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.

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EXAMPLES

• Linear models

• The log-linear model

• Semilog models

• Reciprocal models

• The logarithmic reciprocal model

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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!

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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)

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• 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

Alternative Functional Forms (cont.)

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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:

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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

1 2 ii 0 1i 2iY X X e

i 0 1 1i 2 2i iln Y ln lnX lnX

i 1 1i 2 2i iln Y lnX lnX

ln oB

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• 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:

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.

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1 2

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

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Y AL K

• 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.

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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)

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

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0 (1 )ttY Y r

tY

0Y

• Taking natural logs» (1)

• We can rewrite (1) as

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0ln ln(1 )tlnY Y t r 1 0 2ln (1 )let Y ln r

1 2t tlnY t u

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.

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2

22

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)}

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2

ln 8.3226 0.00705

(0.0016) (0.00018) 0.9919

(5201.6) (39.1667)

tEXT t

se r

t

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%

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2

2

2

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”.

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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

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1283.912 257.2700lnFoodExpi TotalExpi

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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

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Figure 7.4 Polynomial Functions

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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

Properties of reciprocal forms

• As the regressor increases indefinitely the regressand approaches its limiting or asymptotic value (the intercept).

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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.

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1ˆ 81.79436 27,237.17i

CMPGNP

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Table 7.1 Summary of Alternative Functional Forms

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