2014 04 14

Ch.8: Nonlinear Regression Functions

and 11.1: Linear Probability Model

Econ 141 Spring 2014

Lecture: April 14, 2014

Bart Hobijn

4/14/2014 Econ 141, Spring 2014 1

The views expressed in these lecture notes are solely those of the instructor and do not necessarily

reflect those of the UC Berkeley, or other institutions with which he is affiliated.

Threats to internal validity

Source of bias Solution Where

covered?

Internal invalidity: 𝐸 𝑿′𝒖 𝑿 ≠ 𝟎 … biased and inconsistent estimates

Omitted variable bias • Include control variables in

regression

• Fixed effects in panel data

Ch.7

Ch. 10

Misspecification of functional form • Choose functional form

that better fits the data

Ch.8

Errors-in-variables bias • Get more precisely

measured data

• Instrumental variables

Data source

Ch. 12

Sample selection bias • Tobit model

• Heckman correction, etc.

Beyond the

scope of class

Simultaneous causality bias • Instrumental variables

• Set up an experiment

Ch. 12

Ch. 13

4/14/2014 Econ 141, Spring 2014 2

Outline of lecture

• Population regression equation as a Taylor

approximation

– Polynomials (S&W page 263)

– Interactions between explanatory variables (S&W page 274)

• Binary variables in the Taylor approximation context

– Dummy variables

– Interactions that include dummy variables

• Using binary variable as dependent variable

• Levels versus logarithms

4/14/2014 Econ 141, Spring 2014 3

A general framework

Consider a general relationship between the

dependent variable, 𝑌𝑖, and the explanatory

variables, 𝑋𝑗𝑖, where 𝑗 = 1,… , 𝑘 and 𝑖 = 1,… , 𝑛.

𝑌𝑖 = 𝑓 𝑋1𝑖 , … , 𝑋𝑘𝑖 + 𝑢𝑖

Where we assume that the function 𝑓 . is well-

behaved (continuously differentiable)

4/14/2014 Econ 141, Spring 2014 4

Linear regression model as a

first-order Taylor approximation

The first-order Taylor approximation of this

expression around the sample mean gives

𝑌𝑖 ≈ 𝑓 𝑋 1, … , 𝑋 𝑘 + 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋𝑗𝑖 − 𝑋 𝑗

𝑘

𝑗=1+ 𝑢𝑖

= 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1

+ 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋𝑗𝑖

𝑘

𝑗=1+ 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 5





𝑌𝑖 ≈ 𝑓 𝑋 1, … , 𝑋 𝑘 + 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋𝑗𝑖 − 𝑋 𝑗

𝑘

𝑗=1+ 𝑢𝑖

= 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1

+ 𝜕


𝑘

𝑗=1+ 𝑢𝑖

= 𝛽0 + 𝛽𝑗𝑋𝑗𝑖

𝑘

𝑗=1+ 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 6





𝑌𝑖 ≈ 𝑓 𝑋 1, … , 𝑋 𝑘 + 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋𝑗𝑖 − 𝑋 𝑗

𝑘

𝑗=1+ 𝑢𝑖

= 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1

+ 𝜕


𝑘

𝑗=1+ 𝑢𝑖


𝑘

𝑗=1+ 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 7

𝜷𝒋 measures the

marginal effect

of 𝑿𝒋𝒊 on 𝒀𝒊

When does this work?

First-order Taylor approximation is accurate

when

• Second-order derivatives 𝜕2

𝜕𝑋 𝑗𝑋 𝑙𝑓 𝑋 1, … , 𝑋 𝑘

are relatively small compared to the range

over which 𝑋𝑗𝑖 − 𝑋 𝑗 is evaluated.

If not, higher-order approximation necessary.

4/14/2014 Econ 141, Spring 2014 8

Example: A Mincer regression

Consider the relationship between the log of a

worker’s hourly wage, 𝑤𝑖, and his or her

potential work experience, 𝑋𝑖, defined by age

minus years of education minus 6.

