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
83. Compare the two slope coefficients: B In the linear model In the log-linear model How to compare the elasticity of the two models n og-linear model: the elasticity( which is the slope coefficient remains the same, that is, B, no matter at what price the elasticity is measured. So it is called constant elasticity model inear model E、% change In Y △Y/Y·100 △YⅩ Slope %0 change inⅩ △X/X·100 △XY The elasticity changes from point to point on the linear curve because the ratio X/Y changes from point to point. In practice, the elasticity coefficient for the linear model is often computed by average elasticity △YⅩ Average elasticity (8.9) △XY
❖ 3. Compare the two slope coefficients: B2 In the linear model In the log-linear model How to compare the elasticity of the two models? In log-linear model: the elasticity（which is the slope coefficient） remains the same, that is, B2, no matter at what price the elasticity is measured. So it is called constant elasticity model. In linear model: = = = slope· The elasticity changes from point to point on the linear curve because the ratio X/Y changes from point to point. In practice, the elasticity coefficient for the linear model is often computed by average elasticity. Average elasticity = (8.9) % change in X % change in Y E = X X 100 Y Y 100 • • Y X X Y • Y X Y X X Y
8.3 Multiple Log-Linear Regression Models A three-variable log-linear model nY=B1+B2nx21+B3nx31+u(8.10) B, B Partial elasticity coefficients 62: measures the elasticity of Y with respect to X, holding the influence of X, constant that is it measures the percentage change in Y for a percentage change in X2 holding the influence of X, constant measures the(Partial) elasticity of Y with respect to X3, holding the influence of X, constant
8.3 Multiple Log-Linear Regression Models A three-variable log-linear model: lnYi=B1+B2 lnX2i+B3 lnX3i+ui (8.10) B2,B3 : Partial elasticity coefficients. B2 : measures the elasticity of Y with respect to X2 holding the influence of X3 constant; that is ,it measures the percentage change in Y for a percentage change in X2 , holding the influence of X3 constant. B3 : measures the (Partial) elasticity of Y with respect to X3 , holding the influence of X2 constant
2. Instantaneous versus Compound Rate of Growth b,the estimate of B,=In(1+r) antilog (b2)=(1+r) r= antilog(b2)-1(8.21) r-the compound (over a period of time rate of growth o b, in semilog model: the instantaneous ( at a point in time) owth rate In practice, we generally use the instantaneous growth rate 3. The Linear Trend Model Y=B1+B2t+u1(8.23) time variable t: the trend variable IfB,>0. there is an upward in y IfB.<o. there is a downward in Y 1) Compare to linear trend model, the growth model is more useful 2) We cannot compare r2 values of the two models because the dependent variables in the two models are not the same
2. Instantaneous versus Compound Rate of Growth b2=the estimate of B2=ln(1+r) antilog (b2 ) = (1+r) r = antilog(b2 ) –1 (8.21) r－－the compound （over a period of time）rate of growth b2 in semilog model: the instantaneous（at a point in time） growth rate. In practice, we generally use the instantaneous growth rate. 3. The Linear Trend Model Yt=B1+B2 t +ut (8.23) time variable t: the trend variable. If B2＞0, there is an upward in Y If B2＜0, there is a downward in Y. （1）Compare to linear trend model, the growth model is more useful. （2）We cannot compare r 2 values of the two models because the dependent variables in the two models are not the same
8. 4 How to Measure the Growth Rate. The semilog model The Semilog Model ● Nonlinear model: Y=Y0(1+r) (8.13) InY= InYotIn(1+r) (8.14) Then Iny=B, +B t (8.17) Semilog Model, growth model- Linear model InY B+B,t+u. (8.18) Estimation (1) Under CLrM, the semi-log model can be estimated by using OLS method. and the ols estimators are BLUE 2) In a semilog model, the slope coefficient measures the proportional or relative change in Y for a given absolute change in the explanatory variable (3) Semilog models are called growth models because they are routinely used to measure the growth rate of many variables
8.4 How to Measure the Growth Rate: The Semilog Model 1. The Semilog Model ●Nonlinear model: Yt=Y0 (1+r)t (8.13) lnYt = lnY0+tln (1+r) (8.14) Then lnYt=B1+B2 t (8.17) Semilog Model, growth model－Linear model: lnYt = B1+B2 t + ut (8.18) ● Estimation (1) Under CLRM, the semi-log model can be estimated by using OLS method, and the OLS estimators are BLUE. (2) In a semilog model, the slope coefficient measures the proportional or relative change in Y for a given absolute change in the explanatory variable (3) Semilog models are called growth models because they are routinely used to measure the growth rate of many variables
8. 5The Lin-log Model: When the Explanatory Variable is Logarithmic Y:B1+B2lnx21+u(8.25) the slope coefficient B, measures absolute change in y B 8.27) relative change in △X/X where A Y and AX represent(small) changes in Y and X equivalently △X △Y=B2X (8.28)
8.5The Lin-log Model: When the Explanatory Variable is Logarithmic Yt=B1+B2 lnX2t+ut (8.25) the slope coefficient B2 measures: B2 = = （8.27） where Δ Y and ΔX represent (small) changes in Y and X. equivalently, △Y =B2 (8.28) relative change in X absolute change in X X Y X X