ELASTICITIES AND LOGARITHMIC MODELS 1 This sequence defines elasticities and shows how one may fit nonlinear models with constant elasticities. First,

ELASTICITIES AND LOGARITHMIC MODELS

1

This sequence defines elasticities and shows how one may fit nonlinear models with constant elasticities. First, the general definition of an elasticity.

Y

X

A

0 52O

Definition:

The elasticity of Y with respect to X is the proportional change in Y per proportional change in X.

XYdXdY

XdXYdY

elasticity

XY

2

Re-arranging the expression for the elasticity, we can obtain a graphical interpretation.


Y

X

A

0 52O

Definition:


XYdXdY

XdXYdY

elasticity

XY

slope of the tangent at Aslope of OA

Y

X

A

3

The elasticity at any point on the curve is the ratio of the slope of the tangent at that point to the slope of the line joining the point to the origin.

0 52O

Definition:


XYdXdY

XdXYdY

elasticity


XY


4

In this case it is clear that the tangent at A is flatter than the line OA and so the elasticity must be less than 1.


Y

X

A

0 52O

Definition:


XYdXdY

XdXYdY

elasticity

XY


elasticity < 1

0 52

5

In this case the tangent at A is steeper than OA and the elasticity is greater than 1.

A

O

Y

X

elasticity > 1


Definition:


XYdXdY

XdXYdY

elasticity


6

In general the elasticity will be different at different points on the function relating Y to X.


XY 21

21

2

21

2

)/(

/)(

X

XX


XYdXdY

elasticity

x

A

O

Y

X

7

In the example above, Y is a linear function of X.


x

A

O

Y

X

XY 21

21

2

21

2

)/(

/)(

X

XX


XYdXdY

elasticity

8

The tangent at any point is coincidental with the line itself, so in this case its slope is always 2. The elasticity depends on the slope of the line joining the point to the origin.


x

A

O

Y

X

XY 21

21

2

21

2

)/(

/)(

X

XX


XYdXdY

elasticity

9

OB is flatter than OA, so the elasticity is greater at B than at A. (This ties in with the mathematical expression: (1/ X) + 2 is smaller at B than at A, assuming that 1 is positive.)

x

A

O

B

Y

X


XY 21

21

2

21

2

)/(

/)(

X

XX


XYdXdY

elasticity

10

However, a function of the type shown above has the same elasticity for all values of X.


21

XY

11

For the numerator of the elasticity expression, we need the derivative of Y with respect to X.


21

XY

121

2 XdXdY

12

For the denominator, we need Y/X.


21

XY

121

2 XdXdY

11

1 2

2

X

XX

XY

13

Hence we obtain the expression for the elasticity. This simplifies to 2 and is therefore constant.

21

XY

121

2 XdXdY

11

1 2

2

X

XX

XY

elasticity 211

121

2

2

X

XXYdXdY


14

By way of illustration, the function will be plotted for a range of values of 2. We will start with a very low value, 0.25.

Y

X


21

XY 25.02

15

Y

X


21

XY 50.02

We will increase 2 in steps of 0.25 and see how the shape of the function changes.

16

Y

X


21

XY

2 = 0.75.

75.02

17

Y

X


21

XY

When 2 is equal to 1, the curve becomes a straight line through the origin.

00.12

18

Y

X


21

XY

2 = 1.25.

25.12

19

Y

X


21

XY

2 = 1.50.

50.12

20

Y

X


21

XY

2 = 1.75. Note that the curvature can be quite gentle over wide ranges of X.

75.12

21

Y

X


21

XY 75.12

This means that even if the true model is of the constant elasticity form, a linear model may be a good approximation over a limited range.

22

It is easy to fit a constant elasticity function using a sample of observations. You can linearize the model by taking the logarithms of both sides.


X

X

XY

loglog

loglog

loglog

21

1

1

2

2

21

XY

23

You thus obtain a linear relationship between Y' and X', as defined. All serious regression applications allow you to generate logarithmic variables from existing ones.


X

X

XY

loglog

loglog

loglog

21

1

1

2

2

'' 2'1 XY

1'1 log

log'

,log'

XX

YYwhere

21

XY

24

The coefficient of X' will be a direct estimate of the elasticity, 2.


X

X

XY

loglog

loglog

loglog

21

1

1

2

2

21

XY

'' 2'1 XY

1'1 log

log'

,log'

XX

YYwhere

25

The constant term will be an estimate of log 1. To obtain an estimate of 1, you calculate exp(b1'), where b1' is the estimate of 1'. (This assumes that you have used natural logarithms, that is, logarithms to base e, to transform the model.)


