elasticity of y with respect to x is the proportional change in y per proportional change in x:...
Post on 17-Dec-2015
220 Views
Preview:
TRANSCRIPT
Elasticity of Y with respect toX is the proportional change inY per proportional change in X:
ELASTICITIES AND DOUBLE-LOGARITHMIC MODELS
0 52
XYdXdY
XdXYdY
elasticity Y
X
A
Ox
y
1
This sequence defines elasticities and shows how one may fit nonlinear models with constant elasticities. First, the general definition of an elasticity.
Elasticity of Y with respect toX is the proportional change inY per proportional change in X:
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
0 52
XYdXdY
XdXYdY
elasticity
OAA
of slopeat tangent the of slope
Y
X
A
Ox
y
2
Re-arranging the expression for the elasticity, we can obtain a graphical interpretation.
Elasticity of Y with respect toX is the proportional change inY per proportional change in X:
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
0 52
XYdXdY
XdXYdY
elasticity
OAA
of slopeat tangent the of slope
Y
X
A
Ox
y
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.
Elasticity of Y with respect toX is the proportional change inY per proportional change in X:
4
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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.
0 52
XYdXdY
XdXYdY
elasticity
OAA
of slopeat tangent the of slope
Y
X
A
O
1elasticity
XY
0 52
5
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
In this case the tangent at A is steeper than OA and the elasticity is greater than 1.
A
O
1elasticity
Elasticity of Y with respect toX is the proportional change inY per proportional change in X:
XYdXdY
XdXYdY
elasticity
OAA
of slopeat tangent the of slope
Y
X
6
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
In general the elasticity will be different at different points on the function relating Y to X.
XYdXdY elasticity
OAA
of slopeat tangent the of slope
Y
xO X
XY 21
21
2
21
2
)/(
/)(
X
XX
A
7
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
In the example above, Y is a linear function of X.
xO
A
XY 21
XYdXdY elasticity
OAA
of slopeat tangent the of slope
21
2
21
2
)/(
/)(
X
XX
Y
X
8
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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.
xO
A
XY 21
XYdXdY elasticity
OAA
of slopeat tangent the of slope
21
2
21
2
)/(
/)(
X
XX
Y
X
9
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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
XY 21
XYdXdY elasticity
OAA
of slopeat tangent the of slope
21
2
21
2
)/(
/)(
X
XX
Y
X
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
21
XY
10
However, a function of the type shown above has the same elasticity for all values of X.
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
21
XY
121
2 XdXdY
11
For the numerator of the elasticity expression, we need the derivative of Y with respect to X.
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
21
XY
121
2 XdXdY
11
1 2
2
X
XX
XY
12
For the denominator, we need Y/X.
13
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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
211
121
2
2
elasticity
X
XXYdXdY
14
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
We will increase 2 in steps of 0.25 and see how the shape of the function changes.
21
XY 50.02
Y
X
16
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
21
XY 75.02
Y
X
17
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
When 2 is equal to 1, the curve becomes a straight line through the origin.
21
XY 00.12
Y
X
18
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
21
XY 25.12
Y
X
19
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
21
XY 50.12
Y
X
20
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
Note that the curvature can be quite gentle over wide ranges of X.
21
XY 75.12
Y
X
21
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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.
21
XY 00.22
Y
X
22
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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.
21
XY
X
X
XY
loglog
loglog
loglog
22
1
1
2
2
23
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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.
21
XY
X
X
XY
loglog
loglog
loglog
22
1
1
2
2
'' 2'1 XY
1'1 log
log'
,log' where
XX
YY
24
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
The coefficient of X' will be a direct estimate of the elasticity, 2.
21
XY
X
X
XY
loglog
loglog
loglog
22
1
1
2
2
'' 2'1 XY
1'1 log
log'
,log' where
XX
YY
25
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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.)
21
XY
X
X
XY
loglog
loglog
loglog
22
1
1
2
2
'' 2'1 XY
1'1 log
log'
,log' where
XX
YY
26
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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.
. 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------------------------------------------------------------------------------
28
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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
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------------------------------------------------------------------------------
29
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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.
30
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
Here is the regression line plotted on the scatter diagram
0
2000
4000
6000
8000
10000
12000
14000
16000
0 20000 40000 60000 80000 100000 120000 140000 160000EXP
FDHO
31
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
Here is the result of regressing LGFDHO on LGEXP. The first two commands generate the logarithmic variables.
. 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------------------------------------------------------------------------------
33
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
The estimate of the elasticity is 0.48. Does this seem plausible?
. 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------------------------------------------------------------------------------
34
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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 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------------------------------------------------------------------------------
35
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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
36
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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 13.00
LGFDHO
LGEXP
37
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
Here is the regression line from the logarithmic regression plotted in the original scatter diagram, together with the linear regression line for comparison.
0
2000
4000
6000
8000
10000
12000
14000
16000
0 20000 40000 60000 80000 100000 120000 140000 160000EXP
FDHO
38
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
You can see that the logarithmic regression line gives a somewhat better fit, especially at low levels of expenditure.
0
2000
4000
6000
8000
10000
12000
14000
16000
0 20000 40000 60000 80000 100000 120000 140000 160000EXP
FDHO
39
ELASTICITIES ANDDOUBLE-LOGARITHMIC MODELS
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
2000
4000
6000
8000
10000
12000
14000
16000
0 20000 40000 60000 80000 100000 120000 140000 160000EXP
FDHO
Copyright Christopher Dougherty 1999-2001. This slideshow may be freely copied for personal use.
top related