May 2004 Prof. Himayatullah 1 Basic Econometrics Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL

May 2004 Prof. Himayatullah 2 Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-1. Regression through the origin The SRF form of regression: Y i =  ^ 2 X i + u^ i (6.1.5)  Comparison two types of regressions: * Regression through-origin model and * Regression with intercept

May 2004 Prof. Himayatullah 3 Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-1. Regression through the origin Comparison two types of regressions:  ^ 2 =  X i Y i /  X 2 i (6.1.6) O  ^ 2 =  x i y i /  x 2 i (3.1.6) I var(  ^ 2 ) =  2 /  X 2 i (6.1.7)O var(  ^ 2 ) =  2 /  x 2 i (3.3.1) I  ^ 2 =  u^ i ) 2 /(n-1) (6.1.8)O  ^ 2 =  u^ i ) 2 /(n-2) (3.3.5) I

May 2004 Prof. Himayatullah 4 Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-1. Regression through the origin r 2 for regression through-origin model Raw r 2 = (  X i Y i ) 2 /  X 2 i  Y 2 i (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

May 2004 Prof. Himayatullah 5 Chapter 6 EXTENSIONS 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)

May 2004 Prof. Himayatullah 6 Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELS 6-2. Scaling and units of measurement Let Y i =  ^ 1 +  ^ 2 X i + u^ i (6.2.1) Define Y* i =w 1 Y i and X* i =w 2 X i then:  ^ 2 = (w 1 /w 2 )  ^ 2 (6.2.15)  ^ 1 = w 1  ^ 1 (6.2.16)  *^ 2 = w 1 2  ^ 2 (6.2.17) Var(  ^ 1 ) = w 2 1 Var(  ^ 1 )(6.2.18) Var(  ^ 2 ) = (w 1 /w 2 ) 2 Var(  ^ 2 ) (6.2.19) r 2 xy = r 2 x*y* (6.2.20)

May 2004 Prof. Himayatullah 7 Chapter 6 EXTENSIONS 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 w 1 = w 2 the intercept and standard error are both multiplied by w 1. If w 2 =1 and scale of Y changed by w 1, then all coefficients and standard errors are all multiplied by w 1. If w 1 =1 and scale of X changed by w 2, then only slope coefficient and its standard error are multiplied by 1/w 2. Transformation from (Y,X) to (Y*,X*) scale does not affect the properties of OLS Estimators  A numerical example: (pages 161, )

May 2004 Prof. Himayatullah Functional form of regression model The log-linear model Semi-log model Reciprocal model

May 2004 Prof. Himayatullah How to measure elasticity The log-linear model Exponential regression model: Y i =  1 X i   e u i (6.4.1) By taking log to the base e of both side: lnY i = ln  1 +  2 lnX i + u i, by setting ln  1 =   lnY i =  +  2 lnX i + u i (6.4.3) (log-log, or double-log, or log-linear model) This can be estimated by OLS by letting Y* i =  +  2 X* i + u i, where Y* i =lnY i, X* i =lnX i ;   2 measures the ELASTICITY of Y respect to X, that is, percentage change in Y for a given (small) percentage change in X.

May 2004 Prof. Himayatullah 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 )

May 2004 Prof. Himayatullah Semi-log model: Log-lin and Lin-log Models How to measure the growth rate: The log-lin model Y t = Y 0 (1+r) t (6.5.1) lnY t = lnY 0 + t ln(1+r) (6.5.2) lnY t =    +  2 t, called constant growth model (6.5.5) where  1 = lnY 0 ;  2 = ln(1+r) lnY t =    +  2 t + u i (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)  2 = (Relative change in regressand)/(Absolute change in regressor) (6.5.7)

May 2004 Prof. Himayatullah Semi-log model: Log-lin and Lin-log Models Instantaneous Vs. compound rate of growth  2 is instantaneous rate of growth antilog(  2 ) – 1 is compound rate of growth The linear trend model Y t =    +  2 t + u t (6.5.9) If  2 >  there is an upward trend in Y If  2 <  there is an downward trend in Y Note: (i) Cannot compare the r 2 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.

May 2004 Prof. Himayatullah Semi-log model: Log-lin and Lin-log Models The lin-log model: Y i =  1 +  2 lnX i + u i  (6.5.11)  2 = (Change in Y) / Change in lnX = (Change in Y)/(Relative change in X) ~ (  Y)/(  X/X) (6.5.12) or  Y =  2 (  X/X) (6.5.13) That is, the absolute change in Y equal to  2 times the relative change in X.

May 2004 Prof. Himayatullah Reciprocal Models: Log-lin and Lin-log Models The reciprocal model: Y i =  1 +  2 ( 1/X i ) + u i  (6.5.14) As X increases definitely, the term  2 ( 1/X i ) approaches to zero and Y i approaches the limiting or asymptotic value  1 (See figure 6.5 in page 174) An Illustrative example: The Phillips Curve for the United Kingdom

May 2004 Prof. Himayatullah Summary of Functional Forms Table 6.5 (page 178) ModelEquationSlope = dY/dX Elasticity = (dY/dX).(X/Y) Linear Y =      X    (X/Y) */ Log-linear (log-log) lnY =      lnX    (Y  X)  Log-lin lnY =      X    Y    X */ Lin-log Y =      lnX  2 (1/X)    Y) */ Reciprocal Y =      X)-  2 (1/X 2 )-    XY) */

May 2004 Prof. Himayatullah 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) A note on the stochastic error term 6-9. Summary and conclusions (pages )