• 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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• Linear models
• The log-linear model
• Semilog models
• Reciprocal models
• The logarithmic reciprocal model
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• 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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• 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 = β
0
+ β
1
X + ε is linear in the variables but Equation 7.2:
Y = β
0
+ β
1
X 2 + ε
(7.1)
(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 β
0 and β
1
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• 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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• Assume the following: Y i
  
1

X X e
0 1i 2i
 i
• Taking nat. logs Yields: ln Y i ln
0 1 ln X
1i

2 lnX
2i
  i
• Or ln Y i
  
1 ln X
1i

2 lnX
2i
  i
• Where
  ln B o
• 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
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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:
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• 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.
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  
Y AL K
• 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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• 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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• 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: Y i
= β
0
+ β
1 lnX
1i
+ β
2
X
2i
+ ε i
(7.7)
• Or it can be on the left-hand side, as in: log-lin: lnY = β
0
+ β
1
X
1
+ β
2
X
2
+ ε (7.9)
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• May be interested in estimating the growth rate of population, GNP, Money supply, etc.
• Recall the compound interest formula
Y t

Y
0
(1
 r ) t
• Where r=compound rate of growth of Y , Y t
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• Taking natural logs lnY t
 ln Y
0
 t ln(1
 r ) let

1
 ln Y
0
• We can rewrite (1) as lnY t
 
1 2 t u t
(1)

2
 ln (1
 r )
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
2 proportional or relative change in Y for a given absolute change in the value of the regressor (in this case t )

2 
2
• 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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Consider the following reg. results for expenditure on services over the quarterly period 2003-I to 2006-III t ln EXT t

8.3226

0.00705
t se

(0.0016) (0.00018)

(5201.6) (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)}
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•

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

2

2
• ie. [exp(0.00705)-1]*100=0.708%
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i
0
1
1i
2
2i
i]
• 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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• 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 :
Y i
= β
0
+ β
1
X
1i
+ β
2
(X
1i
) 2 + β
3
X
2i
+ ε i
• The slope of Y with respect to X
1 in Equation 7.10 is:
(7.10)
(7.11)
• Note that the slope depends on the level of X
1
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• 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, X
1
):
Y i
= β
0
+ β
1
(1/X
1i
) + β
2
X
2i
+ ε i
(7.13)
• So X
1 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 X
1 is:
(7.14)
• The slopes for X
1 of β
1 fall into two categories , depending on the sign
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7-21

• As the regressor increases indefinitely the regressand approaches its limiting or asymptotic value (the intercept).
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• Now
 


1
PGNP i



• As
PGNP increases indefinitely CM reaches its asymptotic value of 82 deaths per thousand.
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