Let’s run a linear regression of the type

𝑤𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 9

Example: A Mincer regression . regress lnhrwage exper

Source | SS df MS Number of obs = 3854

-------------+------------------------------ F( 1, 3852) = 13.16

Model | 6.40584203 1 6.40584203 Prob > F = 0.0003

Residual | 1875.08007 3852 .486780912 R-squared = 0.0034

-------------+------------------------------ Adj R-squared = 0.0031

Total | 1881.48592 3853 .488317134 Root MSE = .6977

------------------------------------------------------------------------------

lnhrwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

exper | .0035066 .0009666 3.63 0.000 .0016114 .0054017

_cons | 2.91992 .0273969 106.58 0.000 2.866206 2.973634

------------------------------------------------------------------------------

Though coefficient on experience is significant, plot of

fitted regression line reveals non-linearity.

4/14/2014 Econ 141, Spring 2014 10


4/14/2014 Econ 141, Spring 2014 11

12

34

5

log

hou

rly w

age

0 20 40 60years of potential experience

Log of the hourly wage Fitted values


4/14/2014 Econ 141, Spring 2014 12

12

34

5

log

hou

rly w

age



Example: second-order approximation

𝑌𝑖 ≈ 𝑓 𝑋 1, … , 𝑋 𝑘 + 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋𝑗𝑖 − 𝑋 𝑗

𝑘

𝑗=1+

1

2

𝜕2

𝜕𝑋 𝑗𝑋 𝑙𝑓 𝑋 1, … , 𝑋 𝑘 𝑋𝑗𝑖 − 𝑋 𝑗 𝑋𝑙𝑖 − 𝑋 𝑙

𝑘

𝑙=1

𝑘

𝑗=1+ 𝑢𝑖

= 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1+

1

2

𝜕2

𝜕𝑋 𝑗𝑋 𝑙𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗𝑋 𝑙

𝑘

𝑙=1

𝑘

𝑗=1

+ 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 −

𝜕2

𝜕𝑋 𝑗𝑋 𝑙𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑙

𝑘

𝑙=1𝑋𝑗𝑖

𝑘

𝑗=1

+ 1

2

𝜕2

𝜕𝑋 𝑗𝑋 𝑙𝑓 𝑋 1, … , 𝑋 𝑘 𝑋𝑗𝑖𝑋𝑙𝑖

𝑘

𝑙=1+ 𝑢𝑖

𝑘

𝑗=1

= 𝛽0 + 𝛽𝑗

𝑘

𝑗=1𝑋𝑗𝑖 + 𝛾𝑗𝑙𝑋𝑗𝑖𝑋𝑙𝑖

𝑘

𝑙=1

𝑘

𝑗=1+ 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 13


𝑌𝑖 ≈ 𝑓 𝑋 1, … , 𝑋 𝑘 + 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋𝑗𝑖 − 𝑋 𝑗

𝑘

𝑗=1+

1

2

𝜕2


𝑘

𝑙=1

𝑘

𝑗=1+ 𝑢𝑖

= 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1+

1

2

𝜕2


𝑘

𝑙=1

𝑘

𝑗=1

+ 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 −

𝜕2


𝑘

𝑙=1𝑋𝑗𝑖

𝑘

𝑗=1

+ 1

2

𝜕2


𝑘

𝑙=1+ 𝑢𝑖

𝑘

𝑗=1

= 𝛽0 + 𝛽𝑗

𝑘


𝑘

𝑙=1

𝑘

𝑗=1+ 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 14


𝑌𝑖 ≈ 𝑓 𝑋 1, … , 𝑋 𝑘 + 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋𝑗𝑖 − 𝑋 𝑗

𝑘

𝑗=1+

1

2

𝜕2


𝑘

𝑙=1

𝑘

𝑗=1+ 𝑢𝑖

= 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1+

1

2

𝜕2


𝑘

𝑙=1

𝑘

𝑗=1

+ 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 −

𝜕2


𝑘

𝑙=1𝑋𝑗𝑖

𝑘

𝑗=1

+ 1

2

𝜕2


𝑘

𝑙=1+ 𝑢𝑖

𝑘

𝑗=1

= 𝛽0 + 𝛽𝑗

𝑘


𝑘

𝑙=1

𝑘

𝑗=1+ 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 15

Interactions between

explanatory

variables naturally

occur in higher-order

multivariate

approximations


𝑌𝑖 ≈ 𝑓 𝑋 1, … , 𝑋 𝑘 + 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋𝑗𝑖 − 𝑋 𝑗

