FUNCTIONAL FORMS OF REGRESSION MODELS CHAPTER 2 FUNCTIONAL FORMS OF REGRESSION MODELS Damodar Gujarati Econometrics by Example, second edition

LOG-LINEAR, DOUBLE LOG, OR CONSTANT ELASTICITY MODELS The Cobb-Douglas Production Function: can be transformed into a linear model by taking natural logs of both sides: The slope coefficients can be interpreted as elasticities. If (B2 + B3) = 1, we have constant returns to scale. If (B2 + B3) > 1, we have increasing returns to scale. If (B2 + B3) < 1, we have decreasing returns to scale. Damodar Gujarati Econometrics by Example, second edition

LOG-LIN OR GROWTH MODELS The rate of growth of real GDP: can be transformed into a linear model by taking natural logs of both sides: Letting B1 = ln RGDP1960 and B2 = ln (l+r), this can be rewritten as: ln RGDPt = B1 +B2 t B2 is considered a semi-elasticity or an instantaneous growth rate. The compound growth rate (r) is equal to (eB2 – 1). Damodar Gujarati Econometrics by Example, second edition

LIN-LOG MODELS Lin-log models follow this general form: Note that B2 is the absolute change in Y responding to a percentage (or relative) change in X If X increases by 100%, predicted Y increases by B2 units Used in Engel expenditure functions: “The total expenditure that is devoted to food tends to increase in arithmetic progression as total expenditure increases in geometric proportion.” Damodar Gujarati Econometrics by Example, second edition

RECIPROCAL MODELS Lin-log models follow this general form: Note that: As X increases indefinitely, the term approaches zero and Y approaches the limiting or asymptotic value B1. The slope is: Therefore, if B2 is positive, the slope is negative throughout, and if B2 is negative, the slope is positive throughout. Damodar Gujarati Econometrics by Example, second edition

POLYNOMIAL REGRESSION MODELS The following regression predicting GDP is an example of a quadratic function, or more generally, a second-degree polynomial in the variable time: The slope is nonlinear and equal to: Damodar Gujarati Econometrics by Example, second edition

SUMMARY OF FUNCTIONAL FORMS Damodar Gujarati Econometrics by Example, second edition

COMPARING ON BASIS OF R2 We cannot directly compare two models that have different dependent variables. We can transform the models as follows and compare RSS: Step 1: Compute the geometric mean (GM) of the dependent variable, call it Y*. Step 2: Divide Yi by Y* to obtain: Step 3: Estimate the equation with lnYi as the dependent variable using in lieu of Yi as the dependent variable (i.e., use ln as the dependent variable). Step 4: Estimate the equation with Yi as the dependent variable using as the dependent variable instead of Yi. Damodar Gujarati Econometrics by Example, second edition

STANDARDIZED VARIABLES We can avoid the problem of having variables measured in different units by expressing them in standardized form: where SY and SX are the sample standard deviations and and are the sample means of Y and X, respectively The mean value of a standardized variable is always zero and its standard deviation value is always 1. Damodar Gujarati Econometrics by Example, second edition

MEASURES OF GOODNESS OF FIT R2: Measures the proportion of the variation in the regressand explained by the regressors. Adjusted R2: Denoted as , it takes degrees of freedom into account: Akaike’s Information Criterion (AIC): Adds harsher penalty for adding more variables to the model, defined as: The model with the lowest AIC is usually chosen. Schwarz’s Information Criterion (SIC): Alternative to the AIC criterion, expressed as: The penalty factor here is harsher than that of AIC. Damodar Gujarati Econometrics by Example, second edition

REGRESSION THROUGH THE ORIGIN Also known as a zero intercept model. Example is the well-known capital asset pricing model (CAPM) of portfolio theory: where ERi = expected rate of return on security i, ERm = expected rate of return on a market portfolio, rf = risk-free rate of return, β = the Beta coefficient, a measure of systematic risk that cannot be eliminated through portfolio diversification. Beta coefficient greater than 1: Suggests a volatile security Beta coefficient of less than 1: Suggests a defensive security Sums of squares and cross-product terms are raw terms here: var(b2) = Damodar Gujarati Econometrics by Example, second edition