Chapter 7 - Ken Farr (GCSU)

Transcript Chapter 7 - Ken Farr (GCSU)

Chapter 7

Specification: Choosing a Functional Form

Slides by Niels-Hugo Blunch Washington and Lee University

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. • KISS Principle • Let theory be your guide! Not the data!

The Use and Interpretation of the Constant Term

• An estimate of β 0 has at least

three components

: 1. the true β 0 2. the constant impact of any specification errors (an omitted variable, for example) 3. the mean of ε for the correctly specified equation (if not equal to zero) • Unfortunately, these components can’t be distinguished from one another because we can observe only β 0 , the sum of the three components • As a result of this, we usually

don’t interpret

the constant term • On the other hand, we should

not suppress

either, as illustrated by Figure 7.1 the constant term,

Figure 7.1 The Harmful Effect of Suppressing the Constant Term

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Alternative Functional Forms

• • • An equation is

linear

in the

variables

and Y generates a

straight line

if plotting the function in terms of X For example, Equation 7.1: Y = β 0 + β 1 X + ε

linear in the variables but Equation 7.2: Y = β 0 + β 1 X 2 + ε is

not

linear in the variables (7.1) (7.2) Similarly, an equation is

linear

in the

coefficients

appear in their simplest form —they: only if the

coefficients

– are

not raised

to any powers (other than one) – are

not multiplied

divided

by other coefficients – do

not

themselves

include

some sort of

function

(like

logs

exponents

)

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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 β 0 and β 1

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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What Is a Logarithm?

• • • • • If

e log

constant

of x: equal to 2.71828) to the “

th power” produces x, then b is the b is the log of x to the base e if: e b = x Thus, a log (or logarithm) is the exponent to which a given base must be taken in order to produce a specific number While logs come in more than one variety, we’ll use only natural logs (logs to the base e) in this text The symbol for a natural log is “ln,” so ln(x) = b means that (2.71828) more simply, b = x or, ln(x) = b

means that

e b = x For example, since e 2 = (2.71828) 2 = 7.389, we can state that: ln(7.389) = 2

Double-Log Form

• Assume the following: Y i   0  X X e 1i 1  2i 2  i • Yields: ln Y i ln 0 1 ln X 1i  2 X 2i   i • 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 not

and the

slopes

are • This is in

contrast

to the

linear model

, in which the

slopes constant

but the

elasticities

are

not

are

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Figure 7.2 Double-Log Functions

7-9

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: Y i = β 0 + β 1 lnX 1i + β 2 X 2i + ε i • Or it can be on the

left-hand

side, as in: lnY = β 0 + β 1 X 1 + β 2 X 2 + ε • Figure 7.3 illustrates these two different cases (7.7) (7.9)

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Figure 7.3 Semilog Functions

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Polynomial Form

•

Polynomial functional

forms express Y as a function of 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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Figure 7.4 Polynomial Functions

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Inverse 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, 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

(illustrated in Figure 7.5) , depending on the sign

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Figure 7.5 Inverse Functions

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Table 7.1 Summary of Alternative Functional Forms

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Lagged Independent Variables

• Virtually all the regressions we’ve studied so far have been

“instantaneous”

in nature • In other words, they have included independent and dependent variables from the

same time period

, as in: Y t = β 0 + β 1 X 1t + β 2 X 2t + ε t (7.15) • Many econometric equations include one or more

lagged independent variables

the observation of X 1 like X 1t-1 where “t–1” indicates that is from the time period previous to time period t, as in the following equation: Y t = β 0 + β 1 X 1t-1 + β 2 X 2t + ε t

(7.16)

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Using Dummy Variables

• A dummy variable is a variable that takes on the values of 0 or 1, depending on whether a condition for a qualitative attribute (such as gender) is met • These conditions take the

general form:

(7.18) • This is an example of an

intercept dummy

(as opposed to a

slope dummy

, which is discussed in Section 7.5) • Figure 7.6 illustrates the consequences of including an intercept dummy in a linear regression model

7-18

Figure 7.6 An Intercept Dummy

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Slope Dummy Variables

• Contrary to the

intercept dummy

, which changed only the intercept (and not the slope), the

slope dummy

changes both the intercept and the slope • The

general form

of a

slope dummy equation

is: Y i = β 0 + β 1 X i + β 2 D i + β 3 X i D i + ε i • The slope depends on the value of D: When D = 0, ΔY/ΔX = β 1 When D = 1, ΔY/ΔX = (β 1 + β 3 ) • Graphical illustration of how this works in Figure 7.7

(7.20)

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Figure 7.7 Slope and Intercept Dummies

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Problems with Incorrect Functional Forms

• If functional forms are similar, and if theory does not specify exactly which form to use, there are at least two reasons why we should avoid using goodness of fit over the sample to determine which equation to use: 1.

Fits are difficult to compare if the dependent variable is transformed 2.

An incorrect function form may provide a reasonable fit

within the sample

have the potential to make large forecast errors when used

outside

but the range of

the sample

• The first of these is essentially due to the fact that when the dependent variable is transformed, the

total sum of squares

(TSS)

changes

as well • The second is essentially die to the fact that using an incorrect functional amounts to a specification error similar to the omitted variables bias discussed in Section 6.1

• This second case is illustrated in Figure 7.8

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Key Terms from Chapter 7

• Elasticity • Double-log functional form • Semilog functional form • Polynomial functional form • Inverse functional form • Slope dummy • Natural log • Omitted condition • Interaction term • Linear in the variables • Linear in the coefficients

Chapter 7 - Ken Farr (GCSU)

Transcript Chapter 7 - Ken Farr (GCSU)

Chapter 7

Specification: Choosing a Functional Form

Choosing a Functional Form

The Use and Interpretation of the Constant Term

Figure 7.1 The Harmful Effect of Suppressing the Constant Term

Alternative Functional Forms

Alternative Functional Forms (cont.)

Linear Form

What Is a Logarithm?

Double-Log Form

Figure 7.2 Double-Log Functions

Semilog Form

Figure 7.3 Semilog Functions

Polynomial Form

Figure 7.4 Polynomial Functions

Inverse Form

Figure 7.5 Inverse Functions

Table 7.1 Summary of Alternative Functional Forms

Lagged Independent Variables

Using Dummy Variables

Figure 7.6 An Intercept Dummy

Slope Dummy Variables

Figure 7.7 Slope and Intercept Dummies

Problems with Incorrect Functional Forms

Key Terms from Chapter 7

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