𝑘

𝑗=1+

1

2

𝜕2


𝑘

𝑙=1

𝑘

𝑗=1+ 𝑢𝑖

= 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1+

1

2

𝜕2


𝑘

𝑙=1

𝑘

𝑗=1

+ 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 −

𝜕2


𝑘

𝑙=1𝑋𝑗𝑖

𝑘

𝑗=1

+ 1

2

𝜕2


𝑘

𝑙=1+ 𝑢𝑖

𝑘

𝑗=1

= 𝛽0 + 𝛽𝑗

𝑘


𝑘

𝑙=1

𝑘

𝑗=1+ 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 16

Marginal effect of 𝑿𝒋𝒊

on 𝒀𝒊 depends on

interaction terms with

other variables


The relationship between the log of a worker’s

hourly wage, 𝑤𝑖, and his or her potential work

experience, 𝑋𝑖, is not approximately linear over

the range of 𝑋𝑖 observed in the data.

Let’s add a second-order to to the regression

𝑤𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝛽2𝑋𝑖2 + 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 17

Example: A Mincer regression . regress lnhrwage exper exper2


-------------+------------------------------ F( 2, 3851) = 72.20

Model | 68.0004527 2 34.0002264 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.0356

Total | 1881.48592 3853 .488317134 Root MSE = .68623

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

exper | .0504624 .0042144 11.97 0.000 .0421998 .058725

exper2 | -.000834 .0000729 -11.44 0.000 -.000977 -.0006911

_cons | 2.376179 .0546489 43.48 0.000 2.269036 2.483323

------------------------------------------------------------------------------

Second-order term highly significant

4/14/2014 Econ 141, Spring 2014 18


4/14/2014 Econ 141, Spring 2014 19

12

34

5

log

hou

rly w

age




But a worker’s hourly wage, 𝑤𝑖, does not only

depend on potential work experience, 𝑋𝑖. It also

depends on the years of schooling, 𝑆𝑖. Let’s

estimate a second order polynomial in the two

explanatory variables

Let’s add a second-order to to the regression 𝑤𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝛽2𝑆𝑖 + 𝛽3𝑋𝑖

2 + 𝛽4𝑋𝑖𝑆𝑖 + 𝛽5𝑆𝑖2 + 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 20

Example: A Mincer regression . regress lnhrwage exper schoolyrs exper2 schoolyrs2 schoolyrsexper


-------------+------------------------------ F( 5, 3848) = 244.13

Model | 453.102378 5 90.6204756 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.2398

Total | 1881.48592 3853 .488317134 Root MSE = .60926

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

exper | .0657975 .0066534 9.89 0.000 .0527529 .078842

schoolyrs | .0831624 .0211874 3.93 0.000 .0416229 .1247019

exper2 | -.0006885 .0000698 -9.87 0.000 -.0008252 -.0005517

schoolyrs2 | .0022569 .0005858 3.85 0.000 .0011084 .0034054

schoolyrse~r | -.0011937 .0002909 -4.10 0.000 -.001764 -.0006234

_cons | .5708724 .2155551 2.65 0.008 .1482593 .9934856

------------------------------------------------------------------------------

4/14/2014 Econ 141, Spring 2014 21


Following S&W page 264 and test for joint

insignificance of higher-order terms.

. test (exper2 = 0) (schoolyrs2 = 0) (schoolyrsexper = 0)

( 1) exper2 = 0

( 2) schoolyrs2 = 0

( 3) schoolyrsexper = 0

F( 3, 3848) = 43.71

Prob > F = 0.0000

They are very significant!

