For 1 2 1 2 * 1 2 = b b b b b b x
