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FUNCTIONAL FORMS OF REGRESSION MODELS • A functional form refers to the algebraic form of the relationship between a dependent variable and the regressor(s). • The simplest functional form is the linear functional form, where the relationship between the dependent variable and an independent variable is graphically represented by a straight line. 7-0 EXAMPLES • Linear models • The log-linear model • Semilog models • Reciprocal models • The logarithmic reciprocal model © 2011 Pearson Addison-Wesley. All rights reserved. 7-1 Choosing a Functional Form • After the independent variables are chosen, the next step is to choose the functional form of the relationship between the dependent variable and each of the independent variables. • Let theory be your guide! Not the data! © 2011 Pearson Addison-Wesley. All rights reserved. 7-2 Alternative Functional Forms • An equation is linear in the variables if plotting the function in terms of X and Y generates a straight line • For example, Equation 7.1: Y = β0 + β1X + ε (7.1) is linear in the variables but Equation 7.2: Y = β0 + β1X2 + ε (7.2) is not linear in the variables • Similarly, an equation is linear in the coefficients only if the coefficients appear in their simplest form—they: – are not raised to any powers (other than one) – are not multiplied or divided by other coefficients – do not themselves include some sort of function (like logs or exponents) © 2011 Pearson Addison-Wesley. All rights reserved. 7-3 Alternative Functional Forms (cont.) • For example, Equations 7.1 and 7.2 are linear in the coefficients, while Equation 7:3: (7.3) is not linear in the coefficients • In fact, of all possible equations for a single explanatory variable, only functions of the general form: (7.4) are linear in the coefficients β0 and β1 © 2011 Pearson Addison-Wesley. All rights reserved. 7-4 Linear Form • This is based on the assumption that the slope of the relationship between the independent variable and the dependent variable is constant: • For the linear case, the elasticity of Y with respect to X (the percentage change in the dependent variable caused by a 1-percent increase in the independent variable, holding the other variables in the equation constant) is: © 2011 Pearson Addison-Wesley. All rights reserved. 7-5 Double-Log Form • Assume the following: • Taking nat. logs Yields: ln Yi ln 0 1 ln X1i 2lnX2i i ln Yi 1 ln X1i 2lnX2i i • Or • Where Yi 0X1i1 X2i2 ei ln Bo • this is linear in the parameters and linear in the logarithms of the explanatory variables hence the names log-log, double-log or loglinear models 7-6 • Here, the natural log of Y is the dependent variable and the natural log of X is the independent variable: • In a double-log equation, an individual regression coefficient can be interpreted as an elasticity because: • Note that the elasticities of the model are constant and the slopes are not • This is in contrast to the linear model, in which the slopes are constant but the elasticities are not • Interpretation: © 2011 Pearson Addison-Wesley. All rights reserved. 7-7 Interpretation of double-log functions • In this functional form elasticity coefficients. 1 and 2 are the • A one percent change in x will cause a % change in y, – e.g., if the estimated coefficient is -2 that means that a 1% increase in x will generate a 2% decrease in y. 7-8 C-D production function • where: Y AL K • Y = total production (the monetary value of all goods produced in a year) • L = labour input (the total number of person-hours worked in a year) • K = capital input (the monetary worth of all machinery, equipment, and buildings) • A = total factor productivity © 2011 Pearson Addison-Wesley. All rights reserved. 7-9 • α and β are the output elasticities of labour and capital, respectively. These values are constants determined by available technology. • Output elasticity measures the responsiveness of output to a change in levels of either labour or capital used in production, ceteris paribus. For example if α = 0.15, a 1% increase in labour would lead to approximately a 0.15% increase in output. • Further, if: • α + β = 1, the production function has constant returns to scale: Doubling capital K and labour L will also double output Y. If • α + β < 1, returns to scale are decreasing, and if • α + β > 1 returns to scale are increasing. © 2011 Pearson Addison-Wesley. All rights reserved. 