4/14/2014 Econ 141, Spring 2014 22


Use the lincom command to estimate

marginal effect. Return to a year more

experience for someone with 20 years of

experience and 12 years of education. Evaluate

𝛽1 + 2𝛽3𝑋𝑖 + 𝛽4𝑆𝑖 at 𝑋𝑖 = 12 and 𝑆𝑖 = 20

. lincom (exper + 2*exper2*20 + schoolyrsexper*12 )

( 1) exper + 40*exper2 + 12*schoolyrsexper = 0

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

(1) | .0239336 .0017308 13.83 0.000 .0205403 .027327

------------------------------------------------------------------------------

4/14/2014 Econ 141, Spring 2014 23

Some points related to polynomials

Second-order approximation is only special case. Higher order

approximations lead to higher order polynomial regression

equations.

• Always include lower order terms when including higher

order terms of polynomial.

• Can use standard tests to figure out right order of

polynomials (S&W page 264)

• Though higher-order polynomials lead to better

approximations, they have harder to interpret marginal

effects and estimation has lower number of degrees of

freedom.

• Beware of the units of measurement related to the higher-

order coefficients. Often best to present marginal effects.

4/14/2014 Econ 141, Spring 2014 24

Dummies in Taylor approximation

Revisit the first-order Taylor approximation underlying

the linear regression model

𝑌𝑖 = 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1+

𝜕


𝑘

𝑗=1+ 𝑢𝑖


𝑘

𝑗=1+ 𝑢𝑖

Suppose we have two different subgroups. Let’s

consider two cases

1. They have different 𝑓 𝑋 1, … , 𝑋 𝑘

2. They have different 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 for some 𝑗.

4/14/2014 Econ 141, Spring 2014 25




𝑌𝑖 = 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1+

𝜕


𝑘

𝑗=1+ 𝑢𝑖


𝑘

𝑗=1+ 𝑢𝑖


consider two cases




4/14/2014 Econ 141, Spring 2014 26

Case 1:

Different 𝑓 𝑋 1, … , 𝑋 𝑘 for different groups.

This would mean that the intercept is

different across groups.

Include dummy to allow intercept to vary

by group.

(S&W equation 8.32)




𝑌𝑖 = 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1+

𝜕


𝑘

𝑗=1+ 𝑢𝑖


𝑘

𝑗=1+ 𝑢𝑖


consider two cases




4/14/2014 Econ 141, Spring 2014 27

Case 2:

Different 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 for some 𝒋 for

different groups.

This would mean that the slope

coefficient is different across groups.

Include dummy explanatory variable

interaction to allow slope coefficient to

vary by group.

(S&W equation 8.32)




𝑌𝑖 = 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1+

𝜕


𝑘

𝑗=1+ 𝑢𝑖


𝑘

𝑗=1+ 𝑢𝑖


consider two cases




4/14/2014 Econ 141, Spring 2014 28

Case 2:

However, 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 also affects the

intercept.

Thus, you would also include dummy to

allow intercept to vary by group.

(S&W equation 8.32)




𝑌𝑖 = 𝑓 𝑋 1, … , 𝑋 𝑘 − 𝜕

𝜕𝑋 𝑗𝑓 𝑋 1, … , 𝑋 𝑘 𝑋 𝑗

𝑘

𝑗=1+

𝜕


𝑘

𝑗=1+ 𝑢𝑖


𝑘

𝑗=1+ 𝑢𝑖


consider two cases




4/14/2014 Econ 141, Spring 2014 29

Stock and Watons equation (8.33) that only includes

dummy explanatory variable interaction is not often

used

Generally, in that case, there is also just the dummy

to allow for a different constant term

Panel (c) from figure 8.8 and bullet 3 from Key

Concept 8.4 are not often applied. Nested in panel (b)

and bullet 2


Is the return to education different for women?

Let 𝐷𝑖 = 1 if individual 𝑖 is female.

