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.

EXAMPLES Linear models The log-linear model Semilog models Reciprocal models The logarithmic reciprocal model © 2011 Pearson Addison-Wesley. All rights reserved.

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.

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. 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. 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. 5

Double-Log Form Assume the following: Taking nat. logs Yields: Or Where this is linear in the parameters and linear in the logarithms of the explanatory variables hence the names log-log, double-log or log-linear models 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.

Interpretation of double-log functions   Interpretation of double-log functions In this functional form and are the elasticity coefficients. 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.

C-D production function where: 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.

α + β < 1, returns to scale are decreasing, and if 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.

Semilog Form The semilog functional form is a variant of the double-log 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) 11

Measuring growth rate (log-lin model) May be interested in estimating the growth rate of population, GNP, Money supply, etc. Recall the compound interest formula Where r=compound rate of growth of Y, Is the value at time t and is the initial value

Taking natural logs (1) We can rewrite (1) as

interpretation The slope coefficient ( )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( ) is interpreted as follows. A one unit change in x will cause a (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. ©

Consider the following reg Consider the following reg. results for expenditure on services over the quarterly period 2003-I to 2006-III -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)}

Instantaneous vs. compound rate of growth 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 )-1]*100 or [(exp )-1]*100 ie. [exp(0.00705)-1]*100=0.708%

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”.

Consider results of food expenditure India See A 1% increase in total expenditure leads to 2.57 rupees increase in food expenditure Ie. Slope divided by 100

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. 19

Figure 7.4 Polynomial Functions © 2011 Pearson Addison-Wesley. All rights reserved. 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. 21

Properties of reciprocal forms As the regressor increases indefinitely the regressand approaches its limiting or asymptotic value (the intercept).

Example: relationship b/n child mortality (CM) & per capita GNP (PGNP) Now As PGNP increases indefinitely CM reaches its asymptotic value of 82 deaths per thousand.

Table 7.1 Summary of Alternative Functional Forms © 2011 Pearson Addison-Wesley. All rights reserved. 24