7-10 Semilog Form • The semilog functional form is a variant of the doublelog equation in which some but not all of the variables (dependent and independent) are expressed in terms of their natural logs. • It can be on the right-hand side, as in: lin-log model: Yi = β0 + β1lnX1i + β2X2i + εi (7.7) • Or it can be on the left-hand side, as in: log-lin: lnY = β0 + β1X1 + β2X2 + ε (7.9) 7-11 Measuring growth rate (loglin model) • May be interested in estimating the growth rate of population, GNP, Money supply, etc. • Recall the compound interest formula Yt Y0 (1 r) t • Where r=compound rate of growth of Y, Yt • Is the value at time t and Y0 is the initial value 7-12 • Taking natural logs lnYt » ln Y0 t ln(1 r ) let 1 ln Y0 • We can rewrite (1) as (1) 2 ln(1 r) lnYt 1 2t ut 7-13 interpretation • The slope coefficient (2 )measures the constant proportional or relative change in Y for a given absolute change in the value of the regressor (in this case t) • In this functional form(2 ) is interpreted as follows. A one unit change in x will cause a 2(100)% change in y, • This is the growth rate or sem-ielasticity • e.g., – if the estimated coefficient is 0.05 that means that a one unit increase in x will generate a 5% increase in y. © 7-14 Consider the following reg. results for expenditure on services over the quarterly period 2003-I to 2006-III ln EXTt 8.3226 se t (0.0016) (5201.6) 0.00705t (0.00018) (39.1667) r 2 0.9919 • -Expenditure on services grow at a quarterly rate of 0.705% {ie. (0.00705)*100} • Service expenditure at the start of 2003 is $4115.96 billion {ie. antilog of the intercept (8.3226)} 7-15 Instantaneous vs. compound rate of growth • 2 Gives the instantaneous (at a point in time)rate of growth and not compound rate of growth (ie. Growth over a period of time). • We can get the compound growth rate as • [(Antilog 2 )-1]*100 • or [(exp2 )-1]*100 • ie. [exp(0.00705)-1]*100=0.708% 7-16 Lin-log models [Yi = β0 + β1lnX1i + β2X2i + εi] • Divide slope coefficient by 100 to interpret • Application: Engel expenditure model • Engel postulated that; “the total expenditure that is devoted to food tends to increase in arithmatic progression as total expenditure increases in geometric progression”. 7-17 Consider results of food expenditure India • See FoodExpi 1283.912 257.2700ln TotalExpi • A 1% increase in total expenditure leads to 2.57 rupees increase in food expenditure • Ie. Slope divided by 100 7-18 Polynomial Form • Polynomial functional forms express Y as a function of the independent variables, some of which are raised to powers other than 1 • For example, in a second-degree polynomial (also called a quadratic) equation, at least one independent variable is squared: Yi = β0 + β1X1i + β2(X1i)2 + β3X2i + εi (7.10) • The slope of Y with respect to X1 in Equation 7.10 is: (7.11) • Note that the slope depends on the level of X1 © 2011 Pearson Addison-Wesley. All rights reserved. 7-19 Figure 7.4 Polynomial Functions © 2011 Pearson Addison-Wesley. All rights reserved. 7-20 Inverse (reciprocal) Form • The inverse functional form expresses Y as a function of the reciprocal (or inverse) of one or more of the independent variables (in this case, X1): Yi = β0 + β1(1/X1i) + β2X2i + εi (7.13) • So X1 cannot equal zero • This functional form is relevant when the impact of a particular independent variable is expected to approach zero as that independent variable approaches infinity • The slope with respect to X1 is: (7.14) • The slopes for X1 fall into two categories, depending on the sign of β1 © 2011 Pearson Addison-Wesley. All rights reserved. 7-21 Properties of reciprocal forms • As the regressor increases indefinitely the regressand approaches its limiting or asymptotic value (the intercept). 7-22 Example: relationship b/n child mortality (CM) & per capita GNP (PGNP) • Now ˆ 1 CM 81.79436 27, 237.17 PGNPi • As PGNP increases indefinitely CM reaches its asymptotic value of 82 deaths per thousand. 7-23 Table 7.1 Summary of Alternative Functional Forms © 2011 Pearson Addison-Wesley. All rights reserved. 7-24 7-25