Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Econ 251 Notes 9: Specication
Utku Suleymanoglu
These set of notes go over some of the material in the Chapter 13 of the Gujarati.
We have so far seen many dierent ways we can include regressors to regression equations.
In this chapter, we will discuss th
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section 6
Omitted Variable Bias
Oct 14th, 2016
Dataset: GPA2.dta
In class we talked about the simple regression model relating a students college GPA (colgpa)
and total hours of study per day during the term (tothrs):
colgpa = 0 + 1tothrs + u.
You have th
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section 5
SOLUTIONS
Simple Linear Regression and OLS
I. Algebra of OLS
Throughout this class, we are going to be using STATA to calculate regression lines, but it is good to
compute one regression line by hand as this gives you a better understanding of w
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section 8
Algebra of hypotheses testing
SOLUTIONS
This problem is meant to summarize the algebra behind hypotheses testing in a regression
context. It is a hard problem and you are not expected to be able to fully replicate it, only to
be able to calculat
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section 9
Algebra of hypotheses testing
Nov 4th, 2016
This problem is meant to summarize the algebra behind hypotheses testing in a regression
context. It is a hard problem and you are not expected to be able to fully replicate it, only to
be able to calc
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section 12
Instrumental variable estimation
Dec 2nd, 2016
The Demand for Fish
This problem uses the dataset FISH.RAW, which comes from a paper by Angrist, Imbens and
Graddy. We are going to use it to estimate a demand function for fish (apologies for the
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section 11
Functional form and effect of changing the units of measurement
Nov 18th, 2016
Dataset: WAGE2.dta
This problem illustrates the effect of changing the units of measurement of an explanatory variable.
Suppose that the population model relating lo
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section 8
Models with dummy variables
Oct 28th, 2016
Data and Variable description:
This problem uses data on 445 men, who were unemployed in 1978. Part of these men took part
in a job training program in 1978 aiming at increasing the likelihood that they
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section 10
Nov 11th, 2016
Regression with time series data
The effect of birth control pill and personal exemption tax on fertility
(based on J. Wooldridge)
In this section we are going to use dataset called FERTIL4.dta.
FERTIL4.dta is based on a dataset
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Properties of the sample mean
0. Reminder: the Normal distribution
(Appendix B, section B.5)
Aside: a probability density function (pdf) of a continuous random variable, is a function that
describes the relative likelihood for this random variable to take
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
CHARACTERISTICS OF PROBABILITY DISTRIBUTIONS
(Appendix B, section B.1 and B.3)
Reminder
Discrete random variable: a random variable that takes on countably many values (they can be
finite or infinite, but we can count them)
Continuous random variable: a r
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section 8
Models with dummy variables
SOLUTIONS
Data and Variable description:
This problem uses data on 445 men, who were unemployed in 1978. Part of these men took part
in a job training program in 1978 aiming at increasing the likelihood that they find
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section #4
Testing hypotheses about the population mean
Friday, Sep 30th 2016
IMPORTANT NOTE:
This is exactly how a problem on testing hypotheses about the population mean(s) will be
formulated on the midterm you will be given a STATA output, and asked to
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Econ 251 Notes 5: Multiple Linear Regression Model:
Estimation and Inference
Utku Suleymanoglu
These set of notes go over some of the material in the Chapter 7 and Chapter 8 of the
Gujarati.
We have studied the simple linear regression model with one inde
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Econ 251 Notes 7: Dummy Variables
Utku Suleymanoglu
These set of notes go over some of the material in the Chapter 9 of the Gujarati.
As we have introduced them earlier, dummy variables are special kind of variables which
contain binary information. They
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Econ 251 Notes 6: Functional Forms
Utku Suleymanoglu
These set of notes go over some of the material in the Chapter 6.46.6, 7.9, 7.10 of the
Gujarati.
We have mentioned that multiple linear regression model is capable of handling models which
are nonline
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section 3
SOLUTIONS
You would like to know the average monthly spending of all students at the University of
Michigan. You find it too costly in terms of time and money to go around asking everyone
about their spending, so you randomly select 1,000 studen
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Econ 251
Fall 2016
Problem Set #3
(40 points in total)
SOLUTIONS
Part I: Testing hypotheses in STATA (15 points in total)
This part of the problem set introduces you to using STATA for testing hypotheses about
the population parameters.
Instructions:
Foll
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Econ 251
Fall 2016
Problem Set #1
SOLUTIONS
Part I: Expected value, variance and covariance (8 points in total)
This exercise illustrates the properties of expected value, variance and covariance that we are
going to use further in this course.
Note: We s
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Econ 251
Fall 2016
Problem Set #2
SOLUTIONS
Part II: Statistical theory (24 points in total)
Note: Please, refer to your lecture notes and notes on section 2 when solving Part II of
this homework. These problems are very similar to some examples you saw i
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
ECON251
Fall 2016
Section 2
SOLUTIONS
This section introduces you to using STATA for simple data analysis.
WAGE2.dta contains information on monthly earnings, employment history, education,
demographic characteristics, and two test scores for 935 men in y
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section #4
Testing hypotheses
SOLUTIONS
IMPORTANT NOTE:
This is exactly how a problem on testing hypotheses about the population mean(s) will be
formulated on the midterm you will be given a STATA output, and asked to answer the
same type of questions.
Pr
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
EXTRA PRACTICE MIDTERM
(This was part of a midterm exam question from Winter 2016, section 100).
You wish to investigate whether submitting homework in an Econometrics class affects a students
final exam performance. You have access to a data set of 674 s
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Section #1
Topics covered:
Properties of expected value, variance and covariance
Properties of summations
Problem 1
This exercise illustrates the properties of expected value, variance and covariance that we
are going to use further in this course.
Suppos
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
LECTURE 1
POPULATIONS, SAMPLES, AND STATISTICAL INFERENCE
(Appendix C, sections 1.A and 2.A)
DEFINITION: The population consists of all units (everything/everyone) we want to measure.
E.g. You want to learn what is the average height of all UM undergradua
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
TESTING HYPOTHESES ABOUT THE POPULATION MEAN
I. ONE POPULATION TESTS
Example
In our dataset, the sample mean IQ test score equals 101.28 (see STATA output below).
. sum IQ
Variable
Obs
Mean
IQ
935
101.2824
Std. Dev.
Min
Max
15.05264
50
145
Question: Based
Basic Introduction to Statistics and Econometrics II
ECON 251

