Chapter 8 Functional forms of Regression Model
Chapter 8 Functional Forms of Regression Model
The models we discussed are models that are linear ir parameters, variables Y and Xs do not necessarily have to be linear The price elasticity of demand the log-linear models The rate of growth semilog model Functional forms of regression models which are linear in parameters, but not necessarily linear in variables 1. Log-linear of constant elasticity models. ( Section 8.1) 2. Semilog models(Sections 8.4 and 8.5) 3. Reciprocal models(Section 8.6) 4. Polynomial regression models(Section 8.7 For a regression model linear in explanatory variable(s), for a unit change in the explanatory variable, the rate of change(i. e,the slope)of the dependent variable remains constant For regression models nonlinear in explanatory variable(s), the slope does not remain constant
The models we discussed are models that are linear in parameters; variables Y and Xs do not necessarily have to be linear. The price elasticity of demand～the log-linear models The rate of growth～semilog model ❖ Functional forms of regression models which are linear in parameters, but not necessarily linear in variables: 1. Log-linear of constant elasticity models. (Section 8.1) 2. Semilog models (Sections 8.4 and 8.5) 3. Reciprocal models (Section 8.6) 4. Polynomial regression models (Section 8.7) For a regression model linear in explanatory variable(s), for a unit change in the explanatory variable, the rate of change(i.e., the slope) of the dependent variable remains constant; For regression models nonlinear in explanatory variable(s), the slope does not remain constant
8.7 How to Measure Elasticity. The Log-Linear Model 1 Model and its transformations Nonlinear model AX (8.1) InY.=na +binX (8.2) B =InA (8.3) InY B, +BInX (8.4) Double-log or log-linear model----linear model InY B+B,Inx, +u (8.5) letting Y;=InY. and X'=InY en Y=B,+BX+u (86)
8.1 How to Measure Elasticity: The Log-Linear Model ❖ 1. Model and its transformations: Nonlinear model: (8.1) (8.2) B1=lnA (8.3) lnYi=B1+B2 lnXi (8.4) Double-log or log-linear model----linear model: lnYi = B1+B2 lnXi+ui (8.5) letting and then (8.6) B2 Yi = AXi i 2 i lnY = lnA + B lnX i * i Y = lnY i * i X = lnY i * 1 2 i * Yi = B + B X + u
2. Estimation (1) Under the CLRM assumptions, we can get OLS estimators of the log-linear model, and they are blue (2) Slope coefficient B, measures the elasticity of Y with respect to X, that is, the percentage change in Y for a given (small) percentage change in X How to compute the elasticity coefficient. E let△ Y stand for a small change in Y and△ X for a small change inⅩ %o change in Y △Y/Y●100 E change in X △X/X●100 △XY slope.Y (8.7) 3) Hypothesis Testing in log-Linear Models- the same as linear models
2. Estimation （1）Under the CLRM assumptions, we can get OLS estimators of the log-linear model,and they are BLUE. （2）Slope coefficient B2 measures the elasticity of Y with respect to X, that is , the percentage change in Y for a given (small) percentage change in X. How to compute the elasticity coefficient, E let △Y stand for a small change in Y and △X for a small change in X = = = slope· (8.7) （3）Hypothesis Testing in log-Linear Models — the same as linear models % change in X % change in Y E = X X 100 Y Y 100 • • Y X X Y • Y X
2 Comparing Linear and Log-Linear Regression Models Question: which model is better?) InY=B+B,Inx, +u (8.5) Y B+B,X+u Use the scattergram plot the data: YX, if they are nonlinear, then plot the log of Y against the log of X to find which model is the best estimate of the Pre Problem: this principle works only in the two-variable regression models 2. Compare the two models on the basis of r2 Problem () To compare the r2 values of two models, the dependent variable must be in the same form by adding more explanatory variables to the mode s be increased 2)High r2 value criterion: An r2(R2) can al way
8.2 Comparing Linear and Log-Linear Regression Models ——Question: which model is better?） lnYi = B1+B2 lnXi+ui (8.5) Yi = B1+B2Xi+ui 1.Use the scattergram plot the data: Y～X, if they are nonlinear, then plot the log of Y against the log of X. －－to find which model is the best estimate of the PRF. Problem: this principle works only in the two-variable regression models 2. Compare the two models on the basis of r2 Problem: （1）To compare the r2 values of two models, the dependent variable must be in the same form. （2）High r2 value criterion: An r2 (=R2 ) can always be increased by adding more explanatory variables to the model