may 2004 prof. himayatullah 1 basic econometrics chapter 6 extensions of the two-variable linear...

May 2004Prof. Himayatullah1

Basic EconometricsBasic Econometrics

Chapter 6

EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL


Chapter 6Chapter 6 EXTENSIONS OF THE TWO-VARIABLE LINEAR EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELSREGRESSION MODELS

6-1. Regression through the origin The SRF form of regression: Yi = ^2X i + u^ i (6.1.5) Comparison two types of regressions: * Regression through-origin model and * Regression with intercept



6-1. Regression through the origin Comparison two types of regressions:

^2 = XiYi/X2i (6.1.6) O

^2 = xiyi/x2i (3.1.6) I

var(^2) = 2/X2i (6.1.7) O

var(^2) = 2/x2i (3.3.1) I

^2 = u^i)2/(n-1) (6.1.8) O

^2 = u^i)2/(n-2) (3.3.5) I



6-1. Regression through the origin r2 for regression through-origin model

Raw r2 = (XiYi)2 /X2

iY2i (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


Chapter 6Chapter 6 EXTENSIONS OF THE TWO-VARIABLE EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELSLINEAR 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)



6-2. Scaling and units of measurement

Let Yi = ^1 + ^2Xi + u^ i (6.2.1) Define Y*i=w 1 Y i and X*i=w 2 X i then: ^2 = (w1/w2)^2

(6.2.15) ^1 = w1^1

(6.2.16) *^2 = w1

2^2

(6.2.17) Var(^1) = w2

1 Var(^1) (6.2.18) Var(^2) = (w1/w2)

2 Var(^2) (6.2.19)

r2xy = r2

x*y* (6.2.20)


Chapter 6Chapter 6 EXTENSIONS OF THE TWO-VARIABLE EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODELSLINEAR 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 OLSEstimators

A numerical example: (pages 161, 163-165)


6-3. Functional form of regression model6-3. Functional form of regression model

The log-linear model Semi-log model Reciprocal model


6-4. How to measure elasticity6-4. How to measure elasticity

The log-linear model Exponential regression model: Yi= 1Xi

e u i (6.4.1)

By taking log to the base e of both side: lnYi = ln1 +2lnXi + ui , by setting ln1 =

lnYi = +2lnXi + ui (6.4.3) (log-log, or double-log, or log-linear model) This can be estimated by OLS by letting Y*i = +2X*i + ui , where Y*i=lnYi, X*i=lnXi ; 2 measures the ELASTICITY of Y respect to X, that is,

percentage change in Y for a given (small) percentage change in X.


6-4. How to measure elasticity6-4. How to measure elasticity

The log-linear modelThe 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)


6-5. Semi-log model6-5. Semi-log model: : Log-lin and Lin-log ModelsLog-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 = + 2t , called constant growth model (6.5.5) where 1 = lnY0 ; 2 = ln(1+r) lnYt = + 2t + 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)

2 = (Relative change in regressand)/(Absolute change in regressor) (6.5.7)



Instantaneous Vs. compound rate of growth 2 is instantaneous rate of growth antilog(2) – 1 is compound rate of growth

The linear trend model Yt = + 2t + ut (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 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.



The lin-log model: Yi = 1 +2lnXi + ui (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.


6-6. Reciprocal Models6-6. Reciprocal Models:: Log-lin and Lin-log ModelsLog-lin and Lin-log Models

The reciprocal model: Yi = 1 + 2( 1/Xi ) + ui (6.5.14) As X increases definitely, the term

2( 1/Xi ) approaches to zero and Yi

approaches the limiting or asymptotic value 1 (See figure 6.5 in page 174)

An Illustrative example: The Phillips Curve for the United Kingdom 1950-1966


6-7. Summary of Functional Forms6-7. Summary of Functional Forms

Table 6.5 (page 178)

Model Equation Slope = dY/dX

Elasticity = (dY/dX).(X/Y)

Linear Y = X (X/Y) */

Log-linear (log-log)

lnY = lnX

(YX)

Log-lin lnY = X Y X */

Lin-log Y = lnX 2(1/X) Y) */

Reciprocal Y = X)

- 2(1/X2) - XY) */


6-7. Summary of Functional Forms6-7. 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 term6-9. Summary and conclusions (pages 179-180)

may 2004 prof. himayatullah 1 basic econometrics chapter 6 extensions of the two-variable linear...

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