X

X

XY

loglog

loglog

loglog

21

1

1

2

2

21

XY

'' 2'1 XY

1'1 log

log'

,log'

XX

YYwhere

26

Here is a scatter diagram showing annual household expenditure on FDHO, food eaten at home, and EXP, total annual household expenditure, both measured in dollars, for 1995 for a sample of 869 households in the United States (Consumer Expenditure Survey data).

0

2000

4000

6000

8000

10000

12000

14000

16000

0 20000 40000 60000 80000 100000 120000 140000 160000

FDHO

EXP


. reg FDHO EXP

Source | SS df MS Number of obs = 869---------+------------------------------ F( 1, 867) = 381.47 Model | 915843574 1 915843574 Prob > F = 0.0000Residual | 2.0815e+09 867 2400831.16 R-squared = 0.3055---------+------------------------------ Adj R-squared = 0.3047 Total | 2.9974e+09 868 3453184.55 Root MSE = 1549.5

------------------------------------------------------------------------------ FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- EXP | .0528427 .0027055 19.531 0.000 .0475325 .0581529 _cons | 1916.143 96.54591 19.847 0.000 1726.652 2105.634------------------------------------------------------------------------------

27

Here is a regression of FDHO on EXP. It is usual to relate types of consumer expenditure to total expenditure, rather than income, when using household data. Household income data tend to be relatively erratic.


28

The regression implies that, at the margin, 5 cents out of each dollar of expenditure is spent on food at home. Does this seem plausible? Probably, though possibly a little low.


. reg FDHO EXP



29

It also suggests that $1,916 would be spent on food at home if total expenditure were zero. Obviously this is impossible. It might be possible to interpret it somehow as baseline expenditure, but we would need to take into account family size and composition.


. reg FDHO EXP



30

Here is the regression line plotted on the scatter diagram

0

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12000

14000

16000

0 20000 40000 60000 80000 100000 120000 140000 160000EXP


FDHO

31

We will now fit a constant elasticity function using the same data. The scatter diagram shows the logarithm of FDHO plotted against the logarithm of EXP.

5.00

6.00

7.00

8.00

9.00

10.00

7.00 8.00 9.00 10.00 11.00 12.00 13.00

LGFDHO

LGEXP


. g LGFDHO = ln(FDHO)

. g LGEXP = ln(EXP)

. reg LGFDHO LGEXP

Source | SS df MS Number of obs = 868---------+------------------------------ F( 1, 866) = 396.06 Model | 84.4161692 1 84.4161692 Prob > F = 0.0000Residual | 184.579612 866 .213140429 R-squared = 0.3138---------+------------------------------ Adj R-squared = 0.3130 Total | 268.995781 867 .310260416 Root MSE = .46167

------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- LGEXP | .4800417 .0241212 19.901 0.000 .4326988 .5273846 _cons | 3.166271 .244297 12.961 0.000 2.686787 3.645754------------------------------------------------------------------------------

32

Here is the result of regressing LGFDHO on LGEXP. The first two commands generate the logarithmic variables.


33

The estimate of the elasticity is 0.48. Does this seem plausible?



. g LGEXP = ln(EXP)

. reg LGFDHO LGEXP



34

Yes, definitely. Food is a normal good, so its elasticity should be positive, but it is a basic necessity. Expenditure on it should grow less rapidly than expenditure generally, so its elasticity should be less than 1.



. g LGEXP = ln(EXP)

. reg LGFDHO LGEXP



35

The intercept has no substantive meaning. To obtain an estimate of 1, we calculate e3.16, which is 23.8.

48.08.23ˆ48.017.3ˆ EXPOHFDLGEXPHODLGF



. g LGEXP = ln(EXP)

. reg LGFDHO LGEXP



36

Here is the scatter diagram with the regression line plotted.

5.00

6.00

7.00

8.00

9.00

10.00

7.00 8.00 9.00 10.00 11.00 12.00 13.00LGEXP


LGFDHO

37

Here is the regression line from the logarithmic regression plotted in the original scatter diagram, together with the linear regression line for comparison.

0

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4000

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8000

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12000

14000

16000

0 20000 40000 60000 80000 100000 120000 140000 160000EXP


FDHO

38

You can see that the logarithmic regression line gives a somewhat better fit, especially at low levels of expenditure.

0

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4000

6000

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10000

12000

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0 20000 40000 60000 80000 100000 120000 140000 160000EXP


FDHO

39

However, the difference in the fit is not dramatic. The main reason for preferring the constant elasticity model is that it makes more sense theoretically. It also has a technical advantage that we will come to later on (when discussing heteroscedasticity).

0

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12000

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0 20000 40000 60000 80000 100000 120000 140000 160000EXP


FDHO

Copyright Christopher Dougherty 2012.

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ELASTICITIES AND LOGARITHMIC MODELS 1 This sequence defines elasticities and shows how one may fit nonlinear models with constant elasticities. First,

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