7-2 Choosing a Functional Form

Report
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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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!
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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)
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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
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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:
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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  0X1i1 X2i2 ei
  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
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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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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.
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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
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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.
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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)
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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
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• 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
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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)}
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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 [(exp2 )-1]*100
• ie. [exp(0.00705)-1]*100=0.708%
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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
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
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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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7-19
Figure 7.4
Polynomial Functions
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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
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7-21
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 ˆ
 1 
CM  81.79436  27, 237.17 

 PGNPi 
• As PGNP increases indefinitely CM reaches its
asymptotic value of 82 deaths per thousand.
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Table 7.1 Summary of
Alternative Functional Forms
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