Add appropriate dummy variable and dummy

interaction variables to test this hypothesis 𝑤𝑖 = 𝛽0 + 𝛽1𝐷𝑖 + 𝛽2𝑋𝑖 + 𝛽3𝑆𝑖 + 𝛽4𝑆𝑖𝐷𝑖 + 𝛽5𝑋𝑖

2 + 𝛽6𝑋𝑖𝑆𝑖

+ 𝛽7𝑋𝑖𝑆𝑖𝐷𝑖 + 𝛽8𝑆𝑖2 + 𝛽8𝑆𝑖

2𝐷𝑖 + 𝑢𝑖

4/14/2014 Econ 141, Spring 2014 30

Example: A Mincer regression . regress lnhrwage female exper schoolyrs schoolyrsfem exper2 schoolyrs2 schoolyrsfem2

schoolyrsexper schoolyrsfemexper


-------------+------------------------------ F( 9, 3844) = 148.19

Model | 484.646812 9 53.8496458 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.2558

Total | 1881.48592 3853 .488317134 Root MSE = .60281

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

female | -.2583408 .1758799 -1.47 0.142 -.6031675 .086486

exper | .0697711 .0066445 10.50 0.000 .0567441 .0827981

schoolyrs | .0814035 .0231805 3.51 0.000 .0359563 .1268507

schoolyrsfem | .0421913 .0265727 1.59 0.112 -.0099066 .0942893

exper2 | -.000702 .0000694 -10.12 0.000 -.000838 -.0005659

schoolyrs2 | .0026695 .0007058 3.78 0.000 .0012858 .0040533

schoolyrsf~2 | -.0019102 .0010155 -1.88 0.060 -.0039012 .0000808

schoolyrse~r | -.0013209 .0002921 -4.52 0.000 -.0018936 -.0007481

schoolyrsf~r | -.0002773 .0001192 -2.33 0.020 -.000511 -.0000435

_cons | .5316936 .2220393 2.39 0.017 .0963676 .9670197

------------------------------------------------------------------------------

4/14/2014 Econ 141, Spring 2014 31


𝑤𝑖 = 𝛽0 + 𝛽1𝐷𝑖 + 𝛽2𝑋𝑖 + 𝛽3𝑆𝑖 + 𝛽4𝑆𝑖𝐷𝑖 + 𝛽5𝑋𝑖2 + 𝛽6𝑋𝑖𝑆𝑖

+ 𝛽7𝑋𝑖𝑆𝑖𝐷𝑖 + 𝛽8𝑆𝑖2 + 𝛽8𝑆𝑖

2𝐷𝑖 + 𝑢𝑖

Are the added terms insignificant and can we not reject

the null-hypothesis that returns to education are the same

for women and men?

. test (schoolyrsfem = 0) (schoolyrsfem2 = 0) (schoolyrsfemexp = 0)

( 1) schoolyrsfem = 0

( 2) schoolyrsfem2 = 0

( 3) schoolyrsfemexper = 0

F( 3, 3844) = 3.72

Prob > F = 0.0110

4/14/2014 Econ 141, Spring 2014 32

Linear probability model

How about having a dummy as a dependent

variable, 𝑌𝑖?

• This is called the Linear Probability Model. (S&W 11.1)

• OLS gives unbiased and consistent estimates

of the parameters in that case.

• Problem is that this often results in fitted

values 𝑌 𝑖 > 1 or 𝑌 𝑖 < 0.

• Logit and Probit models are popular non-

linear alternatives. (S&W 11.2-5 use MLE. Not covered in this class)

4/14/2014 Econ 141, Spring 2014 33

Logs or levels?

Two things to bear in mind when deciding

between logs or levels in regression

1. Quality of the fit of the regression for chosen

functional form.

2. Interpretation of the regression coefficients

– Sensitivity to units of measurement

– Level effect versus elasticity (S&W 265-273)

Will use Mincer regression to discuss these

considerations here

4/14/2014 Econ 141, Spring 2014 34

Logs and percentage changes

Another first order Taylor approximation!

ln 𝑥 ≈ ln 1 +𝜕 ln 𝑥

𝜕𝑥𝑥=1

𝑥 − 1

= 0 +𝑥 − 1

1 for 𝑥 ≈ 1

Let 𝑥 =𝑢

𝑣 ≈ 1, then

ln 𝑢 − ln 𝑣 = ln𝑢

𝑣≈

𝑢𝑣− 1

1=

𝑢 − 𝑣

𝑣

So, differences in logarithms are

• Approximate percentage changes.

• Do not depend on the units of measurement of 𝑢 and 𝑣 as long as

it is the same.

4/14/2014 Econ 141, Spring 2014 35


Consider the level versus the log cases here:

• Level of hourly wage, 𝑊𝑖.