Summer 2015
Problem Set 1 Solutions
Economics 251 Spring 2015
University of Michigan
Utku Suleymanoglu
Due: Friday, May 15th
1. Suppose x and y are known real numbers. Take the following function: f (b) =
(y bx)2 . First, plot this function for y = 4 and x = 1 as a f
Basic Introduction to Statistics and Econometrics II
ECON 251

Summer 2015
Problem Set 1
Economics 251 Spring 2015
University of Michigan
Utku Suleymanoglu
Due: Friday, May 15th
1. Suppose x and y are known real numbers. Take the following function: f (b) =
(y bx)2 . First, plot this function for y = 4 and x = 1 as a function of
Basic Introduction to Statistics and Econometrics II
ECON 251

Summer 2015
Midterm Solutions
Economics 251
Spring 2015
University of Michigan
Utku Suleymanoglu
Multiple Choice: Make sure you see all the options
1. Which of the following assumptions are not needed to show the OLS estimators
for the simple linear regression model
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Econ 251
FALL 2016
Problem Set #4
Due: Thursday, Oct 20th in class
(30 points in total)
This problem set introduces you to multiple linear regression in STATA, simple
hypotheses tests about the parameters, and omitted variable bias.
Instructions:
Followin
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Econ 251
Fall 2016
Problem Set #3
(40 points in total)
Due: Tuesday, Oct 11th, in class
Part I: Testing hypotheses in STATA (15 points in total)
This part of the problem set introduces you to using STATA for testing hypotheses about
the population paramet
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
ECON 251
FALL 2016
Problem Set #2
Due: Thursday, Sep 29th in class
(38 points)
Part I: Data analysis (14 points in total)
This part of the problem set introduces you to using STATA for simple data analysis.
Note: Please, refer to your notes on section 1.
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
Econ 251
Fall 2016
Problem Set #1
(16 points)
Due: Thursday, Sep 15th in class
Part I: Expected value, variance and covariance (8 points in total)
This exercise illustrates the properties of expected value, variance and covariance that we are going
to use
Basic Introduction to Statistics and Econometrics II
ECON 251

Winter 2015
THE SIMPLE LINEAR REGRESSION MODEL (SLR)
INTRODUCTION TO THE SIMPLE LINEAR REGRESSION MODEL
So far: when we analyzed data, we have tried to determine whether or not there is a relationship
between two variables X and Y. E.g. we have looked at scatter plot