Albany College of Pharmacy CH1 Log Polynomial and Interaction Models HW MUST know how to use RIn addition to the Bailey reading (sections 7.1, 7.2 and 12.1

Albany College of Pharmacy CH1 Log Polynomial and Interaction Models HW MUST know how to use RIn addition to the Bailey reading (sections 7.1, 7.2 and 12.1) please read this document. It uses the Simpsons data we first saw in Bailey Ch 1 to estimate some of the more advanced models discussed in this week’s assignment. This script carries out the analysis of the Simpsons data described in the document.In your submission, please answer all the questions that appear on the last page of this document. In addition, please try running the script and describe your experience. (What happens? Are you able to recreate the tables or did you have trouble running it? If so, what was the problem?) ???:Gary Mongiovi ??? ?????????? ?????? ??????? ??
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Exercises Using The Simpson’s Data from Real
Econometrics
Matthew J. Holian?
July 23, 2020
Abstract
This document includes some exercises for introductory econometrics courses I’ve
taught at San Jose State University that use Michael Bailey’s Real Econometrics textbook
Keywords: multiple regression, causal inference.
JEL Codes: TBA
?
Corresponding author: matthew.holian@sjsu.edu. Please email me comments or corrections.
1
Nonlinear Models and Linear Probability Models
This document describes nonlinear models, estimated by transforming variables in one of
three ways: with logarithms, polynomials or interactions. It’s ok if you don’t know what
these terms mean yet; I’ll introduce them below.
We begin by describing the natural logarithm function. Remember log base 10 from
algebra? Well the natural log is log base 2.718 (where 2.718 is the value of the natural
exponent e to three decimal places.) Using software, it’s very easy take the natural log of
a variable, such as income, and then to use the logged value of the variable as either an
independent or dependent variable. There are two advantages of doing this. First, enables
regression to estimate a specific type of nonlinear relationship. Second, it facilitates easy
interpretation of regression results.
Interpreting models with logs can be confusing. Therefore I suggest you consult the
tables of rules for interpreting different log models. Table 1 reproduces the descriptions from
Bailey’s textbook, however, I actually prefer the table that appears in Stock and Watson’s
textbook, as it does not require us to change the scale of percentages in our interpretation.
I also reproduce Stock and Watson’s version of this as Table 2, because my preference is to
always talk about percents on a 0-100 scale.
Table 1: How to interpret logged models, table adapted from Bailey’s textbook
model
equation
interpretation
Log-linear
ln Yi = ?0 + ?1 Xi + ei
Linear-log
Yi = ?0 + ?1 ln Xi + ei
Log-log
ln Yi = ?0 + ?1 ln Xi + ei
A one-unit increase in X is associated with a ?1
percent change in Y (on a 0-1 scale).
A one percent increase in X is associated with a
?1 /100 change in Y .
A one-percent increase in X is associated with a
?1 percent change in Y (on a 0-100 scale)
Two other types of model that are nonlinear in the variables are a polynomial modes (e.g.
Y = B0 + B1 ? X + B2 ? X 2 ) and interaction models (e.g. Y = B0 + B1 X + B2 Z + B3 (X ? Z)).
1
Table 2: How to interpret logged models, Table adapted from Stock and Watson’s textbook
model
equation
interpretation
Log-linear
ln Yi = ?0 + ?1 Xi + ei
Linear-log
Yi = ?0 + ?1 ln Xi + ei
Log-log
ln Yi = ?0 + ?1 ln Xi + ei
A change in X by one unit (?X = 1) is associated
with a 100?1 % change in Y.
A one percent change in X is associated with a
change in Y of 0.01?1
A one-percent increase in X is associated with a
?1 percent change in Y , so ?1 is the elasticity of Y
with respect to X.
These models are straightforward to estimate, but like with log models, interpreting them
can be a challenge. The best way to interpret such models is to evaluate them at two sets
of X values. With the polynomial, find fitted values of the model when X equals 1 and then
when X equals 2, and compare the predictions. It is also possible to take a derivative and
evaluate that, but this requires some familiarity with calculus.
Chapters 6 and 7 of Bailey’s textbook describe nonlinear (log and polynomial) models
and models with dummy variables. To illustrate these concepts, I use the Simpsons dataset
from Chapter 1. This is kind of a silly example but it can be used to illustrate the types of
models. Table 3 presents data from Bailey’s Table 1.1, plus an additional variable indicating
gender. Table 4 presents variable descriptions, and Table 5 presents summary statistics.
Table 6 presents estimates of regression models. Column on contains a regression of
WEIGHT on FEMALE. This is a bivariate regression model with continuous dependent and
independent variables. All together Table 6 contains five regression models; models 2-5 are
various nonlinear models:
1. A linear-linear model (regress WEIGHT on DOUGHNUTS)
2. A polynomial model (regress WEIGHT on DOUGHNUTS and DOUGHNUTS2 )
3. A linear-log model (regress WEIGHT on logDOUGHNUTS)
4. A log-linear model (regress logWEIGHT on DOUGHNUTS )
2
Table 3: Simpsons Variable Descriptions
NAME
WEIGHT
DOUGHNTS
FEMALE
275
141
70
75
310
80
160
263
205
185
170
155
145
14
0
0
5
20
0.75
0.25
16
3
2
0.8
5
4
0
1
1
0
0
0
0
0
0
0
0
1
1
Homer
Marge
Lisa
Bart
Comic Book Guy
Mr. Burns
Smithers
Chief Wiggum
Principal Skinner
Rev. Lovejoy
Ned Flanders
Patty
Selma
Table 4: Simpsons Variable Descriptions
Variable
Description
WEIGHT
DOUGHNUTS
FEMALE
logDOUGHNUTS
logWEIGHT
DOUGHNUTS2
INTERACT
Weight in pounds
Weekly doughnut consumption
A dummy equal to one if respondent is female
the natural log of DOUGHNUTS
the natural log of WEIGHT
the square of DOUGHNUTS
the product of WEIGHT and FEMALE
Table 5: Simpsons Summary Statistics
Statistic
N
Mean
St. Dev.
Min
Max
WEIGHT
DOUGHNUTS
FEMALE
INTERACT
logWEIGHT
logDOUGHNUTS
DOUGHNUTS2
13
13
13
13
13
13
13
171.846
5.446
0.308
0.692
5.046
1.356
71.713
76.155
6.750
0.480
1.702
0.485
1.063
128.679
70
0
0
0
4.248
0.000
0
310
20
1
5
5.737
3.045
400
3
5. A log-log model (regress logWEIGHT on logDOUGHNUTS)
Here are the results:
Table 6: Regression of Doughnuts on Weight, Linear and Nonlinear Models
Dependent variable:
WEIGHT
DOUGHNUTS
(1)
(2)
9.224???
(1.105)
5.143
(8.150)
DOUGHNUTS2
logWEIGHT
(3)
(4)
(5)
0.049???
(0.009)
0.221
(0.412)
52.855???
(11.995)
logDOUGHNUTS
0.294???
(0.085)
Constant
121.613???
(17.256)
128.018???
(20.809)
100.150???
(20.515)
4.776???
(0.144)
4.647???
(0.173)
Observations
R2
Adjusted R2
Residual Std. Error
F Statistic
13
0.668
0.638
45.814
22.158???
13
0.676
0.612
47.457
10.451???
13
0.544
0.503
53.700
13.134???
13
0.475
0.428
0.367
9.963???
13
0.415
0.362
0.387
7.810??
?
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