• Log level of hourly wage, 𝑤𝑖 = ln 𝑊𝑖

We can run the regression:

• In levels: 𝑊𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖

• In logs: 𝑤𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖

Why would we choose to run it in logs?

4/14/2014 Econ 141, Spring 2014 36

Level regression yields dubious fit

4/14/2014 Econ 141, Spring 2014 37

05

01

00

150

200

leve

l of h

ou

rly w

age


Hourly wage Fitted values

Parameter estimate hard to interpret

. regress wage exper


-------------+------------------------------ F( 1, 3852) = 15.73

Model | 9951.00275 1 9951.00275 Prob > F = 0.0001

Residual | 2436801.34 3852 632.606786 R-squared = 0.0041

-------------+------------------------------ Adj R-squared = 0.0038

Total | 2446752.34 3853 635.025264 Root MSE = 25.152

------------------------------------------------------------------------------

wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

exper | .1382063 .0348467 3.97 0.000 .0698866 .206526

_cons | 23.052 .9876471 23.34 0.000 21.11564 24.98836

------------------------------------------------------------------------------

4/14/2014 Econ 141, Spring 2014 38

Parameter estimate hard to interpret

. regress wage exper


-------------+------------------------------ F( 1, 3852) = 15.73

Model | 9951.00275 1 9951.00275 Prob > F = 0.0001

Residual | 2436801.34 3852 632.606786 R-squared = 0.0041

-------------+------------------------------ Adj R-squared = 0.0038

Total | 2446752.34 3853 635.025264 Root MSE = 25.152

------------------------------------------------------------------------------

wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

exper | .1382063 .0348467 3.97 0.000 .0698866 .206526

_cons | 23.052 .9876471 23.34 0.000 21.11564 24.98836

------------------------------------------------------------------------------

4/14/2014 Econ 141, Spring 2014 39

What does this 0.13 mean?

It depends on the units of measurement of the

hourly wage.

Is this additional dollars or cents earned per hour

per year of experience?

For what year is this data? A dollar in 2000 is a

different thing from a dollar in 2014 (inflation)

Units of measurement affect

levels of variables

Suppose we measure 𝑊𝑖 not in dollars but in cents.

Such that 𝑊𝑖∗ = 100𝑊𝑖. Then regression equation in

levels goes from

𝑊𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖

To

𝑊𝑖∗ = 𝛽0

∗ + 𝛽1∗𝑋𝑖 + 𝑢𝑖

∗

Where 𝛽0∗ = 100𝛽0, 𝛽1

∗ = 100𝛽1, and 𝑢𝑖∗ = 100𝑢𝑖

4/14/2014 Econ 141, Spring 2014 40

Units of measurement affect

levels of variables

Suppose we measure 𝑊𝑖 not in dollars but in cents.

Such that 𝑊𝑖∗ = 100𝑊𝑖. Then regression equation in

levels goes from

𝑊𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖

To

𝑊𝑖∗ = 𝛽0

∗ + 𝛽1∗𝑋𝑖 + 𝑢𝑖

∗

Where 𝛽0∗ = 100𝛽0, 𝛽1

∗ = 100𝛽1, and 𝑢𝑖∗ = 100𝑢𝑖

4/14/2014 Econ 141, Spring 2014 41

The magnitude of the slope coefficients

and the residuals thus depends on the

units of measurement of the dependent

(or explanatory) variables if they are

included in levels

Logs not subject to units of

measurement issue

Suppose we measure 𝑊𝑖 not in dollars but in cents. Such

that 𝑤𝑖

∗ = ln𝑊𝑖∗ = ln 100𝑊𝑖 = ln 100 + ln 𝑊𝑖 = ln 100 + 𝑤𝑖

Then regression equation in logs goes from

𝑤𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖

To

𝑤𝑖∗ = 𝛽0

∗ + 𝛽1𝑋𝑖 + 𝑢𝑖

Where 𝛽0∗ = ln 100 + 𝛽0.

Thus, slope coefficient and residuals not affected units of

measurement when log is included rather than level.

4/14/2014 Econ 141, Spring 2014 42

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