CHAPTER 6: EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL

ü  Regression Through the Origin

Y_i = B₂X_i  +  u_i

In this model the intercept term is absent or zero, hence the name regression through the origin.

ü  r² for Regression-through-Origin Model

raw r² =   (∑X_iY_i)²

                 ∑X_i²∑Y_i²

ü  Scaling and Units of Measurement

ü  A Word About Interpretation

Since the slope coefficient β₂ is simply the rate of change, it is measured in the units of the ratio.

Units of the dependent variable

Units of the explanatory variable

ü  Regression on Standardized Variables

An interesting property of a standard variable is that its mean value is always zero and its standard deviation is always 1.

ü  Functional Forms of Regression Models

1.      The log-linear model

2.      Semilog models

3.      Reciprocal models

4.      The logarithmic reciprocal model

ü  How to Measure Elasticity: Log-Linear Model

lnY_i = α + β₂lnX_i + u_i

ü  Semi Log Models: Log-lin and lin-log models

ü  How to Measure the Growth Rate: the Log-lin model

lnY_i = β₁ + β₂t + u_i

In this model, the slope coefficient measures the constant proportional or relative change in Y for a given absolute change in the value of the regressor.

β₂ =  relative change in regressand

        absolute change in regressor

ü  Linear Trend Model

Y_i = β₁ + β₂t + u_i

ü  The Lin-Log Model

Y_i = β₁ + β₂lnX_i + u_i

β2 = change in Y

         change in X

                                                                       = change in Y

                                                                           relative change in X

ü  Reciprocal Models

Y_i = β₁ + β₂    1    + u_i

              X_i

ü  Log Hyperbola or Logarithmic Reciprocal Model

lnY_i = β₁ – β₂   1    + u_i

                 X_i

ü  Choice of Functional Form

1.      The underlying theory may suggest a particular functional form.

2.      It is good practice to find out the rate of change of the regressand with respect to the regressor as well as to find out the elasticity of the regressand with respect to the regressor.

3.      The coefficients of the model chosen should satisfy certain a priori expectations.

4.      But make sure that in comparing r² values the dependent variable, or the regressand, of the two models is the same the regressor(s) can take any form.

5.      In general one should not overemphasize the r² measure in the sense that the higher the r² the better the model.