Extensions of the Two-Variable Linear Regression Models

1. Basic Econometrics Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL Prof. Himayatullah

2. Chapter 6EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-1. Regression through the origin The SRF form of regression: Yi = b^2X i + u^ i(6.1.5) Comparison two types of regressions: * Regression through-origin model and * Regression with intercept Prof. Himayatullah

3. Chapter 6EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-1. Regression through the origin Comparison two types of regressions: b^2 = SXiYi/SX2i(6.1.6) O b^2 = Sxiyi/Sx2i(3.1.6) I var(b^2) = s2/ SX2i (6.1.7) O var(b^2) = s2/ Sx2i (3.3.1) I s^2 = S(u^i)2/(n-1) (6.1.8) O s^2 = S(u^i)2/(n-2) (3.3.5) I Prof. Himayatullah

4. Chapter 6EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-1. Regression through the origin r2 for regression through-origin model Raw r2 = (SXiYi)2 /SX2i SY2i(6.1.9) Note: Without very strong a priory expectation, well advise is sticking to the conventional, intercept-present model. If intercept equals to zero statistically, for practical purposes we have a regression through the origin. If in fact there is an intercept in the model but we insist on fitting a regression through the origin, we would be committing a specification error Prof. Himayatullah

5. Chapter 6EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-1. Regression through the origin Illustrative Examples: 1) Capital Asset Pricing Model - CAPM (page 156) 2) Market Model (page 157) 3) The Characteristic Line of Portfolio Theory (page 159) Prof. Himayatullah

6. Chapter 6EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-2. Scaling and units of measurement Let Yi = b^1 + b^2Xi + u^ i(6.2.1) Define Y*i=w 1 Y i and X*i=w 2 X i then: b*^2 = (w1/w2) b^2(6.2.15) b*^1 = w1b^1 (6.2.16) s*^2= w12s^2(6.2.17) Var(b*^1)= w21 Var(b^1)(6.2.18) Var(b*^2)= (w1/w2)2Var(b^2) (6.2.19) r2xy = r2x*y* (6.2.20) Prof. Himayatullah

7. Chapter 6EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-2. Scaling and units of measurement From one scale of measurement, one can derive the results based on another scale of measurement. If w1= w2 the intercept and standard error are both multiplied by w1. If w2=1 and scale of Y changed by w1, then all coefficients and standard errors are all multiplied by w1. If w1=1 and scale of X changed by w2, then only slope coefficient and its standard error are multiplied by 1/w2. Transformation from (Y,X) to (Y*,X*) scale does not affect the properties of OLS Estimators A numerical example: (pages 161, 163-165) Prof. Himayatullah

8. 6-3. Functional form of regression model The log-linear model Semi-log model Reciprocal model Prof. Himayatullah

9. 6-4. How to measure elasticity The log-linear model Exponential regression model: Yi= b1Xi b2 eu i (6.4.1) By taking log to the base e of both side: lnYi = lnb1 +b2lnXi + ui , by setting lnb1 = a => lnYi = a +b2lnXi + ui (6.4.3) (log-log, or double-log, or log-linear model) This can be estimated by OLS by letting Y*i = a +b2X*i + ui , where Y*i=lnYi, X*i=lnXi ; b2 measures the ELASTICITY of Y respect to X, that is, percentage change in Y for a given (small) percentage change in X. Prof. Himayatullah

10. 6-4. How to measure elasticity The log-linear model The elasticity E of a variable Y with respect to variable X is defined as: E=dY/dX=(% change in Y)/(% change in X) ~ [(Y/Y) x 100] / [(X/X) x100]= = (Y/X)x (X/Y) = slope x (X/Y) An illustrative example: The coffee demand function (pages 167-168) Prof. Himayatullah

11. 6-5. Semi-log model: Log-lin and Lin-log Models How to measure the growth rate: The log-lin model Y t = Y0 (1+r) t(6.5.1) lnYt = lnY0 + t ln(1+r) (6.5.2) lnYt = b1+ b2t , called constant growth model (6.5.5) where b1 = lnY0 ; b2 = ln(1+r) lnYt = b1+ b2t + ui (6.5.6) It is Semi-log model, or log-lin model. The slope coefficient measures the constant proportional or relative change in Y for a given absolute change in the value of the regressor (t) b2 = (Relative change in regressand)/(Absolute change in regressor) (6.5.7) Prof. Himayatullah

12. 6-5. Semi-log model: Log-lin and Lin-log Models Instantaneous Vs. compound rate of growth b2 is instantaneous rate of growth antilog(b2) – 1 is compound rate of growth The linear trend model Yt = b1+ b2t + ut (6.5.9) If b2 > 0, there is an upward trend in Y If b2 < 0, there is an downward trend in Y Note: (i) Cannot compare the r2 values of models (6.5.5) and (6.5.9) because the regressands in the two models are different, (ii) Such models may be appropriate only if a time series is stationary. Prof. Himayatullah

13. 6-5. Semi-log model: Log-lin and Lin-log Models The lin-log model: Yi = b1 +b2lnXi + ui (6.5.11) b2 = (Change in Y) / Change in lnX = (Change in Y)/(Relative change in X) ~ (Y)/(X/X) (6.5.12) or Y = b2 (X/X) (6.5.13) That is, the absolute change in Y equal to b2 times the relative change in X. Prof. Himayatullah

14. 6-6. Reciprocal Models:Log-lin and Lin-log Models The reciprocal model: Yi = b1 + b2( 1/Xi ) + ui (6.5.14) As X increases definitely, the term b2( 1/Xi ) approaches to zero and Yi approaches the limiting or asymptotic value b1 (See figure 6.5 in page 174) An Illustrative example: The Phillips Curve for the United Kingdom 1950-1966 Prof. Himayatullah

15. 6-7. Summary of Functional Forms Table 6.5 (page 178) Prof. Himayatullah

16. 6-7. Summary of Functional Forms Note: */ indicates that the elasticity coefficient is variable, depending on the value taken by X or Y or both. when no X and Y values are specified, in practice, very often these elasticities are measured at the mean values E(X) and E(Y). ----------------------------------------------- 6-8. A note on the stochastic error term 6-9. Summary and conclusions (pages 179-180) Prof. Himayatullah